Adequate links in thickened surfaces and the generalized Tait conjectures

TL;DR

This paper develops a skein algebra-based theory of adequate links in thickened surfaces, proving the Tait conjectures for such links and extending classical results to a broader setting.

Contribution

It introduces a new notion of skein adequacy for links in surfaces, proving the Tait conjectures and extending classical crossing number results.

Findings

Alternating links on surfaces are skein adequate.

The span of the skein bracket bounds crossing number, with equality for weakly alternating links.

Crossing number is additive under connected sum for adequate links in thickened surfaces.

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.