[K-OS] Knot Online Seminar — 2022–2023

  • Speaker: Kyle Hyden (Rutgers University-Newark)
  • Title: Doubled Disks and Satellite Surfaces
  • Abstract: Satellite operations are ubiquitous in the study of knots in dimension three, but also have natural four-dimensional extensions. This talk will discuss joint work with Gary Guth, Sungkyung Kang, and JungHwan Park on satellites of slice disks, emphasizing the behavior of 4-dimensional surface invariants under satellite operations. In particular, we will see that these satellite operations behave very differently in the smooth and continuous settings, giving rise to many new examples of exotically knotted surfaces, including the first infinite family of pairwise exotic slice disks.

  • Speaker: Effie Kalfagianni (Michigan State University)
  • Title: Jones diameter and crossing numbers of satellite knots
  • Abstract: It has been long known that the quadratic term in the degree of the colored Jones polynomial of knot provides a lower bound of the crossing number the knot.
    I’ll discuss work, some joint with Christine Lee (2021) and some joint with Rob McConkey (2023), where we determine the class of knots for which this bound is sharp and give applications to computing crossing numbers of satellite knots.

  • Speaker: Ivan Dynnikov (Steklov Mathematical Institute)
  • Title: An algorithm for comparing Legendrian links (2309.05087)
  • Abstract: The talk is based on my joint works with Maxim Prasolov and Vladimir Shastin, where we studied the relation between rectangular diagrams of links and Legendrian links. This relation allows for a complete classification of exchange classes of rectangular diagrams in terms of equivalence classes of Legendrian links and their symmetry groups. Since all rectangular diagrams of given complexity can be searched, this yields a method to algorithmically compare Legendrian links. Of course, the general algorithm has too high complexity for a practical implementation, but in some situations, the most time-consuming parts can be bypassed, which allows us to confirm the non-equivalence of Legendrian knots in several previously unresolved cases.

