The Tait conjecture in g(S¹xS²)

The Tait conjecture states that alternating reduced diagrams of links in S^3 have the minimal number of crossings. It has been proved in 1987 by M. Thistlethwaite, L. Kauffman and K. Murasugi studying the Jones polynomial. The author proved an analogous result for alternating links in S^1xS^2 giving a complete answer to this problem. In this paper we extend the result to alternating links in the connected sum #_g(S^1xS^2) of g copies of S^1xS^2. In S^1xS^2 and #_2(S^1xS^2) the appropriate version of the statement is true for \Z_2-homologically trivial links, and the proof also uses the Jones polynomial. Unfortunately in the general case the method provides just a partial result and we are not able to say if the appropriate statement is true.