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 suggests this is a one day event, with three talks and lots of time for discussions, taking place either in Paris or in Dijon.

Past events

6 Mar 2026 (Dijon)

Julia Bierent: The skein partition function
Abstract
The skein partition function is a generating function for the dimension of the $\operatorname{GL}_N$-skein module of a manifold for all $N$. We will present it for the genus one mapping torus, depending on the mapping class group element $\gamma\in \operatorname{SL}_2(\mathbb{Z})$. This computation will lead us to some evidence for a conjectural deep relation between cohomological Donaldson Thomas-invariants and skein modules.

Thiago Brevidelli: Lower bounds for faithful linear representations of subgroups of the mapping class group
Abstract
The mapping class group of a surface is a fundamental group in low-dimensional topology. Yet, basic questions about its linear representations remain unanswered. For example, the question of linearity remains wide open for genus $g \ge 3$ surfaces. In this talk, we will review the recent progress on the classification of low-dimensional representations of the mapping class group. We will conclude with a result of mine, stating that the (pure) mapping class group of a genus $g \ge 7$ surface has no faithful linear representations in dimension $\le 4g - 4$.

Ben-Michael Kohli: Alexander and Jones-type properties of the Links-Gould invariant of links
Abstract
The Links-Gould polynomial is a two-variable link invariant derived from the quantum supergroup $U_q(\mathfrak{sl}(2|1))$. As such, LG has a "hybrid" flavor between the Jones and Alexander polynomials. Thus one can wonder what properties of the Jones polynomial and what properties of the Alexander polynomial are inherited by LG. Several recent results have shown that the Links-Gould invariant shares some of the Alexander polynomial's most geometric features—a surprising fact for a quantum invariant. In this talk, we will review these properties, then focus on proving that the Links-Gould polynomial and its colored counterparts provide lower bounds for the 3-genus of a knot that are quite precise. In particular, the 2-colored Links-Gould polynomial provides a genus bound that is sharp for all 58 million prime knots with up to 18 crossings.

6 Oct 2025 (Paris)

Nivedita: 'Categorifying' Von Neumann Algebras and their Modules
Abstract
The need to categorify Hilbert spaces and von Neumann algebras arose from looking for higher categorical targets for unitary topological and conformal field theories in the functorial description. We will describe models for 2-Hilb (arXiv:2411.01678) and 3-Hilb (work in progress). We work with Conformal Nets and give construction of fully extended functorial chiral CFTs that land in these target categories.

Jules Martel: Quantum groups in nature
Abstract
Where nature is twisted homologies of configuration spaces of points decorated by simple roots, and quantum groups are symmetrizable quantum Kac-Moody algebras. They constitute algebraic structures extensively used in constructions of TQFTs, quantum invariants etc, and we give them an initial topological flavor. (joint with S. Bigelow)

Sebastian Baader: Sparse curve systems on surfaces
Abstract
A system of simple closed curves on a surface of genus $g$ is said to be sparse if their average pairwise intersection number does not exceed one. We show that the maximal size of a sparse curve systems grows roughly like a function of type $c^{\sqrt{g}}$, with $c$ between 2 and 81938. This work is in collaboration with Jasmin Joerg and Danica Kosanovic.

26 May 2025 (Dijon)

Federica Gavazzi: On the Topology of Virtual Artin Groups
Abstract
Virtual Artin groups were introduced a few years ago by Bellingeri, Paris, and Thiel, with the aim of generalizing the well-studied structure of virtual braid groups to the broader context of Artin groups. These fascinating objects possess a rich algebraic structure that encompasses both Coxeter groups and classical Artin groups. In this talk, we will explore the topology of virtual Artin groups, focusing in particular on the construction of cell complexes that serve as promising candidates for classifying spaces of certain remarkable subgroups. We will also highlight a connection between the topological properties of these spaces and a well-known problem in the theory of Artin groups: the $K(\pi,1)$ conjecture.

Jackson Van Dyke: The moduli space of projective 3d TQFTs
Abstract
Three-dimensional Witten-Reshetikhin-Turaev TQFTs are a source of rich topological invariants. I will discuss a certain moduli space of such theories, built from the classifying spaces of higher groups of automorphisms of ribbon categories. Various symmetries of these theories, and their anomalies, are encoded by the homotopy theory of this moduli space. The emphasis will be on studying the impact of gauging on the associated mapping class group representations.

