Quasi-alternating links and odd homology: computations and conjectures

TL;DR

This paper computationally explores quasi-alternating links and odd homology, listing examples up to 12 crossings, identifying non-quasi-alternating homologically thin links, and providing new insights into their properties.

Contribution

It offers the first comprehensive list of quasi-alternating links up to 12 crossings and identifies homologically thin links that are not quasi-alternating, with new computational methods.

Findings

Listed quasi-alternating links up to 12 crossings

Identified homologically thin links that are not quasi-alternating

Provided examples of links with multiple minimal diagrams

Abstract

We present computational results about quasi-alternating knots and links and odd homology obtained by looking at link families in the Conway notation. More precisely, we list quasi-alternating links up to 12 crossings and the first examples of quasi-alternating knots and links with at least two different minimal diagrams, where one is quasi-alternating and the other is not. We provide examples of knots and links with $n\le 12$ crossings which are homologically thin and have no minimal quasi-alternating diagrams. These links are candidates for homologically thin links that are not quasi-alternating. For one of our candidates [JaSa1], knot $11n_{50}$, J. Greene proved that it is not quasi-alternating, so this is the first example of homologically thin knot which is not quasi-alternating [Gr]. Computations were performed by A. Shumakovitch's program \emph{KhoHo}, the program \emph{Knotscape}, and our program \emph{LinKnot}.