Invariants of Legendrian Knots and the Legendrian...
Transcript of Invariants of Legendrian Knots and the Legendrian...
Invariants of Legendrian Knotsand the Legendrian Mirror Problem
Joshua M. Sabloff
Joint with G. Civan, J. Etnyre, P. Koprowski, A. Walker
Wesleyan University, April 2009
Goals
• Introduce an intriguing problem in Legendrian knot theory• Show you how to use combinatorial and algebraic techniques to
produce a series of “non-classical” invariants of Legendrian knots• Give infinitely many solutions to the problem
Goals
• Introduce an intriguing problem in Legendrian knot theory• Show you how to use combinatorial and algebraic techniques to
produce a series of “non-classical” invariants of Legendrian knots• Give infinitely many solutions to the problem
Goals
• Introduce an intriguing problem in Legendrian knot theory• Show you how to use combinatorial and algebraic techniques to
produce a series of “non-classical” invariants of Legendrian knots• Give infinitely many solutions to the problem
Outline
1 Geometric Notions and Questions
2 The First Invariant: Augmentations
3 The Second Invariant: Linearized Contact Homology
4 Products
Outline
1 Geometric Notions and Questions
2 The First Invariant: Augmentations
3 The Second Invariant: Linearized Contact Homology
4 Products
Outline
1 Geometric Notions and Questions
2 The First Invariant: Augmentations
3 The Second Invariant: Linearized Contact Homology
4 Products
Outline
1 Geometric Notions and Questions
2 The First Invariant: Augmentations
3 The Second Invariant: Linearized Contact Homology
4 Products
Where Are We?
1 Geometric Notions and Questions
2 The First Invariant: Augmentations
3 The Second Invariant: Linearized Contact Homology
4 Products
Geometric Notions and Questions
Knots in R3
A knot is a smooth embedding γ : S1 → R3.
Two knots are equivalent if one can be deformed into the other throughother knots.
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 5 / 33
Geometric Notions and Questions
Knots in R3
A knot is a smooth embedding γ : S1 → R3.
Two knots are equivalent if one can be deformed into the other throughother knots.
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 5 / 33
Geometric Notions and Questions
Mirrors
QuestionWhen is a knot equivalent to its mirror image (under(x , y , z) → (x , y ,−z))?
Sometimes . . . Sometimes not!
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 6 / 33
Geometric Notions and Questions
Mirrors
QuestionWhen is a knot equivalent to its mirror image (under(x , y , z) → (x , y ,−z))?
Sometimes . . .
Sometimes not!
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 6 / 33
Geometric Notions and Questions
Mirrors
QuestionWhen is a knot equivalent to its mirror image (under(x , y , z) → (x , y ,−z))?
Sometimes . . . Sometimes not!
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 6 / 33
Geometric Notions and Questions
Legendrian Knots
Legendrian knots arise in the study of contact topology.
• A contact structure ξ on R3 (or
any smooth 3-manifold) is acompletely non-integrable2-plane field, such as the onespanned by:
{j, i + yk}.
• A knot γ is Legendrian if it isalways tangent to ξ:
z ′(t) = y(t) x ′(t)
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 7 / 33
Geometric Notions and Questions
Legendrian Knots
Legendrian knots arise in the study of contact topology.
• A contact structure ξ on R3 (or
any smooth 3-manifold) is acompletely non-integrable2-plane field, such as the onespanned by:
{j, i + yk}.
• A knot γ is Legendrian if it isalways tangent to ξ:
z ′(t) = y(t) x ′(t)
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 7 / 33
Geometric Notions and Questions
Legendrian Knots
Legendrian knots arise in the study of contact topology.
• A contact structure ξ on R3 (or
any smooth 3-manifold) is acompletely non-integrable2-plane field, such as the onespanned by:
{j, i + yk}.
• A knot γ is Legendrian if it isalways tangent to ξ:
z ′(t) = y(t) x ′(t)
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 7 / 33
Geometric Notions and Questions
VisualizationFront Projections
Project the knot to the xz plane.
