Minimum numbers of Dehn colors of knots and $\mathcal{R}$-palette graphs

TL;DR

This paper investigates the minimum number of colors needed for Dehn colorings of knots, establishing lower bounds for prime p and introducing $ $-palette graphs to identify potential color sets.

Contribution

It provides a lower bound on the minimum colors for Dehn p-colorable knots and introduces $ $-palette graphs as a tool to find candidate color sets.

Findings

Minimum number of colors for Dehn p-colorable knots is at least $loor{\log_2 p} + 2$.

For Dehn 5-colorable knots, the minimum number of colors is exactly 4.

$ $-palette graphs help identify candidate color sets for nontrivial Dehn colorings.

Abstract

In this paper, we consider minimum numbers of colors of knots for Dehn colorings. In particular, we will show that for any odd prime number $p$ and any Dehn $p$-colorable knot $K$, the minimum number of colors for $K$ is at least $\lfloor \log_2 p \rfloor +2$. Moreover, we will define the $\R$-palette graph for a set of colors. The $\R$-palette graphs are quite useful to give candidates of sets of colors which might realize a nontrivially Dehn $p$-colored diagram. In Appendix, we also prove that for Dehn $5$-colorable knot, the minimum number of colors is $4$.