Document Open Access Logo

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

Authors Brannon Basilio, Chaeryn Lee, Joseph Malionek

PDF

File

Thumbnail PDF
PDF
LIPIcs.SoCG.2024.14.pdf
  • Filesize: 2.24 MB
  • 19 pages

Document Identifiers

Related Versions

Subject Classification

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

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

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 Get BibTex

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.

Supplementary Materials

References

  1. Colin Adams and Eric Schoenfeld. Totally geodesic Seifert surfaces in hyperbolic knot and link complements i. Geometriae Dedicata, 116:237-247, December 2005. URL: https://doi.org/10.1007/s10711-005-9018-z.
  2. Colin C. Adams. Thrice-punctured spheres in hyperbolic 3-manifolds. Transactions of the American Mathematical Society, 287(2):645-656, 1985. URL: http://www.jstor.org/stable/1999666.
  3. Colin C. Adams and Alan W. Reid. Quasi-fuchsian surfaces in hyperbolic knot complements. Journal of the Australian Mathematical Society, 55(1):116-131, 1993. URL: https://doi.org/10.1017/S1446788700031967.
  4. Riccardo Benedetti and Carlo Petronio. Lectures on Hyperbolic Geometry. Universitext (Berlin. Print). Springer Berlin Heidelberg, 1992. URL: https://books.google.com/books?id=iTbmytIqdpcC.
  5. Benjamin A. Burton. The complexity of the normal surface solution space. In Proceedings of the Twenty-Sixth Annual Symposium on Computational Geometry, SoCG '10, pages 201-209, New York, NY, USA, 2010. Association for Computing Machinery. URL: https://doi.org/10.1145/1810959.1810995.
  6. Benjamin A. Burton. Enumerating fundamental normal surfaces: Algorithms, experiments and invariants. In 2014 Proceedings of the Sixteenth Workshop on Algorithm Engineering and Experiments (ALENEX), pages 112-124. Society for Industrial and Applied Mathematics, December 2013. URL: https://doi.org/10.1137/1.9781611973198.11.
  7. Benjamin A. Burton. The cusped hyperbolic census is complete. ArXiv, abs/1405.2695, 2014. URL: https://api.semanticscholar.org/CorpusID:13640292.
  8. Benjamin A. Burton, Ryan Budney, William Pettersson, et al. Regina: Software for low-dimensional topology. http:// regina-normal. github. io/, 1999-2023. Google Scholar
  9. Henri Cohen. A Course in Computational Algebraic Number Theory. Springer Publishing Company, Incorporated, 2010. Google Scholar
  10. David Coulson, Oliver A. Goodman, Craig D. Hodgson, and Walter D. Neumann. Computing arithmetic invariants of 3-manifolds. Experimental Mathematics, 9(1):127-152, 2000. Google Scholar
  11. Marc Culler. Lifting representations to covering groups. Advances in Mathematics, 59(1):64-70, 1986. URL: https://doi.org/10.1016/0001-8708(86)90037-X.
  12. Marc Culler, Nathan M. Dunfield, Matthias Goerner, and Jeffrey R. Weeks. SnapPy, a computer program for studying the geometry and topology of 3-manifolds. Available at http://snappy.computop.org (17/06/2023).
  13. Nathan Dunfield, Neil Hoffman, and Joan Licata. Asymmetric hyperbolic L-spaces, Heegaard genus, and Dehn filling. Mathematical Research Letters, 22, July 2014. URL: https://doi.org/10.4310/MRL.2015.v22.n6.a7.
  14. Nathan M. Dunfield, Stavros Garoufalidis, and J. Hyam Rubinstein. Counting essential surfaces in 3-manifolds. Inventiones mathematicae, 228(2):717-775, January 2022. URL: https://doi.org/10.1007/s00222-021-01090-w.
