Independence of Whitehead Doubles of Torus Knots in the Smooth Concordance Group

Geometry Topology Seminar
Monday, March 30, 2015 - 14:00
1 hour (actually 50 minutes)
Skiles 006
University of Georgia
In the 1980’s Furuta and Fintushel-Stern applied the theory of instantons and Chern-Simons invariants to develop a criterion for a collection of Seifert fibred homology spheres to be independent in the homology cobordism group of oriented homology 3-spheres. In turn, using the fact that the 2-fold cover of S^3 branched over the Whitehead double of a positive torus knot is negatively cobordant to a Seifert fibred homology sphere, Hedden-Kirk establish conditions under which an infinite family of Whitehead doubles of positive torus knots are independent in the smooth concordance group. In the talk, I will review some of the definitions and constructions involved in the proof by Hedden and Kirk and I will introduce some topological constructions that greatly simplify their argument. Time permiting I will mention some ways in which the result could be generalized to include a larger set of knots.