Knot TheorySenior Seminar by Tim Wylie

December 3, 2002

Outline

•Introduction with brief history•Lord Kelvin’s atom

•Defining a knot

•History and Progress•Individual advances in the field

•Applications

•Conclusion

In 1867 Lord Kelvin proposed his theory of the vortex atom.

He was inspired by a paper by Helmholtz on vortices, and by a paper byRiemann on Abelian functions.

His theory stated that atoms are vortex rings, the movement of the vortexgiving the illusion of matter.

It also stated that chemical properties of elements were related to knottingthat occurs between atoms.

Peter G. Tait

In 1877 published the first paper addressingthe enumeration of knots.

Over the next few years he began working with C.N. Little and together they completedall the enumerations of knots past 10crossings by 1900.

Problems

When his work began, the formal mathematics needed to address the subjectwas unavailable.

He found the distinct knots but didn’t have the math needed to prove theywere distinct.

However, work at the beginning of the 20th century placed the subject of topology on firm mathematical ground.

Now it was possible to precisely define knots and to prove theorems aboutthem.

Algebraic methods introduced into the subject became especially important,providing the means to prove distinct knots.

Topology

The Tait conjectures-conjectures about knot projections that were unable to be proven at the time.

So, what is a knot?So, what is a knot?

A simple definition is that a knot is a continuous simple closed curvein three-dimensional Euclidean space R3

This however, is not an entirely accurate definition because it isextremely difficult to deal with deformations, and it allows someinfinite wild knots that are improbable.

So, we define a knot as a simple closed polygonal curve in R3.

For any two distinct points in 3-space, p and q, let [p,q] denote theline segment joining them. For an ordered set of distinct points(p1, p2,……, pn), the union of the segments [p1, p2], [p2,p3],….[pn-1, pn] and [pn,p1] is called a closed polygonal curve. If eachsegment intersects exactly two other segments, intersecting each onlyat an endpoint, then the curve is said to be simple.

Right, so now some pictures.

Knots are “stick knots” but are usuallydrawn and thought of as smooth.

Intuitively, we realize that a smoothknot would be closely approximatedwith a very large number of segmentsin a polygonal curve.

This observation leads us to a useful definition of minimizing the pointsin a knot.

If the ordered set (p1, p2, ….,pn) defines a knot, and no properordered subset defines the same knot, the elements of the set{pi} are called vertices of the knot.

The two simplest knots are the right and left trefoil, which are each distinct.

In 1914 Max Dehn was the first to prove that two knotswere distinct. He proved the right and left trefoils weredistinct.

Two knots are viewed as equivalent, or of the same type,if one can be deformed into the other knot. So tounderstand equivalence you have to understand deformations.

Definition: A knot J is called an elementary deformation of the knot K if oneof the two knots is determined by a sequence of points (p1, p2,…,pn) and theother is determined by the sequence (p0, p1, p2,…,pn), where 1. p0 is a point which is not collinear with p1 and pn, and2. The triangle spanned by (p0, p1, pn) intersects the knot determined by (p1, p2,….,pn) only in the segment [p1, pn].

Definition: Knots K and J are called equivalent if there is a sequence of knotsK=K0, K1,…,Kn=J with each K(i+1) an elementary deformation of Ki, for igreater than 0.

Combinatorial Methods

The techniques of knot theory based on the study of knot diagrams.

1. The Reidemeister Moves2. Knot Colorings3. The Alexander polynomial

A knot diagram is the projection (or shadow) of a knot from 3-space to a plane.It can be proven that if two knots have the same projection they are equivalentregardless of their dimensions in R3.

The Reidemeister MovesThe Reidemeister Moves

In 1932 K. Reidemeister invented the Reidemeister moves.

Theorem: If two knots are equivalent, their diagrams are related by a sequenceof Reidemeister moves.

In theory these tools are enough to distinguish any pair of distinct knots, however,for knots with complicated diagrams the calculations are often too lengthy to beof use.

Knot ColoringsKnot Colorings

The method of distinguishing knots using the “colorability” of their diagramswas invented by Ralph Fox.

A knot is called colorable if each arc can be drawn using one of three colorsin such a way that:1. At least two of the colors are used 2. At any crossing at which two colors appear, all three appear.

