On Conway mutation and link homology

Abstract

We give a new, elementary proof that Khovanov homology with Z/2Z–coefficients is invariant under Conway mutation. This proof also gives a strategy to prove Baldwin and Levine's conjecture that δ–graded knot Floer homology is mutation–invariant. Using the Clifford module structure on HFK˜ induced by basepoint maps, we carry out this strategy for mutations on a large class of tangles. Let L′ be a link obtained from L by mutating the tangle T. Suppose some rational closure of T corresponding to the mutation is the unlink on any number of components. Then L and L′ have isomorphic δ–graded HFKˆ groups over Z/2Z as well as isomorphic Khovanov homology over Q. We apply these results to establish mutation–invariance for the infinite families of Kinoshita-Terasaka and Conway knots. Finally, we give sufficient conditions for a general Khovanov-Floer theory to be mutation–invariant.