On the minimum number of Fox Colorings of Knots

TL;DR

This paper proves that any knot that can be colored with 17 colors can be represented with a diagram using exactly 6 of those colors, advancing understanding of minimal colorings in knot theory.

Contribution

It establishes a precise minimal coloring number for 17-colorable knots, showing a specific diagrammatic coloring constraint.

Findings

Any 17-colorable knot has a diagram with exactly 6 colors used.

The result refines the understanding of minimal Fox colorings for specific color sets.

Provides a new bound on the number of colors needed for certain knot colorings.

Abstract

We investigate Fox colorings of knots that are 17-colorable. Precisely, we prove that any 17-colorable knot has a diagram such that exactly 6 among the seventeen colors are assigned to the arcs of the diagram.