Theorem: If a diagram of a knot, K, is colorable, then every diagram of Kis colorable.

The Alexander PolynomialIn 1928 James Alexander described a method of associating to each knot apolynomial such that if one knot can be deformed into another, both will havethe same associated polynomial.

Invariant: A quantity which remains unchanged under certain classes of transformations. Invariants are extremely useful for classifying mathematical objects because they usually reflect intrinsic properties of the object of study.

Calculating the Alexander Polynomial…….

1. Pick a diagram of the knot K2. Number the arcs of the diagram3. Separately, number the crossings4. Define an N x N matrix where N is the number of crossings (and arcs)

5. Take a crossing numbered L. If it is right-handed with arc i passing overarcs j and k enter a (1-t) in column i of row L, enter a (-1) in column j ofthat row, and enter a (t) in column k of that row.If the crossing is left-handed, enter a (1-t) in column i of row L, enter a(t) in column j and enter a (-1) in column k of row L.All other entries in row L are 0.

6. Remove the last column and row of the N x N matrix.7. Take the determinant of the (N-1) x (N-1) matrix

AND WE’RE DONE!!!

The most simple knot to determine the alexander polynomialof is the trefoil.

Applying the first several steps we end up with this matrix.

(1-t) t -1-1 (1-t) t-1 t (1-t)

Deleting the bottom row and the last column gives a 2x2 matrix.Taking the determinant gives the Alexander polynomial A(t)=t^2 –t +1

Theorem: If the Alexander polynomial for a knot is computed using two different sets of choices for diagrams and labelings, the two polynomials willdiffer by a multiple of +-t^k for some integer k.

More History

In the mid 1930’s H. Seifert demonstrated that if a knot is the boundary of asurface in 3-space, then that surface can be used to study the knot.

This discovery laid the foundation for the use of geometric methods in knottheory.

In 1947 H. Schubert used geometric methods to prove that there are primeknots. A knot is called prime if it cannot be decomposed as a connected sum of nontrivial knots. Any knot can be decomposed uniquely as the connected sum of prime knots.

F. Waldhausen proved in 1968 that two knots are equivalent if and only if certainalgebraic properties are the same.

In 1957 C. Papakyriakopoulos succeeded in proving the Dehn Lemma, which saysthat if a knot were indistinguishable from the trivial knot using algebraic methods,then the knot is in fact trivial.

William Thurston proved in 1978 that the complements of knots in 3-space havea complete hyberbolic structures.

The Tait conjectures were finally proven in the late 80’s based on a completelydifferent polynomial invariant using the theory of operator algebras.The invariant was discovered in 1987 by Vaughan Jones.

Applications

The first place knot theory was seriously used wasin the study and manipulation of DNA. It was discovered in 1953 by Watson and Crick. Theyalso discovered that DNA can become knottedwhich makes it difficult to carry out its function.

Knot theory is also used in molecular chemistry and statistical mechanics.A recent use of knot theory is applying it to quantum computing.

The biggest contributions of knot theory have just recently developed thanksTo the work of Vaughan Jones and his invariants.

1) all 3-manifolds can be describe in terms of knots and links via an operation called Dehn surgery; 2) there exists a set of moves, the Kirby calculus, that allow one to move between differing Dehn surgery descriptions of the same homeomorphic 3-manifold.

Edward Witten has discovered that knots are connected to quantum field theorythrough generalized 3-manifold invariants.

Distinct knots

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Acknowledgements

God

My Family

My Wife, Rachel

My Teachers

My Friends

SourcesLivingston, Charles. Knot theory. Washington, DC: Mathematical Association of America, 1993.

Johnson, Scholz, et al. History of Topology. Amsterdam: ElsevierScience B.V., 1999.

Crowell, Richard H., and Ralph H. Fox. Introduction to Knot Theory. New York, NY. Blaisdell Publishing Company, 1963

A Circular History of Knot Theoryhttp://www.math.buffalo.edu/~menasco/Knottheory.html

The Knotplot sitehttp://www.pims.math.ca/knotplot/

Knot Theory: An Introductionhttp://www.yucc.yorku.ca/~mouse/knots/intro.html#what

Knot Theoryhttp://library.thinkquest.org/12295/main.html

Knot Theory Onlinehttp://www.freelearning.com/knots/

Questions?