Mutation-invariance of Khovanov-Floer theories

TL;DR

This paper proves that certain strong Khovanov-Floer theories, including Szab{'}o}s homology and singular instanton homology, are invariant under Conway mutation, confirming existing conjectures and advancing understanding of link invariants.

Contribution

It demonstrates mutation invariance for specific Khovanov-Floer theories and proves related conjectures about Szab{'}o}s homology structure.

Findings

Szab{'}o}s homology is mutation invariant.

Singular instanton homology is mutation invariant.

Two conjectures about Szab{'}o}s homology structure are proven.

Abstract

Khovanov-Floer theories are a class of homological link invariants which admit spectral sequences from Khovanov homology. They include Khovanov homology, Szab{\'o}'s geometric link homology, singular instanton homology, and various Floer theories applied to branched double covers. In this short note we show that certain strong Khovanov-Floer theories, including Szab{\'o} homology and singular instanton homology, are invariant under Conway mutation. This confirms conjectures of Seed and Lambert-Cole. Along the way we prove two other conjectures about the structure of Szab{\'o} homology.