Mini-courses

Jean-Baptiste Meilhan : A combinatorial approach to knotted surfaces in 4-space

Part 1. Surfaces in 4-space and cut-diagrams
Part 2. Concordance invariants of knotted surfaces
Part 3. Link-homotopy invariants of link maps

Mark Powell : Surfaces dans les 4-variétés

Les exposés porteront sur les trois aspects suivants.
1) Existence d'un plongement d'une surface dans une classe d'homotopie.
2) Unicité des plongements à isotopie près.
3) Concordance des surfaces.

Talks

Rhea Palak Bakshi: The Quantum A-polynomial for twist knots and double twist knots

The discovery of the Jones polynomials by Vaughan Jones revolutionised knot theory and started the field of quantum topology. The coloured Jones polynomial is a generalisation of the Jones polynomial and is the Reshetikhin-Turaev invariant associated with the (N+1)-irreducible representation of the quantum group U_q(sl_2). On the other hand, the A-polynomial of a knot, introduced 1994 by Cooper, Culler, Gillett, Long, and Shalen, describes more or less the representation space of the knot group into SL(2,C), and has been fundamental in geometric topology. The AJ conjecture by Garoufalidis relates these two polynomials and was proved for twist knots by Thang Le in 2006. However, no explicit formulas for the two polynomials are computed. In this talk we compute a recurrence formula for the coloured Jones polynomial for twist and double twist knots. We also find a closed formula for the quantum A-polynomial for twist knots and double twist knots.

Léo Bénard: Zeta functions counting triangles

In a joint work with Yann Chaubet, Nguyen Viet Dang and Thomas Schick, we recover various topological invariants, such as (L2)-Betti numbers and linking numbers, as special values of some combinatorial zeta functions.

Younes El Maamoun Benyahia: Exotic 2-spheres and 2-links

Two smoothly embedded surfaces in a 4-manifold are called exotic if they are topologically isotopic but smoothly not. In 1996, Fintushel and Stern constructed the first examples of exotic surfaces in some 4-manifolds. Since then, there has been many constructions of exotic surfaces in other settings, in particular, ones closer to the smooth unknotting conjecture. In this talk, we give a knot surgery construction of an infinite exotic family of nullhomotopic 2-spheres with knot group Z and add that with a few tweaks we can also get infinite exotic families of nullhomotopic 2-links. This is based on a joint work with Bais, Malech and Torres (see also https://arxiv.org/abs/2206.09659).

Filippo Callegaro: Cohomology of quasi-abelianized braid groups

We study the rational cohomology of the quotient of (generalized) braid groups by the commutator subgroup of the pure braid groups. We provide a combinatorial description of it using isomorphism classes of certain families of graphs. Several properties like Poincaré dualities and stabilization appear. This is a joint work with Ivan Marin.

Jacques Darné: Centers of braid groups on surfaces

The computation of centers of braid groups on surfaces has been completed quite recently, with the cases of the Mœbius strip and of the Klein bottle, computer by Guaschi and Pereiro in 2020. This was the last one in a series of results spanning over several decades, beginning with Artin's original result for braids on the plane in 1945, followed by Gillette and Van Buskirk in 1968 for the sphere, Murasagi in 1982 for the projective plane, and Paris and Rolfsen in 1999 for the cylinder, the torus and the generic case. In this talk, I will explain how all these results can be recovered easily by exploiting a beautiful topological argument from Birman's thesis.

Marco De Renzi: Homological construction of quantum representations of mapping class groups

For a connected surface Σ with connected boundary, I will explain how to recover representations of Mod(Σ) arising from the quantum group of sl(2) using twisted homology groups of configuration spaces of Σ. This model sheds new light on these constructions: indeed, it naturally pinpoints integral bases for these actions, while also developing the analogue of the tools that enabled Bigelow to establish linearity of braid groups. This is a joint work with Jules Martel.

Alessio Di Prisa: Every knot is strongly invertible up to S-equivalence

A knot K in the 3-sphere is said to be strongly invertible if there is an orientation-preserving involution of the 3-sphere which leaves K invariant and reverses the orientation on K. In 1983 Livingston proved that there exist knots which are not concordant to their reverses, and in particular which are not concordant to any strongly invertible knots. On the other hand, a knot and its reverse are indistinguishable in the algebraic concordance group, but it was unclear if every algebraic concordance class contains a strongly invertible knot. In fact, if we consider an invariant Seifert surface for a strongly invertible knot, the associated Seifert form has a particular symmetry induced by the strong inversion, and at first sight this could impose some restrictive condition on the Seifert form. In this talk we will prove that this is not the case, showing that every Seifert form (up to S-equivalence) admits this type of symmetry and can be realized explicitly by a strongly invertible knot. As a consequence it follows that every knot is S-equivalent (and in particular algebraically concordant) to a strongly invertible knot. Finally, we deduce some results on the realizability of the classes in the equivariant algebraic concordance group of strongly invertible knots. Joint work with C. Collari, G. Framba and A. Merz.

