Adequate links in thickened surfaces and the generalized Tait conjectures

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 $$\mathrm{span}([D{]}_{\Sigma })\le 4c(D)+4|D|-4g(\Sigma ),$$ 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.

27 pages, 15 figures. Includes many corrections and improvements. Section 7.3 has been completely revised

References in corpus (6)

• The Jones polynomial and graphs on surfaces
• Minimal crossing number implies minimal supporting genus
• A characterization of alternating links in thickened surfaces
• TG-Hyperbolicity of Virtual Links
• Homological Polynomial Coefficients and the Twist Number of Alternating Surface Links
• Twist Number and the Alternating Volume of Knots

Cited by in corpus (3)

• A characterization of alternating links in thickened surfaces
• A volumish theorem for alternating virtual links
• Addendum to: Guts, Volume and Skein Modules of 3-Manifolds