Twisted virtual biracks and their twisted virtual link invariants

TL;DR

This paper introduces twisted virtual biracks, algebraic structures that extend virtual link invariants to non-orientable surfaces, enabling classification of twist structures on virtual links.

Contribution

It defines twisted virtual biracks and extends existing methods to create computable invariants for twisted virtual links, including classification of twist structures.

Findings

Defined twisted virtual biracks with axioms from Reidemeister moves

Extended invariants to twisted virtual links using finite twisted virtual biracks

Classified twist structures on the virtual Hopf link

Abstract

A virtual link can be understood as a link in a trivial I-bundle over an orientable compact surface with genus. A twisted virtual link is a link in a trivial I-bundle over a not-necessarily orientable compact surface. A twisted virtual birack is an algebraic structure with axioms derived from the twisted virtual Reidemeister moves. We extend a method previously used with racks and biracks to the twisted case to define computable invariants of twisted virtual links using finite twisted virtual biracks with birack rank $N\ge 1$. As an application, we classify twist structures on the virtual Hopf link.