The minimum number of Fox colors modulo 13 is 5

TL;DR

This paper proves that for knots colored modulo 13, the minimum number of colors needed for a non-trivial coloring is exactly five, by demonstrating the existence of an equivalent diagram with only five colors.

Contribution

The paper establishes that the minimum number of Fox colors modulo 13 is five, improving understanding of colorings in knot theory.

Findings

Minimum number of Fox colors modulo 13 is 5

Existence of a diagram with only 5 colors for non-trivial colorings

Implication for knot coloring classification

Abstract

In this article we show that if a knot diagram admits a non-trivial coloring modulo 13 then there is an equivalent diagram which can be colored with 5 colors. Leaning on known results, this implies that the minimum number of colors modulo 13 is 5.