• Since y(t) = z′(t)x ′(t) , the y coordinate
is the slope of the curve• . . . so no vertical tangents, only
cusps• . . . so crossings are always like:
xy
zz
x
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 8 / 33
Geometric Notions and Questions
VisualizationFront Projections
Project the knot to the xz plane.
• Since y(t) = z′(t)x ′(t) , the y coordinate
is the slope of the curve
• . . . so no vertical tangents, onlycusps
• . . . so crossings are always like:
xy
zz
x
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 8 / 33
Geometric Notions and Questions
VisualizationFront Projections
Project the knot to the xz plane.
• Since y(t) = z′(t)x ′(t) , the y coordinate
is the slope of the curve• . . . so no vertical tangents, only
cusps
• . . . so crossings are always like:
xy
zz
x
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 8 / 33
Geometric Notions and Questions
VisualizationFront Projections
Project the knot to the xz plane.
• Since y(t) = z′(t)x ′(t) , the y coordinate
is the slope of the curve• . . . so no vertical tangents, only
cusps• . . . so crossings are always like:
xy
zz
x
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 8 / 33
Geometric Notions and Questions
VisualizationLagrangian Projections
Project the knot to the xy plane.
• Since z(t) =∫ t
0 y(t)x ′(t) dt , the zcoordinate can be recovered
• . . . and by Green’s theorem, theprojection must bound zero signedarea
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 9 / 33
Geometric Notions and Questions
VisualizationLagrangian Projections
Project the knot to the xy plane.
• Since z(t) =∫ t
0 y(t)x ′(t) dt , the zcoordinate can be recovered
• . . . and by Green’s theorem, theprojection must bound zero signedarea
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 9 / 33
Geometric Notions and Questions
VisualizationLagrangian Projections
Project the knot to the xy plane.
• Since z(t) =∫ t
0 y(t)x ′(t) dt , the zcoordinate can be recovered
• . . . and by Green’s theorem, theprojection must bound zero signedarea
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 9 / 33
Geometric Notions and Questions
Classical Invariants
• The Thurston-Bennequin number tbmeasures the linking between theknot and a push-off transverse to ξ
• tb may be computed from thewrithe of the xy diagram
• . . . or writhe−#right cusps in thexz diagram
• The rotation number r measurestwisting of γ′(t) inside a trivializationof ξ
• r may be computed from therotation number of the xy diagram
• . . . or #down cusps−#up cuspsin the xz diagram
• tb = −2• r = ±1
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 10 / 33
Geometric Notions and Questions
Classical Invariants
• The Thurston-Bennequin number tbmeasures the linking between theknot and a push-off transverse to ξ
• tb may be computed from thewrithe of the xy diagram
• . . . or writhe−#right cusps in thexz diagram
• The rotation number r measurestwisting of γ′(t) inside a trivializationof ξ
• r may be computed from therotation number of the xy diagram
• . . . or #down cusps−#up cuspsin the xz diagram
• tb = −2
• r = ±1
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 10 / 33
Geometric Notions and Questions
Classical Invariants
• The Thurston-Bennequin number tbmeasures the linking between theknot and a push-off transverse to ξ
• tb may be computed from thewrithe of the xy diagram
• . . . or writhe−#right cusps in thexz diagram
• The rotation number r measurestwisting of γ′(t) inside a trivializationof ξ
• r may be computed from therotation number of the xy diagram
• . . . or #down cusps−#up cuspsin the xz diagram
• tb = −2
• r = ±1
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 10 / 33
Geometric Notions and Questions
Classical Invariants
• The Thurston-Bennequin number tbmeasures the linking between theknot and a push-off transverse to ξ
• tb may be computed from thewrithe of the xy diagram
• . . . or writhe−#right cusps in thexz diagram
• The rotation number r measurestwisting of γ′(t) inside a trivializationof ξ
• r may be computed from therotation number of the xy diagram
• . . . or #down cusps−#up cuspsin the xz diagram
• tb = −2
• r = ±1
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 10 / 33
Geometric Notions and Questions
Classical Invariants
• The Thurston-Bennequin number tbmeasures the linking between theknot and a push-off transverse to ξ
• tb may be computed from thewrithe of the xy diagram
• . . . or writhe−#right cusps in thexz diagram
• The rotation number r measurestwisting of γ′(t) inside a trivializationof ξ
• r may be computed from therotation number of the xy diagram
• . . . or #down cusps−#up cuspsin the xz diagram
• tb = −2• r = ±1
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 10 / 33
Geometric Notions and Questions
Classical Invariants
• The Thurston-Bennequin number tbmeasures the linking between theknot and a push-off transverse to ξ
• tb may be computed from thewrithe of the xy diagram
• . . . or writhe−#right cusps in thexz diagram
• The rotation number r measurestwisting of γ′(t) inside a trivializationof ξ
• r may be computed from therotation number of the xy diagram
• . . . or #down cusps−#up cuspsin the xz diagram
• tb = −2• r = ±1
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 10 / 33
Geometric Notions and Questions
Classification?
