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Hopf Arborescent Links, Minor Theory, and Decidability of the Genus Defect

Authors Pierre Dehornoy , Corentin Lunel , Arnaud de Mesmay

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Author Details

Pierre Dehornoy
  • Aix-Marseille Université, CNRS, I2M, Marseille, France
Corentin Lunel
  • LIGM, CNRS, Univ. Gustave Eiffel, ESIEE Paris, F-77454 Marne-la-Vallée, France
Arnaud de Mesmay
  • LIGM, CNRS, Univ. Gustave Eiffel, ESIEE Paris, F-77454 Marne-la-Vallée, France

Acknowledgements

We would like to thank Sebastian Baader for helpful discussions, and the anonymous reviewers for their questions and suggestions which allowed us to significantly improve the paper.

Cite AsGet BibTex

Pierre Dehornoy, Corentin Lunel, and Arnaud de Mesmay. Hopf Arborescent Links, Minor Theory, and Decidability of the Genus Defect. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 48:1-48:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.SoCG.2024.48

Abstract

While the problem of computing the genus of a knot is now fairly well understood, no algorithm is known for its four-dimensional variants, both in the smooth and in the topological locally flat category. In this article, we investigate a class of knots and links called Hopf arborescent links, which are obtained as the boundaries of some iterated plumbings of Hopf bands. We show that for such links, computing the genus defects, which measure how much the four-dimensional genera differ from the classical genus, is decidable. Our proof is non-constructive, and is obtained by proving that Seifert surfaces of Hopf arborescent links under a relation of minors defined by containment of their Seifert surfaces form a well-quasi-order.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Geometric topology
  • Theory of computation → Computational geometry
Keywords
  • Knot Theory
  • Genus
  • Slice Genus
  • Hopf Arborescent Links
  • Well-Quasi-Order

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