Brussels-London XXIV

Low dimensional topology

University College London, 18th September 2025

The event will take place in the Mathematics Department of University College London. Talks are in room 500, and coffee and lunch is in room 502, both on the 5th floor of the mathematics department, 25 Gordon Street. Click here for travel instructions. Click here for a Google Maps link.

The event will begin with coffee from 10am in Room 500. The talks are in Room 502, at 11.30, 2.00 and 3.30, with lunch from 12.30 to 2.00, again in Room 500. Lunch is free for all registered attendees.

To attend this event, you should register in advance.

Cristina Anghel. Universal Link Invariants Via Configuration Spaces.

Coloured Jones and Alexander polynomials are quantum invariants originating in representation theory. They are building blocks for 3 -manifold quantum invariants, and their geometric information is an important open problem in quantum topology.

We will describe them from a unified topological viewpoint. For a fixed level N, we define new link invariants: Nth Unified Jones invariant Nth Unified Alexander invariant. They globalise topologically all coloured Jones and all ADO link polynomials with (multi-)colours bounded by N. This shows that all coloured Jonse and coloured Alexander polynomials at bounded (multi-)level are encoded by the same Lagrangian intersections in a fixed configuration space.

Then, asymptotically, we define geometrically a universal ADO link invariant and universal Jones link invariant. The question of providing a universal invariant recovering all ADO link polynotmials was an open problem. A parallel question about semi-simple knot invariants is the subject of Habiro's famous universal invariant. Our universal Jones invariant recovers all coloured Jones polynomials, providing a new semi-simple universal link invariant. The first non semi-simple universal link invariant that we construct unifies geometrically all ADO link invariants.

Andras Juhasz. Reinforcement learning and knot theory.

Several hard problems in knot theory can be formulated as single-player games, including computing the unknotting number and the 4-ball genus, making these amenable to reinforcement learning algorithms. I will discuss some of the latest results and remaining challenges.

Alice Merz. Equivariant sliceness and the extension of involutions on lens spaces to rational homology balls.

A knot in the three-sphere is called slice if it bounds a smoothly embedded disk in the four-ball. Deciding which knots are slice is a central question in low-dimensional topology, closely tied to fundamental conjectures such as the smooth 4-dimensional Poincaré conjecture and the Slice–Ribbon conjecture, and more in general to the study of 4-manifold topology.

One obstruction to sliceness arises from analyzing the double branched cover of the knot and determining whether the branching involution extends to a rational homology 4-ball.

In this talk, I will discuss the problem of extending involutions to rational homology balls and explain its relation to the question of whether a knot is equivariantly slice. I will then focus on the case of involutions on lens spaces and revisit the equivariant sliceness of two-bridge knots, giving a new proof of a theorem of Di Prisa–Framba. This is a joint work with Antony Fung and Lisa Lokteva.