An algorithm to construct one-vertex triangulations of Heegaard splittings
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866915398726713344 |
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| author | He, Alexander Morgan, James Thompson, Em K. |
| author_facet | He, Alexander Morgan, James Thompson, Em K. |
| contents | Following work of Jaco and Rubinstein (2006), which (non-constructively) proved that any 3-manifold admits a one-vertex layered triangulation, we present an algorithm, with implementation using Regina, that uses a combinatorial presentation of a Heegaard diagram to construct a generalised notion of a layered triangulation. We show that work of Huszár and Spreer (2019) extends to our construction: given a genus-$g$ Heegaard splitting, our algorithm generates a triangulation with cutwidth bounded above by $4g-2$. Beyond Heegaard splittings, our construction actually extends to a natural generalisation of Dehn fillings: given a one-vertex triangulation with a genus-$g$ boundary component $B$, we can construct a one-vertex triangulation of any 3-manifold obtained by filling $B$ with a handlebody. To demonstrate the usefulness of our algorithm, we present findings from preliminary computer searches using this algorithm. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_17556 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | An algorithm to construct one-vertex triangulations of Heegaard splittings He, Alexander Morgan, James Thompson, Em K. Geometric Topology 57Q15, 57N10 Following work of Jaco and Rubinstein (2006), which (non-constructively) proved that any 3-manifold admits a one-vertex layered triangulation, we present an algorithm, with implementation using Regina, that uses a combinatorial presentation of a Heegaard diagram to construct a generalised notion of a layered triangulation. We show that work of Huszár and Spreer (2019) extends to our construction: given a genus-$g$ Heegaard splitting, our algorithm generates a triangulation with cutwidth bounded above by $4g-2$. Beyond Heegaard splittings, our construction actually extends to a natural generalisation of Dehn fillings: given a one-vertex triangulation with a genus-$g$ boundary component $B$, we can construct a one-vertex triangulation of any 3-manifold obtained by filling $B$ with a handlebody. To demonstrate the usefulness of our algorithm, we present findings from preliminary computer searches using this algorithm. |
| title | An algorithm to construct one-vertex triangulations of Heegaard splittings |
| topic | Geometric Topology 57Q15, 57N10 |
| url | https://arxiv.org/abs/2312.17556 |