Adequate links in thickened surfaces and the generalized Tait conjectures

We apply Kauffman bracket skein algebras to develop a theory of skein adequate links in thickened surfaces. We show that any alternating link diagram on a surface is skein adequate. We apply our theory to establish the first and second Tait conjectures for adequate links in thickened surfaces. Our notion of skein adequacy is broader and more powerful than the corresponding notions of adequacy previously considered for link diagrams in surfaces. For a link diagram D on a surface † of minimal genus g.†/, we show that span(formula presented) where OED† is its skein bracket, jDj is the number of connected components of D, and c.D/ is the number of crossings. 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 weakly alternating. This is a generalization of a well-known result for classical links due to Thistlethwaite. Thus, the skein bracket detects the crossing number for weakly alternating links. As an application, we show that the crossing number is additive under connected sum for adequate links in thickened surfaces.