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[K-OS] Knot Online Seminar — Spring semester 2021

  • Speaker: Raphael Zentner (Universität Regensburg)
  • Title: Toroidal homology 3-spheres and SU(2)-representations.
  • Abstract: We will review some of the history of SU(2)-representations of the fundamental groups of 3-manifolds, and we will then focus on the following result: Every integral homology 3-sphere with an incompressible torus admits a SU(2)-representation of its fundamental group with non-abelian image. The proof uses instanton gauge theory, but also some more classical 3-manifold topology. This is joint work with Tye Lidman and Juanita Pinzon-Caicedo.
  • Handout.

  • Speaker: Yuta Nozaki (Hiroshima University)
  • Title: Abelian quotients of the Y-filtration on the homology cylinders via the LMO functor.
  • Abstract: We construct a series of homomorphisms from the Y-filtration on the monoid of homology cylinders to torsion modules via the mod ℤ reduction of the LMO functor. The restrictions of our homomorphisms to the lower central series of the Torelli group do not factor through Morita’s refinement of the Johnson homomorphism. We use it to show that the abelianization of the Johnson kernel of a closed surface has torsion elements. This is joint work with Masatoshi Sato and Masaaki Suzuki.
  • Slides.

  • Speaker: Agnès Gadbled (Université Paris–Saclay)
  • Title: Weinstein handlebody diagrams for complements of smoothed toric divisors.
  • Abstract: We study so-called toric hypersurfaces of symplectic toric manifolds in real dimension 4, or rather the complement of these hypersurfaces and some of their smoothings. If the complement of the smoothings corresponds topologically to handle gluing, we need an extra condition to make sense of this symplectically and construct a Weinstein structure on the complement. Moreover, in dimension 4, these Weinstein structures can be described via handlebody diagrams from which we can extract some topological and symplectic invariants. We also give an algorithm that produces, out of a toric manifold and compatible smoothing data (encoded in a polytope), a handlebody diagram of the Weinstein structure of the complement of the smoothing.
    In this talk I will introduce the different notions involved and illustrate the algorithm with examples. If time allows, I will mention some ingredients for our constructions and proofs as well as consequences and perspectives for our algorithm. This is a joint work with Acu, Capovilla-Searle, Marinkovic, Murphy, Starkston and Wu.
  • Handout.

  • Speaker: Delphine Moussard (Aix–Marseille Université)
  • Title: A triple point knot invariant and the slice and ribbon genera.
  • Abstract: The T-genus of a knot is the minimal number of borromean-type triple points on a normal singular disk with no clasp bounded by the knot; it is an upper bound for the slice genus. Kawauchi, Shibuya and Suzuki characterized the slice knots by the vanishing of their T-genus. I will explain how this generalizes to provide a 3-dimensional characterization of the slice genus. Further, I will show that the difference between the T-genus and the slice genus can be arbitrarily large. Finally, I will introduce the ribbon counterpart of the T-genus, which is an upper bound for the ribbon genus, and we will see that the T-genus and the ribbon T-genus coincide for all knots if and only if all slice knots are ribbon.
  • Handout.

  • Speaker: William Ballinger (Princeton University)
  • Title: Concordance invariants from Khovanov homology.
  • Abstract: The Lee differential and Rasmussen's E(-1) differential acting on Khovanov homology combine to give a pair of cancelling differentials, an algebraic structure that has been studied in the context of knot Floer homology. I will describe some concordance invariants that come from this structure, with applications to nonorientable genus bounds and linear independence in the concordance group.
  • Slides.

  • Speaker: Jacob Rasmussen (University of Cambridge)
  • Title: An SL2(ℝ) Casson–Lin invariant.
  • Abstract: In the early 90's, X.S. Lin defined a Casson-type invariant of knots in 𝕊3 by counting representations π1(𝕊3K) → SU(2) with fixed holonomy around the meridian. This invariant was subsequently shown to be equivalent to the Levine–Tristram signature of K. I'll describe a similar construction, using representations to SL2(ℝ) and discuss some applications and connections to other known invariants. This is joint work with Nathan Dunfield.
  • Slides.

