On the Classification of Quandles of Low Order

Racks and quandles are prominent set-theoretical solutions of the Yang-Baxter equation. We enumerate racks and quandles of orders n ≤ 13 up to isomorphism, improving upon the previously known results for n ≤ 8 and n ≤ 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 ≤ 11 and quandles of order ≤ 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.1991 Mathematics Subject Classification. 16T25, 20N05, 57M27.

show abstract

“…Whenever an oriented knot K is colored by a quandle X, the elements of X actually used in the coloring form a connected subquandle of X. Consequently, for the purposes of quandle colorings, it is sufficient to consider connected quandles. Although far from settled, the theory of connected quandles is better understood than the general case [11], and connected quandles have been classified up to size 47 [11,19].…”

Section: Introductionmentioning

confidence: 99%

Simply connected latin quandles

Bonatto

Vojtěchovský

2018

J. Knot Theory Ramifications

View full text Add to dashboard Cite

A (left) quandle is connected if its left translations generate a group that acts transitively on the underlying set. In 2014, Eisermann introduced the concept of quandle coverings, corresponding to constant quandle cocycles of Andruskiewitsch and Graña. A connected quandle is simply connected if it has no nontrivial coverings, or, equivalently, if all its second constant cohomology sets with coefficients in symmetric groups are trivial.In this paper we develop a combinatorial approach to constant cohomology. We prove that connected quandles that are affine over cyclic groups are simply connected (extending a result of Graña for quandles of prime size) and that finite doubly transitive quandles of order different from 4 are simply connected. We also consider constant cohomology with coefficients in arbitrary groups.

show abstract

On the Classification of Quandles of Low Order

Cited by 36 publications

References 20 publications

Conjugacy Classes ofP-Cycles of Type D in Alternating Groups

Conjugacy Classes ofP-Cycles of Type D in Alternating Groups

Enumeration of racks and quandles up to isomorphism

Simply connected latin quandles

Contact Info

Product

Resources

About