  • Speaker: Søren Galatius (University of Copenhagen and Columbia University)
  • Title: The Alexander trick for homology spheres
  • Abstract: The boundary of any contractible compact d-manifold is a homology (d-1)-sphere, but the boundary need not be simply connected. I will discuss joint work with Randal-Williams (2308.15607) in which we show that in dimension d > 5, the topological group of homeomorphisms of any such contractible d-manifold relative to it boundary is contractible. In the case of the d-disk bounding the (d-1)-sphere, this was proved 100 years ago by writing down an explicit contraction, now known as the Alexander trick.
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  • Speaker: Mark Hughes (Brigham Young University)
  • Title: Nonisotopic splitting spheres for split surface links
  • Abstract: Given a split link in the 3-sphere, it can be shown that any two splitting spheres for the link are isotopic in its complement. In this talk I will outline a construction showing that this fact fails in dimension 4; I will show how to construct a surface unlink with nonisotopic splitting 3-spheres. Along the way I will show how a related construction can be used to obtain 3-dimensional handlebodies in the 4-sphere that are not isotopic rel boundary, even when pushed into the 5-ball. This is joint work with Seungwon Kim and Maggie Miller.
  • Slides.
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  • Speaker: Lisa Piccirillo (MIT)
  • Title: Building closed exotic manifolds by hand.
  • Abstract: Historically, closed exotic 4-manifolds are built using cut and paste constructions, and their gauge theoretic invariants are computed using gluing formulae. In this talk, I'll define some new smooth 4-manifold invariants which we can compute by hand using a (Dehn) surgery formula. Armed with this I'll build explicit closed exotic 4-manifolds out of elementary handle cobordisms. We'll see some new phenomena, like closed exotic definite 4-manifolds (with fundamental group ℤ), closed 4-manifolds with homologically essential square 0 spheres and nonvanishing invariants, and instances when knot surgery on an Alexander polynomial 1 knot can change the smooth structure. We'll try not to work too hard. This is joint work with Adam Levine and Tye Lidman.
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  • Speaker: Lvzhou Chen (Purdue University)
  • Title: The Kervaire conjecture and the minimal complexity of surfaces (about 2302.09811)
  • Abstract: We use topological methods to solve special cases of a fundamental problem in group theory, the Kervaire conjecture, which has connection to various problems in topology. The conjecture asserts that, for any nontrivial group G and any element w in the free product G⋆ℤ, the quotient (G⋆ℤ)/<<w>> is still nontrivial. We interpret this as a problem of estimating the minimal complexity (in terms of Euler characteristic) of surface maps to certain spaces. This gives a conceptually simple proof of Klyachko's theorem that confirms the Kervaire conjecture for any G torsion-free. We also obtain results on injectivity of the map G→(G⋆ℤ)/<<w>> when w is a proper power for arbitrary G. Both results generalize to certain HNN extensions.
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  • Speaker: Jonathan Simone (Georgia Tech)
  • Title: Lefschetz fibrations, rational blowdowns, and exotic 4-manifolds
  • Abstract: We will define and explore the use of Lefschetz fibrations, fiber sums, and rational blowdowns to construct symplectic exotic 4-manifolds. In particular, we start by constructing Lefschetz fibrations with singular fibers containing "clustered nodes." We will then apply the rational blowdown operation on particular collections of embedded spheres in these Lefschetz fibrations to produce small exotic 4-manifolds (and, in particular, an exotic ℂℙ2#-5ℂℙ2) and to produce symplectic 4-manifolds that populate a portion of the so-called symplectic geography plane, which is a symplectic extension of the classical complex geography plane. This work is joint with Inanc Baykur and Mustafa Korkmaz.
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  • Speaker: Daniel López-Neumann (Indiana University)
  • Title: Genus bounds for non-semisimple quantum invariants of knots (about arXiv:2211.15010)
  • Abstract: In this talk we explain how to define knot polynomials out of finite dimensional N-graded Hopf algebras whose degrees give lower bounds to the Seifert genus. This construction recovers known bounds, such as for twisted Alexander polynomials, but also provides new bounds for "non-semisimple" quantum invariants, such as the Akutsu–Deguchi–Ohtsuki invariants, which are defined out of representation theory of quantum sl(2) at a root of unity. This is joint work with Roland van der Veen.
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  • Speaker: Sungkyung Kang (Institute of Basic Science)
  • Title: One stabilization is not enough for contractible 4-manifolds
  • Abstract: We construct an example of a cork that remains exotic after taking a connected sum with 𝕊2 × 𝕊2. Combined with a work of Akbulut–Ruberman, this implies the existence of an exotic pair of contractible 4-manifolds which remains absolutely exotic after taking a connected sum with 𝕊2 × 𝕊2.
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  • Speaker: Irving Dai (Stanford University)
  • Title: The (2, 1)-cable of the figure-8 knot is not slice
  • Abstract: We prove that the (2, 1)-cable of the figure-eight knot is not smoothly slice by showing that its branched double cover bounds no equivariant homology ball. This is joint work with Sungkyung Kang, Abhishek Mallick, JungHwan Park, and Matthew Stoffregen.
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  • Speaker: Steven Sivek (Imperial College)
  • Title: Floer homology and non-fibered knot detection
  • Abstract: Floer homology can tell whether a knot is fibered, and this has led to proofs that both knot Floer homology and Khovanov homology can positively identify a small handful of knots: the unknot, the trefoils, the figure eight, and the cinquefoil. In this talk, I'll discuss recent work with John Baldwin in which we show for the first time that both invariants can also detect non-fibered knots, including 52, and that HOMFLY homology detects infinitely many knots. The key input is a classification of genus-1 knots which are "nearly fibered" from the perspective of knot Floer homology.
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[K-OS] Knot Online Seminar — spring semester 2022

  • Speaker: Claudius Zibrowius (Universität Regensburg)
  • Title: Towards mutation invariance of Khovanov homology
  • Abstract: Khovanov homology of knots is expected to be preserved under Conway mutation. In 2005, Bar-Natan proposed an almost-proof, which Wehrli subsequently made to work over the field of two elements. However, the conjecture remains open in general. Puzzlingly, it is known to be false for links.
    I will explain how a certain Fukaya category may help in implementing Bar-Natan's original strategy over arbitrary fields. This new perspective has already led to a proof of mutation invariance of Rasmussen invariants. It also gives the first conceptual explanation for the failure of the conjecture for links.
    This is joint work with Artem Kotelskiy and Liam Watson

  • Speaker: JungHwan Park (Korea Advanced Institute of Science and Technology)
  • Title: Definite fillings of lens spaces
  • Abstract: We consider the problem of determining the smallest (as measured by the second Betti number) smooth negative-definite filling of a lens space. We classify those lens spaces for which the associated negative-definite canonical plumbing is minimal. We also show that whenever the plumbing is minimal any other negative-definite filling for the given lens space has, up to stabilisations, the same intersection form. This is joint work with Paolo Aceto and Duncan McCoy.

  • Speaker: Benjamin Audoux (Aix–Marseille Université)
  • Title: Why care about welded theory
  • Abstract: This talk will deal with welded knotted objects. In a first part, I will discuss the different ways they can be defined and the different context they can be applied to. In a second part, I will discuss their connection with Wirtinger presentation; in particular, I will introduce two increasing sequences of quotients, one for finitely normally generated groups (k-reduced quotients) and one for welded objects (self wk-concordance), and show that, up to these quotients, the algebraic Wirtinger and the diagrammatic welded theories are the same. This is joint works and discussions with J-B. Meilhan and A. Yasuhara.

