On the notion of Khovanov A-adequacy

TL;DR

The paper investigates Khovanov A-adequacy by linking extreme Khovanov homology to the independence complexes of Lando graphs derived from a diagram, situating this within circle graphs and bipartite graph theory. It leverages the stable homotopy equivalence between the extreme Khovanov complex and the independence complex $I(D)$ to characterize A-adequacy via non-contractibility of $I(D)$. Through explicit constructions, it exhibits several families of diagrams (e.g., twisted torus links, negative braids, and modified cables) that are Khovanov A-adequate, while illustrating subtleties where classical A-adequacy fails but Khovanov adequacy persists. The work clarifies how graph-theoretic invariants of Lando and circle graphs control extreme Khovanov homology and provides concrete obstructions and examples that deepen understanding of Khovanov adequacy in knot theory.

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.

Figures (11)

• Figure 1: A chord diagram $\mathcal{C}$, its associated circle graph $G_\mathcal{C}$ and the geometric realization of the corresponding independence complex $I(G_\mathcal{C}$) are shown in (a), (b) and (c), respectively.
• Figure 2: Left: a positive (+) and a negative (-) crossing. Right: the smoothing of a crossing, according to its $A$ or $B$ label.
• Figure 3: A triangulation of $\mathbb{R}P^2$ and the associated bipartite graph illustrating Example \ref{['projplane']}.
• Figure 4: Four facets and three parallelograms illustrating proof of Theorem \ref{['Sec2MainThm']}.
• Figure 5: Chord Diagram.

Theorems & Definitions (18)

• Definition 2.1
• Conjecture 2.2
• Proposition 2.3
• Definition 3.1
• Definition 3.2
• Definition 3.3
• Theorem 3.4
• Conjecture 3.5
• Theorem 4.1: Jon Theorem 1
• Corollary 4.2