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

TL;DR

This work investigates the minimum number of region colors required for Dehn $p$-colorings of knots, establishing a universal lower bound ${\rm mincol}^{\rm Dehn}_p(K) \ge \lfloor \log_2 p \rfloor + 2$ for odd primes $p$. The authors develop extended coloring matrices and rank-based arguments, showing that nontrivial Dehn $p$-colorings force color-counts to grow proportionally with $\log_2 p$, via ${\rm rank}_p$ constraints and determinant bounds. They introduce $\mathcal{R}$-palette graphs to translate color-set structure into graph-theoretic constraints, proving that any realizable color set must contain a connected subgraph of size at least $\lfloor \log_2 p \rfloor + 2$; this yields both lower bounds and guidance for constructing colorings. The paper combines theoretical bounds with computational analysis of color sets (up to $p<2^5$) to identify candidate color sets that realize the bound for many primes, while proving nonexistence for certain primes (e.g., $p=13,29$) and furnishing an appendix that establishes ${\rm mincol}^{\rm Dehn}_5(K)=4$ for all Dehn 5-colorable knots. Overall, the work provides a framework to bound and realize the Dehn-coloring minimum colors, linking algebraic colorings to combinatorial palette-graph structures with implications for knot-coloring classifications.

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 $4$.

Figures (37)

• Figure 1: A crossing on $D$ and the one on $(D,C)$ with $C(x_1)=a_1$, $C(x_2)=a_2$, $C(x_3)=a_3$, $C(x_4)=a_1-a_2+a_3$
• Figure 2: A correspondence between Dehn colorings and Fox colorings
• Figure 3:
• Figure 4: $\mathcal{R}$-palette graphs
• Figure 5: $\mathcal{R}$-palette graphs

Theorems & Definitions (25)

• Remark 1.1
• Lemma 1.2
• proof
• Remark 1.3
• Definition 1.4
• Theorem 1.5
• Lemma 2.1
• Lemma 3.1
• proof
• Lemma 3.2