Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical...
Transcript of Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical...
![Page 1: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/1.jpg)
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Khovanov homology of framed and signed chord diagrams.
Oleg Viro
December 2, 2006
![Page 2: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/2.jpg)
Knots and links
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
2 / 47
![Page 3: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/3.jpg)
Classical link diagrams
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
3 / 47
A knot is a smooth simple closed curve in the 3-space.
![Page 4: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/4.jpg)
Classical link diagrams
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
3 / 47
A knot is a smooth simple closed curve in the 3-space.
That is a circle smoothly embedded into R3 .
![Page 5: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/5.jpg)
Classical link diagrams
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
3 / 47
A knot is a smooth simple closed curve in the 3-space.
A link is a union of several disjoint knots.
![Page 6: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/6.jpg)
Classical link diagrams
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
3 / 47
A knot is a smooth simple closed curve in the 3-space.
A link is a union of several disjoint knots.
To describe a knot graphically, project it to a plane
![Page 7: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/7.jpg)
Classical link diagrams
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
3 / 47
A knot is a smooth simple closed curve in the 3-space.
A link is a union of several disjoint knots.
To describe a knot graphically, project it to a plane
and decorate at double points to show over- and
under-passes.
![Page 8: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/8.jpg)
Classical link diagrams
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
3 / 47
A knot is a smooth simple closed curve in the 3-space.
A link is a union of several disjoint knots.
To describe a knot graphically, project it to a plane
and decorate at double points to show over- and
under-passes.
![Page 9: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/9.jpg)
Classical link diagrams
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
3 / 47
A knot is a smooth simple closed curve in the 3-space.
A link is a union of several disjoint knots.
To describe a knot graphically, project it to a plane
and decorate at double points to show over- and
under-passes. This gives rise to a knot diagram:
![Page 10: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/10.jpg)
Classical link diagrams
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
3 / 47
A knot is a smooth simple closed curve in the 3-space.
A link is a union of several disjoint knots.
To describe a knot graphically, project it to a plane
and decorate at double points to show over- and
under-passes. This gives rise to a knot diagram:
![Page 11: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/11.jpg)
Classical link diagrams
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
3 / 47
A knot is a smooth simple closed curve in the 3-space.
A link is a union of several disjoint knots.
To describe a knot graphically, project it to a plane
and decorate at double points to show over- and
under-passes. This gives rise to a knot diagram:
A link diagram:
![Page 12: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/12.jpg)
1D-picture
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
4 / 47
A knot diagram
![Page 13: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/13.jpg)
1D-picture
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
4 / 47
A knot diagram is a 2D picture of knot.
![Page 14: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/14.jpg)
1D-picture
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
4 / 47
A knot diagram is a 2D picture of knot.
In many cases 1D picture serves better.
![Page 15: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/15.jpg)
1D-picture
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
4 / 47
A knot diagram is a 2D picture of knot.
In many cases 1D picture serves better.
1D picture comes from a parameterization.
![Page 16: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/16.jpg)
1D-picture
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
4 / 47
A knot diagram is a 2D picture of knot.
In many cases 1D picture serves better.
1D picture comes from a parameterization.
initial
point
12
3
4
1
23
4
2
1 4
3
![Page 17: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/17.jpg)
Gauss diagram
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
5 / 47
Decorate the source:
initial
point
12
3
4
1
23
4
2
1 4
3
![Page 18: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/18.jpg)
Gauss diagram
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
5 / 47
Decorate the source:
• with arrows from overpass to underpass,
initial
point
12
3
4
1
23
4
2
1 4
3
![Page 19: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/19.jpg)
Gauss diagram
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
5 / 47
Decorate the source:
• with arrows from overpass to underpass,
initial
point
12
3
4
1
23
4
2
1 4
3
![Page 20: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/20.jpg)
Gauss diagram
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
5 / 47
Decorate the source:
• with arrows from overpass to underpass,
• with the signs of crossings
initial
point
12
3
4
1
23
4
2
1 4
3
![Page 21: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/21.jpg)
Gauss diagram
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
5 / 47
Decorate the source:
• with arrows from overpass to underpass,
• with the signs of crossings
1
23
4
2
1 4
3
+−−
+
![Page 22: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/22.jpg)
Gauss diagram
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
5 / 47
Decorate the source:
• with arrows from overpass to underpass,
• with the signs of crossings
1
23
4
2
1 4
3
+−−
+
Signs:
![Page 23: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/23.jpg)
Gauss diagram
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
5 / 47
Decorate the source:
• with arrows from overpass to underpass,
• with the signs of crossings
1
23
4
2
1 4
3
+−−
+
Signs: positive
![Page 24: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/24.jpg)
Gauss diagram
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
5 / 47
Decorate the source:
• with arrows from overpass to underpass,
• with the signs of crossings
1
23
4
2
1 4
3
+−−
+
Signs: positive , negative .
![Page 25: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/25.jpg)
Gauss diagram
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
5 / 47
Decorate the source:
• with arrows from overpass to underpass,
• with the signs of crossings
1
23
4
2
1 4
3
+−−
+
Signs: positive , negative . The result
![Page 26: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/26.jpg)
Gauss diagram
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
5 / 47
Decorate the source:
• with arrows from overpass to underpass,
• with the signs of crossings
1
23
4
2
1 4
3
+−−
+
Signs: positive , negative . The result,
1
23
4
2
1 4
3
+−−
+
![Page 27: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/27.jpg)
Gauss diagram
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
5 / 47
Decorate the source:
• with arrows from overpass to underpass,
• with the signs of crossings
1
23
4
2
1 4
3
+−−
+
Signs: positive , negative . The result,
1
23
4
2
1 4
3
+−−
+, is called a Gauss diagram of the knot.
![Page 28: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/28.jpg)
Reconstruction of knot
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
6 / 47
Take any such diagram, say,
1
+ +
−32
and try to reconstruct the knot.
![Page 29: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/29.jpg)
Reconstruction of knot
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
6 / 47
1
+ +
−32
Start with crossings:
![Page 30: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/30.jpg)
Reconstruction of knot
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
6 / 47
1
+ +
−32
Start with crossings:
2
3
1
![Page 31: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/31.jpg)
Reconstruction of knot
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
6 / 47
1
+ +
−32
Connect them step by step:
2
3
1
![Page 32: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/32.jpg)
Reconstruction of knot
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
6 / 47
1
+ +
−32
Connect them step by step:
2
3
1
![Page 33: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/33.jpg)
Reconstruction of knot
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
6 / 47
1
+ +
−32
The next step does not work!
2
3
1
![Page 34: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/34.jpg)
Reconstruction of knot
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
6 / 47
1
+ +
−32
But let us continue!
2
3
1
![Page 35: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/35.jpg)
Reconstruction of knot
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
6 / 47
1
+ +
−32
Yet another obstruction!
2
3
1
![Page 36: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/36.jpg)
Reconstruction of knot
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
6 / 47
1
+ +
−32
We did it!
2
3
1
![Page 37: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/37.jpg)
Reconstruction of knot
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
6 / 47
1
+ +
−32
We did it! But what is the result?
2
3
1
![Page 38: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/38.jpg)
Reconstruction of knot
Knots and links• Classical linkdiagrams
• 1D-picture
• Gauss diagram
• Reconstruction ofknot
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
6 / 47
1
+ +
−32
We did it! But what is the result?
2
3
1
The result is called a virtual knot diagram.
![Page 39: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/39.jpg)
Virtual links
Knots and links
Virtual links
• Virtual knot diagrams
• Diagram on a surface
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
7 / 47
![Page 40: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/40.jpg)
Virtual knot diagrams
Knots and links
Virtual links
• Virtual knot diagrams
• Diagram on a surface
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
8 / 47
A virtual knot diagram has crossings of 2 types:
![Page 41: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/41.jpg)
Virtual knot diagrams
Knots and links
Virtual links
• Virtual knot diagrams
• Diagram on a surface
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
8 / 47
A virtual knot diagram has crossings of 2 types:classical
![Page 42: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/42.jpg)
Virtual knot diagrams
Knots and links
Virtual links
• Virtual knot diagrams
• Diagram on a surface
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
8 / 47
A virtual knot diagram has crossings of 2 types:classical or real
![Page 43: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/43.jpg)
Virtual knot diagrams
Knots and links
Virtual links
• Virtual knot diagrams
• Diagram on a surface
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
8 / 47
A virtual knot diagram has crossings of 2 types:classical or real decorated like in a knot diagram
![Page 44: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/44.jpg)
Virtual knot diagrams
Knots and links
Virtual links
• Virtual knot diagrams
• Diagram on a surface
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
8 / 47
A virtual knot diagram has crossings of 2 types:classical or real decorated like in a knot diagramand virtual
![Page 45: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/45.jpg)
Virtual knot diagrams
Knots and links
Virtual links
• Virtual knot diagrams
• Diagram on a surface
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
8 / 47
A virtual knot diagram has crossings of 2 types:classical or real decorated like in a knot diagramand virtual not decorated at all.
![Page 46: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/46.jpg)
Virtual knot diagrams
Knots and links
Virtual links
• Virtual knot diagrams
• Diagram on a surface
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
8 / 47
A virtual knot diagram has crossings of 2 types:classical or real decorated like in a knot diagramand virtual not decorated at all.Who can help to get rid of virtual crossings?
![Page 47: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/47.jpg)
Virtual knot diagrams
Knots and links
Virtual links
• Virtual knot diagrams
• Diagram on a surface
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
8 / 47
A virtual knot diagram has crossings of 2 types:classical or real decorated like in a knot diagramand virtual not decorated at all.Who can help to get rid of virtual crossings?Handles!
![Page 48: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/48.jpg)
Virtual knot diagrams
Knots and links
Virtual links
• Virtual knot diagrams
• Diagram on a surface
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
8 / 47
A virtual knot diagram has crossings of 2 types:classical or real decorated like in a knot diagramand virtual not decorated at all.Who can help to get rid of virtual crossings?Handles!
2
3
1
![Page 49: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/49.jpg)
Diagram on a surface
Knots and links
Virtual links
• Virtual knot diagrams
• Diagram on a surface
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
9 / 47
A knot diagram drawn on orientable surface S
![Page 50: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/50.jpg)
Diagram on a surface
Knots and links
Virtual links
• Virtual knot diagrams
• Diagram on a surface
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
9 / 47
A knot diagram drawn on orientable surface S ,instead of the plane
![Page 51: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/51.jpg)
Diagram on a surface
Knots and links
Virtual links
• Virtual knot diagrams
• Diagram on a surface
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
9 / 47
A knot diagram drawn on orientable surface S ,instead of the plane, defines a knotin a thickened surface S × I .
![Page 52: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/52.jpg)
Diagram on a surface
Knots and links
Virtual links
• Virtual knot diagrams
• Diagram on a surface
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
9 / 47
A knot diagram drawn on orientable surface S ,instead of the plane, defines a knotin a thickened surface S × I .It defines also a Gauss diagram.
![Page 53: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/53.jpg)
Diagram on a surface
Knots and links
Virtual links
• Virtual knot diagrams
• Diagram on a surface
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
9 / 47
A knot diagram drawn on orientable surface S ,instead of the plane, defines a knotin a thickened surface S × I .It defines also a Gauss diagram.Any Gauss diagram appears in this way.
![Page 54: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/54.jpg)
Diagram on a surface
Knots and links
Virtual links
• Virtual knot diagrams
• Diagram on a surface
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
9 / 47
A knot diagram drawn on orientable surface S ,instead of the plane, defines a knotin a thickened surface S × I .It defines also a Gauss diagram.Any Gauss diagram appears in this way.For each Gauss diagram there is the smallest surface
![Page 55: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/55.jpg)
Diagram on a surface
Knots and links
Virtual links
• Virtual knot diagrams
• Diagram on a surface
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
9 / 47
A knot diagram drawn on orientable surface S ,instead of the plane, defines a knotin a thickened surface S × I .It defines also a Gauss diagram.Any Gauss diagram appears in this way.For each Gauss diagram there is the smallest surfacewith a knot diagram defining this Gauss diagram.
