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.
@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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