On 15 June 2020 I gave a talk in the topology seminar at the University of Melbourne.
Title: A-polynomials, Ptolemy varieties, and Dehn filling, Melbourne June 2020
Abstract: The A-polynomial is a 2-variable knot polynomial which encodes topological and hyperbolic geometric information about a knot complement. In recent times it has been shown that the A-polynomial can be calculated from Ptolemy equations. Historically reaching back to antiquity, Ptolemy equations arise all across mathematics, often alongside cluster algebras.
In recent work with Howie and Purcell, we showed how to compute A-polynomials by starting with a triangulation of a manifold, then using symplectic properties of the Neumann-Zagier matrix to change basis, eventually arriving at a set of Ptolemy equations. This work refines methods of Dimofte, and the result is similar to certain varieties studied by Zickert and others. Applying this method to families of manifolds obtained by Dehn filling, we find relations between their A-polynomials and the cluster algebra of the cusp torus.20-06_unimelb_talk_web