On The Jones Polynomial of Quasi-alternating Links

TL;DR

This paper demonstrates that twisting quasi-alternating links without gaps in their Jones polynomial preserves this property, leading to a conjecture that most prime quasi-alternating links have no gaps in their Jones polynomial, offering a new characterization.

Contribution

It introduces a method to produce new quasi-alternating links with no gaps in their Jones polynomial and proposes a conjecture relating this property to all prime quasi-alternating links.

Findings

Twisting preserves no-gap property in Jones polynomial for quasi-alternating links.

Conjecture: All prime quasi-alternating links, except (2,n)-torus links, have no gaps in their Jones polynomial.

Confirmed the conjecture for Montesinos links and links with braid index 3.

Abstract

We prove that twisting any quasi-alternating link $L$ with no gaps in its Jones polynomial $V_L(t)$ at the crossing where it is quasi-alternating produces a link $L^{*}$ with no gaps in its Jones polynomial $V_{L^*}(t)$. This leads us to conjecture that the Jones polynomial of any prime quasi-alternating link, other than $(2,n)$-torus links, has no gaps. This would give a new property of quasi-alternating links and a simple obstruction criterion for a link to be quasi-alternating. We prove that the conjecture holds for quasi-alternating Montesinos links as well as quasi-alternating links with braid index 3.