![Page 56: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/56.jpg)
Diagram on a surface
Knots and links
Virtual links
• Virtual knot diagrams
• Diagram on a surface
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
9 / 47
A knot diagram drawn on orientable surface S ,instead of the plane, defines a knotin a thickened surface S × I .It defines also a Gauss diagram.Any Gauss diagram appears in this way.For each Gauss diagram there is the smallest surfacewith a knot diagram defining this Gauss diagram.
Virtual knot diagrams emerge as projections to plane
![Page 57: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/57.jpg)
Diagram on a surface
Knots and links
Virtual links
• Virtual knot diagrams
• Diagram on a surface
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
9 / 47
A knot diagram drawn on orientable surface S ,instead of the plane, defines a knotin a thickened surface S × I .It defines also a Gauss diagram.Any Gauss diagram appears in this way.For each Gauss diagram there is the smallest surfacewith a knot diagram defining this Gauss diagram.
Virtual knot diagrams emerge as projections to plane of knotdiagrams on a surface.
![Page 58: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/58.jpg)
Diagram on a surface
Knots and links
Virtual links
• Virtual knot diagrams
• Diagram on a surface
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
9 / 47
A knot diagram drawn on orientable surface S ,instead of the plane, defines a knotin a thickened surface S × I .It defines also a Gauss diagram.Any Gauss diagram appears in this way.For each Gauss diagram there is the smallest surfacewith a knot diagram defining this Gauss diagram.
Virtual knot diagrams emerge as projections to plane of knotdiagrams on a surface.The surfaces is not unique:
![Page 59: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/59.jpg)
Diagram on a surface
Knots and links
Virtual links
• Virtual knot diagrams
• Diagram on a surface
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
9 / 47
A knot diagram drawn on orientable surface S ,instead of the plane, defines a knotin a thickened surface S × I .It defines also a Gauss diagram.Any Gauss diagram appears in this way.For each Gauss diagram there is the smallest surfacewith a knot diagram defining this Gauss diagram.
Virtual knot diagrams emerge as projections to plane of knotdiagrams on a surface.The surfaces is not unique: one can add handles.
![Page 60: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/60.jpg)
Moves
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
10 / 47
![Page 61: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/61.jpg)
Moves
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
11 / 47
What happens to a link diagram, when the link moves?
![Page 62: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/62.jpg)
Moves
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
11 / 47
What happens to a link diagram, when the link moves?Link diagram moves, too.
![Page 63: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/63.jpg)
Moves
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
11 / 47
What happens to a link diagram, when the link moves?Link diagram moves, too.
Reidemeister moves:
![Page 64: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/64.jpg)
Moves
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
11 / 47
What happens to a link diagram, when the link moves?Link diagram moves, too.
Reidemeister moves:
(R1):
![Page 65: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/65.jpg)
Moves
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
11 / 47
What happens to a link diagram, when the link moves?Link diagram moves, too.
Reidemeister moves:
(R1):
![Page 66: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/66.jpg)
Moves
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
11 / 47
What happens to a link diagram, when the link moves?Link diagram moves, too.
Reidemeister moves:
(R1):
![Page 67: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/67.jpg)
Moves
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
11 / 47
What happens to a link diagram, when the link moves?Link diagram moves, too.
Reidemeister moves:
(R1):
(R2):
![Page 68: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/68.jpg)
Moves
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
11 / 47
What happens to a link diagram, when the link moves?Link diagram moves, too.
Reidemeister moves:
(R1):
(R2):
![Page 69: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/69.jpg)
Moves
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
11 / 47
What happens to a link diagram, when the link moves?Link diagram moves, too.
Reidemeister moves:
(R1):
(R2):
![Page 70: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/70.jpg)
Moves
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
11 / 47
What happens to a link diagram, when the link moves?Link diagram moves, too.
Reidemeister moves:
(R1):
(R2):
(R3):
![Page 71: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/71.jpg)
Moves
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
11 / 47
What happens to a link diagram, when the link moves?Link diagram moves, too.
Reidemeister moves:
(R1):
(R2):
(R3):
![Page 72: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/72.jpg)
Moves
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
11 / 47
What happens to a link diagram, when the link moves?Link diagram moves, too.
Reidemeister moves:
(R1):
(R2):
(R3):
![Page 73: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/73.jpg)
Moves of virtual link diagram
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
12 / 47
A virtual link diagram
(i.e., a plane projection of a link diagram on a surface)
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![Page 74: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/74.jpg)
Moves of virtual link diagram
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
12 / 47
A virtual link diagram moves like this:
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![Page 75: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/75.jpg)
Moves of virtual link diagram
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
12 / 47
A virtual link diagram moves like this:
Reidemeister moves:
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![Page 76: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/76.jpg)
Moves of virtual link diagram
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
12 / 47
A virtual link diagram moves like this:
Reidemeister moves:
Virtual moves:
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![Page 77: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/77.jpg)
Moves of virtual link diagram
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
12 / 47
A virtual link diagram moves like this:
Reidemeister moves:
Virtual moves:
All virtual moves can be replaced by detour moves:
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![Page 78: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/78.jpg)
Moves of Gauss diagrams
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
13 / 47
Gauss diagrams has nothing to do with virtual crossings!
![Page 79: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/79.jpg)
Moves of Gauss diagrams
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
13 / 47
Gauss diagrams has nothing to do with virtual crossings!They do not change under virtual moves.
![Page 80: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/80.jpg)
Moves of Gauss diagrams
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
13 / 47
Reidemeister moves acts on Gauss diagram:
![Page 81: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/81.jpg)
Moves of Gauss diagrams
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
13 / 47
Reidemeister moves acts on Gauss diagram:
Move’sname
Reidemeistermove
Its action on Gauss diagram
Positivefirstmove
Nega-tive firstmove
![Page 82: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/82.jpg)
Moves of Gauss diagrams
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
13 / 47
Reidemeister moves acts on Gauss diagram:
Move’sname
Reidemeistermove
Its action on Gauss diagram
Positivefirstmove
+
Nega-tive firstmove
![Page 83: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/83.jpg)
Moves of Gauss diagrams
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
13 / 47
Reidemeister moves acts on Gauss diagram:
Move’sname
Reidemeistermove
Its action on Gauss diagram
Positivefirstmove
+
Nega-tive firstmove
![Page 84: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/84.jpg)
Moves of Gauss diagrams
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
13 / 47
Reidemeister moves acts on Gauss diagram:
Move’sname
Reidemeistermove
Its action on Gauss diagram
Positivefirstmove
+
Nega-tive firstmove
−
![Page 85: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/85.jpg)
Moves of Gauss diagrams
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
13 / 47
Reidemeister moves acts on Gauss diagram:
Move’sname
Reidemeistermove
Its action on Gauss diagram
Secondmove
Thirdmove
![Page 86: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/86.jpg)
Moves of Gauss diagrams
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
13 / 47
Reidemeister moves acts on Gauss diagram:
Move’sname
Reidemeistermove
Its action on Gauss diagram
Secondmove
−ε
ε
Thirdmove
![Page 87: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/87.jpg)
Moves of Gauss diagrams
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
13 / 47
Reidemeister moves acts on Gauss diagram:
Move’sname
Reidemeistermove
Its action on Gauss diagram
Secondmove
−ε
ε
Thirdmove
![Page 88: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/88.jpg)
Moves of Gauss diagrams
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
13 / 47
Reidemeister moves acts on Gauss diagram:
Move’sname
Reidemeistermove
Its action on Gauss diagram
Secondmove
−ε
ε
Thirdmove β
γ
α
β
γ
α
![Page 89: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/89.jpg)
Combinatorial incarnation of knot theory
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
14 / 47
![Page 90: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/90.jpg)
Combinatorial incarnation of knot theory
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
14 / 47
Classical Links → Link diagrams
![Page 91: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/91.jpg)
Combinatorial incarnation of knot theory
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
14 / 47
Classical Links → Link diagramsIsotopies → Reidemeister moves
![Page 92: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/92.jpg)
Combinatorial incarnation of knot theory
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
14 / 47
Classical Links → Link diagramsIsotopies → Reidemeister moves
Combinatorial incarnations of virtual knot theory
![Page 93: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/93.jpg)
Combinatorial incarnation of knot theory
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
14 / 47
Classical Links → Link diagramsIsotopies → Reidemeister moves
Combinatorial incarnations of virtual knot theory
GaussDiagrams
←Virtual Links(?)
→Virtual LinkDiagrams
ReidemeisterMoves
←VirtualIstopies (?)
→Reidemeisterand Detourmoves
![Page 94: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/94.jpg)
Topological meaning of virtual knot theory
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
15 / 47
Third incarnation of virtual knot theoryis provided by Kuperberg’s theorem.
![Page 95: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/95.jpg)
Topological meaning of virtual knot theory
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
15 / 47
Third incarnation of virtual knot theoryis provided by Kuperberg’s theorem.
Virtual links up tovirtual isotopies
=
Irreducible links inthickened orientablesurfaces up to ori-entation preservinghomeomorphisms.
![Page 96: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/96.jpg)
Topological meaning of virtual knot theory
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
15 / 47
Third incarnation of virtual knot theoryis provided by Kuperberg’s theorem.
Virtual links up tovirtual isotopies
=
Irreducible links inthickened orientablesurfaces up to ori-entation preservinghomeomorphisms.
Implies that virtual links generalize classical ones.
![Page 97: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/97.jpg)
Topological meaning of virtual knot theory
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
15 / 47
Third incarnation of virtual knot theoryis provided by Kuperberg’s theorem.
Virtual links up tovirtual isotopies
=
Irreducible links inthickened orientablesurfaces up to ori-entation preservinghomeomorphisms.
Implies that virtual links generalize classical ones.
Bridges combinatorics
![Page 98: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/98.jpg)
Topological meaning of virtual knot theory
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
15 / 47
Third incarnation of virtual knot theoryis provided by Kuperberg’s theorem.
Virtual links up tovirtual isotopies
=
Irreducible links inthickened orientablesurfaces up to ori-entation preservinghomeomorphisms.
Implies that virtual links generalize classical ones.
Bridges combinatorics (= 1D topology)
![Page 99: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/99.jpg)
Topological meaning of virtual knot theory
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
15 / 47
Third incarnation of virtual knot theoryis provided by Kuperberg’s theorem.
Virtual links up tovirtual isotopies
=
Irreducible links inthickened orientablesurfaces up to ori-entation preservinghomeomorphisms.
Implies that virtual links generalize classical ones.
Bridges combinatorics with (3D-) topology.
![Page 100: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/100.jpg)
Isotopy problem
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
16 / 47
Isotopy Problem:
![Page 101: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/101.jpg)
Isotopy problem
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
16 / 47
Isotopy Problem: Are given two classical links isotopic?
![Page 102: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/102.jpg)
Isotopy problem
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
16 / 47
Isotopy Problem: Are given two classical links isotopic?
Combinatorial reformulation:
![Page 103: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/103.jpg)
Isotopy problem
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
16 / 47
Isotopy Problem: Are given two classical links isotopic?
Combinatorial reformulation:Can given two Gauss diagrams be related by moves?
![Page 104: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/104.jpg)
Isotopy problem
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
16 / 47
Isotopy Problem: Are given two classical links isotopic?
Combinatorial reformulation:Can given two Gauss diagrams be related by moves?
Virtual Isotopy Problem:
![Page 105: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/105.jpg)
Isotopy problem
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
16 / 47
Isotopy Problem: Are given two classical links isotopic?
Combinatorial reformulation:Can given two Gauss diagrams be related by moves?
Virtual Isotopy Problem:Can given two Gauss diagrams be related by moves?
![Page 106: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/106.jpg)
Isotopy problem
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
16 / 47
Isotopy Problem: Are given two classical links isotopic?
Combinatorial reformulation:Can given two Gauss diagrams be related by moves?
Virtual Isotopy Problem:Can given two Gauss diagrams be related by moves?
Invariants needed!
