Enumeration of virtual quandles up to isomorphism

TL;DR

This paper systematically enumerates and classifies small virtual quandles and racks up to isomorphism, using computational methods based on automorphism group structures, contributing to the algebraic understanding of virtual knot invariants.

Contribution

It provides the first comprehensive enumeration and classification of virtual racks and quandles up to order 8, including dihedral and permutation types, using conjugacy class analysis.

Findings

Classified virtual racks and quandles up to order 8.

Enumerated virtual dihedral quandles and permutation racks.

Computed class numbers of holomorphs of cyclic groups.

Abstract

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.