  15. David Eppstein. Dynamic generators of topologically embedded graphs. In Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '03, pages 599-608, USA, 2003. Society for Industrial and Applied Mathematics. Google Scholar
  16. Matthias Goerner. Verified computations for closed hyperbolic 3‐manifolds. Bulletin of the London Mathematical Society, 53, 2019. URL: https://api.semanticscholar.org/CorpusID:139104728.
  17. Luke Harris and Peter Scott. The uniqueness of compact cores for 3-manifolds. Pacific Journal of Mathematics, 172:139-150, 1996. URL: https://api.semanticscholar.org/CorpusID:116949986.
  18. Jim Hoste, Morwen Thistlethwaite, and Jeff Weeks. The first 1,701,936 knots. The Mathematical Intelligencer, 20(4):33-48, 1998. URL: https://doi.org/10.1007/BF03025227.
  19. Kazuhiro Ichihara and Makoto Ozawa. Hyperbolic knot complements without closed embedded totally geodesic surfaces. Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics, 68:379-386, 2000. Google Scholar
  20. William Jaco and Ulrich Oertel. An algorithm to decide if a 3-manifold is a Haken manifold. Topology, 23(2):195-209, 1984. URL: https://doi.org/10.1016/0040-9383(84)90039-9.
  21. Marc Lackenby. An algorithm to determine the Heegaard genus of simple 3–manifolds with nonempty boundary. Algebraic & Geometric Topology, 8:911-934, 2007. URL: https://api.semanticscholar.org/CorpusID:18296965.
  22. Chaeryn Lee, Brannon Basilio, and Joseph Malionek. Totally Geodesic Surfaces in Hyperbolic 3-Manifolds: Algorithms and Examples, 2024. URL: https://doi.org/10.7910/DVN/3YGSTI.
  23. Charles Livingston and Allison H. Moore. Knotinfo: Table of knot invariants. URL: knotinfo.math.indiana.edu, November 2023.
  24. Colin Maclachlan and Alan W. Reid. The Arithmetic of Hyperbolic 3-Manifolds. Graduate Texts in Mathematics. Springer New York, 2013. URL: https://link.springer.com/book/10.1007/978-1-4757-6720-9.
  25. William Menasco and Alan W. Reid. Totally Geodesic Surfaces in Hyperbolic Link Complements, pages 215-226. De Gruyter, Berlin, Boston, 1992. URL: https://doi.org/doi:10.1515/9783110857726.215.
  26. Yosuke Miyamoto. Volumes of hyperbolic manifolds with geodesic boundary. Topology, 33(4):613-629, 1994. URL: https://doi.org/10.1016/0040-9383(94)90001-9.
  27. John W. Morgan and Peter B. Shalen. Valuations, trees, and degenerations of hyperbolic structures, i. Annals of Mathematics, 120(3):401-476, 1984. URL: http://www.jstor.org/stable/1971082.
  28. Ulrich Oertel. Closed incompressible surfaces in complements of star links. Pacific Journal of Mathematics, 111(1):209-230, 1984. Google Scholar
  29. Jessica S. Purcell. Hyperbolic Knot Theory. Graduate Studies in Mathematics. American Mathematical Society, 2020. URL: https://bookstore.ams.org/gsm-209.
  30. Jennifer Schultens. Introduction to 3-Manifolds. American Mathematical Society, 2014. URL: https://api.semanticscholar.org/CorpusID:117609654.
  31. The Sage Developers. SageMath, the Sage Mathematics Software System (Version 9.2), 2023. URL: https://www.sagemath.org.
  32. Joseph Miller Thomas. Sturm’s theorem for multiple roots. National Mathematics Magazine, 15(8):391-394, 1941. URL: http://www.jstor.org/stable/3028551.
  33. Stephan Tillmann. Normal surfaces in topologically finite 3-manifolds. L'Enseignement Mathématique, 54(3):329-380, 2008. Google Scholar
  34. Jeffrey L. Tollefson. Isotopy classes of incompressible surfaces in irreducible 3-manifolds. Osaka Journal of Mathematics, 32(4):1087-1111, 1995. URL: https://doi.org/ojm/1200786486.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact