On linear $n$-colorings for knots

TL;DR

This paper explores the relationship between Alexander polynomial properties and linear n-colorings of knots, providing bounds on minimal quandle orders, with detailed analysis for twist knots.

Contribution

It establishes a connection between Alexander polynomial conditions and linear n-colorings, and investigates minimal quandle orders for twist knots.

Findings

Knots with nontrivial Alexander polynomial are linear n-colorable.

Provides upper bounds for minimal quandle orders based on colorings.

Detailed analysis of minimal quandle orders for twist knots.

Abstract

If a knot has the Alexander polynomial not equal to 1, then it is linear $n$-colorable. By means of such a coloring, such a knot is given an upper bound for the minimal quandle order, i.e., the minimal order of a quandle with which the knot is quandle colorable. For twist knots, we study the minimal quandle orders in detail.