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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

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

span([D]Σ) 4c(D) + 4|D| 4g(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.

Kauffman skein bracket, adequate diagram, alternating link, Tait conjectures
Mathematical Subject Classification
Primary: 57K10, 57K12
Secondary: 57K14, 57K31
Received: 16 June 2021
Revised: 19 January 2022
Accepted: 3 February 2022
Published: 25 July 2023
Hans U Boden
Department of Mathematics and Statistics
McMaster University
Hamilton, ON
Homayun Karimi
Department of Mathematics and Statistics
McMaster University
Hamilton, ON
Adam S Sikora
Department of Mathematics
University at Buffalo
Buffalo, NY
United States

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