On the Jones polynomial of quasi-alternating links, II

TL;DR

This paper extends known properties of the Jones polynomial from alternating links to quasi-alternating links, revealing structural insights and bounds, and characterizing gaps in the polynomial for various classes.

Contribution

It generalizes the structure of the Jones polynomial to quasi-alternating links and characterizes when gaps occur, including bounds and specific cases for prime and non-prime links.

Findings

Jones polynomial of prime quasi-alternating links (not (2,n)-torus) has no gap.

Determinant bounds the breadth of the Jones polynomial.

Non-prime quasi-alternating links have gaps only if they are connected sums of Hopf links.

Abstract

We extend a result of Thistlethwaite [17, Theorem 1(iv)] on the structure of the Jones polynomial of alternating links to the wider class of quasi-alternating links. In particular, we prove that the Jones polynomial of any prime quasi-alternating link that is not a $(2,n)$-torus link has no gap. As an application, we show that the differential grading of the Khovanov homology of any prime quasi-alternating link that is not a $(2,n)$-torus link has no gap. Also, we show that the determinant is an upper bound for the breadth of the Jones polynomial for any quasi-alternating link. Finally, we prove that the Jones polynomial of any non-prime quasi-alternating link $L$ has more than one gap if and only if $L$ is a connected sum of Hopf links.