A Note on Alternating Knots in Handlebodies

We establish a Kauffman–Murasugi–Thistlethwaite-type theorem for alternating knots in a solid torus. Specifically, we show that any dotted-reduced alternating diagram of a knot in a handlebody realises the minimal crossing number. The proof relies on a generalisation of the Jones polynomial to the setting of handlebodies. A stronger version of this result was already proved by Boden et al. [‘Adequate links in thickened surfaces and the generalized Tait conjectures’, Algebr. Geom. Topol. 23 (5) (2023), 2271–2308] using a different generalised Jones polynomial; this paper largely expands on one of the main proof tools.