The Tait conjecture in S¹xS²

The Tait conjecture states that reduced alternating diagrams of links in S^3 have the minimal number of crossings. It has been proved in 1987 by M. Thistlethwaite, L.H. Kauffman and K. Murasugi studying the Jones polynomial. In this paper we prove an analogous result for alternating links in S^1xS^2 giving a complete answer to this problem. In S^1xS^2 we find a dichotomy: the appropriate version of the statement is true for \Z_2-homologically trivial links, and our proof also uses the Jones polynomial. On the other hand, the statement is false for \Z_2-homologically non trivial links, for which the Jones polynomial vanishes.