The classical invariants completely classify Legendrian unknots[Eliashberg-Fraser], torus knots and the figure eight knot [Etnyre-Honda], . . . .Call these knot types simple.
Unknot [Eliashberg-Fraser]
0 1 2 3–1–2–3
–1
–2
–3
–4
tb
r
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 11 / 33
Geometric Notions and Questions
Classification?
The classical invariants completely classify Legendrian unknots[Eliashberg-Fraser], torus knots and the figure eight knot [Etnyre-Honda], . . . .Call these knot types simple.
Unknot [Eliashberg-Fraser]
0 1 2 3–1–2–3
–1
–2
–3
–4
tb
r
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 11 / 33
Geometric Notions and Questions
The Plot Thickens
But the classical invariants do not classify all Legendrian knots!
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 12 / 33
Geometric Notions and Questions
Legendrian Mirrors
QuestionWhen is a Legendrian knot with r = 0 equivalent to its “Legendrianmirror” under (x , y , z) → (x ,−y ,−z)?
Note that this is a rotation in R3, not a reflection, but it reverses theco-orientation of ξ
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 13 / 33
Geometric Notions and Questions
Legendrian Mirrors
QuestionWhen is a Legendrian knot with r = 0 equivalent to its “Legendrianmirror” under (x , y , z) → (x ,−y ,−z)?
Note that this is a rotation in R3, not a reflection, but it reverses theco-orientation of ξ
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 13 / 33
Geometric Notions and Questions
Legendrian Mirrors
QuestionWhen is a Legendrian knot with r = 0 equivalent to its “Legendrianmirror” under (x , y , z) → (x ,−y ,−z)?
Note that this is a rotation in R3, not a reflection, but it reverses theco-orientation of ξThis reflects the front diagram:
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 13 / 33
Geometric Notions and Questions
Legendrian Mirrors
QuestionWhen is a Legendrian knot with r = 0 equivalent to its “Legendrianmirror” under (x , y , z) → (x ,−y ,−z)?
Note that this is a rotation in R3, not a reflection, but it reverses theco-orientation of ξAnd also the Lagrangian diagram (but also switches the crossings):
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 13 / 33
Geometric Notions and Questions
Distinct Legendrian Mirrors?
Ng found examples of 62 knots . . .
TheoremThere exists an infinite family of Legendrian knots that are notLegendrian isotopic to their Legendrian mirrors.
In fact, I’ll show you an infinite family of infinite families, distinguishedby ever-deeper invariants.
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 14 / 33
Geometric Notions and Questions
Distinct Legendrian Mirrors?
Ng found examples of 62 knots . . .
TheoremThere exists an infinite family of Legendrian knots that are notLegendrian isotopic to their Legendrian mirrors.
In fact, I’ll show you an infinite family of infinite families, distinguishedby ever-deeper invariants.
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 14 / 33
Geometric Notions and Questions
Distinct Legendrian Mirrors?
Ng found examples of 62 knots . . .
TheoremThere exists an infinite family of Legendrian knots that are notLegendrian isotopic to their Legendrian mirrors.
In fact, I’ll show you an infinite family of infinite families, distinguishedby ever-deeper invariants.
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 14 / 33
Where Are We?
1 Geometric Notions and Questions
2 The First Invariant: Augmentations
3 The Second Invariant: Linearized Contact Homology
4 Products
The First Invariant: Augmentations
Special Subsets of Crossings
The goal is to pick out special subsets of the crossings in a Lagrangiandiagram.