  • Speaker: Arunima Ray (Max Planck Institute für Mathematik)
  • Title: Slicing knots in definite 4-manifolds
  • Abstract: I will give an overview on questions related to slicing knots in 4-manifolds more general than the 4-ball. Then, I will focus on the ℂℙ2-slicing number of knots, which is by definition the smallest integer m such that the given knot bounds a properly embedded, null-homologous disk in m(ℂℙ2) minus a ball. Specifically, I will give a lower bound on the smooth ℂℙ2-slicing number and an upper bound on the topological ℂℙ2-slicing number, and describe situations where these quantities are large/small/distinct. The talk will be based on joint work with Alexandra Kjuchukova, Allison Miller, and Sumeyra Sakalli.

  • Speaker: Lisa Traynor (Bryn Mawr College)
  • Title: Legendrian torus and cable links
  • Abstract: In contact topology, an important problem is to understand Legendrian submanifolds; these submanifolds are always tangent to the plane field given by the contact structure. In fact, every smooth knot type will have an infinite number of different Legendrian representatives. A basic problem is to give the “Legendrian mountain range” of a smooth knot, which records all Legendrian representatives of the knot type.
    In topology, torus knots, torus links, and cable links form important families of knots and links. The mountain range classifications of all Legendrian torus knots have been established by Etnyre and Honda. I will explain the classification of all Legendrian torus links. In the process, we will see that there are interesting patterns on which tuples of points on the Legendrian mountain range of a torus knot can be realized as the components of a Legendrian torus link. We will also see that there are some Legendrian torus links that have smooth symmetries that cannot be realized by a Legendrian isotopy. I will also explain how these torus link statements have extensions to Legendrian cable links. These results are applications of convex surface theory.

  • Speaker: Kyle Hayden (Columbia University)
  • Title: Where are the complex curves in Khovanov homology?
  • Abstract: Since the advent of gauge theory, many modern tools exhibit a close connection with complex curves and a heightened sensitivity to objects from the complex realm. Surprisingly, this is true even for Khovanov homology, whose construction is combinatorial rather than geometric. I will discuss this in the context of joint work with Isaac Sundberg that uses Khovanov homology to study knotted surfaces in 4-space, especially (compact pieces of) complex curves in the 4-ball.
  • Slides.

  • Speaker: Michel Boileau (Aix–Marseille Université)
  • Title: Epimorphisms between knot groups and the SL(2,ℂ)-character variety
  • Abstract: A partial order on the set of prime knots is given by the existence of an epimorphism between the fundamental groups of the knot complements. In this talk we will survey some basic properties of this order, and discuss some results and questions in connection with the SL(2,ℂ)-character variety. In particular we will study to what extend the SL(2,ℂ)-character variety of the fundamental group of the knot complement helps to determine the knot. This includes some joint work with Teruaki Kitano, Steven Sivek, and Raphael Zentner.
  • Slides.
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  • Speaker: Jean Pierre Mutanguha (Institute for Advanced Studies)
  • Title: Canonical forms for free group automorphisms
  • Abstract: The Nielsen–Thurston theory of surface homeomorphisms can be thought of as a surface analogue to the Jordan Canonical Form. I will discuss my progress in developing a similar decomposition for free group automorphisms. (Un)Fortunately, free group automorphisms can have arbitrarily complicated behaviour. This forms a significant barrier to translating specific arguments that worked for surfaces into the free group setting; nevertheless, the overall ideas/strategies do translate!
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  • Speaker: Quentin Faes (Université de Bourgogne)
  • Title: Triviality of the J4-equivalence among homology 3-spheres
  • Abstract: Redemeister–Singer theorem states that, up to homeomorphism, any compact connected oriented 3-manifold can be obtained by gluing two handlebodies together. This connects the study of 3-manifolds to the study of the mapping class group of surfaces. For instance, one can get all homology 3-spheres by restricting the gluing map to be an element acting trivially on the homology of the surface, i.e. an element of the Torelli group. Another point of view is to say that one can get any homology 3-sphere from 𝕊3 by performing the following surgery : remove a handlebody and glue it back with an element of the Torelli group. Somewhat surprisingly, we shall prove in this talk that we can actually suppose this surgery to be performed with an element of the 4-th term of the Johnson filtration, i.e. an element acting trivially on the 4-th nilpotent quotient of the fundamental group of the surface. This result is an improvement of results obtained successively by Morita and Pitsch. It is obtained by using Goussarov–Habiro clasper calculus, a formula by Morita computing the Casson invariant of homology 3-spheres, and a formula by Kawazumi and Kuno that encodes the action of a Dehn twist on the fundamental group.
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  • Speaker: Charles Livingston (Indiana University)
  • Title: A survey of knot and link symmetry
  • Abstract: I will begin with a discussion of some basic symmetry properties of knots. This will include a proof that the trefoil knot, T(2,3), is not amphichiral (it is not isotopic to its mirror image) and that a specific knot is not reversible (it is not isotopic to its string orientation reverse). I will then discuss how such basic questions about knots and their symmetries are generalized to the setting of links. The talk will conclude with some recent results and open problems regarding link symmetry.
  • Slides.

