Enumeration of racks and quandles up to isomorphism

TL;DR

This paper systematically enumerates racks and quandles up to isomorphism for orders up to 13, improving previous results and providing explicit representatives for smaller orders through computational and theoretical methods.

Contribution

It extends the enumeration of racks and quandles up to order 13, offering explicit representatives for smaller orders and utilizing advanced classification techniques.

Findings

Complete enumeration of racks of order ≤ 11

Complete enumeration of quandles of order ≤ 12

Improved enumeration results for orders 9 to 13

Abstract

Racks and quandles are prominent set-theoretical solutions of the Yang-Baxter equation. We enumerate racks and quandles of orders $n\le 13$ up to isomorphism, improving upon the previously known results for $n\le 8$ and $n\le 9$, respectively. The enumeration is based on the classification of subgroups of small symmetric groups up to conjugation, on a representation of racks and quandles in symmetric groups due to Joyce and Blackburn, and on a number of theoretical and computational observations concerning the representation. We explicitly find representatives of isomorphism types of racks of order $\le 11$ and quandles of order $\le 12$. For the remaining orders we merely count the isomorphism types, relying in part on the enumeration of $2$-reductive racks and $2$-reductive quandles due to Jedli\v{c}ka, Pilitowska, Stanovsk\'y and Zamojska-Dzienio.