  • Speaker: David Leturcq (Research Institute for Mathematical Sciences, Kyōto University).
  • Title: (High-dimensional) Alexander polynomial(s) and diagram counts.
  • Abstract: The main objects of this talk are Bott–Cattaneo–Rossi invariants (Zk)k>1 for long n-knots (embeddings f : ℝn → ℝ n+2 linear near the infinity) with odd n≥3. They were originally defined as combination of configuration space integrals associated to diagrams with 2k vertices, with some of them on the knot. We give a more flexible definition as (signed) counts of such diagrams with constraints on the edges (given by some chains in two-point configuration spaces called "propagators"). First, we will present these definitions, and explain how they adapt in any dimension. Next, we will use some specific propagators to compute our invariant in terms of linking numbers of some cycles in a surface whose boundary is f (ℝn). Eventually, this leads to a formula for Zk in terms of Alexander polynomials of the embedding. In particular, when n=1, this recovers a formula for the Alexander polynomial in terms of diagrams due to Bar-Natan and Garoufalidis. This method extends their formula to null-homologous knots in rational homology spheres, where their original proof (using finite type invariant theory) did not extend.
  • This talk is joint with the second meeting of AlMaRe.
  • Handout.

  • Speaker: Andrew Lobb (Durham University).
  • Title: Four-sided pegs fitting round holes fit all smooth holes.
  • Abstract: Given a smooth Jordan curve and a cyclic quadrilateral (a cyclic quadrilateral is a quadrilateral that can be inscribed in a circle) we show that there exist four points on the Jordan curve forming the vertices of a quadrilateral similar to the one given. The smoothness condition cannot be dropped (since not all cyclic quadrilaterals can be inscribed in all triangles). The proof involves some results in symplectic topology. No prior knowledge assumed. Joint work with Josh Greene.
  • Slides.

  • Speaker: Maciej Borodzik (Uniwersytet Warszawski).
  • Title: Non-rational, non-cuspidal plane curves via Heegaard Floer homology.
  • Abstract: Let C be a complex curve in ℂℙ2. Using d-invariants from Heegaard Floer theory we provide topological constraints for possible singularities of C. The novelty is that we do not need to assume that C is rational, or that all its singularities have one branch, so we generalize previous results of Bodnar, Borodzik, Celoria, Golla, Hedden and Livingston. As an application, we show precise examples of surfaces in ℂℙ2, for which "genus cannot be traded for double points". This is a joint project with Beibei Liu and Ian Zemke.
  • Slides.

  • Speaker: Jean-Baptiste Meilhan (Université Grenoble Alpes).
  • Title: Milnor concordance invariant for knotted surfaces and beyond.
  • Abstract: The purpose of this talk is to propose a general framework for extending Milnor's link invariants to all knotted surfaces in 4-space – and more. Milnor link invariants are numerical concordance invariants that are extracted from the nilpotent quotient of the link group. We shall briefly review their definition, as well as some known generalization/adaptation in dimension 3 and 4. Next, we will introduce "cut-diagrams" of knotted surfaces in 4-space, which encode these objects in a simple combinatorial way. Roughly speaking, for a knotted surface obtained as embedding of the abstract surface S, a cut-diagram is a kind of 1-dimensional diagram on S. Using this language, we generalize Milnor invariants to all types of knotted surface. This construction actually applies in any dimension, and recovers in particular welded Milnor invariants of tangles in dimension 1. Based on a work in progress with Benjamin Audoux and Akira Yasuhara.
  • Handout.

  • Speaker: Ricard Riba Garcia (Universitat Autònoma de Barcelona).
  • Title: Invariants of rational homology 3-spheres and the mod p Torelli group.
  • Abstract: Unlike the integral case, given a prime number p, not all ℤ/p-homology 3-spheres can be contructed as a Heegaard splitting with a gluing map an element of mod p Torelli group, M[p]. Nevertheless, letting p vary we can get any rational homology 3-sphere. This motivated us to study invariants of rational homology 3-spheres that comes from M[p]. In this talk we present an algebraic tool to construct invariants of rational homology 3-spheres from a family of 2-cocycles on M[p]. Then we apply this tool and give all possible invariants that are induced by a lift to M[p] of a family of 2-cocycles on the abelianization of M[p], getting a family of invariants that we did not found in the known literature. Joint work with Wolfgang Pitsch.
  • Handout.

  • Speaker: Akram Alishahi (University of Georgia).
  • Title: Braid invariant related to knot Floer homology and Khovanov homology.
  • Abstract: Knot Floer homology and Khovanov homology are homological knot invariants that are defined using very different methods — the former is a Lagrangian Floer homology, while the latter has roots in representation theory. Despite these differences, the two theories contain a great deal of the same information and were conjectured by Rasmussen to be related by a spectral sequence. This conjecture was recently proved by Dowlin, however, his proof is not computationally effective. In this talk we will sketch a local framework for proving this conjecture. To do that, we will describe an algebraic/glueable braid invariant which using a specific closing up operation results in a knot invariant related by a spectral sequence to Khovanov homology. Moreover, it is chain homotopic to Ozsvath–Szabo's braid invariants which using their closing up operation recovers knot Floer homology. If time permits we will compare the two closing up operations. This is a joint work with Nathan Dowlin.
  • Slides.

