License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2022.37
URN: urn:nbn:de:0030-drops-160457
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/16045/
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Dunfield, Nathan M. ; Obeidin, Malik ; Rudd, Cameron Gates

Computing a Link Diagram from Its Exterior

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LIPIcs-SoCG-2022-37.pdf (3 MB)

Abstract

A knot is a circle piecewise-linearly embedded into the 3-sphere. The topology of a knot is intimately related to that of its exterior, which is the complement of an open regular neighborhood of the knot. Knots are typically encoded by planar diagrams, whereas their exteriors, which are compact 3-manifolds with torus boundary, are encoded by triangulations. Here, we give the first practical algorithm for finding a diagram of a knot given a triangulation of its exterior. Our method applies to links as well as knots, and allows us to recover links with hundreds of crossings. We use it to find the first diagrams known for 23 principal congruence arithmetic link exteriors; the largest has over 2,500 crossings. Other applications include finding pairs of knots with the same 0-surgery, which relates to questions about slice knots and the smooth 4D Poincaré conjecture.

BibTeX - Entry

@InProceedings{dunfield_et_al:LIPIcs.SoCG.2022.37,
  author =	{Dunfield, Nathan M. and Obeidin, Malik and Rudd, Cameron Gates},
  title =	{{Computing a Link Diagram from Its Exterior}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{37:1--37:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-227-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{224},
  editor =	{Goaoc, Xavier and Kerber, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2022/16045},
  URN =		{urn:nbn:de:0030-drops-160457},
  doi =		{10.4230/LIPIcs.SoCG.2022.37},
  annote =	{Keywords: computational topology, low-dimensional topology, knot, knot exterior, knot diagram, link, link exterior, link diagram}
}

Keywords: computational topology, low-dimensional topology, knot, knot exterior, knot diagram, link, link exterior, link diagram
Collection: 38th International Symposium on Computational Geometry (SoCG 2022)
Issue Date: 2022
Date of publication: 01.06.2022
Supplementary Material: Software (Source Code and Data): https://doi.org/10.7910/DVN/BT1M8R

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