Minimum numbers of Dehn colors of knots and symmetric local biquandle cocycle invariants

TL;DR

The paper develops a symmetric local biquandle framework for unoriented knots to study minimum numbers of Dehn $p$-colorings ${\rm mincol}^{\rm Dehn}_p(K)$. By constructing local biquandle (co)homology and associated cocycle invariants, it derives new lower bounds that force larger color sets for certain primes, notably ${\rm mincol}^{\rm Dehn}_p(K) \geq \lfloor \log_2 p \rfloor +3$ for $p=13,29$ and additional primes up to $31$. It provides rigorous proofs for these bounds via explicit color-sets (e.g., $\{0,1,2,4,7\}$ and $\{0,1,2,4,8,15\}$) and shows that $0$ must be excluded from the NT invariant set to enforce the lower bound. The authors also furnish explicit knot diagrams achieving these bounds for several primes, establishing sharpness in many cases and advancing understanding of region-coloring phenomena in knot theory. Overall, the work connects Dehn colorings with symmetric local biquandle cocycle invariants to distinguish knots by Dehn color counts and offers concrete, provable bounds and examples across primes up to 31.

Abstract

In this paper, we give a method to evaluate minimum numbers of Dehn colors for knots by using symmetric local biquandle cocycle invariants. We give answers to some questions arising as a consequence of our previous paper [6]. In particular, we show that there exist knots which are distinguished by minimum numbers of Dehn colors.

Figures (20)

• 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:
• Figure 3:
• Figure 4:
• Figure 5:

Theorems & Definitions (46)

• Lemma 2.1: MatsudoOshiroYamagishi-1
• Remark 2.2
• Definition 2.3
• Theorem 2.4: MatsudoOshiroYamagishi-1
• Theorem 2.5: MatsudoOshiroYamagishi-1
• Proposition 2.6: MatsudoOshiroYamagishi-1
• Remark 2.7: MatsudoOshiroYamagishi-1
• Lemma 3.1
• proof
• Lemma 3.2