A386234 Number of good involutions of all nontrivial core quandles of order n.
1, 4, 1, 3, 1, 72, 2, 3, 1, 31, 1, 3, 1, 10856, 1, 7, 1, 47, 2, 3, 1
Offset: 3
Examples
For n = 4 the only nontrivial core quandle is the dihedral quandle R4 = Core(Z/4Z) of order 4. It is well-known (see Thm. 3.2 of Kamada and Oshiro) that R4 has exactly four good involutions. Hence a(4) = 4. For n = 6 the only nontrivial core quandles are Core(S3) and R6 = Core(Z/6Z), which have one and two good involutions, respectively. Hence a(6) = 3.
References
- Seiichi Kamada, Quandles with good involutions, their homologies and knot invariants, Intelligence of Low Dimensional Topology 2006, World Scientific Publishing Co. Pte. Ltd., 2007, 101-108.
Links
- Seiichi Kamada and Kanako Oshiro, Homology groups of symmetric quandles and cocycle invariants of links and surface-links, Trans. Amer. Math. Soc., 362 (2010), no. 10, 5501-5527.
- Lực Ta, Good involutions of conjugation subquandles, arXiv:2505.08090 [math.GT], 2025. See Table 3.
- Lực Ta, Symmetric-Rack-Classification, GitHub, 2025.
- Index entries for sequences related to quandles and racks
Programs
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GAP
See Ta, GitHub link
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