Primeness of Alternating Virtual Links

We introduce a new tool called lassos, which we use to establish a correspondence between cellular link diagrams on closed surfaces and equivalence classes of virtual link diagrams—this is a diagrammatic analog of a well-known correspondence between the links represented by these diagrams. Under these correspondences, the traditional notion of primeness for virtual links is stricter than the one for links in thickened surfaces. Our main result, generalizing a classical result of Menasco, is that an alternating link in a thickened surface is prime in the stricter sense unless it is obviously composite. (Adams et al and Howie-Purcell previously extended Menasco's result for the other notion of primeness.)