On colorability of knots by rotations, Torus knot and PL trochoid

TL;DR

This paper explores the connection between knot colorability by a plane rotation quandle and the roots of the Alexander polynomial, providing a complete classification for torus knots and their Alexander polynomial factorizations.

Contribution

It establishes a criterion linking knot colorability by plane rotations to roots of the Alexander polynomial and enumerates all non-trivial colorings of torus knots using PL trochoids.

Findings

A knot is colorable by the rotation quandle iff its Alexander polynomial has a root on the unit circle.

Complete enumeration of non-trivial colorings of torus knots by PL trochoids.

Complete factorization of the Alexander polynomial of torus knots.

Abstract

The set consisting of all rotations of the Euclidean plane is equipped with a quandle structure. We show that a knot is colorable by this quandle if and only if its Alexander polynomial has a root on the unit circle in $\mathbb{C}$. Further we enumerate all non-trivial colorings of a torus knot diagram by the quandle using PL trochoids. As an application of these results, we have the complete factorization of the Alexander polynomial of the torus knot.