• First, decorate the crossings.• Look for smoothly immersed
disks with convex cornerswhose boundary lies in theknot diagram. There is one +corner, any # of − corners.
• Select a subset of crossings iffor each fixed corner c, thenumber of disks with a positivecorner at c and all negativecorners in the chosen subsetis even.
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 16 / 33
The First Invariant: Augmentations
Special Subsets of Crossings
The goal is to pick out special subsets of the crossings in a Lagrangiandiagram.
• First, decorate the crossings.
• Look for smoothly immerseddisks with convex cornerswhose boundary lies in theknot diagram. There is one +corner, any # of − corners.
• Select a subset of crossings iffor each fixed corner c, thenumber of disks with a positivecorner at c and all negativecorners in the chosen subsetis even.
1
2
345
+ +––
+ +––
+ +––
+ +––+ +
––
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 16 / 33
The First Invariant: Augmentations
Special Subsets of Crossings
The goal is to pick out special subsets of the crossings in a Lagrangiandiagram.
• First, decorate the crossings.• Look for smoothly immersed
disks with convex cornerswhose boundary lies in theknot diagram. There is one +corner, any # of − corners.
• Select a subset of crossings iffor each fixed corner c, thenumber of disks with a positivecorner at c and all negativecorners in the chosen subsetis even.
1
2
345
+ +––
+ +––
+ +––
+ +––+ +
––
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 16 / 33
The First Invariant: Augmentations
Special Subsets of Crossings
The goal is to pick out special subsets of the crossings in a Lagrangiandiagram.
• First, decorate the crossings.• Look for smoothly immersed
disks with convex cornerswhose boundary lies in theknot diagram. There is one +corner, any # of − corners.
• Select a subset of crossings iffor each fixed corner c, thenumber of disks with a positivecorner at c and all negativecorners in the chosen subsetis even.
1
2
345
+ +––
+ +––
+ +––
+ +––+ +
––
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 16 / 33
The First Invariant: Augmentations
Special Subsets of Crossings
The goal is to pick out special subsets of the crossings in a Lagrangiandiagram.
• First, decorate the crossings.• Look for smoothly immersed
disks with convex cornerswhose boundary lies in theknot diagram. There is one +corner, any # of − corners.
• Select a subset of crossings iffor each fixed corner c, thenumber of disks with a positivecorner at c and all negativecorners in the chosen subsetis even.
1
2
345
+ +––
+ +––
+ +––
+ +––+ +
––
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 16 / 33
The First Invariant: Augmentations
An Example
This trefoil knot has 5 augmentations.
1
2
345
1
2
345
1
2
3451
2
345
1
2
345
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 17 / 33
The First Invariant: Augmentations
Gradings
The crossings are graded by a “Conley-Zehnder index” in Z/2r(K )Z.
Choose a capping path γi for the double point i , beginning at theovercrossing:
γ3
|qi | ≡ 2r(γi)−12
mod 2r(K )
|qi | = 0 here. . .
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 18 / 33
The First Invariant: Augmentations
Gradings
The crossings are graded by a “Conley-Zehnder index” in Z/2r(K )Z.Choose a capping path γi for the double point i , beginning at theovercrossing:
γ3
|qi | ≡ 2r(γi)−12
mod 2r(K )
|qi | = 0 here. . .
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 18 / 33
The First Invariant: Augmentations
Gradings
The crossings are graded by a “Conley-Zehnder index” in Z/2r(K )Z.Choose a capping path γi for the double point i , beginning at theovercrossing:
γ3
|qi | ≡ 2r(γi)−12
mod 2r(K )
|qi | = 0 here. . .
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 18 / 33
The First Invariant: Augmentations
Gradings
The crossings are graded by a “Conley-Zehnder index” in Z/2r(K )Z.Choose a capping path γi for the double point i , beginning at theovercrossing:
γ3
|qi | ≡ 2r(γi)−12
mod 2r(K )
|qi | = 0 here. . .
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 18 / 33
The First Invariant: Augmentations
The Augmentation Invariant
We can form an invariant Aug(D) by forming a (normalized) count of“graded” augmentations of a diagram D.