  • Speaker: Daniel Ruberman (Brandeis University).
  • Title: A Levine–Tristram invariant for knotted tori.
  • Abstract: In 1969, Tristram and Levine independently introduced an integer-valued function of a knot, depending on the choice of a unit complex number. It gives rise to a concordance invariant that in turn shows that the the concordance group is infinitely generated. I will explain a generalization of this invariant to the setting where the 3-sphere is replaced by X, a homology 𝕊1×𝕊3, and the knot is replaced by an embedded torus (that carries the first homology). I will show how to compute this invariant, and discuss its relation to recent work of Echeverria that counts SU(2) connections on the complement of the torus with specified holonomy on the meridian.
  • Handout.

  • Speaker: María Cumplido (Universidad de Sevilla).
  • Title: Braiding trees: A new family of Thompson-like groups.
  • Abstract: There is a generalization of Thompson's groups constructed from the Thompson's group V and Artin's braid group. The braided Thompson's group BV2 was independently introduced by Patrick Dehornoy and Matthew G. Brin in 2006. In this talk we will explain how to extend this concept to a much bigger family of groups by using infinite braids: Infinitely braided Thompson's groups BVn(H), where H is a subgroup of the braid group on n strands. We will prove that they are indeed groups by using braided diagrams and rewriting systems. We will also see that they are finitely generated if H is finitely generated and give an explicit set of generators for BVn(H) and some other cases. This is a joint work with Julio Aroca.
  • Handout.

  • Speaker: Marco De Renzi (Universität Zürich).
  • Title: Non-semisimple quantum invariants of 3-Manifolds from the Kauffman bracket.
  • Abstract: In recent years, several constructions in the field of quantum topology have been extended to the non-semisimple case, producing TQFTs with remarkable new properties. At present, all the different approaches rely on a rather elaborate technical setup, involving either the structure of Hopf algebras, or more abstract categorical machineries. In this talk, we will explain how the family of non-semisimple quantum invariants associated with the small quantum group of 𝖘𝖑2 at odd roots of unity can be reformulated in purely combinatorial and diagrammatic terms, using only Temperley–Lieb categories and Kauffman bracket polynomials. Based on joint work with C. Blanchet and J. Murakami.

  • Speaker: Anthony Conway (Massachusetts Institute of Technology).
  • Title: Knotted surfaces with infinite cyclic knot group.
  • Abstract: This talk will concern embedded surfaces in 4-manifolds for which the fundamental group of the complement is infinite cyclic. Working in the topological category, necessary and sufficient conditions will be given for two such surfaces to be isotopic. This is based on joint work with Mark Powell.
  • Handout.

  • Speaker: Jennifer Hom (Georgia Tech).
  • Title: Infinite order rationally slice knots.
  • Abstract: A knot in 𝕊3 is rationally slice if it bounds a disk in a rational homology ball. We give an infinite family of rationally slice knots that are linearly independent in the knot concordance group. In particular, our examples are all infinite order. All previously known examples of rationally slice knots were order two. The proof relies on bordered and involutive Heegaard–Floer homology. No prior knowledge of Heegaard–Floer homology will be assumed. This is joint work with Sungkyung Kang, Jung Hwan Park, and Matt Stoffregen.
  • Handout.

  • Speaker: Mikhail Khovanov (Columbia University).
  • Title: Bilinear pairings and topological theories.
  • Abstract: The notion of a TQFT admits a generalization where one starts with the values of a topological theory on closed n-manifolds and builds state spaces for (n–1)-manifolds from that data. The resulting structures are nontrivial already in dimension n = 2 and even for n = 1 if defects are allowed. This story will be discussed in the talk.
  • Handout.

  • Speaker: Peter Feller (ETH Zürich).
  • Title: Braids, quasimorphisms, and slice-Bennequin inequalities.
  • Abstract: The writhe of a braid (=#pos crossing - #neg crossings) and the fractional Dehn twist coefficient of a braid (a rational number that measures "how much the braid twists") are the two most prominent examples of what is known as a quasimorphism (a map that fails to be a group homomorphism by at most a bounded amount) from Artin's braid group on n-strands to the reals.
    We consider characterizing properties for such quasimorphisms and talk about relations to the study of knot concordance. For the latter, we consider inequalities for quasimorphism modelled after the following consequence of the local Thom conjecture known as the slice-Bennequin inequality:
    writhe(β) ≤ 2g4(K) – 1 + n
    for all n-stranded braids β with closure a knot K.
  • Handout.

