On bipartite circle graphs and Khovanov homology

TL;DR

This paper proves that the independence complex of bipartite circle graphs is homotopy equivalent to a wedge of spheres, and as a consequence, the extreme Khovanov spectrum also has this property, implying no torsion in extreme Khovanov homology.

Contribution

It resolves a conjecture by Przytycki and Silvero by establishing the homotopy type of independence complexes of bipartite circle graphs.

Findings

Independence complex of bipartite circle graphs is homotopy equivalent to a wedge of spheres.

Extreme Khovanov spectrum is homotopy equivalent to a wedge of spheres.

Extreme Khovanov homology has no torsion.

Abstract

We prove that independence complex of a bipartite circle graph is homotopy equivalent to a wedge of spheres, resolving a conjecture posed by Przytycki and Silvero. As a corollary, we obtain that extreme Khovanov spectrum, $\mathcal{X}_{j_{extreme}}$ is homotopy equivalent to a wedge of spheres. In particular, the extreme Khovanov homology has no torsion.