Kazhdan's
Seminar on Khovanov Theory: Winter 2005
Sunday 4-6 in Maths Bldg 209
Outline of possible topics:
- 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".
Sources for topics (participants should choose a topic and lecture on it!)
Seminars given
- (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.
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Last modified January 25th, 2001.
Comments and questions to ruthel@ma.huji.ac.il