A geometric description of the extreme Khovanov cohomology

TL;DR

This paper establishes a geometric framework linking extreme Khovanov cohomology of links to the cohomology of independence complexes of Lando graphs, and constructs knots with arbitrarily many non-trivial cohomology modules.

Contribution

It introduces a geometric interpretation of extreme Khovanov cohomology via Lando graphs and provides examples of knots with extensive cohomological complexity.

Findings

Extreme Khovanov cohomology equals the cohomology of the independence complex of Lando graphs.

Constructed knots with arbitrarily many non-trivial extreme Khovanov cohomology modules.

Demonstrated the existence of H-thick knots with maximal cohomological complexity.

Abstract

We prove that the hypothetical extreme Khovanov cohomology of a link is the cohomology of the independence simplicial complex of its Lando graph. We also provide a family of knots having as many non-trivial extreme Khovanov cohomology modules as desired, that is, examples of $H$-thick knots which are as far of being $H$-thin as desired.