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The Tait conjecture in S^1xS^2 (1510.01622v3)

Published 6 Oct 2015 in math.GT

Abstract: The Tait conjecture states that reduced alternating diagrams of links in S3 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 S1xS2 giving a complete answer to this problem. In S1xS2 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.

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