On Khovanov Homology of Quasi-Alternating Links

TL;DR

This paper proves that quasi-alternating links have a specific structure in their Khovanov homology and Jones polynomial, providing new bounds and confirming a conjecture for this class of links.

Contribution

It establishes the Knight Move Conjecture for quasi-alternating links and derives bounds on the determinant based on the Jones polynomial.

Findings

Length of gaps in Khovanov homology is one.

Gaps in Jones polynomial are of length one.

Lower bounds for determinants based on Jones polynomial.

Abstract

We prove that the length of any gap in the differential grading of the Khovanov homology of any quasi-alternating link is one. As a consequence, we obtain that the length of any gap in the Jones polynomial of any such link is one. This establishes a weaker version of Conjecture 2.3 in [5]. Moreover, we obtain a lower bound for the determinant of any such link in terms of the breadth of its Jones polynomial. This establishes a weaker version of Conjecture 3.8 in [17]. The main tool in obtaining this result is establishing the Knight Move Conjecture [2,Conjecture 1] for the class of quasi-alternating links.