Double coverings of twisted links

TL;DR

This paper introduces the concept of double coverings for twisted links, generalizing virtual links to non-orientable surfaces, and explores their properties, diagrams, and associated groups.

Contribution

It defines the double covering of twisted links, linking twisted knot groups to virtual knot groups, and discusses their diagrams and stable equivalence classes.

Findings

Double coverings of twisted links are well-defined and relate to orientation double coverings.

Bourgoin's twisted knot group corresponds to the virtual knot group of the double covering.

The paper extends the understanding of links in non-orientable surfaces.

Abstract

Twisted links are a generalization of virtual links. As virtual links correspond to abstract links on orientable surfaces, twisted links correspond to abstract links on (possibly non-orientable) surfaces. In this paper, we introduce the notion of the double covering of a twisted link. It is defined by considering the orientation double covering of an abstract link or alternatively by constructing a diagram called a double covering diagram. We also discuss links in thickened surfaces, their diagrams and their stable equivalence classes. Bourgoin's twisted knot group is understood as the virtual knot group of the double covering.