  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.

Geometric Topology