The Tait conjecture in \(S^1 \times S^2\) (Q2833317)

This paper proves an analogous result to what has come to be known as a ``Tait conjecture'' that every reduced alternating diagram of a link in \(S^3\) has the minimal number of crossings. Here it is shown to be true in \(S^1\times S^2\) for \(\mathbb Z_2\) homologically trivial links only, and a knotted counterexample is exhibited as Figure 4. The author errs, however, in asserting that Tait ``stated'' the three conjectures about alternating links that have been attributed to him as ``inherently'' on his mind in the course of his work [\textit{W. B. R. Lickorish}, An introduction to knot theory. New York, NY: Springer (1997; Zbl 0886.57001)]. In particular, and plainly to the contrary, Tait (along with Dehn and Heegaard) explicitly blessed Little's false proof of the invariance of minimal crossing writhe for all knots, not just the alternating ones [\textit{M. Epple}, Die Entstehung der Knotentheorie. Kontexte und Konstruktionen einer modernen mathematischen Theorie. Wiesbaden: Vieweg (1999; Zbl 0972.57001)] [\textit{A. Stoimenow}, Diagram genus, generators, and applications. Boca Raton, FL: CRC Press (2016; Zbl 1336.57016)].