On mutation and Khovanov homology
arXiv:0801.4937
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.
Revised and expanded with a review of the spanning tree complex. To appear in Communications in Contemporary Mathematics, special volume in memory of Xiao-Song Lin. 18 pages