Crushing Surfaces of Positive Genus
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866915636811137024 |
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| author | Burton, Benjamin A. de Paiva, Thiago He, Alexander Hui, Connie On Yu |
| author_facet | Burton, Benjamin A. de Paiva, Thiago He, Alexander Hui, Connie On Yu |
| contents | The operation of crushing a normal surface has proven to be a powerful tool in computational $3$-manifold topology, with applications both to triangulation complexity and to algorithms. The main difficulty with crushing is that it can drastically change the topology of a triangulation, so applications to date have been limited to relatively simple surfaces: $2$-spheres, discs, annuli, and closed boundary-parallel surfaces. We give the first detailed analysis of the topological effects of crushing closed essential surfaces of positive genus. To showcase the utility of this new analysis, we use it to prove some results about how triangulation complexity interacts with JSJ decompositions and satellite knots; although similar applications can also be obtained using techniques of Matveev, our approach has the advantage that it avoids the machinery of almost simple spines and handle decompositions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_11523 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Crushing Surfaces of Positive Genus Burton, Benjamin A. de Paiva, Thiago He, Alexander Hui, Connie On Yu Geometric Topology 57K30, 57Q15 The operation of crushing a normal surface has proven to be a powerful tool in computational $3$-manifold topology, with applications both to triangulation complexity and to algorithms. The main difficulty with crushing is that it can drastically change the topology of a triangulation, so applications to date have been limited to relatively simple surfaces: $2$-spheres, discs, annuli, and closed boundary-parallel surfaces. We give the first detailed analysis of the topological effects of crushing closed essential surfaces of positive genus. To showcase the utility of this new analysis, we use it to prove some results about how triangulation complexity interacts with JSJ decompositions and satellite knots; although similar applications can also be obtained using techniques of Matveev, our approach has the advantage that it avoids the machinery of almost simple spines and handle decompositions. |
| title | Crushing Surfaces of Positive Genus |
| topic | Geometric Topology 57K30, 57Q15 |
| url | https://arxiv.org/abs/2403.11523 |