![Page 107: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/107.jpg)
Isotopy problem
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
16 / 47
Isotopy Problem: Are given two classical links isotopic?
Combinatorial reformulation:Can given two Gauss diagrams be related by moves?
Virtual Isotopy Problem:Can given two Gauss diagrams be related by moves?
Invariants needed!The most classical link invariant is the link group.
![Page 108: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/108.jpg)
Isotopy problem
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
16 / 47
Isotopy Problem: Are given two classical links isotopic?
Combinatorial reformulation:Can given two Gauss diagrams be related by moves?
Virtual Isotopy Problem:Can given two Gauss diagrams be related by moves?
Invariants needed!The most classical link invariant is the link group,the fundamental group of the link complement R
3r link .
![Page 109: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/109.jpg)
Isotopy problem
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
16 / 47
Isotopy Problem: Are given two classical links isotopic?
Combinatorial reformulation:Can given two Gauss diagrams be related by moves?
Virtual Isotopy Problem:Can given two Gauss diagrams be related by moves?
Invariants needed!The most classical link invariant is the link group.It was generalized.
![Page 110: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/110.jpg)
Isotopy problem
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
16 / 47
Isotopy Problem: Are given two classical links isotopic?
Combinatorial reformulation:Can given two Gauss diagrams be related by moves?
Virtual Isotopy Problem:Can given two Gauss diagrams be related by moves?
Invariants needed!The most classical link invariant is the link group.It was generalized, even in two ways!
![Page 111: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/111.jpg)
Isotopy problem
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
16 / 47
Isotopy Problem: Are given two classical links isotopic?
Combinatorial reformulation:Can given two Gauss diagrams be related by moves?
Virtual Isotopy Problem:Can given two Gauss diagrams be related by moves?
Invariants needed!The most classical link invariant is the link group.It was generalized: upper and lower!
![Page 112: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/112.jpg)
Isotopy problem
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
16 / 47
Isotopy Problem: Are given two classical links isotopic?
Combinatorial reformulation:Can given two Gauss diagrams be related by moves?
Virtual Isotopy Problem:Can given two Gauss diagrams be related by moves?
Invariants needed!The most classical link invariant is the link group.It was generalized: upper and lower!In terms of links in a thickened surface this is the fundamentalgroup of the complement, but with one of two sides of theboundary contracted to a point.
![Page 113: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/113.jpg)
Isotopy problem
Knots and links
Virtual links
Moves
• Moves• Moves of virtual linkdiagram
• Moves of Gaussdiagrams
• Combinatorialincarnation of knottheory• Topological meaningof virtual knot theory
• Isotopy problem
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
16 / 47
Isotopy Problem: Are given two classical links isotopic?
Combinatorial reformulation:Can given two Gauss diagrams be related by moves?
Virtual Isotopy Problem:Can given two Gauss diagrams be related by moves?
Invariants needed!The most classical link invariant is the link group.It was generalized: upper and lower!In terms of links in a thickened surface this is the fundamentalgroup of the complement, but with one of two sides of theboundary contracted to a point.
Kauffman bracket is more practical and elementary invariant.
![Page 114: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/114.jpg)
Kauffman bracket
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
17 / 47
![Page 115: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/115.jpg)
Kauffman bracket
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
18 / 47
〈Link diagram〉 ∈ Z[A,A−1]
![Page 116: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/116.jpg)
Kauffman bracket
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
18 / 47
〈Link diagram〉 ∈ Z[A,A−1]
(a Laurent polynomial in A with integer coefficients).
![Page 117: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/117.jpg)
Kauffman bracket
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
18 / 47
〈Link diagram〉 ∈ Z[A,A−1]
〈unknot〉 =
![Page 118: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/118.jpg)
Kauffman bracket
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
18 / 47
〈Link diagram〉 ∈ Z[A,A−1]
〈unknot〉 = 〈©〉 =
![Page 119: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/119.jpg)
Kauffman bracket
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
18 / 47
〈Link diagram〉 ∈ Z[A,A−1]
〈unknot〉 = 〈©〉 = − A2 − A−2
![Page 120: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/120.jpg)
Kauffman bracket
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
18 / 47
〈Link diagram〉 ∈ Z[A,A−1]
〈unknot〉 = 〈©〉 = − A2 − A−2
〈Hopf link〉 =
![Page 121: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/121.jpg)
Kauffman bracket
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
18 / 47
〈Link diagram〉 ∈ Z[A,A−1]
〈unknot〉 = 〈©〉 = − A2 − A−2
〈Hopf link〉 = 〈 〉 =
![Page 122: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/122.jpg)
Kauffman bracket
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
18 / 47
〈Link diagram〉 ∈ Z[A,A−1]
〈unknot〉 = 〈©〉 = − A2 − A−2
〈Hopf link〉 = 〈 〉 = A6 + A2 + A−2 + A−6
![Page 123: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/123.jpg)
Kauffman bracket
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
18 / 47
〈Link diagram〉 ∈ Z[A,A−1]
〈unknot〉 = 〈©〉 = − A2 − A−2
〈Hopf link〉 = 〈 〉 = A6 + A2 + A−2 + A−6
〈empty link〉 = 〈 〉 =
![Page 124: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/124.jpg)
Kauffman bracket
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
18 / 47
〈Link diagram〉 ∈ Z[A,A−1]
〈unknot〉 = 〈©〉 = − A2 − A−2
〈Hopf link〉 = 〈 〉 = A6 + A2 + A−2 + A−6
〈empty link〉 = 〈 〉 = 1
![Page 125: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/125.jpg)
Kauffman bracket
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
18 / 47
〈Link diagram〉 ∈ Z[A,A−1]
〈unknot〉 = 〈©〉 = − A2 − A−2
〈Hopf link〉 = 〈 〉 = A6 + A2 + A−2 + A−6
〈empty link〉 = 〈 〉 = 1
〈trefoil〉 =
![Page 126: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/126.jpg)
Kauffman bracket
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
18 / 47
〈Link diagram〉 ∈ Z[A,A−1]
〈unknot〉 = 〈©〉 = − A2 − A−2
〈Hopf link〉 = 〈 〉 = A6 + A2 + A−2 + A−6
〈empty link〉 = 〈 〉 = 1
〈trefoil〉 = 〈 〉 =
![Page 127: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/127.jpg)
Kauffman bracket
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
18 / 47
〈Link diagram〉 ∈ Z[A,A−1]
〈unknot〉 = 〈©〉 = − A2 − A−2
〈Hopf link〉 = 〈 〉 = A6 + A2 + A−2 + A−6
〈empty link〉 = 〈 〉 = 1
〈trefoil〉 = 〈 〉 = A7 + A3 + A−1 − A−9
![Page 128: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/128.jpg)
Kauffman bracket
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
18 / 47
〈Link diagram〉 ∈ Z[A,A−1]
〈unknot〉 = 〈©〉 = − A2 − A−2
〈Hopf link〉 = 〈 〉 = A6 + A2 + A−2 + A−6
〈empty link〉 = 〈 〉 = 1
〈trefoil〉 = 〈 〉 = A7 + A3 + A−1 − A−9
〈figure-eight knot〉 =
![Page 129: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/129.jpg)
Kauffman bracket
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
18 / 47
〈Link diagram〉 ∈ Z[A,A−1]
〈unknot〉 = 〈©〉 = − A2 − A−2
〈Hopf link〉 = 〈 〉 = A6 + A2 + A−2 + A−6
〈empty link〉 = 〈 〉 = 1
〈trefoil〉 = 〈 〉 = A7 + A3 + A−1 − A−9
〈figure-eight knot〉 = 〈 〉 =
![Page 130: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/130.jpg)
Kauffman bracket
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
18 / 47
〈Link diagram〉 ∈ Z[A,A−1]
〈unknot〉 = 〈©〉 = − A2 − A−2
〈Hopf link〉 = 〈 〉 = A6 + A2 + A−2 + A−6
〈empty link〉 = 〈 〉 = 1
〈trefoil〉 = 〈 〉 = A7 + A3 + A−1 − A−9
〈figure-eight knot〉 = 〈 〉 = − A10 − A−10
![Page 131: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/131.jpg)
Kauffman bracket
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
18 / 47
〈Link diagram〉 ∈ Z[A,A−1]
〈unknot〉 = 〈©〉 = − A2 − A−2
〈Hopf link〉 = 〈 〉 = A6 + A2 + A−2 + A−6
〈empty link〉 = 〈 〉 = 1
〈trefoil〉 = 〈 〉 = A7 + A3 + A−1 − A−9
〈figure-eight knot〉 = 〈 〉 = − A10 − A−10
Kauffman bracket is defined by the following properties:
![Page 132: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/132.jpg)
Kauffman bracket
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
18 / 47
〈Link diagram〉 ∈ Z[A,A−1]
〈unknot〉 = 〈©〉 = − A2 − A−2
〈Hopf link〉 = 〈 〉 = A6 + A2 + A−2 + A−6
〈empty link〉 = 〈 〉 = 1
〈trefoil〉 = 〈 〉 = A7 + A3 + A−1 − A−9
〈figure-eight knot〉 = 〈 〉 = − A10 − A−10
Kauffman bracket is defined by the following properties:1. 〈©〉 = −A2 − A−2 ,
![Page 133: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/133.jpg)
Kauffman bracket
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
18 / 47
〈Link diagram〉 ∈ Z[A,A−1]
〈unknot〉 = 〈©〉 = − A2 − A−2
〈Hopf link〉 = 〈 〉 = A6 + A2 + A−2 + A−6
〈empty link〉 = 〈 〉 = 1
〈trefoil〉 = 〈 〉 = A7 + A3 + A−1 − A−9
〈figure-eight knot〉 = 〈 〉 = − A10 − A−10
Kauffman bracket is defined by the following properties:1. 〈©〉 = −A2 − A−2 ,2. 〈D ∐ ©〉 = (−A2 − A−2)〈D〉 ,
![Page 134: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/134.jpg)
Kauffman bracket
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
18 / 47
〈Link diagram〉 ∈ Z[A,A−1]
〈unknot〉 = 〈©〉 = − A2 − A−2
〈Hopf link〉 = 〈 〉 = A6 + A2 + A−2 + A−6
〈empty link〉 = 〈 〉 = 1
〈trefoil〉 = 〈 〉 = A7 + A3 + A−1 − A−9
〈figure-eight knot〉 = 〈 〉 = − A10 − A−10
Kauffman bracket is defined by the following properties:1. 〈©〉 = −A2 − A−2 ,2. 〈D ∐ ©〉 = (−A2 − A−2)〈D〉 ,3. 〈 〉 = A〈 〉+ A−1〈 〉 (Kauffman Skein Relation) .
![Page 135: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/135.jpg)
Kauffman bracket
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
18 / 47
〈Link diagram〉 ∈ Z[A,A−1]
〈unknot〉 = 〈©〉 = − A2 − A−2
〈Hopf link〉 = 〈 〉 = A6 + A2 + A−2 + A−6
〈empty link〉 = 〈 〉 = 1
〈trefoil〉 = 〈 〉 = A7 + A3 + A−1 − A−9
〈figure-eight knot〉 = 〈 〉 = − A10 − A−10
Kauffman bracket is defined by the following properties:1. 〈©〉 = −A2 − A−2 ,2. 〈D ∐ ©〉 = (−A2 − A−2)〈D〉 ,3. 〈 〉 = A〈 〉+ A−1〈 〉 (Kauffman Skein Relation) .Uniqueness is obvious.
![Page 136: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/136.jpg)
Kauffman bracket
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
18 / 47
〈Link diagram〉 ∈ Z[A,A−1]
〈unknot〉 = 〈©〉 = − A2 − A−2
〈Hopf link〉 = 〈 〉 = A6 + A2 + A−2 + A−6
〈empty link〉 = 〈 〉 = 1
〈trefoil〉 = 〈 〉 = A7 + A3 + A−1 − A−9
〈figure-eight knot〉 = 〈 〉 = − A10 − A−10
Kauffman bracket is defined by the following properties:1. 〈©〉 = −A2 − A−2 ,2. 〈D ∐ ©〉 = (−A2 − A−2)〈D〉 ,3. 〈 〉 = A〈 〉+ A−1〈 〉 (Kauffman Skein Relation) .Uniqueness is obvious.Invariant under R2 and R3, under R1 multiplies by −A±3.