TheoremAug(D) is a Legendrian invariant (now denoted Aug(K ))
Example
The augmentation number can distinguish the 52 knots, but not a knotfrom its Legendrian mirror.
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 19 / 33
The First Invariant: Augmentations
The Augmentation Invariant
We can form an invariant Aug(D) by forming a (normalized) count of“graded” augmentations of a diagram D.
TheoremAug(D) is a Legendrian invariant (now denoted Aug(K ))
Example
The augmentation number can distinguish the 52 knots, but not a knotfrom its Legendrian mirror.
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 19 / 33
The First Invariant: Augmentations
The Augmentation Invariant
We can form an invariant Aug(D) by forming a (normalized) count of“graded” augmentations of a diagram D.
TheoremAug(D) is a Legendrian invariant (now denoted Aug(K ))
Example
The augmentation number can distinguish the 52 knots, but not a knotfrom its Legendrian mirror.
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 19 / 33
Where Are We?
1 Geometric Notions and Questions
2 The First Invariant: Augmentations
3 The Second Invariant: Linearized Contact Homology
4 Products
The Second Invariant: Linearized Contact Homology
Linearized CohomologyThe Vector Space
Label the crossings in an Lagrangian diagram with {1, . . . , n}. Define avector space A over Z/2Z generated by labels {q1, . . . , qn}.
We can split A as a direct sum according to the gradings of thegenerators.
Example
For the trefoil, A0 = 〈q3, q4, q5〉 and A1 = 〈q1, q2〉.
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 21 / 33
The Second Invariant: Linearized Contact Homology
Linearized CohomologyThe Vector Space
Label the crossings in an Lagrangian diagram with {1, . . . , n}. Define avector space A over Z/2Z generated by labels {q1, . . . , qn}.
We can split A as a direct sum according to the gradings of thegenerators.
Example
For the trefoil, A0 = 〈q3, q4, q5〉 and A1 = 〈q1, q2〉.
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 21 / 33
The Second Invariant: Linearized Contact Homology
Linearized CohomologyThe Vector Space
Label the crossings in an Lagrangian diagram with {1, . . . , n}. Define avector space A over Z/2Z generated by labels {q1, . . . , qn}.
We can split A as a direct sum according to the gradings of thegenerators.
Example
For the trefoil, A0 = 〈q3, q4, q5〉 and A1 = 〈q1, q2〉.
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 21 / 33
The Second Invariant: Linearized Contact Homology
Linearized CohomologyThe Differential
Given an augmentation ε, let δε : A → A be the linear map defined asfollows:
The contribution of qkto δεqi is the number ofdisks with − at qi , + atqk , and maybe other −corners in ε.
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 22 / 33
The Second Invariant: Linearized Contact Homology
Linearized CohomologyThe Differential
Given an augmentation ε, let δε : A → A be the linear map defined asfollows:
The contribution of qkto δεqi is the number ofdisks with − at qi , + atqk , and maybe other −corners in ε.
1
2
345
+ +––
+ +––
+ +––
+ +––+ +
––
δεq5 = q1 + q2
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 22 / 33
The Second Invariant: Linearized Contact Homology
Linearized CohomologyThe Differential
Given an augmentation ε, let δε : A → A be the linear map defined asfollows:
The contribution of qkto δεqi is the number ofdisks with − at qi , + atqk , and maybe other −corners in ε.
1
2
345
+ +––
+ +––
+ +––
+ +––+ +
––
δεq5 = 2q1 + 2q2 = 0
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 22 / 33
The Second Invariant: Linearized Contact Homology
The Algebra in the Linearized Invariant
Theorem (Chekanov)
δε ◦ δε = 0
Thus im δε ⊂ ker δε, and we can form the linearized homologyLCH∗
ε = ker δε/im δε.
Example
For the trefoil above, the only nonzero differential is
δεq4 = q1 + q2
Thus, we have LCH∗ε = 〈[q1], [q3], [q5]〉.