Vincent Florens: Topological invariants of line arrangements

A line arrangement is a finite set of lines in the complex projective plane CP². They are particular cases of plane algebraic curves where the components have all degree 1. We are interested in the influence of the combinatorial data of an arrangement on its embedded topology in CP².
The boundary manifold of an arrangement is the common boundary of a regular neighborhood of the arrangement and its exterior. This is a graph three-manifold, in the sense of Waldhausen, whose topology is completly determined by the combinatorics. We use the inclusion map of the boundary manifold in the exterior to construct a new topological invariant of arrangements.
This is a joint work with E.Artal and A.Rodau.

Noémie Legout: Obstruction to Lagrangian cobordisms between Legendrian links

Given two Legendrian links in a contact manifold, one can ask if there exists an embedded Lagrangian cobordism from one to the other. This is a very studied question and a lot of obstructions to the existence of Lagrangian cobordisms have been developed. In this talk I will discuss one of them using tools of contact and symplectic geometry, and show that it is also useful to obstruct the existence of some immersed cobordisms.
If time permits I will on the other hand mention concrete examples of Legendrian knots for which the known obstructions coming from contact geometry do not apply to exclude the existence of a Lagrangian cobordism. In this case low dimensional topology tools are used instead in order to show the non existence of a cobordism.
This talk reports on joint work with Orsola Capovilla-Searle, Maÿlis Limouzineau, Emmy Murphy, Yu Pan and Lisa Traynor, and on joint work with Baptiste Chantraine.

Lisa Lokteva: Constructing Graph Manifolds that Bound Rational Homology Balls

We discuss results and conjectures on graph manifolds bounding rational homology balls.

Jules Martel: Knot invariants and Verma modules

Quantum groups are (more than) algebras from which one extracts tools such as R-matrices and Markov traces allowing the construction of respectively braid groups representations and knot invariants. In the sl2 case, one obtains the famous family of Jones polynomials for knots, but their algebraic construction makes their topological content hard to catch. Even more mysterious is the family of ADO polynomials constructed from quantum sl2 at roots of unity. Fortunately, the latter caught up while the two families were recently unified by S. Willetts' invariant. We will rebuild this unifying invariant using braid action on quantum Verma modules and a Markov trace (on infinite dim. space), recover the unification of Jones and ADO families and eventually deduce a symmetry on variables nicely generalizing a famous one for the Alexander polynomial. It relies on an algebraic study, but if time permits I'll say a word on how to recover all the construction from geometry. (j.w. S. Willetts)

Bruno Martelli: A curious 4-dimensional aspherical manifold

I introduce a 4-manifold that arises as the fiber of a hyperbolic 5-manifold fibering over the circle. The 4-manifold may be realised as a double branched covering over a 4-manifold of very simple kind. This 4-manifold has various curious properties, many of which are still not properly understood. 

Oscar Ocampo: Subgroup separability of surface braid groups

A group G is said subgroup separable or locally extended residually finite (LERF) if each finitely generated subgroup H of G is the intersection of finite index subgroups of G. Subgroup separability is a powerful property introduced by M. Hall in 1949, important for group theory and low-dimensional topology, but established either positively or negatively for very few classes of groups. It is known that, for finitely presented groups, Subgroup Separability implies the generalized word problem is solvable. The purpose of this talk is to determine under which conditions surface braid groups are subgroup separable. This is a joint work with Kisnney Almeida (UEFS-Brazil) and Igor Lima (UnB-Brazil).

Anthony Saint-Criq: Non-empty ovals of odd degree flexible curbes

By studying the geometry of CP(2) and the behaviour of an odd degree flexible curve under different branched covers, we show an upper bound on the number of non-empty ovals of such a curve. This method generalizes to other manifolds, such that flexible curves on a quadric.

Mélanie Theillière: The hyperbolic plane in E³

From a theorem of Nash and Kuiper, we know it is possible to isometrically embed (preserving lengths) the hyperbolic plane into the Euclidean 3-space. However such an embedding only exists in C^1-regularity. By a theorem of Hilbert-Efimov, the regularity can not be enhanced to be C^2. In this talk, we will explicitly build such an embedding and we will explore its geometry. This is a common work with the Hevea team.