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

  • Speaker: Joshua Sussan (City University of New York).
  • Title: p-dg structures in link homology.
  • Abstract: Categorification of quantum groups, their representations, and associated link invariants at generic values of q have been constructed in many ways. In order to categorify these objects when q is a prime root of unity, one should look for p-differentials on existing categorifications. We will consider one such case leading to a categorification of the Jones polynomial at a prime root of unity.
  • Handout.

  • Speaker: Celeste Damiani (Queen Mary University of London).
  • Title: Generalisations of Hecke Algebras from Loop Braid Groups.
  • Abstract: This work takes inspiration by from the braid group revolution ignited by Jones in the early 80s, to study representations of the motion group of the free unlinked circles in the 3 dimensional space, the loop braid group LBn. Since LBn contains a copy of the braid group Bn as a subgroup, a natural approach to look for linear representations is to extend known representations of the braid group Bn. Another possible strategy is to look for finite dimensional quotients of the group algebra, mimicking the braid group / Iwahori–Hecke algebra / Temperley–Lieb algebra paradigm. Here we combine the two in a hybrid approach: starting from the loop braid group LBn we quotient its group algebra by the ideal generated by (σi + 1)(σi – 1) as in classical Iwahori–Hecke algebras. We then add certain quadratic relations, satisfied by the extended Burau representation, to obtain a finite dimensional quotient that we denote by LHn. We proceed then to analyse this structure. Our hope is that this work could be one of the first steps to find invariants à la Jones for knotted objects related to loop braid groups.
  • Handout.

  • Speaker: Léo Bénard (Georg-August-Universität Göttingen).
  • Title: Twisted L2 torsion on character varieties of 3-manifolds.
  • Abstract: Given a representation ρ of the fundamental group of a 3-manifold M, we study the associated real-valued invariant given by the L2 torsion of the pair (M,ρ). In the case M is hyperbolic and ρ is the holonomy representation, a recent extension by Wasserman of former work of Luck–Schick states that this is explicitly related with the hyperbolic volume. Our main result shows that the L2 torsion defines a real-analytic function on a neighborhood of this holonomy representation in the SL(2,ℂ) character variety of M. In the non-hyperbolic setting, we also obtain simple formulas for this function in the case M is a graph manifold. This is a joint work with Jean Raimbault (University of Toulouse).
  • Handout.

  • Speaker: Caterina Campagnolo (ENS Lyon).
  • Title: The simplicial volume of surface bundles over surfaces, and other invariants.
  • Abstract: Surface bundles over surfaces form an easy to define and well studied family of 4-manifolds. Nevertheless, many questions about them remain unanswered to this day, from their possible geometric structures to their numerical invariants, while only few explicit constructions of such bundles exist.
    In this talk we will be mainly concerned with the simplicial volume of surface bundles over surfaces, and its relation with more classical invariants, such as the Euler characteristic and the signature. We will recall all necessary definitions and present new inequalities obtained jointly with Michelle Bucher.
  • Notes.

  • Speaker: Lisa Piccirillo (MIT).
  • Title: A users guide to straightforward exotica.
  • Abstract: We say a smooth 4-manifold W is exotic if there exists some smooth 4-manifold W' which is homeomorphic but not diffeomorphic to W. Early proofs that compact exotica exists tended to feature some wining combination of manifolds with complicated algebraic topology, manifolds with enormous handle decompositions, and tricky hands-on gauge theory, even in the (simpler) setting where W has boundary. In the last 20 years, a suite of tricks has emerged which make constructing and detecting exotica (with boundary) much more user friendly. I'll survey recent work of several authors constructing simple exotica (eg. with the homotopy type of a point, 𝕊1, or 𝕊2), while emphasizing the techniques that keep these proofs straightforward (and perhaps even gauge-theory free).
  • Handout.

  • Speaker: Hugo Parlier (University of Luxembourg).
  • Title: Where the orthogeodesics roam.
  • Abstract: The lengths of geodesics on hyperbolic surfaces satisfy intriguing equations, known as identities, relating these lengths to geometric quantities of the surface. The talk will be about a family of identities that relate lengths of closed geodesics and orthogeodesics to boundary lengths or the number of cusps. These include, as particular cases, identities due to Basmajian, to McShane and to Mirzakhani and Tan–Wong–Zhang. In contrast to previously studied cases, the new identities include lengths taken among all closed geodesics.
  • Handout.

