On Simply Connected Quandles

TL;DR

The paper addresses the problem of characterizing and classifying simply connected finite quandles. It develops an abelian-cocycle perspective on quandle covers and shows how $H^2(Q,S)$ vanishes for simply connected cases, enabling a prime-power size strategy. The authors classify simply connected quandles of size $p^2$ and $p^3$ (for $p>3$), and they extend the analysis to nilpotent latin and core involutory subclasses, providing an algorithmic framework to decide simple connectivity for $p^n$ using data on groups of size $p^{n+1}$. These results illuminate the structure of quandle covers and have implications for knot invariants and Yang–Baxter-related algebraic structures by clarifying when covers collapse to trivial extensions.

Abstract

In this paper we provide an alternative characterization of finite simply connected quandles involving only cocycles with values in abelian groups of prime size. As a corollary of such a characterization and the classification of connected quandles of size $p^2$ and $p^3$ we obtain a classification of simply connected quandles of size $p^2$ (already obtained with a different method in \cite{VV}) and $p^3$ for $p>3$ using a method that works for quandles of size $p^n$ for arbitrary $n$. We also classify the simply connected quandles within two subclasses of finite involutory quandles: nilpotent latin quandles and core quandles.

Theorems & Definitions (90)

• Example 1.1
• Proposition 1.2
• Proposition 1.3
• Proposition 1.4
• proof
• Proposition 1.5: GB, Proposition 1.6
• Lemma 1.6
• proof
• Lemma 1.7
• proof