The Tait conjecture in S1 × S2

Abstract

The Tait conjecture states that reduced alternating diagrams of links in [Formula: see text] 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 [Formula: see text] giving a complete answer to this problem. In [Formula: see text] we find a dichotomy: the appropriate version of the statement is true for [Formula: see text]-homologically trivial links, and our proof also uses the Jones polynomial. On the other hand, the statement is false for [Formula: see text]-homologically non-trivial links, for which the Jones polynomial vanishes.