On Conway mutation and link homology

TL;DR

This paper provides elementary proofs of mutation invariance for Khovanov homology and proposes a strategy for proving mutation invariance of knot Floer homology, with applications to specific knot families.

Contribution

It offers a new elementary proof for Khovanov homology's mutation invariance and a strategy to prove mutation invariance for knot Floer homology, extending understanding of link invariants.

Findings

Khovanov homology with Z/2Z coefficients is mutation-invariant.

Under certain conditions, delta-graded knot Floer homology is mutation-invariant.

Established mutation-invariance for Kinoshita-Terasaka and Conway knots.

Abstract

We give a new, elementary proof that Khovanov homology with $\mathbb{Z}/2\mathbb{Z}$--coefficients is invariant under Conway mutation. This proof also gives a strategy to prove Baldwin and Levine's conjecture that $\delta$--graded knot Floer homology is mutation--invariant. Using the Clifford module structure on $\widetilde{\text{HFK}}$ 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 $\delta$--graded $\widehat{\text{HFK}}$-groups over $\mathbb{Z}/2\mathbb{Z}$ as well as isomorphic Khovanov homology over $\mathbb{Q}$. 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.