Enumeration of virtual quandles up to isomorphism

Virtual racks and virtual quandles are nonassociative algebraic structures based on the Reidemeister moves of virtual knots. In this note, we enumerate virtual dihedral quandles and several families of virtual permutation racks and virtual conjugation quandles up to isomorphism. We also classify virtual racks and virtual quandles up to order 8 using a computer search. These classifications are based on the conjugacy class structures of rack automorphism groups. In particular, we compute class numbers of holomorphs of finite cyclic groups, which may be of independent interest.

💡 Research Summary

This paper addresses the classification problem for virtual racks and virtual quandles, which are non-associative algebraic structures arising from the study of virtual knots via Reidemeister moves. The central objective is to enumerate isomorphism classes of these structures, particularly for specific families like virtual dihedral quandles.

The authors establish a key theoretical framework: classifying virtual racks (R, f) with a fixed underlying rack R is equivalent to classifying the conjugacy classes of the rack’s automorphism group Aut(R). This reduces the algebraic topology problem to a group-theoretic one. They define an invariant v(R) as the number of isomorphism classes of virtual racks with underlying rack R, which equals the class number k(Aut(R)).

This principle is applied in three main directions. First, using the computational algebra system GAP and existing libraries of finite racks/quandles, the authors perform an exhaustive computer search to classify all virtual racks and virtual quandles up to order 8. The resulting data is made publicly available.

Second, the invariant v(R) is computed for several important families. For a permutation rack (X, σ), v(R) equals the number of conjugacy classes in the centralizer of σ in the symmetric group. For conjugation quandles Conj(G) of a centerless group G, they leverage a result stating Aut(Conj(G)) is isomorphic to the group automorphism group Aut_Grp(G), allowing the calculation of v for symmetric groups S_n (yielding the partition number p(n), except for n=6), alternating groups A_n, and projective special linear groups PSL(2,p).

The third and most detailed application is the complete classification of virtual dihedral quandles. The dihedral quandle R_n of order n has an automorphism group isomorphic to the holomorph G = Z/nZ ⋊ (Z/nZ)×. The authors characterize when two virtual structures (corresponding to affine maps T_{a,u} and T_{b,v}) are conjugate in G. This leads to the paper’s main enumerative result (Theorem 1): the number of isomorphism classes of virtual quandles with underlying quandle R_n is given by the formula v(R_n) = φ(n) Σ_{d|n} 1/φ(d), where φ is Euler’s totient function. They further show that this sequence coincides with the class numbers of the holomorphs of finite cyclic groups and is listed in the OEIS.

In summary, the paper successfully bridges virtual knot theory, group theory, and combinatorics. It provides both a general method for classification based on automorphism group conjugacy and concrete enumerations for significant families of virtual racks and quandles, supported by computational data and precise mathematical formulas.