Properties and applications of the annular filtration on Khovanov homology

Abstract

The first part of this thesis is on properties of annular Khovanov homology. We prove a connection between the Euler characteristic of annular Khovanov homology and the classical Burau representation for closed braids. This yields a straightforward method for distinguishing, in some cases, the annular Khovanov homologies of two closed braids. As a corollary, we obtain the main result of the first project: that annular Khovanov homology is not invariant under a certain type of mutation on closed braids that we call axis-preserving. The second project is joint work with Adam Saltz. Plamenevskaya showed in 2006 that the homology class of a certain distinguished element in Khovanov homology is an invariant of transverse links. In this project we define an annular refinement of this element, kappa, and show that while kappa is not an invariant of transverse links, it is a conjugacy class invariant of braids. We first discuss examples that show that kappa is non-trivial. We then prove applications of kappa relating to braid stabilization and spectral sequences, and we prove that kappa provides a new solution to the word problem in the braid group. Finally, we discuss definitions and properties of kappa in the reduced setting.