![Page 137: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/137.jpg)
Kauffman state sum. I
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
19 / 47
A state of diagram is a distribution of markers over allcrossings.
![Page 138: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/138.jpg)
Kauffman state sum. I
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
19 / 47
A state of diagram is a distribution of markers over allcrossings.
Knot diagram:
![Page 139: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/139.jpg)
Kauffman state sum. I
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
19 / 47
A state of diagram is a distribution of markers over allcrossings.
Knot diagram: and its states:
![Page 140: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/140.jpg)
Kauffman state sum. I
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
19 / 47
A state of diagram is a distribution of markers over allcrossings.
Knot diagram: and its states:
,
![Page 141: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/141.jpg)
Kauffman state sum. I
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
19 / 47
A state of diagram is a distribution of markers over allcrossings.
Knot diagram: and its states:
, ,
![Page 142: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/142.jpg)
Kauffman state sum. I
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
19 / 47
A state of diagram is a distribution of markers over allcrossings.
Knot diagram: and its states:
, , ,
![Page 143: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/143.jpg)
Kauffman state sum. I
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
19 / 47
A state of diagram is a distribution of markers over allcrossings.
Knot diagram: and its states:
, , , ,
![Page 144: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/144.jpg)
Kauffman state sum. I
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
19 / 47
A state of diagram is a distribution of markers over allcrossings.
Knot diagram: and its states:
, , , , . . .
![Page 145: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/145.jpg)
Kauffman state sum. I
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
19 / 47
A state of diagram is a distribution of markers over allcrossings.
Knot diagram: and its states:
, , , , . . .
Totally 2c states, where c is the number of crossings.
![Page 146: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/146.jpg)
Kauffman state sum. II
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
20 / 47
Three numbers associated to a state s :
![Page 147: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/147.jpg)
Kauffman state sum. II
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
20 / 47
Three numbers associated to a state s :
1. the number a(s) of positive markers ,
![Page 148: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/148.jpg)
Kauffman state sum. II
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
20 / 47
Three numbers associated to a state s :
1. the number a(s) of positive markers ,
2. the number b(s) of negative markers ,
![Page 149: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/149.jpg)
Kauffman state sum. II
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
20 / 47
Three numbers associated to a state s :
1. the number a(s) of positive markers ,
2. the number b(s) of negative markers ,
3. the number |s| of components of the curve obtained bysmoothing along the markers:
![Page 150: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/150.jpg)
Kauffman state sum. II
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
20 / 47
Three numbers associated to a state s :
1. the number a(s) of positive markers ,
2. the number b(s) of negative markers ,
3. the number |s| of components of the curve obtained bysmoothing along the markers:
s = 7→
![Page 151: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/151.jpg)
Kauffman state sum. II
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
20 / 47
Three numbers associated to a state s :
1. the number a(s) of positive markers ,
2. the number b(s) of negative markers ,
3. the number |s| of components of the curve obtained bysmoothing along the markers:
s = 7→ smoothing(s) =
![Page 152: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/152.jpg)
Kauffman state sum. II
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
20 / 47
Three numbers associated to a state s :
1. the number a(s) of positive markers ,
2. the number b(s) of negative markers ,
3. the number |s| of components of the curve obtained bysmoothing along the markers:
s = 7→ smoothing(s) =
|s| = 2
![Page 153: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/153.jpg)
Kauffman state sum. II
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
20 / 47
Three numbers associated to a state s :
1. the number a(s) of positive markers ,
2. the number b(s) of negative markers ,
3. the number |s| of components of the curve obtained bysmoothing along the markers:
s = 7→ smoothing(s) =
|s| = 2
State Sum: 〈D〉 =∑
s state of D Aa(s)−b(s)(−A2 − A−2)|s|
![Page 154: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/154.jpg)
Example
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
21 / 47
Hopf link,
![Page 155: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/155.jpg)
Example
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
21 / 47
Hopf link,⟨ ⟩
=
![Page 156: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/156.jpg)
Example
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
21 / 47
Hopf link,⟨ ⟩
=
⟨ ⟩
+
⟨ ⟩
+
⟨ ⟩
+
⟨ ⟩
=
![Page 157: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/157.jpg)
Example
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
21 / 47
Hopf link,⟨ ⟩
=
⟨ ⟩
+
⟨ ⟩
+
⟨ ⟩
+
⟨ ⟩
=
A2(−A2−A−2)2 + 2(−A2−A−2) + A−2(−A2−A−2)2 =
![Page 158: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/158.jpg)
Example
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
21 / 47
Hopf link,⟨ ⟩
=
⟨ ⟩
+
⟨ ⟩
+
⟨ ⟩
+
⟨ ⟩
=
A2(−A2−A−2)2 + 2(−A2−A−2) + A−2(−A2−A−2)2 =
A6 + A2 + A−2 + A−6
![Page 159: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/159.jpg)
Kauffman state sum model for Gauss diagrams
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
22 / 47
Crossing 7→ arrow.
7→−
7→+
![Page 160: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/160.jpg)
Kauffman state sum model for Gauss diagrams
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
22 / 47
Crossing 7→ arrow.
7→−
7→+
Smoothing of a crossing 7→ a surgery along the arrow.
![Page 161: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/161.jpg)
Kauffman state sum model for Gauss diagrams
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
22 / 47
Crossing 7→ arrow.
7→−
7→+
Smoothing of a crossing 7→ a surgery along the arrow.
+7→
−7→
positive marker, positive crossing negative marker, negative crossing
![Page 162: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/162.jpg)
Kauffman state sum model for Gauss diagrams
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
22 / 47
Crossing 7→ arrow.
7→−
7→+
Smoothing of a crossing 7→ a surgery along the arrow.
+7→
−7→
positive marker, positive crossing negative marker, negative crossing
+7→
−7→
positive marker, negative crossingnegative marker, positive crossing
![Page 163: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/163.jpg)
Kauffman state sum model for Gauss diagrams
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
22 / 47
Crossing 7→ arrow.
7→−
7→+
Smoothing of a crossing 7→ a surgery along the arrow.
+7→
−7→
positive marker, positive crossing negative marker, negative crossing
+7→
−7→
positive marker, negative crossingnegative marker, positive crossing
Smoothing depends only of the signs of marker and crossing.
![Page 164: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/164.jpg)
Kauffman state sum model for Gauss diagrams
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
22 / 47
Crossing 7→ arrow.
7→−
7→+
Smoothing of a crossing 7→ a surgery along the arrow.
+7→
−7→
positive marker, positive crossing negative marker, negative crossing
+7→
−7→
positive marker, negative crossingnegative marker, positive crossing
Smoothing depends only of the signs of marker and crossing.No need in direction of the arrow!
![Page 165: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/165.jpg)
Kauffman state sum model for Gauss diagrams
Knots and links
Virtual links
Moves
Kauffman bracket
• Kauffman bracket
• Kauffman state sum. I• Kauffman state sum.II
• Example• Kauffman state summodel for Gaussdiagrams
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
22 / 47
Crossing 7→ arrow.
7→−
7→+
Smoothing of a crossing 7→ a surgery along the arrow.
+7→
−7→
positive marker, positive crossing negative marker, negative crossing
+7→
−7→
positive marker, negative crossingnegative marker, positive crossing
Smoothing depends only of the signs of marker and crossing.No need in direction of the arrow!Kauffman state sum is defined for signed chord diagrams.
![Page 166: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/166.jpg)
Gauss diagrams of a poorman
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
23 / 47
![Page 167: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/167.jpg)
Signed chord diagrams
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
24 / 47
A chord diagram (B, c1, . . . , cn)
![Page 168: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/168.jpg)
Signed chord diagrams
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
24 / 47
A chord diagram (B, c1, . . . , cn)( a closed 1-manifold B (base), and
B
![Page 169: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/169.jpg)
Signed chord diagrams
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
24 / 47
A chord diagram (B, c1, . . . , cn)( a closed 1-manifold B (base), anddisjoint chords c1, . . . , cn with end points on the base.)
![Page 170: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/170.jpg)
Signed chord diagrams
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
24 / 47
A chord diagram (B, c1, . . . , cn)( a closed 1-manifold B (base), anddisjoint chords c1, . . . , cn with end points on the base.)• in which B is oriented and
![Page 171: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/171.jpg)
Signed chord diagrams
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
24 / 47
A chord diagram (B, c1, . . . , cn)( a closed 1-manifold B (base), anddisjoint chords c1, . . . , cn with end points on the base.)• in which B is oriented and• each chord is equipped with a sign
−+
−−−
+ +
![Page 172: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/172.jpg)
Signed chord diagrams
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
24 / 47
A chord diagram (B, c1, . . . , cn)( a closed 1-manifold B (base), anddisjoint chords c1, . . . , cn with end points on the base.)• in which B is oriented and• each chord is equipped with a signis called a signed chord diagram.
−+
−−−
+ +
![Page 173: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/173.jpg)
State of signed chord diagram
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
25 / 47
A state of the signed chord diagramis a distribution of another collection of signs over the set of allchords.
![Page 174: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/174.jpg)
State of signed chord diagram
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
25 / 47
A state of the signed chord diagramis a distribution of another collection of signs over the set of allchords.These are marker signs,
![Page 175: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/175.jpg)
State of signed chord diagram
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
25 / 47
A state of the signed chord diagramis a distribution of another collection of signs over the set of allchords.These are marker signs, the original signs are structure signs.
![Page 176: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/176.jpg)
Smoothing of a signed chord diagram
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
26 / 47
A smoothing of a chord diagram (B, c1, . . . , cn) is the resultof Morse modifications of index 1 performed on B along eachof its chords.
![Page 177: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/177.jpg)
Smoothing of a signed chord diagram
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
26 / 47
A smoothing of a chord diagram (B, c1, . . . , cn) is the resultof Morse modifications of index 1 performed on B along eachof its chords.
+ +
(−)
(+)
(+) (−)
−
−
![Page 178: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/178.jpg)
Smoothing of a signed chord diagram
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
26 / 47
A smoothing of a chord diagram (B, c1, . . . , cn) is the resultof Morse modifications of index 1 performed on B along eachof its chords.
+ +
(−)
(+)
(+) (−)
−
−
+ +
(−)
(+)
(+) (−)
−
−
![Page 179: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/179.jpg)
Smoothing of a signed chord diagram
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
26 / 47
A smoothing of a chord diagram (B, c1, . . . , cn) is the resultof Morse modifications of index 1 performed on B along eachof its chords.
+ +
(−)
(+)
(+) (−)
−
−
+ +
(−)
(+)
(+) (−)
−
−
![Page 180: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/180.jpg)
Smoothing of a signed chord diagram
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
26 / 47
A smoothing of a chord diagram (B, c1, . . . , cn) is the resultof Morse modifications of index 1 performed on B along eachof its chords.
+ +
(−)
(+)
(+) (−)
−
−
+ +
(−)
(+)
(+) (−)
−
−
![Page 181: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/181.jpg)
Smoothing of a signed chord diagram
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
26 / 47
A smoothing of a chord diagram (B, c1, . . . , cn) is the resultof Morse modifications of index 1 performed on B along eachof its chords.
+ +
(−)
(+)
(+) (−)
−
−
+ +
(−)
(+)
(+) (−)
−
−
Morse modification at a chord depends on its signs.
![Page 182: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/182.jpg)
Smoothing of a signed chord diagram
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
26 / 47
A smoothing of a chord diagram (B, c1, . . . , cn) is the resultof Morse modifications of index 1 performed on B along eachof its chords.
+ +
(−)
(+)
(+) (−)
−
−
+ +
(−)
(+)
(+) (−)
−
−
Morse modification at a chord depends on its signs.Denote by σ the product of the structure and the marker signs.
