# Journées de topologie quantique

The “journées de topologie quantique” (quantum topology days) are a regular event aiming at bringing together, in a somewhat informal setting, people thinking about quantum topology in a broad sense. As the name suggest this is a one day event, with three talks and lots of time for discussions, taking place either in Paris or in Dijon.

## Next session: 13 November 2023

### Location: Dijon

Precise location and schedule tba

### Program

- ?h -
**Emmanuel Graff**:*The Homotopy Braid Group is Torsion-Free*

V. Lin, in the 'Kourkova notebook', questions the existence of a non-trivial epimorphism of the braid group onto a non-abelian torsion-free group. The homotopy braid group studied by Goldsmith in 1974 an known to be nilpotent, naturally appears as a potential candidate. In 2001, Humphries showed that this homotopy braid group is torsion-free for less than 6 strands. In this presentation, we will see a new approach based on the broader concept of welded braids, as well as algebraic techniques. This will allow us to demonstrate that the homotopy braid group is torsion-free for any number of strands.

## Past events

### 21 November 2022 (Paris)

**Pedro Vaz**:*De la catégorification des modules de Verma aux homologies d'entrelacs*## Abstract

Dans cet exposé j'expliquerai la catégorification des modules de Verma et leur utilisation dans la construction d'invariants d’entrelacs dans l'espace de dimension 3 et dans le tore solide. Les résultats présentés sont issus de collaborations avec G. Naisse et A. Lacabanne.

**Cristina Palmer-Anghel**:*A globalisation of the Jones and Alexander polynomials from configurations on arcs and ovals in the punctured disc*## Abstract

The Jones and Alexander polynomials are two important knot invariants and our aim is to see them from a topological model given by a graded intersection in a configuration space. Bigelow and Lawrence showed a topological model for the Jones polynomial, using arcs and figure eights in the punctured disc. On the other hand, the Alexander polynomial can be obtained from intersections between ovals and arcs. We present a common topological viewpoint which sees both invariants, based on ovals and arcs in the punctured disc. The model is constructed from a graded intersection between two explicit Lagrangians in a configuration space. It is a polynomial in two variables, recovering the Jones and Alexander polynomials through specialisations of coefficients. Then, we prove that the intersection before specialisation is (up to a quotient) an invariant which globalises these two invariants, given by an explicit interpolation between the Jones polynomial and Alexander polynomial. We also show how to obtain the quantum generalisation, coloured Jones and coloured Alexander polynomials, from a graded intersection between two Lagrangians in a symmetric power of a surface.

**Julien Korinman**:*Représentation d'algèbres d'écheveaux*## Abstract

Le but de cet exposé est de présenter des progrès récents vers la classification des représentations des algèbres d'écheveaux aux racines de l'unité. Les représentations indécomposables de ces algèbres forment (conjecturalement) les briques de bases d'objets algébriques appelés TQFT qui contiennent des invariants de nœuds et de 3-variétés et des représentations des groupes modulaires de surfaces. Je définirai d'abord les algèbres d'écheveaux, les relierait aux variétés de caractères et exposerai ensuite des méthodes générales de théorie des représentations (domaine Azumaya, théorie des ordres de Poisson), qui permettent de relier la théorie de représentations des algèbres d'écheveaux à la géométrie de Poisson des variétés de caractères. Au final, on obtiendra une description 'presque' complète des représentations à poids des algèbres d'écheveaux. C'est un travail en collaboration avec H.Karuo.

### 13 March 2023 (Dijon)

**Rinat Kashaev**:*Generalized TQFT's from local fields*## Abstract

Based on the theory of quantum dilogarithms over locally compact Abelian groups, I will talk about a particular example of a quantum dilogarithm associated with a local field $F$ which leads to a generalized 3d TQFT based on the combinatorial input of ordered $\Delta$-complexes. The associated invariants of 3-manifolds are expected to be specific counting invariants of representations of $\pi_1$ into the group $PSL_2(F)$. This is an ongoing project in collaboration with Stavros Garoufalidis.

