Adequate Links In Thickened Surfaces And The Generalized Tait Conjectures

We develop a theory of adequacy for link diagrams on surfaces and show that any alternating link diagram on a surface is adequate. We apply our theory to establish the first and second Tait conjectures for adequate link diagrams on surfaces, stating that any adequate link diagram has minimal crossing number, and any two adequate diagrams of the same link have the same writhe.
Let us denote the (Kauffman) skein bracket of a diagram D on a surface Σ by [D]. If D has minimal genus, we show that span([D]) ≤ 4c(D) + 4|D| − 4g(Σ), where |D| is the number of connected components of D, c(D) is the number of crossings, and g(Σ) is the genus of Σ. This extends a classical result of Kauffman, Murasugi, and Thistlethwaite. We further show that the above inequality is an equality if and only if D is alternating. This generalizes a well-known result for classical links and it implies that the skein bracket detects the crossing number for alternating links. As an application, we show that the crossing number is additive under connected sum for adequate links in thickened surfaces. This is joint work with H. Boden and H. Karimi.