On the knot quandle of the twist-spun trefoil

TL;DR

This paper investigates the knot quandle of twist-spun trefoils, revealing their isomorphism to certain cell-related quandles and characterizing when these quandles are finite based on the twist parameter.

Contribution

It establishes a connection between the knot quandle of twist-spun trefoils and regular tessellations, and characterizes the finiteness of these quandles for specific twist values.

Findings

Knot quandle of 3-, 4-, 5-twist-spun trefoil is isomorphic to quandles related to 16-, 24-, 600-cells.

Finiteness of the knot quandle occurs if and only if 1 ≤ m ≤ 5.

Regular tessellation {3, m} is infinite for m ≥ 6.

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 $1 \leq m \leq 5$. This phenomenon is attributable to the fact that the regular tessellation $\{ 3, m \}$, in the sense of the Schl\"{a}fli symbol, consists of infinite triangles if $m$ is greater than or equal to 6.