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A conjecture on superconnected quandles

Marco Bonatto

TL;DR

This work addresses the problem of understanding Mal\'cev-type properties in finite quandles by focusing on superconnected and superfaithful structures, with a emphasis on simple and primitive cases. It combines group-theoretic constructions (conjugation and coset quandles), star-property criteria, and normalizer analyses to derive both classifications and counterexamples to conjectures that predicted solvability or abelianness. A key contribution is the demonstration that finite simple superconnected quandles can be non-abelian, including primitive instances, thereby refining the landscape of finite superconnected varieties. The results offer concrete criteria for when conjugation and simple group-based quandles exhibit superfaithful or superconnected behavior and illuminate primitivity constraints for simple and primitive quandle families. Overall, the paper advances the structural understanding of quandles in relation to Mal\'cev conditions and knot-theoretic invariants, with concrete group-theoretic templates for building and recognizing these objects.

Abstract

We study simple superfaithful and superconnected quandles and we found counterexamples to a conjecture suggested by computational data. We provide also examples of superconnected quandles built using group theoretical results and investigate primitive quandles.

A conjecture on superconnected quandles

TL;DR

This work addresses the problem of understanding Mal\'cev-type properties in finite quandles by focusing on superconnected and superfaithful structures, with a emphasis on simple and primitive cases. It combines group-theoretic constructions (conjugation and coset quandles), star-property criteria, and normalizer analyses to derive both classifications and counterexamples to conjectures that predicted solvability or abelianness. A key contribution is the demonstration that finite simple superconnected quandles can be non-abelian, including primitive instances, thereby refining the landscape of finite superconnected varieties. The results offer concrete criteria for when conjugation and simple group-based quandles exhibit superfaithful or superconnected behavior and illuminate primitivity constraints for simple and primitive quandle families. Overall, the paper advances the structural understanding of quandles in relation to Mal\'cev conditions and knot-theoretic invariants, with concrete group-theoretic templates for building and recognizing these objects.

Abstract

We study simple superfaithful and superconnected quandles and we found counterexamples to a conjecture suggested by computational data. We provide also examples of superconnected quandles built using group theoretical results and investigate primitive quandles.
Paper Structure (10 sections, 30 theorems, 28 equations)

This paper contains 10 sections, 30 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Left quasigroups and quandles
  4. Invariant equivalence relations and congruences
  5. Conjugation quandles
  6. Group theoretical results
  7. Superfaithful and superconnected conjugation quandles
  8. Simple Quandles
  9. Superfaithful and superconnected simple quandles
  10. Primitive quandles

Key Result

Lemma 2.1

Let $G$ be a group, $\theta\in\mathrm{Aut}(G)$ and $H=G\rtimes \langle\theta\rangle$. Then and $Conj((1,\theta)^H)\cong \mathcal{Q}(G,Fix(\theta),\theta).$

Theorems & Definitions (60)

  • Conjecture 1.1
  • Lemma 2.1
  • proof
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Lemma 2.5
  • Corollary 2.6
  • proof
  • Lemma 2.7
  • ...and 50 more