On a Conjecture by Hayashi on Finite Connected Quandles

TL;DR

This paper investigates a conjecture about the cycle structure of finite connected quandles, proving it holds for profiles with up to five lengths, thus advancing understanding of their algebraic properties.

Contribution

The paper proves Hayashi's conjecture regarding the cycle lengths in the profile of finite connected quandles for cases with up to five lengths.

Findings

Confirmed the conjecture for profiles with up to five lengths

Established the uniformity of cycle length multiples in finite connected quandles

Enhanced understanding of the algebraic structure of 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.