On mutation and Khovanov homology

TL;DR

This paper explores the invariance of Khovanov homology under mutation by using matroid theory and spanning tree complexes, providing a new perspective on a longstanding conjecture.

Contribution

It reformulates the mutation invariance conjecture for Khovanov homology using matroids derived from Tait graphs, showing the E_2-term is a matroid invariant.

Findings

E_2-term of the spectral sequence is a matroid invariant

The matroid invariant property implies invariance under mutation

Provides a new framework connecting matroid theory and knot invariants

Abstract

It is conjectured that the Khovanov homology of a knot is invariant under mutation. In this paper, we review the spanning tree complex for Khovanov homology, and reformulate this conjecture using a matroid obtained from the Tait graph (checkerboard graph) G of a knot diagram K. The spanning trees of G provide a filtration and a spectral sequence that converges to the reduced Khovanov homology of K. We show that the E_2-term of this spectral sequence is a matroid invariant and hence invariant under mutation.