  • Speaker: Louis Funar (Université de Grenoble).
  • Title: Braided surfaces and their characteristic maps.
  • Abstract: Our aim is to show that branched coverings of surfaces of large enough genus arise as characteristic maps of braided surfaces. In the reverse direction we show that any nonabelian surface group has infinitely many finite simple nonabelian groups quotients with characteristic kernels which do not contain any simple loop and hence the quotient maps do not factor through free groups. Eventually we discuss about topological invariants of braided surfaces arising from finite dimensional Hermitian representations of braid groups. Joint work with Pablo Pagotto.
  • Handout.

  • Speaker: Tara Brendle (University of Glasgow).
  • Title: The mapping class group of connect sums of 𝕊2 × 𝕊1.
  • Abstract: Let Mn denote the connect sum of n copies of 𝕊2 × 𝕊1. Laudenbach showed that the mapping class group Mod(Mn) is an extension of the group Out(Fn) by (ℤ/2)n, where the latter group is the "sphere twist" subgroup of Mod(Mn). In joint work with N. Broaddus and A. Putman, we have shown that in fact this extension splits. In this talk, we will describe the splitting and discuss some simplifications of Laudenbach's original proof that arise from our techniques.
  • Handout.

  • Speaker: Livio Liechti (Université de Fribourg).
  • Title: Bi-Perron numbers and the Alexander polynomial.
  • Abstract: A bi-Perron number is a positive real algebraic unit all of whose Galois conjugates are contained in the annulus with outer radius the bi-Perron number itself and inner radius its inverse, with at most one Galois conjugate on either boundary of the annulus. Among bi-Perron numbers, we characterise those all of whose Galois conjugates are real or unimodular as the ones that have a power that, up to sign, equals the maximal root (in absolute value) of the Alexander polynomial of a link of certain type. We propose the class of links that admit an upper diagonal block Seifert matrix, where the diagonal blocks are identity matrices. Hopefully, this choice can be modified into a more geometric one. This is a variation of joint work with J. Pankau.
  • Handout.

  • Speaker: András Stipsicz (Rényi Institute of Mathematics).
  • Title: Cosmetic surgery conjecture for pretzel knots.
  • Abstract: We describe the (purely) cosmetic surgery conjecture, and show how it can be proved for pretzel knots. Along the way we also give a method for estimating the thickness of a knot from one of its diagrams.
  • Slides.

  • Speaker: Fathi Ben Aribi (Université Catholique de Louvain).
  • Title: The Teichmüller TQFT volume conjecture for twist knots.
  • Abstract: In 2011, Andersen and Kashaev defined an infinite-dimensional TQFT from quantum Teichmüller theory. This Teichmüller TQFT yields an invariant of triangulated 3-manifolds, in particular knot complements.
    The associated volume conjecture states that the Teichmüller TQFT of an hyperbolic knot complement contains the hyperbolic volume of the knot as a certain asymptotical coefficient, and Andersen–Kashaev proved this conjecture for the first two hyperbolic knots.
    In this talk, after a brief history of quantum knot invariants and volume conjectures, I will present the construction of the Teichmüller TQFT and how we proved its volume conjecture for the infinite family of twist knots, by constructing new geometric triangulations of the knot complements. No prerequisites in quantum topology or hyperbolic geometry are needed.
    (joint project with E. Piguet–Nakazawa and F. Guéritaud)
  • Slides.

  • Speaker: Stefan Friedl (Universität Regensburg).
  • Title: Reidemeister torsion and topological link concordance.
  • Abstract: We will explain how Reidemeister torsion can be used to address questions in topological link concordance. This talk is based on an earlier paper with Jae Choon Cha and recent discussions with Matthias Nagel, Patrick Orson and Mark Powell.
  • Board, Stefan's Topology lecture notes.

  • Speaker: Marco Golla (Université de Nantes — CNRS).
  • Title: Symplectic rational cuspidal curves.
  • Abstract: I will talk about symplectic rational cuspidal curves in the complex projective plane and their isotopies. These are PL-embedded spheres whose singular points are cones on algebraic knots, and that are symplectic away from the singular point. I will talk about existence and obstructions, especially in low-degrees, borrowing ideas from complex algebraic geometry and 3.5-dimensional topology. This is based on joint work with Laura Starkston and with Fabien Kütle.
  • Notes.

  • Speaker: Jun Murakami (Waseda University).
  • Title: Some applications of the volume conjecture.
  • Abstract: The volume conjecture for the colored Jones polynomial was improved by Q. Chen and T. Yang, and they proposed similar conjectures for the Turaev–Viro invariants and Witten–Reshetikhin–Turaev invariants of three-manifolds. These conjectures are not proved yet, but there are some by-products which I would like to explain in this talk.
  • Handout.

[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 (Universitá di Pisa).
  • Title: Convex hyperbolic 4-manifolds.
  • Abstract: Due to work of Poincaré 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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