We prove a Kauffman-Murasugi-Thistlethwaite theorem for alternating links in
thickened surfaces. It states that any reduced alternating diagram of a link in
a thickened surface has minimal crossing number, and any two reduced
alternating diagrams of the same link have the same writhe. This result is
proved more generally for link diagrams that are adequate, and the proof
involves a two-variable generalization of the Jones polynomial for surface
links defined by Krushkal. The main result is used to establish the first and
second Tait conjectures for links in thickened surfaces and for virtual links.Comment: 32 pages, 20 figures, and 1 tabl