![Page 183: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/183.jpg)
Smoothing of a signed chord diagram
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
26 / 47
A smoothing of a chord diagram (B, c1, . . . , cn) is the resultof Morse modifications of index 1 performed on B along eachof its chords.
+ +
(−)
(+)
(+) (−)
−
−
+ +
(−)
(+)
(+) (−)
−
−
Morse modification at a chord depends on its signs.Denote by σ the product of the structure and the marker signs.If σ = + , the Morse modification preserves the structureorientation.
![Page 184: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/184.jpg)
Smoothing of a signed chord diagram
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
26 / 47
A smoothing of a chord diagram (B, c1, . . . , cn) is the resultof Morse modifications of index 1 performed on B along eachof its chords.
+ +
(−)
(+)
(+) (−)
−
−
+ +
(−)
(+)
(+) (−)
−
−
Morse modification at a chord depends on its signs.Denote by σ the product of the structure and the marker signs.If σ = + , the Morse modification preserves the structureorientation.If σ = − , the Morse modification destroys the orientation.
![Page 185: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/185.jpg)
Framing
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
27 / 47
A sign of an arrow in Gauss diagram of a classical linkdepends on orientation of the link.
![Page 186: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/186.jpg)
Framing
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
27 / 47
A sign of an arrow in Gauss diagram of a classical linkdepends on orientation of the link.
−
+
![Page 187: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/187.jpg)
Framing
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
27 / 47
A sign of an arrow in Gauss diagram of a classical linkdepends on orientation of the link.
−
+
+
−
![Page 188: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/188.jpg)
Framing
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
27 / 47
A sign of an arrow in Gauss diagram of a classical linkdepends on orientation of the link.
−
+
If the link is not oriented, specify the framing on the chordsgiving positive smoothing.
![Page 189: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/189.jpg)
Framing
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
27 / 47
A sign of an arrow in Gauss diagram of a classical linkdepends on orientation of the link.
−
+
If the link is not oriented, specify the framing on the chordsgiving positive smoothing.
![Page 190: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/190.jpg)
Framing
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
27 / 47
A sign of an arrow in Gauss diagram of a classical linkdepends on orientation of the link.
−
+
If the link is not oriented, specify the framing on the chordsgiving positive smoothing.
shorthand notation
![Page 191: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/191.jpg)
Framed chord diagrams
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
28 / 47
A chord diagram (B, c1, . . . , cn)
![Page 192: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/192.jpg)
Framed chord diagrams
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
28 / 47
A chord diagram (B, c1, . . . , cn)
![Page 193: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/193.jpg)
Framed chord diagrams
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
28 / 47
A chord diagram (B, c1, . . . , cn)in which each chord is equipped with a framing
![Page 194: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/194.jpg)
Framed chord diagrams
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
28 / 47
A chord diagram (B, c1, . . . , cn)in which each chord is equipped with a framing
![Page 195: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/195.jpg)
Framed chord diagrams
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
28 / 47
A chord diagram (B, c1, . . . , cn)in which each chord is equipped with a framingis called a framed chord diagram.
![Page 196: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/196.jpg)
Framed chord diagrams
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
28 / 47
A chord diagram (B, c1, . . . , cn)in which each chord is equipped with a framingis called a framed chord diagram.
Kauffman bracket state sum is defined for a framed chorddiagram.
![Page 197: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/197.jpg)
Framed chord diagrams
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
28 / 47
A chord diagram (B, c1, . . . , cn)in which each chord is equipped with a framingis called a framed chord diagram.
−−
+
−
+−
+
Kauffman bracket state sum is defined for a framed chorddiagram.A state is a distribution of signs over the set of chords.
![Page 198: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/198.jpg)
Framed chord diagrams
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
28 / 47
A chord diagram (B, c1, . . . , cn)in which each chord is equipped with a framingis called a framed chord diagram.
−−
+
−
+−
+
Kauffman bracket state sum is defined for a framed chorddiagram.A state is a distribution of signs over the set of chords.The smoothing defined by a state is according to the famingalong the chords marked with + and the opposite oneotherwise.
![Page 199: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/199.jpg)
Signed to framed
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
29 / 47
A signed chord diagram turns canonically to a framed one:
![Page 200: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/200.jpg)
Signed to framed
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
29 / 47
A signed chord diagram turns canonically to a framed one:On a chord with + take the framing surgery along whichpreserves the orientation
![Page 201: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/201.jpg)
Signed to framed
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
29 / 47
A signed chord diagram turns canonically to a framed one:On a chord with + take the framing surgery along whichpreserves the orientation,on a chord with − take the framing surgery along whichreverses the orientation.
![Page 202: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/202.jpg)
Signed to framed
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
29 / 47
A signed chord diagram turns canonically to a framed one:On a chord with + take the framing surgery along whichpreserves the orientation,on a chord with − take the framing surgery along whichreverses the orientation.Forget the orientation.
![Page 203: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/203.jpg)
Orientable thickenings of non-orientable surfaces
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
30 / 47
Non-orientable surface can be thickenedto an oriented 3-manifold!
![Page 204: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/204.jpg)
Orientable thickenings of non-orientable surfaces
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
30 / 47
Non-orientable surface can be thickenedto an oriented 3-manifold!Example:Thicken a Mobius band M in R
3 .
![Page 205: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/205.jpg)
Orientable thickenings of non-orientable surfaces
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
30 / 47
Non-orientable surface can be thickenedto an oriented 3-manifold!Example:Thicken a Mobius band M in R
3 .
![Page 206: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/206.jpg)
Orientable thickenings of non-orientable surfaces
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
30 / 47
Non-orientable surface can be thickenedto an oriented 3-manifold!Example:Thicken a Mobius band M in R
3 .
A neighborhood of M in R3 is orientable and fibers over M .
![Page 207: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/207.jpg)
Abstract construction of an orientable thickening
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
31 / 47
Thicken a non-orientable surface S :
![Page 208: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/208.jpg)
Abstract construction of an orientable thickening
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
31 / 47
Thicken a non-orientable surface S :1. Find an orientation change line C (like International dateline) on S .
![Page 209: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/209.jpg)
Abstract construction of an orientable thickening
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
31 / 47
Thicken a non-orientable surface S :1. Find an orientation change line C (like International dateline) on S .
C
![Page 210: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/210.jpg)
Abstract construction of an orientable thickening
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
31 / 47
Thicken a non-orientable surface S :1. Find an orientation change line C (like International dateline) on S .
C
2. Cut S along C : S 7→ S C
![Page 211: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/211.jpg)
Abstract construction of an orientable thickening
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
31 / 47
Thicken a non-orientable surface S :1. Find an orientation change line C (like International dateline) on S .
C
2. Cut S along C : S 7→ S C
3. Thicken: (S C)× R .
![Page 212: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/212.jpg)
Abstract construction of an orientable thickening
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
31 / 47
Thicken a non-orientable surface S :1. Find an orientation change line C (like International dateline) on S .
C
2. Cut S along C : S 7→ S C
3. Thicken: (S C)× R .4. Paste over the sides of the cut (x+, t) ∼ (x−,−t) .
![Page 213: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/213.jpg)
A link in orientable thickening of a non-orientablesurface
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
32 / 47
A diagram on the surface.
![Page 214: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/214.jpg)
A link in orientable thickening of a non-orientablesurface
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
32 / 47
A diagram on the surface.Reidemeister moves plus two more moves:
and
![Page 215: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/215.jpg)
A link in orientable thickening of a non-orientablesurface
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
32 / 47
A diagram on the surface.Reidemeister moves plus two more moves:
and
Twisted Gaussdiagram
=Gauss diagram with a finiteset of dots marked on thecircle.
![Page 216: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/216.jpg)
A link in orientable thickening of a non-orientablesurface
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
32 / 47
A diagram on the surface.Reidemeister moves plus two more moves:
and
Twisted Gaussdiagram
=Gauss diagram with a finiteset of dots marked on thecircle.
Two more moves:
and ε ε
![Page 217: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/217.jpg)
A link in orientable thickening of a non-orientablesurface
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
32 / 47
A diagram on the surface.Reidemeister moves plus two more moves:
and
Twisted Gaussdiagram
=Gauss diagram with a finiteset of dots marked on thecircle.
Two more moves:
and ε ε
Forgetting dots and arrows turns a twisted Gauss diagram intoa signed chord diagram.
![Page 218: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/218.jpg)
A link in orientable thickening of a non-orientablesurface
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
32 / 47
A diagram on the surface.Reidemeister moves plus two more moves:
and
Twisted Gaussdiagram
=Gauss diagram with a finiteset of dots marked on thecircle.
Two more moves:
and ε ε
Forgetting dots and arrows turns a twisted Gauss diagram intoa signed chord diagram.(together with moves)
![Page 219: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/219.jpg)
A link in orientable thickening of a non-orientablesurface
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
• Signed chorddiagrams
• State of signed chorddiagram
• Smoothing of asigned chord diagram
• Framing
• Framed chorddiagrams
• Signed to framed
• Orientablethickenings ofnon-orientable surfaces• Abstract constructionof an orientablethickening
• A link in orientablethickening of anon-orientable surface
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
32 / 47
A diagram on the surface.Reidemeister moves plus two more moves:
and
Twisted Gaussdiagram
=Gauss diagram with a finiteset of dots marked on thecircle.
Two more moves:
and ε ε
Forgetting dots and arrows turns a twisted Gauss diagram intoa signed chord diagram.Corollary (Bourgoin). Links in orientable thickenings ofsurfaces have well-defined Kauffman bracket.
![Page 220: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/220.jpg)
Khovanov homology
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
33 / 47
![Page 221: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/221.jpg)
Khovanov homology
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
34 / 47
Khovanov homology categorifies Jones polynomial.
![Page 222: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/222.jpg)
Khovanov homology
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
34 / 47
Khovanov homology categorifies Jones polynomial.Here we will deal with a version of Khovanov homology, whichcategorifies Kauffman bracket.
![Page 223: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/223.jpg)
Khovanov homology
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
34 / 47
Khovanov homology categorifies Jones polynomial.Here we will deal with a version of Khovanov homology, whichcategorifies Kauffman bracket.
D 7→ Hp,q(D) , 〈D〉 =∑
p,q(−1)pAq rk Hp,q(D) .
![Page 224: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/224.jpg)
Khovanov homology
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
34 / 47
Khovanov homology categorifies Jones polynomial.Here we will deal with a version of Khovanov homology, whichcategorifies Kauffman bracket.
D 7→ Hp,q(D) , 〈D〉 =∑
p,q(−1)pAq rk Hp,q(D) . Relationto the original Khovanov homology:
![Page 225: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/225.jpg)
Khovanov homology
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
34 / 47
Khovanov homology categorifies Jones polynomial.Here we will deal with a version of Khovanov homology, whichcategorifies Kauffman bracket.
D 7→ Hp,q(D) , 〈D〉 =∑
p,q(−1)pAq rk Hp,q(D) . Relationto the original Khovanov homology:
Hp,q(D) = Hw(D)−q−2p
2,3w(D)−q
2 (D)
![Page 226: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/226.jpg)
Khovanov homology
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
34 / 47
Khovanov homology categorifies Jones polynomial.Here we will deal with a version of Khovanov homology, whichcategorifies Kauffman bracket.
D 7→ Hp,q(D) , 〈D〉 =∑
p,q(−1)pAq rk Hp,q(D) . Relationto the original Khovanov homology:
Hp,q(D) = Hw(D)−q−2p
2,3w(D)−q
2 (D) , or
Hi,j(D) = Hj−i−w(D),3w(D)−2j(D).
![Page 227: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/227.jpg)
Khovanov homology
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
34 / 47
Khovanov homology categorifies Jones polynomial.Here we will deal with a version of Khovanov homology, whichcategorifies Kauffman bracket.
D 7→ Hp,q(D) , 〈D〉 =∑
p,q(−1)pAq rk Hp,q(D) . Relationto the original Khovanov homology:
Hp,q(D) = Hw(D)−q−2p
2,3w(D)−q
2 (D) , or
Hi,j(D) = Hj−i−w(D),3w(D)−2j(D).
