Extremal Khovanov homology of Turaev genus one links

TL;DR

This paper investigates the Khovanov homology of Turaev genus one links, showing it has a specific algebraic structure in extremal gradings and applying this to compute maximal Thurston-Bennequin numbers.

Contribution

It establishes that Turaev genus one links have Khovanov homology isomorphic to  in extremal gradings and applies this to Thurston-Bennequin number calculations.

Findings

Khovanov homology is  in at least one extremal quantum grading.

Provides formulas or bounds for the maximal Thurston-Bennequin number of Turaev genus one links.

Extends understanding of the relationship between link genus, homology, and contact invariants.

Abstract

The Turaev genus of a link can be thought of as a way of measuring how non-alternating a link is. A link is Turaev genus zero if and only if it is alternating, and in this viewpoint, links with large Turaev genus are very non-alternating. In this paper, we study Turaev genus one links, a class of links which includes almost alternating links. We prove that the Khovanov homology of a Turaev genus one link is isomorphic to $\mathbb{Z}$ in at least one of its extremal quantum gradings. As an application, we compute or nearly compute the maximal Thurston Bennequin number of a Turaev genus one link.