Khovanov Homology and Mutation

Abstract

Khovanov homology is a knot invariant that categorifies the Jones polynomial. In Chapter 1, we will go through the construction of Khovanov homology, and survey some of its connections to knot Floer homology. In Chapter 2, we will discuss the conjecture that Khovanov homology is invariant under Conway mutation. We will consider two independent proofs of the fact that Khovanov homology over $\mathbb{F}_2$ is mutation invariant. Finally, in Chapter 3, we will look at the behavior of Khovanov homology under cabled mutation, which is a generalization of Conway mutation and a special case of genus-2 mutation.