A note on quasi-alternating Montesinos links

TL;DR

This paper investigates conditions under which Montesinos links are quasi-alternating, extending previous classifications of pretzel links and exploring the relationship with Heegaard Floer homology and left-orderable groups.

Contribution

It generalizes the classification of quasi-alternating pretzel links to Montesinos links using rational parameters and establishes criteria for non-quasi-alternating cases.

Findings

Derived conditions for quasi-alternating Montesinos links.

Established criteria for non-quasi-alternating Montesinos links.

Discussed examples outside the main classification results.

Abstract

Quasi-alternating links are a generalization of alternating links. They are homologically thin for both Khovanov homology and knot Floer homology. Recent work of Greene and joint work of the first author with Kofman resulted in the classification of quasi-alternating pretzel links in terms of their integer tassel parameters. Replacing tassels by rational tangles generalizes pretzel links to Montesinos links. In this paper we establish conditions on the rational parameters of a Montesinos link to be quasi-alternating. Using recent results on left-orderable groups and Heegaard Floer L-spaces, we also establish conditions on the rational parameters of a Montesinos link to be non-quasi-alternating. We discuss examples which are not covered by the above results.