Complexity of 3-manifolds obtained by Dehn filling
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866910868185284608 |
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| author | Jaco, William Rubinstein, J. Hyam Spreer, Jonathan Tillmann, Stephan |
| author_facet | Jaco, William Rubinstein, J. Hyam Spreer, Jonathan Tillmann, Stephan |
| contents | Let $M$ be a compact 3--manifold with boundary a single torus. We present upper and lower complexity bounds for closed 3--manifolds obtained as even Dehn fillings of $M.$ As an application, we characterise some infinite families of even Dehn fillings of $M$ for which our method determines the complexity of its members up to an additive constant. The constant only depends on the size of a chosen triangulation of $M$, and the isotopy class of its boundary.
We then show that, given a triangulation $\mathcal T$ of $M$ with $2$--triangle torus boundary, there exist infinite families of even Dehn fillings of $M$ for which we can determine the complexity of the filled manifolds with a gap between upper and lower bound of at most $13 |\mathcal T| + 7.$ This result is bootstrapped to obtain the gap as a function of the size of an ideal triangulation of the interior of $M$, or the number of crossings of a knot diagram. We also show how to compute the gap for explicit families of fillings of knot complements in the three-sphere. The practicability of our approach is demonstrated by determining the complexity up to a gap of at most 10 for several infinite families of even fillings of the figure eight knot, the pretzel knot $P(-2,3,7)$, and the trefoil. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_12066 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Complexity of 3-manifolds obtained by Dehn filling Jaco, William Rubinstein, J. Hyam Spreer, Jonathan Tillmann, Stephan Geometric Topology 57M25, 57N10 Let $M$ be a compact 3--manifold with boundary a single torus. We present upper and lower complexity bounds for closed 3--manifolds obtained as even Dehn fillings of $M.$ As an application, we characterise some infinite families of even Dehn fillings of $M$ for which our method determines the complexity of its members up to an additive constant. The constant only depends on the size of a chosen triangulation of $M$, and the isotopy class of its boundary. We then show that, given a triangulation $\mathcal T$ of $M$ with $2$--triangle torus boundary, there exist infinite families of even Dehn fillings of $M$ for which we can determine the complexity of the filled manifolds with a gap between upper and lower bound of at most $13 |\mathcal T| + 7.$ This result is bootstrapped to obtain the gap as a function of the size of an ideal triangulation of the interior of $M$, or the number of crossings of a knot diagram. We also show how to compute the gap for explicit families of fillings of knot complements in the three-sphere. The practicability of our approach is demonstrated by determining the complexity up to a gap of at most 10 for several infinite families of even fillings of the figure eight knot, the pretzel knot $P(-2,3,7)$, and the trefoil. |
| title | Complexity of 3-manifolds obtained by Dehn filling |
| topic | Geometric Topology 57M25, 57N10 |
| url | https://arxiv.org/abs/2207.12066 |