Totally Geodesic Surfaces in Hyperbolic 3-Manifolds: Algorithms and Examples
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909141629403136 |
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| author | Basilio, Brannon Lee, Chaeryn Malionek, Joseph |
| author_facet | Basilio, Brannon Lee, Chaeryn Malionek, Joseph |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_12397 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Totally Geodesic Surfaces in Hyperbolic 3-Manifolds: Algorithms and Examples Basilio, Brannon Lee, Chaeryn Malionek, Joseph Geometric Topology 57K32 (Primary), 57K10 (Secondary) G.4 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. |
| title | Totally Geodesic Surfaces in Hyperbolic 3-Manifolds: Algorithms and Examples |
| topic | Geometric Topology 57K32 (Primary), 57K10 (Secondary) G.4 |
| url | https://arxiv.org/abs/2403.12397 |