On the notion of Khovanov A-adequacy

TL;DR

This paper investigates Khovanov adequacy, a property related to the non-triviality of certain homologies in Khovanov homology, extending the concept of adequacy from classical links to the Khovanov setting.

Contribution

It introduces a new perspective on Khovanov adequacy using independence complexes and homotopy types of extreme Khovanov spectra.

Findings

Khovanov-adequate diagrams have non-trivial homology at extreme gradings

The framework connects Khovanov adequacy with independence complexes

Homotopy types of extreme Khovanov spectra are characterized

Abstract

The concept of adequate links, introduced by Lickorish and Thistlethwaite as a generalization of alternating links, has recently gained interest among knot theorists in the context of Khovanov homology. Przytycki and Silvero introduced the more general concept of Khovanov adequacy: a diagram is Khovanov-adequate if its associated Khovanov chain complexes at both potential maximal and minimal quantum gradings have non-trivial homology. This article explores Khovanov adequacy within the framework of independence complexes and the calculation of the homotopy type of extreme Khovanov spectra.