A refinement of Khovanov homology

Abstract

We refine Khovanov homology in the presence of an involution on the link. This refinement takes the form of a triply-graded theory, arising from a pair of filtrations. We focus primarily on strongly invertible knots and show, for instance, that this refinement is able to detect mutation.

On Khovanov Homology and Related Invariants

“…For example, a prominent open problem was resolved when Piccirillo showed that the Conway knot is not slice, using the s-invariant [44] defined by Rasmussen from the spectral sequence from Khovanov homology to Lee homology. Lobb-Watson's [34] filtered invariant is able to detect mutants in the presence of an involution. In a different direction, one may also consider generalized mutations along genus 2 surfaces from which (Conway) mutation may be recovered.…”