The Tait conjecture in S^1xS^2

TL;DR

This paper proves an analogue of the Tait conjecture for alternating links in S^1xS^2, showing the minimal crossing number property holds for homologically trivial links but not for non-trivial ones, using the Jones polynomial.

Contribution

It extends the Tait conjecture to S^1xS^2, providing a complete characterization based on homology and employing the Jones polynomial.

Findings

The minimal crossing number property holds for Z_2-homologically trivial links in S^1xS^2.

The property fails for Z_2-homologically non-trivial links, where the Jones polynomial vanishes.

The proof distinguishes between homologically trivial and non-trivial links in S^1xS^2.

Abstract

The Tait conjecture states that reduced alternating diagrams of links in S^3 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 S^1xS^2 giving a complete answer to this problem. In S^1xS^2 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.