Martin Palmer: The homology of Thompson-like groups via algebraic K-theory and étale groupoids
Abstract
The Higman-Thompson groups $V_d$ are groups of automorphisms of Cantor sets respecting a certain tree-like structure. They, as well as many families of 'Thompson-like' groups building on these prototypes, are important objects of study in group theory, providing examples of groups with unexpected properties. Recently, the integral homology of $V_d$ has been calculated by M. Szymik and N. Wahl using homological stability and algebraic K-theory techniques: in particular $V_2$ is integrally acyclic and each $V_d$ is rationally acyclic. Even more recently, these acyclicity results have been re-proved by X. Li using étale groupoids. I will explain the ideas of these two approaches and how each of them may be extended to calculate the homology of other families of Thompson-like groups. As an application, we will see the existence of acyclic groups of intermediate finiteness types. This represents joint work in progress with Xiaolei Wu.

10 Feb 2025 (Dijon)

Juan-Ramón Gómez-García: Turaev's coproduct and parabolic restriction
Abstract
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 fundamental representation from $\operatorname{GL}_{m+n}$ to $\operatorname{GL}_m \times \operatorname{GL}_n$. The restriction functor is, however, not braided hence it cannot be extended to the skein category of an arbitrary surface. So there was a priori no reason for Turaev's coproduct to be well-defined. In this talk, I will explain how to construct a universal version of parabolic restriction on framed surfaces, using skein theory with defects. Precisely, parabolic restriction yields a morphism (bimodule) between the $\operatorname{GL}_t$-skein category and the $(\operatorname{GL}_t \times \operatorname{GL}_t)$-skein category of $S$. This construction depends on the choice of the framing, is compatible with gluing of surfaces and recovers Turaev's coproduct when applied to links, justifying why this is well-defined.

Pierre Godfard: Hodge structures on conformal blocks
Abstract
Modular functors are families of finite-dimensional representations of Mapping Class Groups of surfaces, with strong compatibility conditions. As Mapping Class Groups of surfaces are isomorphic to fundamental groups of moduli spaces of curves, modular functors can alternatively be seen as families of vector bundles with flat connection on (twisted) moduli spaces of curves, with strong compatibility conditions with respect to some natural maps between the moduli spaces. In this talk, we will discuss Hodge structures on such flat bundles. If these flat bundles where rigid, a result of Simpson in non-Abelian Hodge theory would imply that they support Hodge structures. However, that is not the case in general. We will explain how a different kind of rigidity for modular functors can be used to prove an existence and uniqueness result for such Hodge structures. Finally, we will discuss the computation of Hodge numbers for $\mathfrak{sl}_2$ modular functors (of odd level) and how these numbers are part of a cohomological field theory (CohFT).

Laura Marino: Homologies symétriques d'entrelacs: le cas $\mathfrak{gl}(1)$
Abstract
Introduites par Robert et Wagner, les homologies symétriques $\mathfrak{gl}(n)$ sont des invariants de noeuds qui fournissent une catégorification d'une certaine famille de polynômes de Reshetikhin-Turaev. Ces homologies, en général difficiles à calculer, admettent une définition nettement plus simple quand $n=1$, et dans ce cas on peut décrire une base pour les espaces qui constituent leur complexe de chaines. Entre autre, cela permet de construire un algorithme qui calcule cette homologie. Je présenterai la construction de l'homologie symétrique $\mathfrak{gl}(1)$ et énoncerai quelques résultats obtenus en la calculant sur les noeuds avec au plus dix croisements.

25 Nov 2024 (Paris)

Danica Kosanović: Configuration spaces of manifolds with vanishing diagonal class
Abstract
In this talk I will present joint work with Pedro Boavida de Brito and Geoffroy Horel, in which we give an explicit geometric description of homology groups of configuration spaces of any manifold whose diagonal class vanishes.

Quentin Faes: Examples of invariants of 4-dimensional 2-handlebodies up to 2-equivalence
Abstract
It is not known whether two diffeomorphic 4-dimensional 2-handlebodies are always equivalent through handle moves of index smaller than 2 (i.e. 2-equivalent). A result of Beliakova, Bobtcheva, de Renzi and Piergallini implies that any so-called BP Hopf algebra in a braided monoidal category yields an invariant of 4-dimensional 2-handlebodies up to 2-equivalence. Unfortunately, it is hard to exhibit such algebras, since we want the category to be non semi-simple and non-factorizable. After explaining all of this in detail, I will list a series of such examples; we proved some of them degenerate to the boundary of the surface, and some of them to the spine of the 2-handlebodies, but there is still some hope for some example: a promising one is an algebra of dimension 4096 inspired by $u_q(\mathfrak{sl}_2)$ at 8-th root of unity. (Joint work in progress with A. Beliakova and M. Manko)

