Computing a Link Diagram from Its Exterior

Authors Nathan M. Dunfield , Malik Obeidin, Cameron Gates Rudd

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Author Details

Nathan M. Dunfield
  • Dept. of Math., University of Illinois at Urbana-Champaign, IL, USA
Malik Obeidin
  • Google, Inc., Mountain View, CA, USA
Cameron Gates Rudd
  • Dept. of Math., University of Illinois at Urbana-Champaign, IL, USA


We thank Matthias Goerner and Henry Segerman for helpful correspondence, and thank the referees for their detailed comments which helped improve this paper.

Cite AsGet BibTex

Nathan M. Dunfield, Malik Obeidin, and Cameron Gates Rudd. Computing a Link Diagram from Its Exterior. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 37:1-37:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Geometric topology
  • computational topology
  • low-dimensional topology
  • knot
  • knot exterior
  • knot diagram
  • link
  • link exterior
  • link diagram


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