A-polynomials, Ptolemy equations and Dehn filling
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arXiv
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| Format: | Preprint |
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2020
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| author | Howie, Joshua A. Mathews, Daniel V. Purcell, Jessica S. |
| author_facet | Howie, Joshua A. Mathews, Daniel V. Purcell, Jessica S. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2002_10356 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | A-polynomials, Ptolemy equations and Dehn filling Howie, Joshua A. Mathews, Daniel V. Purcell, Jessica S. Geometric Topology Symplectic Geometry 57K31, 57K32, 57K14 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. |
| title | A-polynomials, Ptolemy equations and Dehn filling |
| topic | Geometric Topology Symplectic Geometry 57K31, 57K32, 57K14 |
| url | https://arxiv.org/abs/2002.10356 |