Roland van der Veen: Quantum invariants of fibred links
Abstract
The Alexander polynomial is the oldest and most fundamental quantum knot invariant. To deepen our understanding of other quantum invariants we ask whether they share or generalize a property that the Alexander polynomial has. For example the Alexander of a fibred knot is monic of degree twice the genus of the fiber surface. Our main result is a generalization of this property for both the ADO-invariants and the Links-Gould invariant. This time the top coefficient relates to the Hopf invariant of the plane field in the three-sphere associated to the fiber surface. This is joint work with Daniel López Neumann, https://arxiv.org/abs/2407.15561 and a handout for this talk will be available at http://www.rolandvdv.nl/Talks/jdtq24

18 Jun 2024 (Paris)

Matthieu Faitg: Quantum moduli algebras and skein algebras
Abstract
Quantum moduli algebras are a quantization of character varieties of surfaces which have been introduced by Alekseev-Grosse-Schomerus and Buffenoir-Roche in the 90's. I will explain the definition of these algebras (based on quantum groups) and give structure results, namely they are finitely generated, Noetherian and do not have zero divisors. Then I will relate quantum moduli algebras with skein algebras, which is one the motivation for these results. I will finish by a few words on the specialization of quantum moduli algebras at roots of unity. Joint work with S. Baseilhac and P. Roche.

Benjamin Haïoun: Non-semisimple skein (3+1)-TQFTs
Abstract
I will recall some tools from non-semisimple skein theory. The key players are modfied traces, which were introduced to obtain invariants of links with a projective strand for which the usual Reshetikhin–Turaev theory gives trivial invariants. I will explain how they give the first step into the definition of a skein-theoretic (3+1) Topological Quantum Field Theory, and give some elementary properties of this TQFT. This is joint work with Francesco Costantino, Nathan Geer and Bertrand Patureau-Mirand.

Anna Beliakova: On algebraisation of low-dimensional Topology
Abstract
The categories of $n$-cobordisms are among the most studied objects in low dimensional topology. For $n=2$ we know that $2\operatorname{Cob}$ is a monoidal category freely generated by its commutative Frobenius algebra object: the circle. This result also classifies all TQFT functors on $2\operatorname{Cob}$. In this talk I will present similar classification results for special categories of 3- and 4-cobordisms. Here Frobenius algebra is replaced by a so-called Bobtcheva-Piergallini Hopf algebra. The results are obtained in collaboration with Marco De Renzi, Ivelina Bobtcheva and Riccardo Piergallini.

13 Nov 2023 (Dijon)

Ramanujan Santharoubane: On the kernel of $SO(3)$-Witten-Reshetikhin-Turaev quantum representations
Abstract
Given a surface $S$, $SO(3)$-Witten-Reshetikhin-Turaev TQFT's provide a sequence, indexed by odd integers, of complex finite dimensional representations of the mapping class group of $S$ called quantum representations. One major and difficult problem is to find the kernels of these quantum representations. For prime index, Gilmer and Masbaum proved that the representation preserves a lattice over the ring of integers of a cyclotomic field. This allows to approximate each quantum representation by homomorphisms from the mapping class group in to finite groups. In this talk, we will see that certain of these approximations can be completely understood and are related to the Johnson filtration of the mapping class group. As a consequence we can find some non trivial 'upper bound' for the kernels of quantum representations. This is a joint work with Renaud Detcherry.

Anna-Katharina Hirmer: Generalised Kitaev models from Hopf monoids: topological invariance and examples
Abstract
Quantum double models were introduced by Kitaev to obtain a realistic model for a topological quantum computer. They are based on a directed ribbon graph and a finite-dimensional semisimple Hopf algebra. The ground state of these models is a topological invariant of a surface, i.e. only depends on the homeomorphism class of the oriented surface but not the ribbon graph. Meusburger and Voß generalised part of the construction from Hopf algebras to pivotal Hopf monoids in symmetric monoidal categories. We explain the construction of the ground state for involutive Hopf monoids and show that it is topological invariant. We explicitly describe this construction for Hopf monoids in Set, Top, Cat and SSet.

Emmanuel Graff: The Homotopy Braid Group is Torsion-Free
Abstract
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.

13 Jun 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 Jun 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.

21 Nov 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.

Organizers

Adrien Brochier
Renaud Detcherry
Emmanuel Wagner
Lukas Woike