The Tait conjecture in #g(S1 × S2)

Abstract

The Tait conjecture states that alternating reduced diagrams of links in [Formula: see text] have the minimal number of crossings. It has been proved in 1987 by Thistlethwaite, Kauffman and Murasugi studying the Jones polynomial. In [A. Carrega, The Tait conjecture in [Formula: see text], J. Knot Theory Ramifications 25(11) (2016) 1650063], the author proved an analogous result for alternating links in [Formula: see text] giving a complete answer to this problem. In this paper, we extend the result to alternating links in the connected sum [Formula: see text] of [Formula: see text] copies of [Formula: see text]. In [Formula: see text] and [Formula: see text], the appropriate version of the statement is true for [Formula: see text]-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.