Adequate links in thickened surfaces and the generalized Tait
conjectures
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abstract
In this paper, 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 $\Sigma$ of minimal genus $g(\Sigma)$, we
show that $${\rm span}([D]_\Sigma) \leq 4c(D) + 4 |D|-4g(\Sigma),$$ where
$[D]_\Sigma$ is its skein bracket, $|D|$ 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.
