On the Maximum Number of Colors for Links

TL;DR

This paper investigates the maximum number of colors in non-trivial p-colorings of links, establishing bounds, constructions, and properties for various classes of links, including rational and torus links.

Contribution

It proves the maximum number of colors for non-split links with non-trivial p-colorings equals p, and shows how to construct colorings with any number of colors between minimum and maximum.

Findings

Maximum number of colors equals p for certain links.

Any number of colors between minimum and maximum can be achieved.

Coloring properties are characterized for rational and torus links.

Abstract

For each odd prime p, and for each non-split link admitting non-trivial p-colorings, we prove that the maximum number of Fox colors is p. We also prove that we can assemble a non-trivial p-coloring with any number of colors, from the minimum to the maximum number of colors. Furthermore, for any rational link, we prove that there exists a non-trivial coloring of any Schubert Normal Form of it, modulo its determinant, which uses all colors available. If this determinant is an odd prime, then any non-trivial coloring of this Schubert Normal Form, modulo the determinant, uses all available colors. We prove also that the number of crossings in the Schubert Normal Form equals twice the determinant of the link minus 2. Facts about torus links and their coloring abilities are also proved.