[K-OS] Knot Online Seminar — Spring semester 2020

  • Speaker: Anna Beliakova (Universität Zürich).
  • Title: Non-semisimple quantum invariants.
  • Abstract: Non-semisimple invariants became a hot topic in quantum topology after physicists predicted their categorification. In the talk I will give a gentle introduction to the theory of quantum link and 3-manifold invariants and then focus on the relationship between non-semisimple (ADO) and universal 𝖘𝖑(2) link invariants as well as Costantino–Geer–Patureau (CGP) and Witten–Reshetikhin–Turaev (WRT) 3-manifold ones.
  • Handout.

  • Speaker: Catherine Meusburger (Universität Erlangen–Nürnberg).
  • Title: Mapping class group actions from Hopf monoids.
  • Abstract: We show that pivotal Hopf monoids in symmetric monoidal categories give rise to actions of mapping class groups of oriented surfaces with boundary components. These mapping class group actions are associated with edge slides in embedded ribbon graphs that generalise chord slides in chord diagrams. They can be described simply and concretely in terms of generating Dehn twists. Under certain assumptions on the symmetric monoidal category and the Hopf monoid, they induce actions of mapping class groups of closed surfaces. Based on 2002.04089.
  • Handout.

  • Speaker: Maÿlis Limouzineau (Universität zu Köln).
  • Title: About reversing surgery in immersed Lagrangian fillings of Legendrian knots.
  • Abstract: From an immersed filling of a knot, one can performed surgery on the double points to get an embedded filling of the same knot. Each solved double point increases the genus of the filling by one. A natural question is then: is the converse true? can we trade genus for double points? We are interested in the symplectic-contact version of this question. In this talk, we will see how a typical Legendrian invariant, namely the set of augmentations, can help giving a negative answer in some cases. This is work in progress joint with Capovilla-Searle, Legout, Murphy, Pan and Traynor.
  • Handout.

  • Speaker: Bruno Martelli (Universit√° di Pisa).
  • Title: Convex hyperbolic 4-manifolds.
  • Abstract: Due to work of Poincar√© and Thurston, a generic surface or 3-manifold admits a hyperbolic structure. The role of hyperbolic geometry in dimension 4 or higher is still mysterious. In this talk we expose what is currently known in dimension 4; we then focus on the construction of convex hyperbolic manifolds, some objects that play a fundamental role in dimension 3. We show in particular that many plumbings of surfaces admit such a convex hyperbolic structure, and how this leads to the first construction of closed hyperbolic manifolds without spin structures. (joint with Stefano Riolo and Leone Slavich)
  • Handout.

  • Speaker: Hugh Morton (University of Liverpool).
  • Title: Skeins and algebras.
  • Abstract: I will look at some algebras based on braids with a quadratic relation, which are natural candidates for Homfly skein theory models.
    The main theme will be the use of braids in thickened surfaces to construct algebras. These readily produce the Hecke algebras and the affine Hecke algebras of type A. I will point out a problem in doing the same for the double affine Hecke algebras, along with a way round it, which also allows for extensions to the algebras by the use of closed curves and tangles in addition to braids. This is part of recent work with Peter Samuelson.
  • Handout.

  • Speaker: John Guaschi (Université de Caen).
  • Title: (Almost)-crystallographic quotients of Artin and surface braid groups and their finite subgroups.
  • Abstract: We discuss some recent results regarding quotients of Artin and surface braid groups by elements of the lower central series of the corresponding pure braid group, and the embedding of finite groups in these quotients. This is joint work with D. L. Gonçalves, O. Ocampo and C. Pereiro.
  • Handout.

  • Speaker: Giulio Belletti (Scuola Normale Superiore di Pisa).
  • Title: The Volume Conjecture for Turaev–Viro invariants.
  • Abstract: Several connected "Volume Conjectures" inquire about the relationship between various quantum invariants of links, or 3-manifolds, and their geometric properties. A recent version of Chen–Yang, in particular, relates the asymptotic growth of the Turaev–Viro invariants of compact 3-manifolds and their hyperbolic volume. I will give an overview of the known results and techniques used to attack this conjecture, and then focus on a recent result that proves it for an infinite family of hyperbolic manifolds built from certain right-angled ideal polyhedra. The main new tool used in the proof is a Fourier Transform for quantum invariants first introduced by Barrett. The talk will only assume basic knowledge of low-dimensional topology (in particular, no knowledge of the Turaev–Viro invariants is needed).
  • Handout.

