Primeness of alternating virtual links

Using a new tool called lassos, we establish a new correspondence between cellular link {diagrams} on closed surfaces and equivalence classes of virtual link {diagrams}. This is analogous to a well-known correspondence among the links represented by these diagrams, but with a crucial subtlety. We explain how, under these correspondences, the traditional notion of primeness for virtual links is stricter than the one for links in thickened surfaces. We extend a classical result of Menasco by proving 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.) We describe, given an alternating virtual link diagram, how to determine by inspection whether the virtual link it represents is prime in either sense.