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DOI: 10.4230/LIPIcs.SoCG.2024.48

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

Authors: Pierre Dehornoy, Corentin Lunel, and Arnaud de Mesmay

Published in: LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024)

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.

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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)

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@InProceedings{dehornoy_et_al:LIPIcs.SoCG.2024.48,
  author =	{Dehornoy, Pierre and Lunel, Corentin and de Mesmay, Arnaud},
  title =	{{Hopf Arborescent Links, Minor Theory, and Decidability of the Genus Defect}},
  booktitle =	{40th International Symposium on Computational Geometry (SoCG 2024)},
  pages =	{48:1--48:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-316-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{293},
  editor =	{Mulzer, Wolfgang and Phillips, Jeff M.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.48},
  URN =		{urn:nbn:de:0030-drops-199938},
  doi =		{10.4230/LIPIcs.SoCG.2024.48},
  annote =	{Keywords: Knot Theory, Genus, Slice Genus, Hopf Arborescent Links, Well-Quasi-Order}
}

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

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