A new obstruction of quasi-alternating links

TL;DR

This paper introduces a new obstruction criterion based on polynomial degree for identifying non-quasi-alternating links, leading to classifications of certain knots and links and supporting existing conjectures about their finiteness.

Contribution

It establishes a simple polynomial degree-based obstruction for quasi-alternating links and applies it to classify specific knots and links, supporting conjectures on their finiteness.

Findings

Degree of polynomial is less than the determinant for quasi-alternating links

Identified non-quasi-alternating knots and links with up to 12 crossings

Finiteness of quasi-alternating Kanenobu knots supports conjecture

Abstract

We prove that the degree of the Brandt-Lickorish-Millet polynomial of any quasi-alternating link is less than its determinant. Therefore, we obtain a new and a simple obstruction criterion for quasi-alternateness. As an application, we identify some knots of 12 crossings or less and some links of 9 crossings or less that are not quasi-alternating. Also, we show that there are only finitely many Kanenobu knots which are quasi-alternating. This last result supports Conjecture 3.1 of Greene in [10] which states that there are only finitely many quasi-alternating links with a given determinant. Moreover, we identify an infinite family of non quasi-alternating Montesinos links and this supports Conjecture 3.10 in [20] that characterizes quasi-alternating Montesinos links.