In other words: Hp,q(D) = Hi,j(D) iff
q + 2j = 3w(D) and j − i + p = w(D) .
![Page 228: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/228.jpg)
Enhanced states
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
35 / 47
Enhance states involved in the Kauffman state sum byattaching a sign to each component of the smoothing alongthe state.
![Page 229: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/229.jpg)
Enhanced states
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
35 / 47
Enhance states involved in the Kauffman state sum byattaching a sign to each component of the smoothing alongthe state.
For example: state with smoothing
gives rise to 4 enhanced states
![Page 230: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/230.jpg)
Enhanced states
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
35 / 47
Enhance states involved in the Kauffman state sum byattaching a sign to each component of the smoothing alongthe state.
For example: state with smoothing
gives rise to 4 enhanced states
−
+
+
+
+
−
−
−
![Page 231: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/231.jpg)
Khovanov complex
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
36 / 47
For enhanced state S , set τ(S) = #(pluses)−#(minuses)and 〈S〉 = (−1)τ(S)Aa(S)−b(S)−2τ(S).
![Page 232: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/232.jpg)
Khovanov complex
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
36 / 47
For enhanced state S , set τ(S) = #(pluses)−#(minuses)and 〈S〉 = (−1)τ(S)Aa(S)−b(S)−2τ(S).
〈D〉 =∑
S enhanced state of D〈S〉
![Page 233: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/233.jpg)
Khovanov complex
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
36 / 47
For enhanced state S , set τ(S) = #(pluses)−#(minuses)and 〈S〉 = (−1)τ(S)Aa(S)−b(S)−2τ(S).
〈D〉 =∑
S enhanced state of D〈S〉Let Cp,q(D) be a free abelian group generated by enhancedstates S of D with:
τ(S) = p and a(S)− b(S)− 2τ(S) = q .
![Page 234: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/234.jpg)
Khovanov complex
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
36 / 47
For enhanced state S , set τ(S) = #(pluses)−#(minuses)and 〈S〉 = (−1)τ(S)Aa(S)−b(S)−2τ(S).
〈D〉 =∑
S enhanced state of D〈S〉Let Cp,q(D) be a free abelian group generated by enhancedstates S of D with:
τ(S) = p and a(S)− b(S)− 2τ(S) = q .Then 〈D〉 =
∑
p,q(−1)pAq rk Cp,q(D) .
![Page 235: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/235.jpg)
Khovanov complex
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
36 / 47
For enhanced state S , set τ(S) = #(pluses)−#(minuses)and 〈S〉 = (−1)τ(S)Aa(S)−b(S)−2τ(S).
〈D〉 =∑
S enhanced state of D〈S〉Let Cp,q(D) be a free abelian group generated by enhancedstates S of D with:
τ(S) = p and a(S)− b(S)− 2τ(S) = q .Then 〈D〉 =
∑
p,q(−1)pAq rk Cp,q(D) .Any differential ∂ : Cp,q(D)→ Cp−1,q(D) gives homologyHp,q(D) with 〈D〉 =
∑
p,q(−1)pAq rk Hp,q(D) .
![Page 236: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/236.jpg)
Khovanov complex
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
36 / 47
For enhanced state S , set τ(S) = #(pluses)−#(minuses)and 〈S〉 = (−1)τ(S)Aa(S)−b(S)−2τ(S).
〈D〉 =∑
S enhanced state of D〈S〉Let Cp,q(D) be a free abelian group generated by enhancedstates S of D with:
τ(S) = p and a(S)− b(S)− 2τ(S) = q .Then 〈D〉 =
∑
p,q(−1)pAq rk Cp,q(D) .Any differential ∂ : Cp,q(D)→ Cp−1,q(D) gives homologyHp,q(D) with 〈D〉 =
∑
p,q(−1)pAq rk Hp,q(D) .
Invariance of Hp,q(D) under Reidemeister moves wanted!
![Page 237: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/237.jpg)
Khovanov complex
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
36 / 47
For enhanced state S , set τ(S) = #(pluses)−#(minuses)and 〈S〉 = (−1)τ(S)Aa(S)−b(S)−2τ(S).
〈D〉 =∑
S enhanced state of D〈S〉Let Cp,q(D) be a free abelian group generated by enhancedstates S of D with:
τ(S) = p and a(S)− b(S)− 2τ(S) = q .Then 〈D〉 =
∑
p,q(−1)pAq rk Cp,q(D) .Any differential ∂ : Cp,q(D)→ Cp−1,q(D) gives homologyHp,q(D) with 〈D〉 =
∑
p,q(−1)pAq rk Hp,q(D) .
Invariance of Hp,q(D) under Reidemeister moves wanted!
∂(S) =∑
±T with T , which differ from S by a singlemarker and appropriate signs on the circles passing near thevertex.
![Page 238: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/238.jpg)
Khovanov complex
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
36 / 47
For enhanced state S , set τ(S) = #(pluses)−#(minuses)and 〈S〉 = (−1)τ(S)Aa(S)−b(S)−2τ(S).
〈D〉 =∑
S enhanced state of D〈S〉Let Cp,q(D) be a free abelian group generated by enhancedstates S of D with:
τ(S) = p and a(S)− b(S)− 2τ(S) = q .Then 〈D〉 =
∑
p,q(−1)pAq rk Cp,q(D) .Any differential ∂ : Cp,q(D)→ Cp−1,q(D) gives homologyHp,q(D) with 〈D〉 =
∑
p,q(−1)pAq rk Hp,q(D) .
Invariance of Hp,q(D) under Reidemeister moves wanted!
∂(S) =∑
±T with T , which differ from S by a singlemarker and appropriate signs on the circles passing near thevertex.
(|T | − |S|) = 1 is needed to have τ(T ) = τ(S)− 1 .
![Page 239: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/239.jpg)
More algebraic construction
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
37 / 47
Let A be an algebra over Z generated by 1 and X withX2 = 0 .
![Page 240: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/240.jpg)
More algebraic construction
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
37 / 47
Let A be an algebra over Z generated by 1 and X withX2 = 0 .Grading: deg(1) = 0 , deg(X) = 2 .
![Page 241: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/241.jpg)
More algebraic construction
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
37 / 47
Let A be an algebra over Z generated by 1 and X withX2 = 0 .Grading: deg(1) = 0 , deg(X) = 2 .Comultiplication:
∆ : A → A⊗A , ∆(1) = X ⊗ 1 + 1⊗X ,∆(X) = X ⊗X .
![Page 242: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/242.jpg)
More algebraic construction
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
37 / 47
Let A be an algebra over Z generated by 1 and X withX2 = 0 .Grading: deg(1) = 0 , deg(X) = 2 .Comultiplication:
∆ : A → A⊗A , ∆(1) = X ⊗ 1 + 1⊗X ,∆(X) = X ⊗X .For a state s of a link diagram D
associate a copy of A with each component of Ds .
![Page 243: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/243.jpg)
More algebraic construction
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
37 / 47
Let A be an algebra over Z generated by 1 and X withX2 = 0 .Grading: deg(1) = 0 , deg(X) = 2 .Comultiplication:
∆ : A → A⊗A , ∆(1) = X ⊗ 1 + 1⊗X ,∆(X) = X ⊗X .For a state s of a link diagram D
associate a copy of A with each component of Ds .Denote by Vs the tensor product of these copies of A .
![Page 244: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/244.jpg)
More algebraic construction
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
37 / 47
Let A be an algebra over Z generated by 1 and X withX2 = 0 .Grading: deg(1) = 0 , deg(X) = 2 .Comultiplication:
∆ : A → A⊗A , ∆(1) = X ⊗ 1 + 1⊗X ,∆(X) = X ⊗X .For a state s of a link diagram D
associate a copy of A with each component of Ds .Denote by Vs the tensor product of these copies of A .Equip Vs with the second grading equal to the first gradingshifted by a(s)− b(s)− |s| .
![Page 245: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/245.jpg)
More algebraic construction
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
37 / 47
Let A be an algebra over Z generated by 1 and X withX2 = 0 .Grading: deg(1) = 0 , deg(X) = 2 .Comultiplication:
∆ : A → A⊗A , ∆(1) = X ⊗ 1 + 1⊗X ,∆(X) = X ⊗X .For a state s of a link diagram D
associate a copy of A with each component of Ds .Denote by Vs the tensor product of these copies of A .Equip Vs with the second grading equal to the first gradingshifted by a(s)− b(s)− |s| .
Then⊕p,qCp,q(D) = ⊕sVs
![Page 246: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/246.jpg)
More algebraic construction
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
37 / 47
Let A be an algebra over Z generated by 1 and X withX2 = 0 .Grading: deg(1) = 0 , deg(X) = 2 .Comultiplication:
∆ : A → A⊗A , ∆(1) = X ⊗ 1 + 1⊗X ,∆(X) = X ⊗X .For a state s of a link diagram D
associate a copy of A with each component of Ds .Denote by Vs the tensor product of these copies of A .Equip Vs with the second grading equal to the first gradingshifted by a(s)− b(s)− |s| .
Then⊕p,qCp,q(D) = ⊕sVs
Differentials are defined by the multiplication andco-multiplication in A .
![Page 247: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/247.jpg)
What about virtual links?
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
38 / 47
This works for classical links, but does not for virtual!
![Page 248: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/248.jpg)
What about virtual links?
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
38 / 47
This works for classical links, but does not for virtual!
For virtual links, it works with Z2 coefficients.
![Page 249: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/249.jpg)
What about virtual links?
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
38 / 47
Over integers d2 6= 0!
![Page 250: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/250.jpg)
What about virtual links?
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
38 / 47
Over integers d2 6= 0!
Consider virtual diagram of the unknot:
![Page 251: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/251.jpg)
What about virtual links?
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
38 / 47
Over integers d2 6= 0!
Consider virtual diagram of the unknot:There are 4 states contributing to Kauffman bracket as follows:
![Page 252: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/252.jpg)
What about virtual links?
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
38 / 47
Over integers d2 6= 0!
Consider virtual diagram of the unknot:There are 4 states contributing to Kauffman bracket as follows:
− A4 − 1 −−−→ −A2 − A−2
y
y
A4 + 2 + A−4 −−−→ −1− A−4
![Page 253: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/253.jpg)
What about virtual links?
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
38 / 47
Over integers d2 6= 0!
Consider virtual diagram of the unknot:There are 4 states contributing to Kauffman bracket as follows:
− A4 − 1 −−−→ −A2 − A−2
y
y
A4 + 2 + A−4 −−−→ −1− A−4
Differentials are obvious in all A -components but the onecorresponding to A0 .
![Page 254: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/254.jpg)
What about virtual links?
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
38 / 47
Over integers d2 6= 0!
Consider virtual diagram of the unknot:There are 4 states contributing to Kauffman bracket as follows:
− A4 − 10
−−−→ −A2 − A−2
y0
y
A4 + 2 + A−4 −−−→ −1− A−4
Differentials are obvious in all A -components but the onecorresponding to A0 .
![Page 255: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/255.jpg)
What about virtual links?
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
38 / 47
Over integers d2 6= 0!
Consider virtual diagram of the unknot:There are 4 states contributing to Kauffman bracket as follows:
10
−−−→
y0
y
1⊗X + X ⊗ 1 −−−→ 2× 1
Differentials are obvious in all A -components but the onecorresponding to A0 .
![Page 256: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/256.jpg)
What about virtual links?
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
• Khovanov homology
• Enhanced states
• Khovanov complex• More algebraicconstruction• What about virtuallinks?
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
38 / 47
Over integers d2 6= 0!
Consider virtual diagram of the unknot:There are 4 states contributing to Kauffman bracket as follows:
10
−−−→
y0
y
1⊗X + X ⊗ 1 −−−→ 2× 1
This does not happen if the chord diagram is orientable!
