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A-polynomials, Ptolemy
equations and Dehn filling
Joshua A Howie, Daniel V Mathews and Jessica S Purcell |
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Algebraic & Geometric Topology 25 (2025) 1265–1320
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Abstract |
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The A-polynomial encodes hyperbolic geometric information on knots and related manifolds. Historically, it has been difficult to compute, and particularly difficult to determine A-polynomials of infinite families of knots. Here, we compute A-polynomials by starting with a triangulation of a manifold, then using symplectic properties of the Neumann–Zagier matrix encoding the gluings to change the basis of the computation. The result is a simplification of the defining equations. We apply this method to families of manifolds obtained by Dehn filling, and show that the defining equations of their A-polynomials are Ptolemy equations which, up to signs, are equations between cluster variables in the cluster algebra of the cusp torus. |
Keywords
A-polynomial, gluing equations, triangulations, Ptolemy
equations, Dehn filling, layered solid torus, Farey complex
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Mathematical Subject Classification
Primary: 57K14, 57K31, 57K32
Secondary: 57K10
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References |
Publication
Received: 10 August 2021
Revised: 6 February 2024
Accepted: 25 March 2024
Published: 20 June 2025
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Authors |
| Joshua A Howie | |
| School of Mathematics Monash University Clayton VIC Australia |
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| Daniel V Mathews | |
| School of Mathematics Monash University Clayton VIC Australia |
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| Jessica S Purcell | |
| School of Mathematics Monash University Clayton VIC 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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