Characterization of Quasi-alternating Montesinos Links

TL;DR

This paper constructs an infinite family of quasi-alternating Montesinos links from known examples, expanding the classification and proposing a conjecture that this family encompasses all non-alternating quasi-alternating Montesinos links.

Contribution

It introduces a method to generate an infinite family of quasi-alternating Montesinos links and conjectures this family includes all such links that are non-alternating.

Findings

Constructed an infinite family of quasi-alternating links.

Identified all known quasi-alternating Montesinos links within this family.

Proposed a conjecture that this family characterizes all non-alternating quasi-alternating Montesinos links.

Abstract

We construct an infinite family of quasi-alternating links from a given quasi-alternating link by replacing a crossing by a product of rational tangles each of which extends that crossing. Consequently, we determine an infinite family of quasi-alternating Montesinos links. This family contains all the classes of quasi-alternating Montesinos links that have been detected by Widmar in \cite{W}. We conjecture that this family contains all quasi-alternating Montesinos links up to mirror image that are not alternating and this will characterize all quasi-alternating Montesinos links.