Coloring Torus Knots by Conjugation Quandles

TL;DR

This paper investigates how torus knots can be colored using conjugation quandles over various groups, providing general results and specific numerical criteria for certain groups like matrix groups, dihedral, and symmetric groups.

Contribution

It offers new theoretical insights into the colorability of torus knots with conjugation quandles and provides explicit numerical characterizations for specific groups.

Findings

General results on torus knot colorability with conjugation quandles

Numerical criteria for colorability over $GL(2,q)$, $SL(2,q)$, dihedral, and symmetric groups

Enhanced understanding of knot invariants via group-based colorings

Abstract

In the first part of this paper, we present general results concerning the colorability of torus knots using conjugation quandles over any abstract group. Subsequently, we offer a numerical characterization for the colorability of torus knots using conjugation quandles over some particular groups, such as the matrix groups $GL(2,q)$ and $SL(2,q)$, the dihedral group, and the symmetric group.