By Alexander's theorem, every link in the 3-sphere can be represented as the closure of a braid. Lorenz links and twisted torus links are two families that have been extensively studied and are well-described in terms of braids. In this talk, we will present a natural generalization of Lorenz links and twisted torus links that produces all links in the 3-sphere. This provides a simpler braid description for all links in the 3-sphere.

I will review two results pertaining to 3-dimensional Reshetikhin–Turaev TQFTs, defined from modular tensor categories M. These theories were not constructed as “fully local” TQFTs (in the framework of Lurie’s Cobordism Hypothesis): no algebraic structures were assigned to points. (The obstruction was the Witt class of M.) Kevin Walker solved the locality problem in the setting of anomalous theories. A ‘no-go’ theorem (joint with Dan Freed) showed that, if localized as linear theories, these RT theories did not admit local topological boundary conditions, and could therefore not be generated from a point by this method. (The group-like case had been addressed by Kapustin and Saulina.) In recent work with Freed and Claudia Scheimbauer, we displayed a fully local realization of these theories, by objects in a target 3-category which enlarges that of fusion categories. This allowed us to settle some conjectures relating orientations and spherical structures.

Inspired by Jaeger’s composition formula for the HOMFLY polynomial, Turaev defined a coproduct on the HOMFLY skein algebra of a framed surface S, turning it into a bialgebra. Jaeger’s formula can be viewed as a universal version of the restriction of the defining representation from $\operatorname{GL}_{m+n}$ to $\operatorname{GL}_m × \operatorname{GL}_n$. The restriction functor, however, is not braided, and therefore there is a priori no reason for the induced linear map between the corresponding skein algebras to be multiplicative. In this talk, I will address this problem using defect skein theory and the formalism of parabolic restriction.

In the first part of the talk, I will introduce skein theory for 3-manifolds with both surface and line defects. Local relations near the defects are produced from the algebraic data of a central algebra (codimension 1) and a centred bimodule (codimension 2). Examples of such structures are provided by the formalism of parabolic restriction. In the second part of the talk, I will explain how to construct a universal version of this formalism. Finally, we will see how Turaev’s coproduct extends to the entire skein category using the previous constructions.

Trisections give a diagrammatic description of smooth 4-manifolds, similar to Heegaard splittings in dimension three. In this talk, I will describe new 4-manifold invariants defined from trisection diagrams using categorical data. The input consists of three spherical fusion categories, a semisimple bimodule category with a bimodule trace, and a pivotal functor into the category of bimodule endofunctors.

The construction works by labelling the trisection diagrams with the categorical data and evaluating them using a diagrammatic calculus for bimodule categories. The details of this procedure ensures that the result is invariant under moves on trisections yielding the same 4-manifold. This construction generalises existing Hopf algebraic trisection invariants due to Chaidez-Cotler-Cui and recovers the Crane-Yetter and Bärenz-Barrett invariants as special cases. I will outline the main ideas of the construction and briefly discuss examples and connections to TQFTs.

Based on the work 2511.19384 with Catherine Meusburger (FAU) and Fiona Torzewska (Bristol).

A well-known folklore theorem classifies 2-dimensional topological quantum field theories (TQFTs) in terms of Frobenius algebras, providing a unifying link between topology, algebra, and physics. In this talk, we explore what happens when the usual cobordism category is replaced by a category of nested cobordisms, in which 2-dimensional surfaces are equipped with embedded 1-dimensional submanifolds. We study symmetric monoidal functors out of this category and the resulting algebraic structures they encode. This talk is based on joint work with R. Hoekzema, L. Murray, N. Pacheco-Tallaj, C. Rovi, and S. Sridhar.