Virtual links which are equivalent as twisted links

TL;DR

This paper investigates the conditions under which virtual links are equivalent as twisted links, providing a complete characterization of their equivalence in the context of links in non-orientable surface thickenings.

Contribution

It offers a necessary and sufficient condition for virtual links to be equivalent as twisted links, advancing the understanding of their relationship.

Findings

Characterization of when virtual links are equivalent as twisted links

Necessary and sufficient condition for virtual link equivalence as twisted links

Connection between virtual links and stable equivalence classes in non-orientable surfaces

Abstract

A virtual link is a generalization of a classical link that is defined as an equivalence class of certain diagrams, called virtual link diagrams. It is further generalized to a twisted link. Twisted links are in one-to-one correspondence with stable equivalence classes of links in oriented thickenings of (possibly non-orientable) closed surfaces. By definition, equivalent virtual links are also equivalent as twisted links. In this paper, we discuss when two virtual links are equivalent as twisted links, and give a necessary and sufficient condition for this to be the case.