  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.