  • Speaker: Mark Powell (Durham University).
  • Title: Shake slice knots.
  • Abstract: The n-trace of a knot is a 4-manifold, homotopy equivalent to the 2-sphere, obtained by attaching a 2-handle to the 4-ball along the knot with framing n. The knot is said to be n shake slice if the generator of second homology of the n-trace can be represented by a locally flat embedded 2-sphere. There is also a smooth version. A slice knot is n-shake slice for every n, but many other statements of this kind that you can think of are either false or open. I will discuss some background, some of these other statements, and my results on this question from work with Peter Feller, Allison Miller, Matthias Nagel, Patrick Orson, and Arunima Ray.
  • Notes.

  • Speaker: Jules Martel (Université de Toulouse).
  • Title: A full homological model for quantum Verma modules and their representations of braid groups.
  • Abstract: Categories of modules on quantum groups produce strong topological invariants (knots, braids, 3-manifolds, TQFTs...), such as the famous Jones polynomial for knots. They rely essentially on this purely algebraic tool, so that their topological content remains often mysterious. In this work we will build relative homology modules from configuration spaces of points. We will endow them with an action of the quantum sl(2) algebra and we will recognize a tensor product of Verma modules. They are provided with (an extension of) Lawrence representations of braid groups which turns out to be the quantum representation of braid groups (given by the R-matrix). We work over the ring of Laurent polynomials – suitable for evaluating variables – and we preserve this structure all along the construction; it yields a full homological model for integral versions of quantum Verma modules. If time allows, we will apply this model to give an interpretation for colored Jones polynomials dealing with Lefschetz numbers.
  • Handout.

  • Speaker: Alexandra Kjuchukova (Max Planck Institute).
  • Title: The meridional rank conjecture: an attack with crayons.
  • Abstract: The meridional rank conjecture posits equality between the bridge number β and meridional rank μ of a link L⊂𝕊3. I will describe a diagrammatic technique – relying on "coloring" knot diagrams – by which we establish the conjecture for new infinite classes of links. We obtain upper bounds for β via the Wirtinger number of L, which is a combinatorial equivalent of the bridge number. Matching lower bounds on μ are found using Coxeter quotients of π1(𝕊3L). As a corollary, we derive formulas for the bridge numbers for the links in question. Based on joint works with Ryan Blair, Sebastian Baader, Filip Misev.
  • Handout.

  • Speaker: Lukas Lewark (Universität Regensburg).
  • Title: Unknotting and cobordism distances.
  • Abstract: We will compare various metrics on the set of knots. These metrics are defined in terms of crossing changes, or of genera of certain cobordisms. The distance of a knot K to the set of knots with Alexander polynomial 1 can be shown to be the same in all those metrics. This distance is thus a knot invariant with four-dimensional, three-dimensional, algebraic characterizations. For example, it can be defined in terms of Seifert matrices or the Blanchfield pairing; or as minimum genus of a Seifert surfaces with boundary K union a Alexander-polynomial-1 knot; or as minimum genus of a topological slice surface of K whose complement has cyclic fundamental group. It also equals the topological super slice genus (minimum genus of a locally flat slice surface for K whose double is an unknotted surface in the 4-sphere) and the ℤ-stabilization number (minimum n such that K is boundary of a locally flat disk whose complement has cyclic fundamental group in the connected sum of the four-ball with n copies of ℂℙ2 # -ℂℙ2). Joint work with Peter Feller, see arXiv:1905.08305.
  • Handout.

  • A technical problem occured during the recording. Only the first 23 minutes of the talk are available.

  • Speaker: Julien Marché (Université Pierre et Marie Curie).
  • Title: Automorphisms of the character variety of a surface.
  • Abstract: The character variety of a surface group into SL2(ℂ) is an affine algebraic variety. We will show that its automorphism group is (up to a finite group) the mapping class group of the surface. To that aim, we will study certain valuations among which we will recognize Thurston's boundary of the Teichmüller space. Joint work with Christopher-Lloyd Simon.
  • Handout, Preprint.

  • Speaker: Pierre Dehornoy (Université Grenoble–Alpes).
  • Title: Almost equivalence for transitive Anosov flows.
  • Abstract: The world of 3-dimensional Anosov flows is still mysterious: for example we do not know which 3-manifold support Anosov flows, or whether some 3-manifold support infinitely many non-equivalent Anosov flows. Fried asked whether all transitive Anosov flows admit genus-one Birkhoff sections and Ghys asked whether all transitive Anosov flows are almost-equivalent (two 3-dimensional flows are almost equivalent if one can go from one to the other by Dehn surgering finitely many periodic orbits). We will motivate these two questions, show that they are equivalent, and give a positive answer in some special cases. This is a joint work with Mario Shannon.
  • Plan, Handout, Barthelmé's notes.

Credit for the jingle: Daniel Maszkowicz.

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