On mutation invariance in Khovanov homology

TL;DR

This paper proves that reduced Khovanov homology over any field remains unchanged under component-preserving Conway mutation, using a new multicurve invariant and homological mirror symmetry techniques.

Contribution

It introduces a new Khovanov multicurve invariant for Conway tangles and proves its invariance under mutation, expanding understanding of Khovanov homology's behavior.

Findings

Reduced Khovanov homology is mutation invariant over any field.

A classification of the components of the multicurve invariant is provided.

The proof employs homological mirror symmetry techniques.

Abstract

We show that reduced Khovanov homology over any field is invariant under component-preserving Conway mutation. Our proof relies on strong geography restrictions for a certain Khovanov multicurve invariant associated with Conway tangles that we introduced in previous work [arXiv:1910.14584]. Applying ideas from homological mirror symmetry, we give a full classification of the components of this invariant.