Biorderability of knot quandles of knots up to eight crossings
Vaishnavi Gupta, Hitesh Raundal
TL;DR
The paper investigates biorderability of knot quandles for prime knots up to eight crossings. It employs explicit quandle presentations derived from knot diagrams and performs an exhaustive, table-driven check of all possible linear orders on generating sets to determine if any could extend to a biorder. The main findings are that knot quandles for $6_3$, $8_7$, $8_8$, $8_{10}$, and $8_{16}$ are not biorderable, while a substantial list of knots including $4_1$, $6_1$, $6_2$, $7_6$, $7_7$, $8_1$, $8_2$, $8_3$, $8_4$, $8_5$, $8_6$, $8_9$, $8_{11}$, $8_{12}$, $8_{13}$, $8_{14}$, $8_{17}$, $8_{18}$, $8_{20}$, and $8_{21}$ could be biorderable, with explicit generating-set orders provided that may extend to biorders. The work complements known results on Montesinos knots and highlights how orderability properties of knot quandles can differ from those of knot groups, offering a detailed, constructive approach to identifying potential biorderings across eight-crossing knots.
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.