Kazhdan's Seminar on Khovanov Theory: Winter 2005

Sunday 4-6 in Maths Bldg 209

- What is knot theory? Kauffman bracket construction of Jones polynomial of knots in a 3-sphere.
- Witten's generalisation of Jones polynomial to the case of knots in an arbitrary 3-manifold and the Reshetikhin-Turaev construction of Witten's invariants. Background on U_qsl_2 and its representation theory.
- Topological quantum field theories, the 2-dimensional case [Frobenius algebras] and Reshetikhin-Turaev's example of a 3-dimensional Topological Quantum Field Theory.
- Definition of Khovanov homology theories for links induced by Frobenius systems (a la Khovanov).
- Universal Bar-Natan morphism between planar algebras of tangle diagrams and of complexes of cobordisms.
- Homotopy of complexes and reduction to Bar-Natan morphism from planar algebra of tangles to planar algebra of complexes up to homotopy type, to cobordisms up to cobordism relations.
- Beginnings of categorification of 3-manifold invariant (equivalent to 4-TQFT) in math.QA/0509083 "Hopfological algebra and categorifications at a root of unity".

- Reshetikhin-Turaev construction of Witten invariants of links in 3-manifolds: surgery description of 3-manifolds, background on U_qsl_2 and its representation theory (Invent. Math. 103 (1991) 547-597)
- Classification of 1+1 dimensional TQFT (Frobenius alg)
- Khovanov's categorification of Jones poly: original Khovanov paper and Bar-Natan reformulation
- Khovanov homology for tangles and cobordisms: Bar-Natan paper, Frobenius extensions.
- Slides from talks of Scott Morrison (UCB) 08/05, Bar-Natan version of Khovanov homology and delooping.
- sl(3) link homology.
- math.QA/0509083 "Hopfological algebra and categorifications at a root of unity".

- (30/10/05) Introduction. Knots via knot diagrams/Reideimeister moves. Kauffman bracket construction of Jones polynomial (slightly modified): transform an oriented knot diagram to a linear combination (coefficients are signed powers of an indeterminate v) of diagrams of non-intersecting loops in the plane, and then obtain a polynomial by replacing a diagram containing d loops by (v+v^{-1})^d. Proof of invariance. Explanation that Jones poly fits into Witten TQFT and categorical definition of TQFT.

Last modified January 25th, 2001.

*Comments and questions to *ruthel@ma.huji.ac.il