The Jones–Krushkal polynomial and minimal diagrams of surface links
Annales de l'Institut Fourier, Online first, 39 p.

We prove a Kauffman–Murasugi–Thistlethwaite theorem for alternating links in thickened surfaces. It implies that any reduced alternating diagram of a link in a thickened surface has minimal crossing number, and any two reduced alternating diagrams of the same link have the same writhe. This result is proved more generally for link diagrams that are adequate, and the proof involves a two-variable generalization of the Jones polynomial for surface links defined by Krushkal. The main result is used to establish the first and second Tait conjectures for links in thickened surfaces and for virtual links.

Nous montrons un théorème de Kauffman-Murasugi-Thistlethwaite pour des entrelacs alternants dans des voisinages tubulaires d’une surface. Il implique que tout diagramme réduit d’entrelacs alternant dans une surface a un nombre minimal de croisements, et que deux diagrammes réduits alternants quelconques du même entrelacs ont le même nombre d’auto-enlacements.

Ce résultat est prouvé plus généralement pour les diagrammes d’entrelacs qui sont adéquats, et sa démonstration utilise une généralisation à deux variables du polynôme de Jones pour les entrelacs sur des surfaces définies par Krushkal. Le résultat principal est utilisé pour établir la première et la deuxième conjecture de Tait pour les entrelacs dans des voisinages tubulaires d’une surface et pour les entrelacs virtuels.

Online First:
DOI: 10.5802/aif.3516
Classification: 57K14,  57K12
Keywords: Kauffman bracket, Jones polynomial, Krushkal polynomial, alternating link diagram, adequate diagram, Tait conjectures, virtual link
Boden, Hans U. 1; Karimi, Homayun 1

1 Mathematics & Statistics McMaster University Hamilton, Ontario (Canada)
     author = {Boden, Hans U. and Karimi, Homayun},
     title = {The {Jones{\textendash}Krushkal} polynomial and minimal diagrams of surface links},
     journal = {Annales de l'Institut Fourier},
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     year = {2022},
     doi = {10.5802/aif.3516},
     language = {en},
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Boden, Hans U.; Karimi, Homayun. The Jones–Krushkal polynomial and minimal diagrams of surface links. Annales de l'Institut Fourier, Online first, 39 p.

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