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Enumeration of racks and quandles up to isomorphism


Authors: Petr Vojtěchovský and Seung Yeop Yang
Journal: Math. Comp. 88 (2019), 2523-2540
MSC (2010): Primary 16T25, 20N05, 57M27
DOI: https://doi.org/10.1090/mcom/3409
Published electronically: January 7, 2019
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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 Jedlička, Pilitowska, Stanovský, and Zamojska-Dzienio.


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Additional Information

Petr Vojtěchovský
Affiliation: Department of Mathematics, University of Denver, 2390 S York St, Denver, Colorado, 80208
Email: petr@math.du.edu

Seung Yeop Yang
Affiliation: Department of Mathematics, Kyungpook National University, Daegu, 41566, Republic of Korea
Email: seungyeop.yang@knu.ac.kr

DOI: https://doi.org/10.1090/mcom/3409
Keywords: Rack, quandle, $2$-reductive rack, medial rack, Yang-Baxter equation, enumeration, isomorphism search, oriented knot, subgroups of symmetric group.
Received by editor(s): May 22, 2018
Received by editor(s) in revised form: September 25, 2018
Published electronically: January 7, 2019
Additional Notes: The first author was supported by a 2015 PROF grant of the University of Denver.
Article copyright: © Copyright 2019 American Mathematical Society