A383831 Number of medial Legendrian quandles of order n, up to isomorphism.

1, 1, 2, 5, 14, 48, 219, 1207, 9042

Offset: 0

Author

Luc Ta, May 16 2025

Keywords

• hard
• more
• nonn

Comments

A rack or quandle is medial if it satisfies the identity (xy)(uv) = (xu)(yv).

A Legendrian quandle is a pair (X,u) where X is a quandle and u is an involutory automorphism of X such that u(yx)=y(u(x)); see Ta, "Generalized Legendrian...," Corollary 3.13.

a(n) is also the number of medial racks X such that the kink map X -> X defined by x -> x(x) is an involution; see Ta, "Equivalences of...," Theorem 1.1.

Links

• Jose Ceniceros, Mohamed Elhamdadi, and Sam Nelson, Legendrian rack invariants of Legendrian knots, Communications of the Korean Mathematical Society, 36 (2021), no. 3, 623-639.
• Lực Ta, Equivalences of racks, Legendrian racks, and symmetric racks, arXiv:2505.08090 [math.GT], 2025.
• Lực Ta, Generalized Legendrian racks: Classification, tensors, and knot coloring invariants, arXiv:2504.12671 [math.GT], 2025.
• Lực Ta, GL-Rack Classification, GitHub, 2025.

Crossrefs

Cf. A383828-A383830, A383145, A383146, A181769, A181770, A178432.

Sequences related to racks and quandles: A383144, A181771, A176077, A179010, A193024, A254434, A177886, A196111, A226173, A236146, A248908, A165200, A242044, A226193, A242275, A243931, A257351, A198147, A225744, A226172, A226174.

Sequences related to Legendrian knots: A374939, A374942, A374943, A374944, A374945, A374946, A374947.

Programs

GAP

# See Ta, GitHub link