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Eisermann has shown that the Jones polynomial of a ncomponent ribbon link L ⊂ S 3 is divided by the Jones polynomial of the trivial n-component link. We improve this theorem by extending its range of application from links in S 3 to colored knotted trivalent graphs in #g(S 2 × S 1 ), the connected sum of g 0 copies of S 2 × S 1 .We show in particular that if the Kauffman bracket of a knot in #g(S 2 × S 1 ) has a pole in q = i of order n, the ribbon genus of the knot is at least n+1 2 . We construct some families of knots in #g(S 2 × S 1 ) for which this lower bound is sharp and arbitrarily big. We prove these estimates using Turaev shadows.

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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.