Generalized Augmented Cellular Alternating Links in Thickened Surfaces are Hyperbolic
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arXiv
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| Hauptverfasser: | , , , , , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2021
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| author | Adams, Colin Capovilla-Searle, Michele Li, Darin Li, Qiao McErlean, Jacob Simons, Alexander Stewart, Natalie Wang, Xiwen |
| author_facet | Adams, Colin Capovilla-Searle, Michele Li, Darin Li, Qiao McErlean, Jacob Simons, Alexander Stewart, Natalie Wang, Xiwen |
| contents | Menasco proved that nontrivial links in the 3-sphere with connected prime alternating non-2-braid projections are hyperbolic. This was further extended to augmented alternating links wherein non-isotopic trivial components bounding disks punctured twice by the alternating link were added. Lackenby proved that the first and second collections of links together form a closed subset of the set of all finite volume hyperbolic 3-manifolds in the geometric topology. Adams showed hyperbolicity for generalized augmented alternating links, which include additional trivial components that bound n-punctured disks for $n \geq 2$. Here we prove that generalized augmented cellular alternating links in I-bundles over closed surfaces are also hyperbolic and that in $S \times I$, the cellular alternating links and the augmented cellular alternating together form a closed subset of finite volume hyperbolic 3-manifolds in the geometric topology. Explicit examples of additional links in $S \times I$ to which these results apply are included. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2107_05406 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Generalized Augmented Cellular Alternating Links in Thickened Surfaces are Hyperbolic Adams, Colin Capovilla-Searle, Michele Li, Darin Li, Qiao McErlean, Jacob Simons, Alexander Stewart, Natalie Wang, Xiwen Geometric Topology 57K32, 57K10 Menasco proved that nontrivial links in the 3-sphere with connected prime alternating non-2-braid projections are hyperbolic. This was further extended to augmented alternating links wherein non-isotopic trivial components bounding disks punctured twice by the alternating link were added. Lackenby proved that the first and second collections of links together form a closed subset of the set of all finite volume hyperbolic 3-manifolds in the geometric topology. Adams showed hyperbolicity for generalized augmented alternating links, which include additional trivial components that bound n-punctured disks for $n \geq 2$. Here we prove that generalized augmented cellular alternating links in I-bundles over closed surfaces are also hyperbolic and that in $S \times I$, the cellular alternating links and the augmented cellular alternating together form a closed subset of finite volume hyperbolic 3-manifolds in the geometric topology. Explicit examples of additional links in $S \times I$ to which these results apply are included. |
| title | Generalized Augmented Cellular Alternating Links in Thickened Surfaces are Hyperbolic |
| topic | Geometric Topology 57K32, 57K10 |
| url | https://arxiv.org/abs/2107.05406 |