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Totally Geodesic Surfaces in Hyperbolic 3-Manifolds: Algorithms and Examples

Authors Brannon Basilio, Chaeryn Lee, Joseph Malionek

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ACM Subject Classification
  • Mathematics of computing → Geometric topology
Keywords
  • totally geodesic
  • Fuchsian group
  • hyperbolic
  • knot complement
  • computational topology
  • low-dimensional topology

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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.

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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) https://doi.org/10.4230/LIPIcs.SoCG.2024.14

Author Details

Brannon Basilio
  • Department of Mathematics, University of Illinois Urbana-Champaign, IL, USA
Chaeryn Lee
  • Department of Mathematics, University of Illinois Urbana-Champaign, IL, USA
Joseph Malionek
  • Department of Mathematics, University of Illinois Urbana-Champaign, IL, USA

Funding

All authors were supported by US National Science Foundation grants DMS-1811156 and DMS-2303572.

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References

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