EIDMA course on Knot Theory

TUE number

EIDMA code

ECTS credit points

Course description

A knot is what you think it is---the only surprise being that the ends are joined, see The KnotPlot site, for an impression. It is quite natural, although mathematically somewhat tricky, to define which knots are to considered the same (or equivalent). A fundamental problem is to decide whether two given knots are the same. See some of the knots in Dror Bar Natan's lecture notes.

Knot theory has many links to important areas of mathematics. In the course, we shall come across Lie algebras, invariant theory, topology, homotopy groups and category theory.

For whom

Notes

Instructors

Exam

The examination will consist of

• Homework assignments.
• A final project assignment, resulting in a report
• An oral exam.

1. March 31, MA 1.46
   • Knots and Links: Both live in 3-space (say, in S³). A link is a finite disjoint union of a number of knots. A knot is a homeomorphic image of the circle S¹, where one takes care to avoid wild embeddings (require "tame" or "smooth" or "piecewise linear" or so).
   • Isotopy: two knots or links are considered equivalent when one is obtained from the other by continuous deformation where at each point in time we have a homeomorphic image of S¹. This relation is called isotopy - it is finer than homotopy.
   • Two knots are isotopic iff their complements in S³ are homemorphic (Gordon-Lueke, 1989)
   • Project a link from a point in general position into a plane to get a shadow diagram. Attach to the shadow diagram information about over/under crossings to get a diagram (link diagram).
   • Two link diagrams belong to isotopic links iff the diagrams are related by a sequence of Reidemeister moves.
   • Isotopy is decidable (Haken, 1961) Isotopy to unknot is in NP (Hass-Lagarias, 1999) Isotopy can be decided in time O(2^cn), space O(n².log n) (Jaco-Tollefson)
   • Two diagrams or links are called regularly isotopic when one can be transformed into the other using only Reidemeister moves II, III. Writhe, rotation, Kauffman bracket are invariant for regular isotopy. Two isotopic links with same writhe and rotation are regularly isotopic (Trace, 1983)
   • Jones invariant is invariant under all Reidemeister moves. It is expressed in A. With t = A^-4 one gets a Laurent polynomial in t (for knots, or, more generally, for links with oddly many components) or sqrt(t) times a Laurent polynomial in t (for links with evenly many components). Interchanging up and down corresponds to replacing t by t^-1. Different knots may have the same Jones polynomial.
2. April 1, AUD 12
   • Oriented reidemeister moves.
   • The skein relation for the Jones polynomial.
   • The linking number of two knots or links sums the sign (as in the definition of writhe) of their crossings. This is an invariant. A splittable link has linking number zero.
   • Let G be an abelian group. A G-colouring of the arcs of a link diagram is a nonconstant labeling of the arcs by elements of G such that at each crossing left under + right under = 2 over. Having a G-colouring is invariant under Reidemeister moves. The translate of a colouring is again a colouring. The unknot cannot be coloured, the trefoil can be coloured when 3 | |G|. A splittable link has a G-colouring for any abelian group G. The Borromean rings do not have a G-colouring for G of odd order. The Whitehead link has linking number zero and non-splittability follows from looking at colourings.
   • If a diagram has n crossings and n arcs, then finding a colouring is a problem of n equations with n unknowns. The equations are dependent, so can omit one. By translation, one of the labels can be set to 0. Left: n-1 equations with n-1 unknowns. The (absolute value of) the determinant of this system is called the determinant of the link (and is invariant). If the determinant is 1, there is no colouring. If it is d, then there is a colouring with G=C_p the cyclic group of order p iff p | d.
   • A G-colouring says that if a strand labeled a passes under one labeled b then it comes out with label 2b-a. One can use other formulas, or even a completely general setup in some noncommutative, nonassociative system with axioms that describe Reidemeister moves. Kauffman has his knot crystals (union of the set of arcs, and the fundamental group of the complement of the knot, with two injections from the first into the second, and a right action of the second on the first), invariant for regular isotopy. Joyce has his quandle, a quotient of the knot crystal, that is invariant for isotopy.
   • Alternating diagrams for knots or links alternate over/under. An alternating diagram is reduced when the shadow does not have a cut point. If a knot has a reduced alternating diagram then no diagram has fewer crossings, and every non-alternating (or non-reduced) diagram has more crossings. The writhe (of a reduced alternating diagram) is an invariant. (The basic tools here are the Kauffman bracket and the chessboard colouring of the diagram.)
   • Given a diagram of a knot, the fundamental group of the complement of the knot has a presentation by generators labeled by the edges of the diagram and relations of length 4 indexed by crossings.
3. April 7, MA 1.46
   • To an oriented knot diagram, we associate a set of Seifert circles in the plane, a Seifert surface in Euclidean space whose boundary is the knot, and a Seifert tree to register regions (separated by the circles) and their adjacencies.
