On the knot quandle of the twist-spun trefoil

Abstract

We show that the knot quandle of the $3$-, $4$-, or $5$-twist-spun trefoil is isomorphic to a quandle related to the $16$-, $24$-, or $600$-cell respectively. We further show that the cardinality of the knot quandle of the $m$-twist-spun trefoil is finite if and only if $1\le m\le 5$. This phenomenon is attributable to the fact that the regular tessellation $\{3,m\}$, in the sense of the Schläfli symbol, consists of infinite triangles if $m$ is greater than or equal to 6.

The contents of this paper was integrated into the paper "The knot quandle of the twist-spun trefoil is a central extension of a Schlafli quandle" (arXiv:2104.13065; Osaka Journal of Mathematics 60 (2023), no. 3, 597--611)