Orderability of link quandles

TL;DR

This paper investigates the orderability properties of link quandles, revealing that many knot quandles are right-orderable while most link quandles of non-trivial torus links are not, highlighting differences from their associated groups.

Contribution

It introduces a general theory of orderability for link quandles, providing new constructions and results on their orderability properties, and compares these with the orderability of their enveloping groups.

Findings

Many fibered prime knot quandles are right-orderable.

Most non-trivial torus link quandles are not right-orderable.

Link quandles of certain non-trivial positive or negative links are not bi-orderable.

Abstract

The paper develops a general theory of orderability of quandles with a focus on link quandles of tame links and gives some general constructions of orderable quandles. We prove that knot quandles of many fibered prime knots are right-orderable, whereas link quandles of most non-trivial torus links are not right-orderable. As a consequence, we deduce that the knot quandle of the trefoil is neither left nor right orderable. Further, it is proved that link quandles of certain non-trivial positive (or negative) links are not bi-orderable, which includes some alternating knots of prime determinant and alternating Montesinos links. The paper also explores interconnections between orderability of quandles and that of their enveloping groups. The results establish that orderability of link quandles behave quite differently than that of corresponding link groups.