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 23 / 33
The Second Invariant: Linearized Contact Homology
The Algebra in the Linearized Invariant
Theorem (Chekanov)
δε ◦ δε = 0
Thus im δε ⊂ ker δε, and we can form the linearized homologyLCH∗
ε = ker δε/im δε.
Example
For the trefoil above, the only nonzero differential is
δεq4 = q1 + q2
Thus, we have LCH∗ε = 〈[q1], [q3], [q5]〉.
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 23 / 33
The Second Invariant: Linearized Contact Homology
The Algebra in the Linearized Invariant
Theorem (Chekanov)
δε ◦ δε = 0
Thus im δε ⊂ ker δε, and we can form the linearized homologyLCH∗
ε = ker δε/im δε.
Example
For the trefoil above, the only nonzero differential is
δεq4 = q1 + q2
Thus, we have LCH∗ε = 〈[q1], [q3], [q5]〉.
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 23 / 33
The Second Invariant: Linearized Contact Homology
The Linearized Invariant
Theorem (Chekanov)
The set{
LCH∗ε
}ε
is an invariant of Legendrian isotopy.
Example
The linearized invariant can also distinguish the 52 knots, but still not aknot from its Legendrian mirror.
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 24 / 33
The Second Invariant: Linearized Contact Homology
The Linearized Invariant
Theorem (Chekanov)
The set{
LCH∗ε
}ε
is an invariant of Legendrian isotopy.
Example
The linearized invariant can also distinguish the 52 knots, but still not aknot from its Legendrian mirror.
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 24 / 33
Where Are We?
1 Geometric Notions and Questions
2 The First Invariant: Augmentations
3 The Second Invariant: Linearized Contact Homology
4 Products
Products
The Cup Product
To capture the mirror example, we need a potentially non-symmetricoperation on the linearized cohomology.
. . . so define a linear map mε : A⊗ A → A as follows:
The contribution of qkto mε(qi , qj) is thenumber of disks with −at qi and qj (in thatcounterclockwiseorder!), + at qk , andmaybe other − cornersin ε.
1
2
345
+ +––
+ +––
+ +––
+ +––+ +
––
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 26 / 33
Products
The Cup Product
To capture the mirror example, we need a potentially non-symmetricoperation on the linearized cohomology.
. . . so define a linear map mε : A⊗ A → A as follows:
The contribution of qkto mε(qi , qj) is thenumber of disks with −at qi and qj (in thatcounterclockwiseorder!), + at qk , andmaybe other − cornersin ε.
1
2
345
+ +––
+ +––
+ +––
+ +––+ +
––
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 26 / 33
Products
The Cup Product
To capture the mirror example, we need a potentially non-symmetricoperation on the linearized cohomology.
. . . so define a linear map mε : A⊗ A → A as follows:
The contribution of qkto mε(qi , qj) is thenumber of disks with −at qi and qj (in thatcounterclockwiseorder!), + at qk , andmaybe other − cornersin ε.
1
2
345
+ +––
+ +––
+ +––
+ +––+ +
––
mε(q5 ⊗ q3) = q1
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 26 / 33
Products
What is Invariant?
The product map mε descends to a map
µε : LCH jε ⊗ LCHk
ε → LCH j+k+1ε
on cohomology, and . . .
TheoremThe set of linearized homology algebras {(LCH∗
ε , µε)}ε is invariantunder Legendrian isotopy.
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 27 / 33
Products
What is Invariant?
The product map mε descends to a map
µε : LCH jε ⊗ LCHk
ε → LCH j+k+1ε
on cohomology, and . . .
TheoremThe set of linearized homology algebras {(LCH∗
ε , µε)}ε is invariantunder Legendrian isotopy.
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 27 / 33
Products
Detecting Mirrors
Want to find a knot (say, with only one ε) with nontrivial
µε : LCH jε ⊗ LCHk
ε → LCH j+k+1ε
but trivialµε : LCHk
ε ⊗ LCH jε → LCH j+k+1
ε
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 28 / 33
Products
Detecting Mirrors
Want to find a knot (say, with only one ε) with nontrivial
µε : LCH jε ⊗ LCHk
ε → LCH j+k+1ε
but trivialµε : LCHk
ε ⊗ LCH jε → LCH j+k+1
ε
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 28 / 33
Products
An Infinite Family of Mirrors
k
lm
+
–
–
–
Nontrivial cup product in the following degrees:
LCH l−m−1 ⊗ LCH1−k+m → LCH1−k+l
. . . but none when order is reversed. Hence it’s not isotopic to itsLegendrian mirror!
