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Crushing surfaces of
positive genus
Benjamin A Burton, Thiago de Paiva, Alexander He and Connie On Yu Hui |
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Algebraic & Geometric Topology 25 (2025) 4949–5012
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DOI: 10.2140/agt.2025.25.4949
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Abstract |
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The operation of crushing a normal surface has proven to be a powerful tool in computational -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: -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. |
Keywords
3-manifold triangulations, normal surfaces, crushing
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Mathematical Subject Classification
Primary: 57K30, 57Q15
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References |
Publication
Received: 10 April 2024
Revised: 9 October 2024
Accepted: 27 November 2024
Published: 20 November 2025
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Authors |
| Benjamin A Burton | |
| School of Mathematics and
Physics University of Queensland Brisbane, QLD Australia |
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| https://people.smp.uq.edu.au/BenjaminBurton/ | |
| Thiago de Paiva | |
| School of Mathematics Monash University Melbourne, VIC Australia |
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| https://sites.google.com/view/thiago-de-paiva | |
| Alexander He | |
| Department of Mathematics Oklahoma State University Stillwater, OK United States |
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| https://sites.google.com/view/alex-he | |
| Connie On Yu Hui | |
| School of Mathematics Monash University Melbourne, VIC Australia |
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| https://sites.google.com/view/oyhui | |
| © 2025 The Author(s), under exclusive license to MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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