In the 1980s, Neumann and Zagier introduced a symplectic vector space associated to an ideal triangulation of a cusped 3-manifold, such as a knot complement. We give a geometric interpretation for this symplectic structure in terms of the topology of the 3-manifold, via intersections of certain curves on a Heegaard surface. We also give an algorithm to construct curves forming a symplectic basis.
Talk at Knots in Washington 49.75
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.
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.
General tips for studying mathematics
I don’t know that I would have anything to say that’s not a platitude, but here are some thoughts.
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.
One line Euler line
A fun fact from Euclidean geometry that I thought was a wonderful enough gem to share. It’s standard, but it’s nowhere near any curriculum. I’ll try not to get too snarky about the curriculum.
I got problems – congruence problems
On 7 December 2020 I gave a (virtual) lecture at the Australian Mathematical Olympiad Committee’s School of Excellence on congruences.
From Here to Hensel
Here’s a nice maths problem, which I thought it would be fun to discuss. The question doesn’t involve any advanced concepts, but it leads on to a very nice result called Hensel’s lemma.