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 29 / 33
Products
An Infinite Family of Mirrors
k
lm
+
–
–
–
Nontrivial cup product in the following degrees:
LCH l−m−1 ⊗ LCH1−k+m → LCH1−k+l
. . . but none when order is reversed. Hence it’s not isotopic to itsLegendrian mirror!
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 29 / 33
Products
An Infinite Family for the Massey Product
Using disks with three negative corners and a little algebra, we candefine a three-fold “Massey product” that can also be used todistinguish knots from their mirrors:
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 30 / 33
Products
An Infinite Family for the Massey Product
Using disks with three negative corners and a little algebra, we candefine a three-fold “Massey product” that can also be used todistinguish knots from their mirrors:
k
l
m n
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 30 / 33
Products
An Infinite Family of Infinite Families
In fact, there is an A∞ structure {mεk} on A∗ that induces higher-order
Massey products on LCH∗ε .
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 31 / 33
Products
An Infinite Family of Infinite Families
In fact, there is an A∞ structure {mεk} on A∗ that induces higher-order
Massey products on LCH∗ε .
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 31 / 33
Summary
Where Do We Come From? What Are We? WhereAre We Going?
What else can we do / say here?
• Cup product? “Poincaré Duality”!• Larger context and motivation: Morse-Witten-Floer theory• Geometric meaning
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 32 / 33
Summary
Where Do We Come From? What Are We? WhereAre We Going?
What else can we do / say here?
• Cup product? “Poincaré Duality”!
• Larger context and motivation: Morse-Witten-Floer theory• Geometric meaning
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 32 / 33
Summary
Where Do We Come From? What Are We? WhereAre We Going?
What else can we do / say here?
• Cup product? “Poincaré Duality”!• Larger context and motivation: Morse-Witten-Floer theory
• Geometric meaning
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 32 / 33
Summary
Where Do We Come From? What Are We? WhereAre We Going?
What else can we do / say here?
• Cup product? “Poincaré Duality”!• Larger context and motivation: Morse-Witten-Floer theory• Geometric meaning
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 32 / 33
Summary
Where Do We Come From? What Are We? WhereAre We Going?
Hopefully, you have seen that the Legendrian mirror problem canmotivate some interesting structure in invariants of Legendrian knotsusing an interplay of:
• Combinatorics (counting disks!)• Algebra (linearized contact homology and its product structure!)• Geometric / analytic motivation (err . . . actually I hid this today, but
you might now be motivated to find out!)
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 33 / 33
Summary
Where Do We Come From? What Are We? WhereAre We Going?
Hopefully, you have seen that the Legendrian mirror problem canmotivate some interesting structure in invariants of Legendrian knotsusing an interplay of:
• Combinatorics (counting disks!)
• Algebra (linearized contact homology and its product structure!)• Geometric / analytic motivation (err . . . actually I hid this today, but
you might now be motivated to find out!)
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 33 / 33
Summary
Where Do We Come From? What Are We? WhereAre We Going?
Hopefully, you have seen that the Legendrian mirror problem canmotivate some interesting structure in invariants of Legendrian knotsusing an interplay of:
• Combinatorics (counting disks!)• Algebra (linearized contact homology and its product structure!)
• Geometric / analytic motivation (err . . . actually I hid this today, butyou might now be motivated to find out!)
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 33 / 33
Summary
Where Do We Come From? What Are We? WhereAre We Going?
Hopefully, you have seen that the Legendrian mirror problem canmotivate some interesting structure in invariants of Legendrian knotsusing an interplay of:
• Combinatorics (counting disks!)• Algebra (linearized contact homology and its product structure!)• Geometric / analytic motivation (err . . . actually I hid this today, but
you might now be motivated to find out!)
Joshua M. Sabloff (et al.) Legendrian Invariants and Mirrors Wesleyan Colloquium 33 / 33