A-polynomials, Ptolemy equations and Dehn filling
Abstract
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.
Cite
@article{arxiv.2002.10356,
title = {A-polynomials, Ptolemy equations and Dehn filling},
author = {Joshua A. Howie and Daniel V. Mathews and Jessica S. Purcell},
journal= {arXiv preprint arXiv:2002.10356},
year = {2025}
}
Comments
46 pages, 18 figures. v4: adjusted orientation of Farey complex, added appendix B with example computations