Khovanov homology and binary dihedral representations for marked links

TL;DR

This paper develops a marked version of Khovanov homology for alternating links inspired by instanton theory, showing spectral sequence collapse and independence of markings for certain classes of links, and explores binary dihedral representations.

Contribution

It introduces a new marked Khovanov homology for alternating links, demonstrating spectral sequence collapse and invariance properties, and studies binary dihedral representations for marked links.

Findings

Spectral sequence from Khovanov to instanton homology collapses at E2 for alternating links.

Marked Khovanov homology is independent of markings for non-split alternating links.

Binary dihedral representations for links of non-zero determinant are independent of markings.

Abstract

We introduce a version of Khovanov homology for alternating links with marking data, $\omega$, inspired by instanton theory. We show that the analogue of the spectral sequence from Khovanov homology to singular instanton homology introduced in \cite{KM_unknot} for this marked Khovanov homology collapses on the $E_2$ page for alternating links. We moreover show that for non-split links the Khovanov homology we introduce for alternating links does not depend on $\omega$; thus, the instanton homology also does not depend on $\omega$ for non-split alternating links. Finally, we study a version of binary dihedral representations for links with markings, and show that for links of non-zero determinant, this also does not depend on $\omega$.