On a Conjecture by Hayashi on Finite Connected Quandles

TL;DR

The paper addresses Hayashi's conjecture on the profiles of finite connected quandles, stating that the largest profile entry $\ell_c$ must be a multiple of each other entry. Leveraging the right-translation (Brieskorn–Fenn–Rourke) framework, the authors analyze the common cycle structure of right translations via cycle-quandle-tables and derive a least common multiple obstruction $\ell=\mathrm{lcm}(\ell_i: \ell_i\nmid \ell_c)$, yielding that either $\ell \mid \ell_c$ or $\ell_c \mid \ell$. They prove the conjecture for $c=3,4,5$ through detailed case analyses and constructive examples (e.g., $Q_{9,4}$, $Q_{12,4}$, $Q_{15,3}$), illustrating the divisibility constraints on the profile components. The results point toward a possible general proof (perhaps computationally aided) for larger $c$ and shed light on the structural constraints governing finite connected quandles.

Abstract

A quandle is an algebraic structure whose binary operation is idempotent, right-invertible and right self-distributive. Right-invertibility ensures right translations are permutations and right self-distributivity ensures further they are automorphisms. For finite connected quandles, all right translations have the same cycle structure, called the profile of the connected quandle. Hayashi conjectured that the longest length in the profile of a finite connected quandle is a multiple of the remaining lengths. We prove that this conjecture is true for profiles with at most five lengths.

Figures (3)

• Figure 1.1: Quandle table for $Q_{9,4}$.
• Figure 2.1: A cycle quandle table for $Q_{9,4}$.
• Figure 3.1: Cycle quandle table for $(X,*)$ when $1=\ell_1=\ell_2<\ell_3<\ell_4<\ell_5$ and $\ell_3\nmid \ell_4$, $\ell_3\nmid \ell_5$, $\ell_4\nmid \ell_5$.

Theorems & Definitions (60)

• Definition 1.1: Quandle
• Example 1.1
• Lemma 1.1
• proof
• Theorem 1.1
• proof
• Theorem 1.2: cf. Lages_Lopes-latin
• proof
• Conjecture 1.1: Hayashi's Conjecture
• Theorem 1.3: Main Theorem