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.