Stable Equivalence of Knots on Surfaces and Virtual Knot Cobordisms
  We introduce an equivalence relation, called stable equivalence, on knot diagrams and closed curves on surfaces. We give bijections between the set of abstract knots, the set of virtual knots, and the set of the stable equivalence classes of knot diagrams on surfaces. Using these bijections, we define concordance and link homology for virtual links. As an application, it is shown that Kauffman's example of a virtual knot diagram is not equivalent to a classical knot diagram. 
J. Scott Carter (University of South Alabama), Seiichi Kamada (Osaka   City University), Masahico Saito (University of South...