11-colored knot diagram with five colors

TL;DR

This paper proves that any 11-colorable knot can be represented with a diagram using exactly five colors, which is the minimum for non-trivial 11-colorings, and extends this result to 11-colorable ribbon 2-knots.

Contribution

It establishes the minimal number of colors needed for non-trivial 11-colorings of knots and ribbon 2-knots, providing new insights into knot coloring theory.

Findings

Any 11-colorable knot has an 11-colored diagram with exactly five colors.

The minimum number of colors for non-trivial 11-colorings is five.

The result extends to 11-colorable ribbon 2-knots.

Abstract

We prove that any $11$-colorable knot is presented by an $11$-colored diagram where exactly five colors of eleven are assigned to the arcs. The number five is the minimum for all non-trivially $11$-colored diagrams of the knot. We also prove a similar result for any $11$-colorable ribbon $2$-knot.