On left-orderability of involutory quandles of links

TL;DR

This paper establishes a criterion to determine non-left-orderability of involutory quandles of links, demonstrating that many classes of links, including non-trivial alternating and augmented alternating links, have involutory quandles that are not left-orderable.

Contribution

The paper introduces a new non-left-orderability criterion for involutory quandles and applies it to various classes of links, extending previous results and proposing a conjecture for all quasi-alternating links.

Findings

Involutory quandles of non-trivial alternating links are not left-orderable.

Involutory quandles of augmented alternating links are not left-orderable.

A new family of links with non-left-orderable involutory quandles is identified.

Abstract

We give a non-left-orderability criterion for involutory quandles of non-split links. We use this criterion to show that the involutory quandle of any non-trivial alternating link is not left-orderable, thus improving Theorem 8.1. proven by Raundal et al. (Proceedings of the Edinburgh Mathematical Society (2021 64), page 646). We also use the criterion to show that the involutory quandles of augmented alternating links are not left-orderable. We introduce a new family of links containing all non-alternating and quasi-alternating 3-braid closures and show that their involutory quandles are not left-orderable. This leads us to conjecture that the involutory quandle of any quasi-alternating link is not left-orderable.