A383146 Number of medial GL-racks of order n, up to isomorphism.

1, 1, 4, 13, 61, 298, 2087, 16941, 187160

Offset: 0

Author

Luc Ta, Apr 17 2025

Keywords

• hard
• more
• nonn

Comments

Generalized Legendrian racks, also called GL-racks or bi-Legendrian racks, are racks equipped with a rack automorphism that commutes with all inner rack automorphisms; see Ta, Definition 3.1 and Proposition 3.12. They are used to distinguish Legendrian links in contact three-space.

GL-racks are precisely virtual racks in which all inner rack automorphisms are virtual rack automorphisms; cf. Cattabriga and Nasybullov, Section 3.2.

A rack or quandle X is medial (also called abelian) if the map X x X -> X defined by (x,y) -> y(x) is a rack homomorphism. Equivalently, the identity (xy)(uv)=(xu)(yv) holds for all elements x, y, u, and v in X.

Links

• Alessia Cattabriga and Timur Nasybullov, Virtual quandle for links in lens spaces, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales, Serie A. Matemáticas, 112 (2018), no. 3, 657-669.
• Biswadeep Karmakar, Deepanshi Saraf, and Mahender Singh, Generalised Legendrian racks of Legendrian links, arXiv: 2301.06854 [math.GT], 2023.
• Naoki Kimura, Bi-Legendrian rack colorings of Legendrian knots, Journal of Knot Theory and its Ramifications, 32 (2023), no. 4, Paper No. 2350029.
• 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. A383145.

Sequences related to medial racks and quandles: A383144, A165200, A242044, A226193, A242275, A243931, A257351.

Other sequences related to racks and quandles: A181769, A181770, A181771, A176077, A178432, A179010, A193024, A254434, A177886, A196111, A226173, A236146, A248908, A198147, A225744, A226172, A226174.

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

Programs

GAP

# see Ta, GitHub link