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DOI: 10.4230/LIPIcs.SoCG.2024.14
Totally Geodesic Surfaces in Hyperbolic 3-Manifolds: Algorithms and Examples

Authors: Brannon Basilio, Chaeryn Lee, and Joseph Malionek

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

Abstract
Finding a totally geodesic surface, an embedded surface where the geodesics in the surface are also geodesics in the surrounding manifold, has been a problem of interest in the study of 3-manifolds. This has especially been of interest in hyperbolic 3-manifolds and knot complements, complements of piecewise-linearly embedded circles in the 3-sphere. This is due to Menasco-Reid’s conjecture stating that hyperbolic knot complements do not contain such surfaces. Here, we present an algorithm that determines whether a given surface is totally geodesic and an algorithm that checks whether a given 3-manifold contains a totally geodesic surface. We applied our algorithm on over 150,000 3-manifolds and discovered nine 3-manifolds with totally geodesic surfaces. Additionally, we verified Menasco-Reid’s conjecture for knots up to 12 crossings.
Cite as

Brannon Basilio, Chaeryn Lee, and Joseph Malionek. Totally Geodesic Surfaces in Hyperbolic 3-Manifolds: Algorithms and Examples. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 14:1-14:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{basilio_et_al:LIPIcs.SoCG.2024.14,
  author =	{Basilio, Brannon and Lee, Chaeryn and Malionek, Joseph},
  title =	{{Totally Geodesic Surfaces in Hyperbolic 3-Manifolds: Algorithms and Examples}},
  booktitle =	{40th International Symposium on Computational Geometry (SoCG 2024)},
  pages =	{14:1--14:19},
  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.2024.14},
  URN =		{urn:nbn:de:0030-drops-199593},
  doi =		{10.4230/LIPIcs.SoCG.2024.14},
  annote =	{Keywords: totally geodesic, Fuchsian group, hyperbolic, knot complement, computational topology, low-dimensional topology}
}
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