# University of Toronto's Symplectic Geometry Seminar

April 13, 2004, Tuesday, 1:10 - 2 PM

SS5017A

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Andrew Kricker

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U of Toronto

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Branched cyclic covers and quantum knot invariants

** Abstract: **
On the one hand the algebraic study of quantum knot and three-manifold
invariants grows increasingly sophisticated (by `quantum' we loosely refer
to the collection of constructions inspired, at some level, by Witten's
Chern-Simons TQFT). On the other, it must be said that these constructions
often have surprisingly little to do with the topological viewpoint on
knots. For example, a venerable construction in classical knot theory is
of the p-fold branched cyclic covers, which associates to any knot a
sequence of three-manifolds. The knot invariants arising in this way from the
Casson-(Walker-Lescop) invariant, which might be seen as the simplest
quantum 3-manifold invariant, for a long time remained mysterious, with
formulae only existing for sporadic values of p and special classes of
knots.
This talk will describe the terms of an equation which relates these
invariants, for any value of p, with a certain piece of the Kontsevich
integral together with a certain classical knot invariant. The audience
will have to take on faith that an important ingredient in the calculation
is the diagrammatic version of the Duflo isomorphism due to Bar-Natan et al.