Homotopy type of circle graphs complexes motivated by extreme Khovanov homology

TL;DR

This paper investigates the homotopy type of complexes derived from circle graphs related to extreme Khovanov homology, proposing a conjecture that these complexes are wedges of spheres, which could imply torsion-free homology for links.

Contribution

It conjectures that the independence complexes of circle graphs are wedges of spheres, proves this in special cases, and explores implications for knot theory and homology.

Findings

Proved the conjecture for cactus, outerplanar, permutation, and non-nested graphs.

Constructed permutation graphs with complexes homotopy equivalent to any finite wedge of spheres.

Computed extreme Khovanov homology for certain torus links, revealing gaps in homology groups.

Abstract

It was proven by Gonz\'alez-Meneses, Manch\'on and Silvero that the extreme Khovanov homology of a link diagram is isomorphic to the reduced (co)homology of the independence simplicial complex obtained from a bipartite circle graph constructed from the diagram. In this paper we conjecture that this simplicial complex is always homotopy equivalent to a wedge of spheres. In particular, its homotopy type, if not contractible, would be a link invariant and it would imply that the extreme Khovanov homology of any link diagram does not contain torsion. We prove the conjecture in many special cases and find it convincing to generalize it to every circle graph (intersection graph of chords in a circle). In particular, we prove it for the families of cactus, outerplanar, permutation and non-nested graphs. Conversely, we also give a method for constructing a permutation graph whose independence simplicial complex is homotopy equivalent to any given finite wedge of spheres. We also present some combinatorial results on the homotopy type of finite simplicial complexes and a theorem generalizing previous results by Csorba, Nagel and Reiner, Jonsson and Barmak. We study the implications of our results to Knot Theory; more precisely, we compute the real-extreme Khovanov homology of torus links $T(3,q)$ and obtain examples of $H$-thick knots whose extreme Khovanov homology groups are separated either by one or two gaps as long as desired.