Adequate links in thickened surfaces and the generalized Tait conjectures

Abstract

The Kauffman bracket of classical links extends to an invariant of links in an arbitrary oriented 3-manifold $M$ with values in the skein module of $M$. In this paper, we consider the skein bracket in case $M$ is a thickened surface. We develop a theory of adequacy for link diagrams on surfaces and 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 link diagrams on surfaces. These are the statements that any adequate link diagram has minimal crossing number, and any two adequate diagrams of the same link have the same writhe. Given a link diagram $D$ on a surface $\Sigma$, we use $[D]_\Sigma$ to denote its skein bracket. If $D$ has minimal genus, we show that $${\rm span}([D]_\Sigma) \leq 4c(D) + 4 |D|-4g(\Sigma),$$ where $|D|$ is the number of connected components of $D$, $c(D)$ is the number of crossings, and $g(\Sigma)$ is the genus of $\Sigma.$ This extends a classical result proved by Kauffman, Murasugi, and Thistlethwaite. We further show that the above inequality is an equality if and only if $D$ is weakly alternating, namely if $D$ is the connected sum of an alternating link diagram on $\Sigma$ with one or more alternating link diagrams on $S^2$. This last statement is a generalization of a well-known result for classical links due to Thistlethwaite, and it implies that 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.