Minimum Number of Colors: the Turk's Head Knots Case Study

TL;DR

This paper investigates the minimum number of colors needed for Turk's head knots on three strands, providing estimates and exact calculations for this specific class of knots.

Contribution

It offers the first detailed analysis and calculations of the minimum number of colors for Turk's head knots, a previously unstudied class in this context.

Findings

Estimated minimum number of colors for Turk's head knots.

Exact calculations for specific Turk's head knots.

Insights into the complexity of determining knot invariants.

Abstract

The minimum number of colors is a challenging knot invariant since, by definition, its calculation requires taking the minimum over infinitely many minima. In this article we estimate and in some cases calculate the minimum number of colors for the Turk's head knots on three strands.