Adequate links in thickened surfaces and the generalized Tait conjectures

Boden, Hans U; Karimi, Homayun; Sikora, Adam S

doi:10.2140/agt.2023.23.2271

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Adequate links in thickened surfaces and the generalized Tait conjectures

Hans U Boden, Homayun Karimi and Adam S Sikora

Algebraic & Geometric Topology 23 (2023) 2271–2308

DOI: 10.2140/agt.2023.23.2271

Abstract

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 $Σ$ of minimal genus $g (Σ)$ , we show that

s p a n ({[D]}_{Σ}) \leq 4 c (D) + 4 | D | - 4 g (Σ),

where ${[D]}_{Σ}$ 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.

Keywords

Kauffman skein bracket, adequate diagram, alternating link, Tait conjectures

Mathematical Subject Classification

Primary: 57K10, 57K12

Secondary: 57K14, 57K31

References

Publication

Received: 16 June 2021

Revised: 19 January 2022

Accepted: 3 February 2022

Published: 25 July 2023

Authors

Hans U Boden
Department of Mathematics and Statistics McMaster University Hamilton, ON Canada

Homayun Karimi
Department of Mathematics and Statistics McMaster University Hamilton, ON Canada

Adam S Sikora
Department of Mathematics University at Buffalo Buffalo, NY United States

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Volume 23, issue 5 (2023)