On 22 April 2022 I gave a (virtual) talk at the 49.75th (!) Knots in Washington conference.
A Symplectic Basis for 3-manifold Triangulations, AustMS 2021
On 8 December 2021 I gave a (virtual) talk in the Topology session of the 2021 meeting of the Australian Mathematical Society.
Five-minute surrealist antiwar exposition of topological data analysis
On Remembrance Day 2021 (11 November) I have a talk at a session of Lightning Talks at a session on “Mathematics for Data Analysis, AI & Machine Learning” organised by the Monash Data Futures Institute. This was a “Lightning Talk”
An Arbitrary-Order Discrete de Rham Complex on Polyhedral Meshes
In this article I am acknowledged for providing an explicit basis for a space, which is used in a software implementation.
Summarise your maths research in one slide, Dan
As part of an upcoming workshop participants were asked to introduce themselves with a one-page slide. I took it as an extreme form of concision: summarise your maths research in one slide, Dan.
A-Polynomials of fillings of the Whitehead sister
Knots obtained by Dehn filling the Whitehead sister include some of the smallest volume twisted torus knots. Here, using results on A-polynomials of Dehn fillings, we give formulas to compute the A-polynomials of these knots. Our methods also apply to more general Dehn fillings of the Whitehead sister.
Ptolemy vs Thurston in Hyperbolic Geometry and Topology, AustMS 2020
On 9 December 2020 I gave a (virtual) talk in the Topology session of the 2020 meeting of the Australian Mathematical Society.
A-polynomials, Ptolemy varieties, and Dehn filling, Melbourne June 2020
On 15 June 2020 I gave a talk in the topology seminar at the University of Melbourne, entitled “A-polynomials, Ptolemy varieties, and Dehn filling.”
Monash topology talk on Circle packings, Lagrangian Grassmannians, and Scattering Diagrams, April 2020
On 1 April 2020 I gave a talk in the Monash topology seminar, entitled “Circle packings, Lagrangian Grassmannians, and scattering diagrams”.
A-polynomials, Ptolemy varieties and Dehn filling
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 show how to compute A-polynomials by starting with a triangulation of a manifold, similar to Champanerkar, then using symplectic properties of the Neumann-Zagier matrix encoding the gluings to change the basis of the computation. The result is a simplicifation of the defining equations. Our methods are a refined version of Dimofte’s symplectic reduction, and we conjecture that the result is equivalent to equations arising from the enhanced Ptolemy variety of Zickert, which would connect these different approaches to the A-polynomial.
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. Thus the change in A-polynomial under Dehn filling is given by an explicit twisted cluster algebra. We compute the equations for Dehn fillings of the Whitehead link.