R-equivalence classes of $\mathrm{Rot} \mathbb{E}^{2}$-colorings of torus knots

TL;DR

This paper introduces R-equivalence classes for quandle colorings of knots, specifically classifying colorings of torus knots by the rotational quandle of the Euclidean plane, under certain conditions.

Contribution

It defines a new R-equivalence relation on knot colorings and determines the classes for torus knots using the rotational Euclidean plane quandle.

Findings

R-equivalence classes of torus knot colorings are explicitly characterized.

The classification depends on specific conditions related to the quandle and knot diagram.

Provides a framework for understanding symmetries in knot colorings.

Abstract

We introduce a new equivalence relation, named R-equivalence relation, on the set of colorings of an oriented knot diagram by a quandle. We determine the R-equivalence classes of colorings of a diagram of a torus knot by a quandle, called $\mathrm{Rot} \mathbb{E}^{2}$, under a certain condition.