Two-fold quasi-alternating links, Khovanov homology and instanton homology

TL;DR

This paper introduces a new class of links extending quasi-alternating links, explores their Khovanov and instanton homologies, and provides computational evidence for a spectral sequence connecting these invariants.

Contribution

It defines a broader class of links with thin mod 2 Khovanov homology and computes their framed instanton homology, advancing understanding of link invariants and their relationships.

Findings

New class of links with thin mod 2 Khovanov homology

Computed framed instanton homology for double branched covers

Evidence for a spectral sequence from twisted Khovanov to instanton homology

Abstract

We introduce a class of links strictly containing quasi-alternating links for which mod 2 reduced Khovanov homology is always thin. We compute the framed instanton homology for double branched covers of such links. Aligning certain dotted markings on a link with bundle data over the branched cover, we also provide many computations of framed instanton homology in the presence of a non-trivial real 3-plane bundle. We discuss evidence for a spectral sequence from the twisted Khovanov homology of a link with mod 2 coefficients to the framed instanton homology of the double branched cover. We also discuss the relevant mod 4 gradings.