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

Brannon Basilio, Chaeryn Lee, Joseph Malionek

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.

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

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.
Paper Structure (23 sections, 14 theorems, 4 equations, 14 figures, 1 table, 3 algorithms)

This paper contains 23 sections, 14 theorems, 4 equations, 14 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Hyperbolic 3-manifolds
  4. Totally Geodesic Surfaces
  5. Normal Surfaces
  6. The Algorithms
  7. Computational Representations of Mathematical Objects
  8. Checking Whether a Surface is Totally Geodesic
  9. Detecting Totally Geodesic Surfaces in a 3-manifold
  10. Computations
  11. Practical Algorithm Considerations
  12. Results
  13. An Example Containing a Totally Geodesic Surface
  14. Future Work
  15. Background
  16. ...and 8 more sections

Key Result

Proposition 2

Suppose $M$ is a finite-volume orientable hyperbolic 3-manifold. Then every closed surface in $M$ is orientable if and only if the inclusion map $i:H_2(\partial M; \mathbb{F}_2)\to H_2(M;\mathbb{F}_2)$ is surjective.

Figures (14)

  • Figure 1: A model of $\mathbb{H}^3$. $\partial\mathbb{H}^3$ is given in green, while two examples of geodesics are given in blue.
  • Figure 2: The shape parameter $z$ for the edge $e$ of the given ideal tetrahedron.
  • Figure 3: The 7 disk types.
  • Figure 4: An example application of Lemma \ref{['fundamental_skeleton_surface']} on a cellulation of a torus by polygons. The blue and red edges together form the dual 1-skeleton, with the blue edges making up the tree $\EuScript{T}$ and the red edges forming the generating set.
  • Figure 5: A partial drawing of a triangulated 3-manifold with its dual 1-skeleton as in Lemma \ref{['fundamental_skeleton_manifold']}. The blue and red edges together form the dual 1-skeleton, with the blue edges belonging to the tree $\EuScript{T}$ and the red edges belonging to the generating set.
  • ...and 9 more figures

Theorems & Definitions (15)

  • Conjecture 1
  • Proposition 2: Proposition 2.4 Dunfield_2022
  • Lemma 3
  • Proposition 4: Purcell2020HyperbolicKT Example 12.11
  • Lemma 5: MorganShalen_part1 Proposition III.1.1
  • Lemma 6: MaclachlanReid Lemma 3.5.3
  • Corollary 7
  • Theorem 8: JACO1984195 Theorem 1.2, see Theorem 5.2.14 in Schultens2014IntroductionT3 for proof
  • Theorem 9: Theorem 3.1.2 and Corollary 3.2.4 of MaclachlanReid
  • Lemma 10: EppsteinFundamentalGroup Lemma 3.2
  • ...and 5 more