![Page 257: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/257.jpg)
Orientation of chorddiagrams
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
• Orientation of a chorddiagram
• Obstruction toorientability
• Orientation of asmoothened chorddiagram
Khovanov complex offramed chord diagram
39 / 47
![Page 258: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/258.jpg)
Orientation of a chord diagram
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
• Orientation of a chorddiagram
• Obstruction toorientability
• Orientation of asmoothened chorddiagram
Khovanov complex offramed chord diagram
40 / 47
= orientations of chords and arcs
![Page 259: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/259.jpg)
Orientation of a chord diagram
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
• Orientation of a chorddiagram
• Obstruction toorientability
• Orientation of asmoothened chorddiagram
Khovanov complex offramed chord diagram
40 / 47
= orientations of chords and arcs
such that the chain with integer coefficients
![Page 260: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/260.jpg)
Orientation of a chord diagram
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
• Orientation of a chorddiagram
• Obstruction toorientability
• Orientation of asmoothened chorddiagram
Khovanov complex offramed chord diagram
40 / 47
= orientations of chords and arcs
such that the chain with integer coefficients∑
arcs +∑
2 chords
![Page 261: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/261.jpg)
Orientation of a chord diagram
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
• Orientation of a chorddiagram
• Obstruction toorientability
• Orientation of asmoothened chorddiagram
Khovanov complex offramed chord diagram
40 / 47
= orientations of chords and arcs
such that the chain with integer coefficients∑
arcs +∑
2 chords is a cycle.
![Page 262: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/262.jpg)
Orientation of a chord diagram
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
• Orientation of a chorddiagram
• Obstruction toorientability
• Orientation of asmoothened chorddiagram
Khovanov complex offramed chord diagram
40 / 47
= orientations of chords and arcs
such that the chain with integer coefficients∑
arcs +∑
2 chords is a cycle.
That is ∂ (∑
arcs +∑
2 chords ) = 0.
![Page 263: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/263.jpg)
Orientation of a chord diagram
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
• Orientation of a chorddiagram
• Obstruction toorientability
• Orientation of asmoothened chorddiagram
Khovanov complex offramed chord diagram
40 / 47
= orientations of chords and arcs
such that the chain with integer coefficients∑
arcs +∑
2 chords is a cycle.
That is ∂ (∑
arcs +∑
2 chords ) = 0.
![Page 264: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/264.jpg)
Orientation of a chord diagram
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
• Orientation of a chorddiagram
• Obstruction toorientability
• Orientation of asmoothened chorddiagram
Khovanov complex offramed chord diagram
40 / 47
= orientations of chords and arcs
such that the chain with integer coefficients∑
arcs +∑
2 chords is a cycle.
That is ∂ (∑
arcs +∑
2 chords ) = 0.
A chord diagram is called orientable if it admits an orientation.
![Page 265: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/265.jpg)
Orientation of a chord diagram
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
• Orientation of a chorddiagram
• Obstruction toorientability
• Orientation of asmoothened chorddiagram
Khovanov complex offramed chord diagram
40 / 47
= orientations of chords and arcs
such that the chain with integer coefficients∑
arcs +∑
2 chords is a cycle.
That is ∂ (∑
arcs +∑
2 chords ) = 0.
A chord diagram is called orientable if it admits an orientation.
Orientability of chord diagram with connected base is
equivalent to the following condition known to K.-F.Gauss:
![Page 266: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/266.jpg)
Orientation of a chord diagram
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
• Orientation of a chorddiagram
• Obstruction toorientability
• Orientation of asmoothened chorddiagram
Khovanov complex offramed chord diagram
40 / 47
= orientations of chords and arcs
such that the chain with integer coefficients∑
arcs +∑
2 chords is a cycle.
That is ∂ (∑
arcs +∑
2 chords ) = 0.
A chord diagram is called orientable if it admits an orientation.
Orientability of chord diagram with connected base is
equivalent to the following condition known to K.-F.Gauss:
The number of endpoints of chords on each arc bounded be
endpoints of a chord is even.
![Page 267: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/267.jpg)
Orientation of a chord diagram
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
• Orientation of a chorddiagram
• Obstruction toorientability
• Orientation of asmoothened chorddiagram
Khovanov complex offramed chord diagram
40 / 47
= orientations of chords and arcs
such that the chain with integer coefficients∑
arcs +∑
2 chords is a cycle.
That is ∂ (∑
arcs +∑
2 chords ) = 0.
A chord diagram is called orientable if it admits an orientation.
Orientability of chord diagram with connected base is
equivalent to the following condition known to K.-F.Gauss:
The number of endpoints of chords on each arc bounded be
endpoints of a chord is even.
The simplest nonorientable chord diagram: ⊗ .
![Page 268: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/268.jpg)
Obstruction to orientability
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
• Orientation of a chorddiagram
• Obstruction toorientability
• Orientation of asmoothened chorddiagram
Khovanov complex offramed chord diagram
41 / 47
Try to orient a chord diagram.
![Page 269: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/269.jpg)
Obstruction to orientability
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
• Orientation of a chorddiagram
• Obstruction toorientability
• Orientation of asmoothened chorddiagram
Khovanov complex offramed chord diagram
41 / 47
Try to orient a chord diagram.
![Page 270: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/270.jpg)
Obstruction to orientability
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
• Orientation of a chorddiagram
• Obstruction toorientability
• Orientation of asmoothened chorddiagram
Khovanov complex offramed chord diagram
41 / 47
Try to orient a chord diagram.
![Page 271: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/271.jpg)
Obstruction to orientability
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
• Orientation of a chorddiagram
• Obstruction toorientability
• Orientation of asmoothened chorddiagram
Khovanov complex offramed chord diagram
41 / 47
Try to orient a chord diagram.
![Page 272: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/272.jpg)
Obstruction to orientability
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
• Orientation of a chorddiagram
• Obstruction toorientability
• Orientation of asmoothened chorddiagram
Khovanov complex offramed chord diagram
41 / 47
Try to orient a chord diagram.
![Page 273: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/273.jpg)
Obstruction to orientability
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
• Orientation of a chorddiagram
• Obstruction toorientability
• Orientation of asmoothened chorddiagram
Khovanov complex offramed chord diagram
41 / 47
Try to orient a chord diagram.
![Page 274: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/274.jpg)
Obstruction to orientability
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
• Orientation of a chorddiagram
• Obstruction toorientability
• Orientation of asmoothened chorddiagram
Khovanov complex offramed chord diagram
41 / 47
Try to orient a chord diagram.
We have met an obstruction.
![Page 275: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/275.jpg)
Obstruction to orientability
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
• Orientation of a chorddiagram
• Obstruction toorientability
• Orientation of asmoothened chorddiagram
Khovanov complex offramed chord diagram
41 / 47
Try to orient a chord diagram.
We have met an obstruction.
The obstruction to orientability of a chord diagram
(B, c1, . . . , cn) is an element of H1(B,∪ni=1∂ci; Z2) .
![Page 276: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/276.jpg)
Obstruction to orientability
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
• Orientation of a chorddiagram
• Obstruction toorientability
• Orientation of asmoothened chorddiagram
Khovanov complex offramed chord diagram
41 / 47
Try to orient a chord diagram.
We have met an obstruction.
The obstruction to orientability of a chord diagram
(B, c1, . . . , cn) is an element of H1(B,∪ni=1∂ci; Z2) .
Dual class belongs to H0(B r ∪ni=1∂ci; Z2) .
![Page 277: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/277.jpg)
Obstruction to orientability
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
• Orientation of a chorddiagram
• Obstruction toorientability
• Orientation of asmoothened chorddiagram
Khovanov complex offramed chord diagram
41 / 47
Try to orient a chord diagram.
We have met an obstruction.
The obstruction to orientability of a chord diagram
(B, c1, . . . , cn) is an element of H1(B,∪ni=1∂ci; Z2) .
Dual class belongs to H0(B r ∪ni=1∂ci; Z2) .
Orient the complement of the 0-cycle realizing it,
to get vice-orientation of the chord diagram.
![Page 278: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/278.jpg)
Orientation of a smoothened chord diagram
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
• Orientation of a chorddiagram
• Obstruction toorientability
• Orientation of asmoothened chorddiagram
Khovanov complex offramed chord diagram
42 / 47
If a chord diagram
![Page 279: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/279.jpg)
Orientation of a smoothened chord diagram
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
• Orientation of a chorddiagram
• Obstruction toorientability
• Orientation of asmoothened chorddiagram
Khovanov complex offramed chord diagram
42 / 47
If a chord diagram
+ +
(−)
(+)
(+) (−)
−
−
![Page 280: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/280.jpg)
Orientation of a smoothened chord diagram
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
• Orientation of a chorddiagram
• Obstruction toorientability
• Orientation of asmoothened chorddiagram
Khovanov complex offramed chord diagram
42 / 47
If a chord diagram is oriented,
+ +
(−)
(+)
(+) (−)
−
−
![Page 281: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/281.jpg)
Orientation of a smoothened chord diagram
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
• Orientation of a chorddiagram
• Obstruction toorientability
• Orientation of asmoothened chorddiagram
Khovanov complex offramed chord diagram
42 / 47
If a chord diagram is oriented,
+ +
(−)
(+)
(+) (−)
−
−
its orientation induces an orientation of each result of itssmoothing.
![Page 282: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/282.jpg)
Orientation of a smoothened chord diagram
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
• Orientation of a chorddiagram
• Obstruction toorientability
• Orientation of asmoothened chorddiagram
Khovanov complex offramed chord diagram
42 / 47
If a chord diagram is oriented,
+ +
(−)
(+)
(+) (−)
−
−
+ +
(−)
(+)
(−)
−
−
(+)
its orientation induces an orientation of each result of itssmoothing.
![Page 283: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/283.jpg)
Orientation of a smoothened chord diagram
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
• Orientation of a chorddiagram
• Obstruction toorientability
• Orientation of asmoothened chorddiagram
Khovanov complex offramed chord diagram
42 / 47
If a chord diagram is oriented,
+ +
(−)
(+)
(+) (−)
−
−
+ +
(−)
(+)
(−)
−
−
(+)
its orientation induces an orientation of each result of itssmoothing.Similarly, a vice-orientation of a signed chord diagram inducesa vice-orientation of each result of its smoothing.
![Page 284: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/284.jpg)
Orientation of a smoothened chord diagram
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
• Orientation of a chorddiagram
• Obstruction toorientability
• Orientation of asmoothened chorddiagram
Khovanov complex offramed chord diagram
42 / 47
If a chord diagram is oriented,
+ +
(−)
(+)
(+) (−)
−
−
+ +
(−)
(+)
(−)
−
−
(+)
its orientation induces an orientation of each result of itssmoothing.Similarly, a vice-orientation of a signed chord diagram inducesa vice-orientation of each result of its smoothing.
Theorem (Manturov, Viro) Definition of the Khovanov complexextended straightforwardly to an oriented framed chorddiagram gives a complex invariant under Reidemeister movespreserving the orientation.
![Page 285: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/285.jpg)
Khovanov complex offramed chord diagram
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
43 / 47
![Page 286: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/286.jpg)
Structure used in the construction
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
44 / 47
1. Framed chord diagram.
![Page 287: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/287.jpg)
Structure used in the construction
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
44 / 47
1. Framed chord diagram.
![Page 288: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/288.jpg)
Structure used in the construction
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
44 / 47
1. Framed chord diagram.2. Vice-orientation of the chord diagram.
![Page 289: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/289.jpg)
Structure used in the construction
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
44 / 47
1. Framed chord diagram.2. Vice-orientation of the chord diagram.
![Page 290: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/290.jpg)
Structure used in the construction
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
44 / 47
1. Framed chord diagram.2. Vice-orientation of the chord diagram.3. At each chord one of two arcs adjacent to its arrowhead ismarked.
![Page 291: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/291.jpg)
Structure used in the construction
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
44 / 47
1. Framed chord diagram.2. Vice-orientation of the chord diagram.3. At each chord one of two arcs adjacent to its arrowhead ismarked.
![Page 292: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/292.jpg)
Structure used in the construction
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
44 / 47
1. Framed chord diagram.2. Vice-orientation of the chord diagram.3. At each chord one of two arcs adjacent to its arrowhead ismarked.