**Eilind Karlsson**:*Deformation quantisation and skein categories*## Abstract

I will start by reviewing deformation quantisation of algebras, and explain how we in a similar spirit can define deformation quantisation of categories. The motivation is to understand how deformation quantisation interacts with categorical factorization homology, or more explicitly: how deformation quantisation interacts with “gluing” local observables to obtain global observables. One important and well-known example of factorization homology is given by skein categories, which I will briefly introduce. We generalise the theory of skein categories to fit into the deformation quantisation-setting, and use it as a running example. This is based on joint work (in progress) with Corina Keller, Lukas Müller and Jan Pulmann.

**Rhea Palak Bakshi**:*Skein modules, torsion, and framing changes of links*## Abstract

Skein modules are invariants of 3-manifolds which were introduced by Józef H. Przytycki (and independently by Vladimir Tuarev) in 1987 as generalisations of the Jones, HOMFLYPT, and Kauffman bracket polynomial link invariants in the 3-sphere to arbitrary 3-manifolds. Over time, skein modules have evolved into one of the most important objects in knot theory and quantum topology, having strong ties with many fields of mathematics such as algebraic geometry, hyperbolic geometry, and the Witten-Reshetikhin-Turaev 3-manifold invariants, to name a few. One avenue in the study of skein modules is determining whether they reflect the geometry or topology of the manifold, for example, whether the module detects the presence of incompressible or non-separating surfaces in the manifold. Interestingly enough this presence manifests itself in the form of torsion in the skein module. In this talk we will discuss various skein modules which detect the presence of non-separating surfaces. We will focus on the framing skein module and show that it detects the presence of non-separating 2-spheres in a 3-manifold by way of torsion.

### 5 June 2023 (Paris)

**Emmanuel Wagner**:*Link homologies and spectral sequences*## Abstract

After giving an overview of various link homologies and spectral sequences relating them, I will concentrate on a spectral sequence allowing for comparison of the triply graded link homology of Khovanov and Rozansky and the Knot Floer homology. The technics used involves in particular quantum Hochschild homology, foams and webs. This is a joint work with A. Beliakova, K. Putyra and L-H. Robert.

**Patrick Kinnear**:*An invertible non-semisimple TQFT varying over the character stack*## Abstract

Lusztig's quantum group at a root of unity comes equipped with a quantum Frobenius map to the classical universal enveloping algebra, allowing us to pull back classical representations to representations of the quantum group. In fact, these pulled-back representations braid trivially with all others in $\operatorname{Rep}_q(G)$, and so they lie in the Müger centre and the functor $\operatorname{Rep}(G) \rightarrow \operatorname{Rep}_q(G)$ makes $\operatorname{Rep}_q(G)$ into a module category for $\operatorname{Rep}(G)$. This makes $\operatorname{Rep}_q(G)$ into an endomorphism of $\operatorname{Rep}(G)$ in the Morita theory SymTens of symmetric tensor categories. In this talk, we will discuss invertibility of this endomorphism and its implications in TQFT.

From the perspective of the Cobordism Hypothesis, this data defines an invertible non-semisimple TQFT varying over the character stack (the moduli space of $G$-local systems) which extends the Crane-Yetter theory. To 3-manifolds this theory assigns a line bundle on the character stack with the fiber over the trivial local system being the non-semisimple Crane-Yetter theory. To surfaces the theory assigns an invertible sheaf of categories on the character stack given by the skein category of the surface, yielding a categorified version of the fact that the skein algebra defines a sheaf on the character variety which is Azumaya over an open dense locus (the locus being precisely determined recently by Karuo-Korinman). Building on recent work of Haïoun on dualizability, we expect to be able to define a non-semisimple $G$-relative version of Witten-Reshetikhin-Turaev theory similar to the contemporary understanding of WRT as relative to CY.

**Nikita Markarian**:*Perturbative Chern-Simons invariants and factorization homology*## Abstract

I will discuss my approach to perturbative Chern-Simons invariants of 3-manifolds and knots, and their generalizations, such as Rozansky-Witten invariants, via the factorization complex of Weyl $e_n$-algebras. I will give a review of some results and will describe possible further applications of them.

## Organizers

- Adrien Brochier
- Gwénaël Massuyeau
- Emmanuel Wagner
- Lukas Woike