Near extremal Khovanov homology of Turaev genus one links

TL;DR

This paper investigates the Khovanov homology of Turaev genus one links, revealing a trivial summand that aids in computing the Rasmussen s invariant and bounding the smooth four genus of certain knots.

Contribution

It provides new insights into the Khovanov homology structure of Turaev genus one links and connects this to invariants like Rasmussen s and four genus bounds.

Findings

A specific summand in Khovanov homology of Turaev genus one links is trivial.

This trivial summand enables computation of Rasmussen s invariant.

Bounds on the smooth four genus are established for certain Turaev genus one knots.

Abstract

The Turaev surface of a link diagram $D$ is a closed, oriented surface constructed from a cobordism between the all-$A$ and all-$B$ Kauffman states of $D$. The Turaev genus of a link $L$ is the minimum genus of the Turaev surface of any diagram $D$ of $L$. A link is alternating if and only if its Turaev genus is zero, and so one can view Turaev genus one links as being close to alternating links. In this paper, we study the Khovanov homology of a Turaev genus one link in the first and last two polynomial gradings where the homology is nontrivial. We show that a particular summand in the Khovanov homology of a Turaev genus one link is trivial. This trivial summand leads to a computation of the Rasmussen $s$ invariant and to bounds on the smooth four genus for certain Turaev genus one knots.