Biorderability of knot quandles of knots up to eight crossings

TL;DR

This paper classifies which prime knots up to eight crossings have knot quandles that are biorderable, providing new insights into the algebraic structure of these knots and establishing criteria for biorderability.

Contribution

The paper determines biorderability of knot quandles for all prime knots up to eight crossings and introduces linear orders on generators that may extend to biorders.

Findings

Knot quandles of knots 6_3, 8_7, 8_8, 8_10, 8_16 are not biorderable.

Knot quandles of specific other knots up to 8 crossings are biorderable.

Linear orders on generators can extend to biorders in some cases.

Abstract

The paper investigates biorderability of knot quandles of prime knots up to eight crossings. We prove that knot quandles of knots $6_3$, $8_7$, $8_8$, $8_{10}$ and $8_{16}$ can not be biorderable. However, we see that knot quandles of knots $4_1$, $6_1$, $6_2$, $7_6$, $7_7$, $8_1$, $8_2$, $8_3$, $8_4$, $8_5$, $8_6$, $8_9$, $8_{11}$, $8_{12}$, $8_{13}$, $8_{14}$, $8_{17}$, $8_{18}$, $8_{20}$ and $8_{21}$ could be biorderable. We also give linear orders on the generating set of the knot quandle of a knot (among these knots) that could be extendable to biorders on the quandle.