   • The genus of a knot is the minimal genus of a Seifert surface over all diagrams for the knot.
   • The braids on n strands form a group, denoted B_n. The closure of a braid is an oriented link.
   • Conversely, by a theorem of Alexander, each oriented link is the closure of a braid. To find such a braid for a given link L, remove the conflicting pairs of arcs from an oriented diagram for L by Reidemeister II. This process terminates. At the end, the R0 move of moving an outer arc to the other side of the diagram, iterated until all Seifert circles are concentric, gives the closure of a braid (up to switching the orientation). This is Vogel's algorithm. See Huggenberger's worked out example.
   • The Markov moves play a role for braids (on an arbitrary number of strands) similar to the Reidemeister moves for diagrams.
4. April 8, AUD 12
• The Alexander polynomial as an extension of the colouring and the determinant of a link. It is a knot invariant.
• The first homology group of S³-K is cyclic, with generator t.
• By means of a Seifert surfaces, the universal cyclic cover of S³-K with t-action can be constructed. This is done by cutting S³-K along the Seifert surface at the boundary, making copies M_i indexed by the integers of this 3-manifold, identifying the right hand boundary of the i-th with the left hand boundary of the i+1-st, and letting t be the shift from one copy to the next.
• Using offsets of homology cycles in a Seifert surface for K of genus g, a 2g×2g matrix M can be computed such that the homology of the universal cyclic cover of S³-K as a Z[t,t^-1]-module is a quotient of (Z[t,t^-1])^2g by the submodule generated by the rows of M.
• The Alexander polynomial is the determinant of M.
5. April 14, MA 1.46
   • The linking number of two curves in S³
   • The Alexander polynomial defined as the determinant of a matrix of linking numbers for bases of the homology of a Seifert surface.
   • The Conway potential
   • s-equivalence of Seifert surfaces
   • The skein relation for the Conway potential
   • The Conway polynomial
6. April 15, AUD 12
   • The signature is an invariant; it distinguishes between the trefoil knot and its mirror.
   • Hecke algebras H_n; embedding of H_n in H_n+1
   • 3-term skein relation for oriented links
   • Braid group representations in the Heck algebra
   • The Markov trace on the Hecke algebras; related to the Jones polynomial.
   • 4-term skein relation for un-oriented links
   • Brauer algebras D_n
   • Dimension of D_n is 1 · 3 · 5 · (2n-1)
   • (n,k)-tangles. The (n,n) tangles form a tangle algebra T_n; embedding T_n into T_n+1 and projecting T_n onto D_n (with specialization of coefficients)
   • BMW algebras BMW_n by generators and relations such that there is a natural homomorphism from BMW_n to T_n
   • Isomorphism BMW and Tangle algebra
   • The Markov trace on the BMW algebras
   • There are natural group homomorphisms from the braid group B_n to the invertible elements of H_n and BMW_n. The former is not know to be faithful (for n lager than 4), the latter is faithful thanks to Krammer and Bigelow.
7. April 21, MA 1.46
   • Vassiliev-Goussarov theory of knot invariants of finite order.
   • Bialgebra of chord diagrams.
   • The 1- and 4-term relations.
   • Weight systems.
   • Milnor-Moore determination of connected, (co)commutative Hopf algebras as symmetric algebras on their primitive elements, and application to the algebra of chord diagrams.
8. April 22, HG 6.96
   • Exercise 1: Define a function v2(K) for knots K as follows: Choose an orientation of K and choose a base point b on K. Let v2(K) be the sum of eps(i).eps(j) over all pairs of crossings i,j with the property that going around K in the chosen direction, starting at b, one meets the crossings i,j in the order ijij where these are of type uoou (under, over, over, under). Here eps(i) is the sign of the crossing (+ if upper comes from the left). Show that v2(K) is independent of the choice of orientation and base point, and invariant for Reidemeister moves. Show that v2(K) is a Vassiliev invariant of order 2.
   • Exercise 2: Show that the Jones polynomial gives rise to Vassiliev invariants: Substitute t = exp(x) and consider the coefficient of x^i. This is a Vassiliev invariant of order i+1.
   • Exercise 3: Show (using the 4-term relations) that the multiplication in the bialgebra of chord diagrams is well defined.
   • The Kontsevich integral as universal Vassiliev invariant.
   • Feynman diagrams and STU relation.
   • Algebra of Chinese characters and IHX relation.
   • Interpreting the IHX relation as Jacobi identity.
   • Getting a weight system (and hence a knot invariant) from a Lie algebra.

Literature

• Brian Sanderson's course
• Dror Bar Natan's course

• Colin Adams, The knot book. An elementary introduction to the mathematical theory of knots. Freeman 1994, 310p. 0-7167-2393-X. 4. [= Das Knotenbuch. Spektrum 1995, 300p. DM 78.]
• T. Ohtsuki, Quantum Invariants, World Scientific, 2002.
• D. Bar-Natan, On the Vassiliev knot invariants, Topology 34 (1995), 423-472.
• V. Kodiyalam and K.N. Raghavan, Picture invariants and the isomorphism problem for complex semisimple Lie algebras, 2004, arXiv.math.RA/0402215
• Lickorish "An Introduction to Knot Theory".

Last update: April 26, 2005