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

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 $\mathbb{R}$-palette graph for a set of colors. The $\mathbb{R}$-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 $4$.