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The
signature and cusp geometry of hyperbolic knots
Alex Davies, András Juhász, Marc Lackenby and Nenad Tomašev |
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Geometry & Topology 28 (2024) 2313–2343
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Abstract |
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We introduce a new real-valued invariant, called the natural slope of a hyperbolic knot in the –sphere, which is defined in terms of its cusp geometry. We show that twice the knot signature and the natural slope differ by at most a constant times the hyperbolic volume divided by the cube of the injectivity radius. This inequality was discovered using machine learning to detect relationships between various knot invariants. It has applications to Dehn surgery and to –ball genus. We also show a refined version of the inequality, where the upper bound is a linear function of the volume, and the slope is corrected by terms corresponding to short geodesics that link the knot an odd number of times. |
Keywords
knot, hyperbolic, signature, cusp, natural slope, machine
learning
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Mathematical Subject Classification
Primary: 57K10, 57K31, 57K32, 68T07
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References |
Publication
Received: 20 April 2022
Revised: 26 August 2022
Accepted: 8 October 2022
Published: 24 August 2024
Proposed: David Gabai
Seconded: Cameron Gordon, Stavros Garoufalidis
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Authors |
| Alex Davies | |
| DeepMind London United Kingdom |
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| András Juhász | |
| Mathematical Institute University of Oxford Oxford United Kingdom |
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| Marc Lackenby | |
| Mathematical Institute University of Oxford Oxford United Kingdom |
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| Nenad Tomašev | |
| DeepMind London United Kingdom |
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| © 2024 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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