The chain groups are the same as in the Khovanovconstruction:
![Page 293: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/293.jpg)
Structure used in the construction
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
44 / 47
1. Framed chord diagram.2. Vice-orientation of the chord diagram.3. At each chord one of two arcs adjacent to its arrowhead ismarked.
The chain groups are the same as in the Khovanovconstruction:
⊕p,qCp,q(D) = ⊕sVs
![Page 294: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/294.jpg)
Structure used in the construction
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
44 / 47
1. Framed chord diagram.2. Vice-orientation of the chord diagram.3. At each chord one of two arcs adjacent to its arrowhead ismarked.
The chain groups are the same as in the Khovanovconstruction:
⊕p,qCp,q(D) = ⊕sVs
algebraically (up to isomorphisms).
![Page 295: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/295.jpg)
Structure used in the construction
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
44 / 47
1. Framed chord diagram.2. Vice-orientation of the chord diagram.3. At each chord one of two arcs adjacent to its arrowhead ismarked.
The chain groups are the same as in the Khovanovconstruction:
⊕p,qCp,q(D) = ⊕sVs
algebraically (up to isomorphisms).The structure is needed for a collection of the isomorphisms
![Page 296: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/296.jpg)
Structure used in the construction
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
44 / 47
1. Framed chord diagram.2. Vice-orientation of the chord diagram.3. At each chord one of two arcs adjacent to its arrowhead ismarked.
The chain groups are the same as in the Khovanovconstruction:
⊕p,qCp,q(D) = ⊕sVs
algebraically (up to isomorphisms).The structure is needed for a collection of the isomorphismsneeded for construction of differentials.
![Page 297: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/297.jpg)
Structure used in the construction
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
44 / 47
1. Framed chord diagram.2. Vice-orientation of the chord diagram.3. At each chord one of two arcs adjacent to its arrowhead ismarked.
The chain groups are the same as in the Khovanovconstruction:
⊕p,qCp,q(D) = ⊕sVs
algebraically (up to isomorphisms).The structure is needed for a collection of the isomorphismsneeded for construction of differentials.Homology does not depend on the structure.
![Page 298: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/298.jpg)
Involution in the Frobenius algebra
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
45 / 47
Remind that A is a Frobenius algebra generated by 1 and X
with X2 = 0 .with Grading: deg(1) = 0 , deg(X) = 2 .and Comultiplication:
∆ : A → A⊗A , ∆(1) = X ⊗ 1 + 1⊗X ,∆(X) = X ⊗X .
![Page 299: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/299.jpg)
Involution in the Frobenius algebra
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
45 / 47
Remind that A is a Frobenius algebra generated by 1 and X
with X2 = 0 .with Grading: deg(1) = 0 , deg(X) = 2 .and Comultiplication:
∆ : A → A⊗A , ∆(1) = X ⊗ 1 + 1⊗X ,∆(X) = X ⊗X .Involution conj : A → A : 1 7→ 1, X 7→ −X .
![Page 300: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/300.jpg)
Involution in the Frobenius algebra
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
45 / 47
Remind that A is a Frobenius algebra generated by 1 and X
with X2 = 0 .with Grading: deg(1) = 0 , deg(X) = 2 .and Comultiplication:
∆ : A → A⊗A , ∆(1) = X ⊗ 1 + 1⊗X ,∆(X) = X ⊗X .Involution conj : A → A : 1 7→ 1, X 7→ −X .Notice: conj(ab) = conj(a) conj(b) .
![Page 301: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/301.jpg)
Involution in the Frobenius algebra
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
45 / 47
Remind that A is a Frobenius algebra generated by 1 and X
with X2 = 0 .with Grading: deg(1) = 0 , deg(X) = 2 .and Comultiplication:
∆ : A → A⊗A , ∆(1) = X ⊗ 1 + 1⊗X ,∆(X) = X ⊗X .Involution conj : A → A : 1 7→ 1, X 7→ −X .Notice: conj(ab) = conj(a) conj(b) .But ∆(conj(1)) = ∆(1) = X ⊗ 1 + 1⊗X
= −∆(X)⊗∆(1)−∆(1)⊗∆(X) .
![Page 302: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/302.jpg)
Involution in the Frobenius algebra
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
45 / 47
Remind that A is a Frobenius algebra generated by 1 and X
with X2 = 0 .with Grading: deg(1) = 0 , deg(X) = 2 .and Comultiplication:
∆ : A → A⊗A , ∆(1) = X ⊗ 1 + 1⊗X ,∆(X) = X ⊗X .Involution conj : A → A : 1 7→ 1, X 7→ −X .Notice: conj(ab) = conj(a) conj(b) .But ∆(conj(1)) = ∆(1) = X ⊗ 1 + 1⊗X
= −∆(X)⊗∆(1)−∆(1)⊗∆(X) .and∆(conj(X)) = ∆(−X) = −X ⊗X = −∆(X)⊗∆(X) .
![Page 303: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/303.jpg)
Space associated to a state
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
46 / 47
Given a state s of a framed chord diagram D .
![Page 304: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/304.jpg)
Space associated to a state
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
46 / 47
Given a state s of a framed chord diagram D .Orient each connected component of Ds .
![Page 305: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/305.jpg)
Space associated to a state
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
46 / 47
Given a state s of a framed chord diagram D .Orient each connected component of Ds .Order the set of components.
![Page 306: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/306.jpg)
Space associated to a state
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
46 / 47
Given a state s of a framed chord diagram D .Orient each connected component of Ds .Order the set of components.Associate a copy of A to each component of Ds .
![Page 307: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/307.jpg)
Space associated to a state
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
46 / 47
Given a state s of a framed chord diagram D .Orient each connected component of Ds .Order the set of components.Associate a copy of A to each component of Ds .Denote by Vs the tensor product of these copies of A .
![Page 308: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/308.jpg)
Space associated to a state
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
46 / 47
Given a state s of a framed chord diagram D .Orient each connected component of Ds .Order the set of components.Associate a copy of A to each component of Ds .Denote by Vs the tensor product of these copies of A .This construction depends on the orientations and ordering.
![Page 309: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/309.jpg)
Space associated to a state
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
46 / 47
Given a state s of a framed chord diagram D .Orient each connected component of Ds .Order the set of components.Associate a copy of A to each component of Ds .Denote by Vs the tensor product of these copies of A .This construction depends on the orientations and ordering.The results corresponding to the different choices of themare related by isomorphisms:
![Page 310: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/310.jpg)
Space associated to a state
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
46 / 47
Given a state s of a framed chord diagram D .Orient each connected component of Ds .Order the set of components.Associate a copy of A to each component of Ds .Denote by Vs the tensor product of these copies of A .This construction depends on the orientations and ordering.The results corresponding to the different choices of themare related by isomorphisms:Reversing of orientation of a component corresponds to conjin the corresponding copy of A .
![Page 311: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/311.jpg)
Space associated to a state
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
46 / 47
Given a state s of a framed chord diagram D .Orient each connected component of Ds .Order the set of components.Associate a copy of A to each component of Ds .Denote by Vs the tensor product of these copies of A .This construction depends on the orientations and ordering.The results corresponding to the different choices of themare related by isomorphisms:Reversing of orientation of a component corresponds to conjin the corresponding copy of A .Permutations of the components corresponds to thepermutation isomorphism of the tensor product multiplied bythe sign of the permutation.
![Page 312: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/312.jpg)
Partial differential
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
47 / 47
Let s and t be adjacent states of a framed chord diagram D
which is equipped with a vice orientation and markers.
![Page 313: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/313.jpg)
Partial differential
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
47 / 47
Let s and t be adjacent states of a framed chord diagram D
which is equipped with a vice orientation and markers.Let t differs from s only by a marker sign at chord c
![Page 314: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/314.jpg)
Partial differential
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
47 / 47
Let s and t be adjacent states of a framed chord diagram D
which is equipped with a vice orientation and markers.Let t differs from s only by a marker sign at chord c ,positive in s and negative at t .
![Page 315: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/315.jpg)
Partial differential
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
47 / 47
Let s and t be adjacent states of a framed chord diagram D
which is equipped with a vice orientation and markers.Let t differs from s only by a marker sign at chord c ,positive in s and negative at t .Construct Vs → Vt .
![Page 316: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/316.jpg)
Partial differential
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
47 / 47
Let s and t be adjacent states of a framed chord diagram D
which is equipped with a vice orientation and markers.Let t differs from s only by a marker sign at chord c ,positive in s and negative at t .Construct Vs → Vt .Put it to be 0 if |s| = |t| .
![Page 317: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/317.jpg)
Partial differential
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
47 / 47
Let s and t be adjacent states of a framed chord diagram D
which is equipped with a vice orientation and markers.Let t differs from s only by a marker sign at chord c ,positive in s and negative at t .Construct Vs → Vt .Put it to be 0 if |s| = |t| .Otherwise, order the components of Ds and Dt so that:
![Page 318: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/318.jpg)
Partial differential
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
47 / 47
Let s and t be adjacent states of a framed chord diagram D
which is equipped with a vice orientation and markers.Let t differs from s only by a marker sign at chord c ,positive in s and negative at t .Construct Vs → Vt .Put it to be 0 if |s| = |t| .Otherwise, order the components of Ds and Dt so that:• The first component passes through the marker at c .
![Page 319: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/319.jpg)
Partial differential
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
47 / 47
Let s and t be adjacent states of a framed chord diagram D
which is equipped with a vice orientation and markers.Let t differs from s only by a marker sign at chord c ,positive in s and negative at t .Construct Vs → Vt .Put it to be 0 if |s| = |t| .Otherwise, order the components of Ds and Dt so that:• The first component passes through the marker at c .• On the second place put the other component passesthough c (if there is one).
![Page 320: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/320.jpg)
Partial differential
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
47 / 47
Let s and t be adjacent states of a framed chord diagram D
which is equipped with a vice orientation and markers.Let t differs from s only by a marker sign at chord c ,positive in s and negative at t .Construct Vs → Vt .Put it to be 0 if |s| = |t| .Otherwise, order the components of Ds and Dt so that:• The first component passes through the marker at c .• On the second place put the other component passesthough c (if there is one).• Other components (which are common for Ds and Dt ) areto be ordered coherently.
![Page 321: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/321.jpg)
Partial differential
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
47 / 47
Let s and t be adjacent states of a framed chord diagram D
which is equipped with a vice orientation and markers.Let t differs from s only by a marker sign at chord c ,positive in s and negative at t .Construct Vs → Vt .Put it to be 0 if |s| = |t| .Otherwise, order the components of Ds and Dt so that:• The first component passes through the marker at c .• On the second place put the other component passesthough c (if there is one).• Other components (which are common for Ds and Dt ) areto be ordered coherently.Orient the first components according to thevice orientation at c .
![Page 322: Oleg Viro - RASolegviro/FrChD.pdfA knot is a smooth simple closed curve in the 3-space. Classical link diagrams Knots and links • Classical link diagrams • 1D-picture • Gauss](https://reader033.fdocuments.net/reader033/viewer/2022060302/5f0899f47e708231d422d0de/html5/thumbnails/322.jpg)
Partial differential
Knots and links
Virtual links
Moves
Kauffman bracket
Gauss diagrams of apoor man
Khovanov homology
Orientation of chorddiagrams
Khovanov complex offramed chord diagram
• Structure used in theconstruction• Involution in theFrobenius algebra
• Space associated toa state
• Partial differential
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Let s and t be adjacent states of a framed chord diagram D
which is equipped with a vice orientation and markers.Let t differs from s only by a marker sign at chord c ,positive in s and negative at t .Construct Vs → Vt .Put it to be 0 if |s| = |t| .Otherwise, order the components of Ds and Dt so that:• The first component passes through the marker at c .• On the second place put the other component passesthough c (if there is one).• Other components (which are common for Ds and Dt ) areto be ordered coherently.Orient the first components according to thevice orientation at c . In these representations of Vs and Vt ,define the map by multiplication or co-multipication.