University of Toronto's Symplectic Geometry Seminar

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.