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DOI: 10.4230/LIPIcs.SoCG.2025.55

A Practical Algorithm for Knot Factorisation

Authors: Alexander He, Eric Sedgwick, and Jonathan Spreer

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract

We present an algorithm for computing the prime factorisation of a knot, which is practical in the following sense: using Regina, we give an implementation that works well for inputs of reasonable size, including prime knots from the 19-crossing census. The main new ingredient in this work is an object that we call an "edge-ideal triangulation", which is what our algorithm uses to represent knots. As other applications, we give an alternative proof that prime knot recognition is in coNP, and present some new complexity results for triangulations. Beyond knots, our work showcases edge-ideal triangulations as a tool for potential applications in 3-manifold topology.

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Alexander He, Eric Sedgwick, and Jonathan Spreer. A Practical Algorithm for Knot Factorisation. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 55:1-55:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{he_et_al:LIPIcs.SoCG.2025.55,
  author =	{He, Alexander and Sedgwick, Eric and Spreer, Jonathan},
  title =	{{A Practical Algorithm for Knot Factorisation}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{55:1--55:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.55},
  URN =		{urn:nbn:de:0030-drops-232075},
  doi =		{10.4230/LIPIcs.SoCG.2025.55},
  annote =	{Keywords: Prime and composite knots, (crushing) normal surfaces, edge-ideal triangulations, co-NP certificate, triangulation complexity}
}

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DOI: 10.4230/LIPIcs.SoCG.2019.49

The Unbearable Hardness of Unknotting

Authors: Arnaud de Mesmay, Yo'av Rieck, Eric Sedgwick, and Martin Tancer

Published in: LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019)

Abstract

We prove that deciding if a diagram of the unknot can be untangled using at most k Reidemeister moves (where k is part of the input) is NP-hard. We also prove that several natural questions regarding links in the 3-sphere are NP-hard, including detecting whether a link contains a trivial sublink with n components, computing the unlinking number of a link, and computing a variety of link invariants related to four-dimensional topology (such as the 4-ball Euler characteristic, the slicing number, and the 4-dimensional clasp number).

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Arnaud de Mesmay, Yo'av Rieck, Eric Sedgwick, and Martin Tancer. The Unbearable Hardness of Unknotting. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 49:1-49:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{demesmay_et_al:LIPIcs.SoCG.2019.49,
  author =	{de Mesmay, Arnaud and Rieck, Yo'av and Sedgwick, Eric and Tancer, Martin},
  title =	{{The Unbearable Hardness of Unknotting}},
  booktitle =	{35th International Symposium on Computational Geometry (SoCG 2019)},
  pages =	{49:1--49:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-104-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{129},
  editor =	{Barequet, Gill and Wang, Yusu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.49},
  URN =		{urn:nbn:de:0030-drops-104530},
  doi =		{10.4230/LIPIcs.SoCG.2019.49},
  annote =	{Keywords: Knot, Link, NP-hard, Reidemeister move, Unknot recognition, Unlinking number, intermediate invariants}
}

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Documents authored by Sedgwick, Eric

A Practical Algorithm for Knot Factorisation

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The Unbearable Hardness of Unknotting

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