University of Virginia Geometry Seminar 2010-11

A pseudo-meridian of the group of a knot in the 3-sphere is an element that normally generates the group, but is not automorphic to a meridian. In this talk, we will see that many knot groups contain an infinite number of pseudo-meridians (up to automorphism), and that the existence of such elements is connected with Simon's conjecture (Problem 1.12 (D) of Rob Kirby's list) and with other questions concerning knot-group epimorphisms.

Let D and D' denote connected, reduced, alternating diagrams for a pair of links, and Y and Y' their branched double-covers. I'll discuss the proof and consequences of the following result: Y and Y' have isomorphic Heegaard Floer homology groups iff Y and Y' are homeomorphic iff D and D' are mutants.

A version of Khovanov link homology categorifies the Reshetikhin-Turaev polynomial corresponding to the fundamental vector representation of su(3). We demonstrate an action of the symmetric group on this homology, and discuss implications for extending the theory to a categorification of polynomials related to higher irreps of su(3).

We will use the knot Floer complex, in particular the
invariant epsilon, to define a new smooth concordance homomorphism.
Applications include a formula for tau of iterated cables, better
bounds (in many cases) on the 4-ball genus than tau alone, and a new
infinite family of smoothly independent topologically slice knots. We
will also discuss various algebraic properties of this construction,
including a total ordering, a ``much greater than'' relation, and a
filtration.