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Complexity of
$3$-manifolds obtained by Dehn filling
William Jaco, Joachim Hyam Rubinstein, Jonathan Spreer and Stephan Tillmann |
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Algebraic & Geometric Topology 25 (2025) 301–327
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Abstract |
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Let be a compact -manifold with boundary a single torus. We present upper and lower complexity bounds for closed -manifolds obtained as even Dehn fillings of . As an application, we characterise some infinite families of even Dehn fillings of for which our method determines the complexity of their members up to an additive constant. The constant only depends on the size of a chosen triangulation of , and the isotopy class of its boundary. We then show that, given a triangulation of with -triangle torus boundary, there exist infinite families of even Dehn fillings of for which we can determine the complexity of the filled manifolds with a gap between upper and lower bounds of at most . This result is bootstrapped to obtain the gap as a function of the size of an ideal triangulation of the interior of , 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 -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 , and the trefoil. |
Keywords
3-manifold, minimal triangulation, layered triangulation,
complexity, Farey tessellation, slope norm
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Mathematical Subject Classification
Primary: 57K10, 57K31, 57K32, 57Q15
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References |
Publication
Received: 28 April 2023
Revised: 6 August 2023
Accepted: 17 September 2023
Published: 5 March 2025
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Authors |
| William Jaco | |
| Department of Mathematics Oklahoma State University Stillwater, OK United States |
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| Joachim Hyam Rubinstein | |
| School of Mathematics and
Statistics The University of Melbourne Melbourne VIC Australia |
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| Jonathan Spreer | |
| School of Mathematics and
Statistics The University of Sydney Sydney NSW Australia |
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| Stephan Tillmann | |
| School of Mathematics and
Statistics The University of Sydney Sydney NSW Australia |
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| © 2025 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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