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

Author

Luc Ta, Jul 21 2025

Keywords

• nonn
• hard
• more

Comments

A good involution f of a quandle Q is an involution that commutes with all inner automorphisms and satisfies the identity f(y)(x) = y^-1(x). We call the pair (Q,f) a symmetric quandle. A symmetric quandle isomorphism is a quandle isomorphism that intertwines good involutions.

A core quandle Core(G) is a group G viewed as a kei (i.e., involutory quandle) under the operation g(h) = g*h^-1*g. Note that Core(G) is nontrivial if and only if exp(G) > 2.

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

Crossrefs

Cf. A000001, A386233, A181770, A178432, A386231, A386232.

Programs

GAP

See Ta, GitHub link

Formula

Let n > 2. Then Ta, Cor. 7.17 implies the following. If n appears in A000040 or A050384, then a(n) = 1. If n appears in A221048, then a(n) = 2. If n > 4 and n appears in A100484, then a(n) = 3.