  • Speaker: Thang T. Q. Le (Georgia Institute of Technology)
  • Title: Quantum trace map for SLn skein algebras of surfaces
  • Abstract: For a punctured surface there are two quantizations of the SLn character variety. The first quantization is the SLn skein algebra, and the second one is the quantization of the higher Teichmüller space. When n=2 Bonahon and Wong showed that there is an algebra homomorphism, called the quantum trace, from the first quantized algebra to the second one. We show for general n a similar quantum trace map exists. The construction of the SLn quantum trace is based on the theory of stated SLn skein algebra, developed in a joint work with A. Sikora. The construction of the quantum trace is a joint work with T. Yu.
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  • Speaker: Marco De Renzi (Universität Zürich)
  • Title: Quantum Invariants of 3-Manifolds and 4-Dimensional 2-Handlebodies
  • Abstract: A 4-dimensional 2-handlebody is a 4-manifold obtained from the 4-ball by attaching a finite number of 1-handles and 2-handles. A 2-deformation is a diffeomorphism implemented by a finite sequence of handle moves that never introduce 3-handles and 4-handles. Whether there exist diffeomorphisms that are not 2-deformations remains an open question, mainly due to the lack of invariants for detecting them. I will explain how to construct quantum invariants of 4-dimensional 2-handlebodies up to 2-deformation using unimodular ribbon categories, such as the category of representations of a unimodular ribbon Hopf algebra. In the case of factorizable ribbon categories, the invariant depends exclusively on the boundary. I will also discuss how this construction relates to a famous question in combinatorial group theory known as the Andrews–Curtis conjecture. Joint work with Anna Beliakova.
  • The talk has not been recorded, we apologize for the inconvenience.

  • Speaker: Paolo Ghiggini (Université de Nantes)
  • Title: Knot Floer homology and surface diffeomorphisms
  • Abstract: I will prove that knot Floer homology of a fibred knot, in the bottom-last Alexander grading, is isomorphic to a version of the fix point Floer homology of an area-preserving representative of the monodromy. This is a joint work with Gilberto Spano.
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  • Speaker: Oliver Singh (Durham University)
  • Title: Pseudo-isotopies and diffeomorphisms of 4-manifolds
  • Abstract: I will talk about pseudo-isotopy, a notion important for understanding self-diffeomorphisms of manifolds up to isotopy. Pseudo-isotopies of manifolds in dimensions 5 and up were understood in the 70s by work of Cerf for simply connected manifolds, and by Hatcher and Wagoner in the non-simply connected case, using invariants from algebraic K-theory. Quinn later proved Cerf’s result topologically in dimension 4, leading to a classification of self-homeomorphisms of simply connected 4-manifolds up to isotopy.
    I will talk about my work on what Hatcher and Wagoner’s K-theoretic invariants can say about pseudo-isotopies of non-simply connected 4-manifolds, and how they can be used to construct diffeomorphisms of certain 4-manifolds which are pseudo-isotopic but not isotopic to the identity.
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  • Speaker: Paula Truöl (ETH Zürch)
  • Title: The alternation number and the Upsilon-invariant at 1 of positive 3-braid knots
  • Abstract: The alternation number of a knot is the minimal number of crossing changes needed to deform the knot into an alternating knot, i.e. a knot with a diagram where the crossings alternate between over- and under-crossings as one travels around the knot. The tau- and the Upsilon-invariant from knot Floer homology give a lower bound on the alternation number of any knot. We use this lower bound and an upper bound by Abe and Kishimoto to determine the alternation numbers of all positive 3-braid knots.
    The key tool and a result of independent interest is an explicit calculation of the Upsilon invariant at 1 of all 3-braid knots. We determine this integer-valued (concordance) invariant – which was defined by Ozsváth, Stipsicz and Szabó – by constructing cobordisms between 3-braid knots and (connected sums of) torus knots. In particular, we will only work with properties of Upsilon and not its definition, so no background in knot Floer homology will be assumed.
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  • Speaker: Linh Truong (University of Michigan)
  • Title: A slicing obstruction from the 10/8+4 theorem
  • Abstract: Donald and Vafaee constructed a knot slicing obstruction for knots in the three-sphere by producing a bound relating the signature and second Betti number of a spin 4-manifold whose boundary is zero-surgery on the knot. Their bound relies on Furuta's 10/8 theorem. In this talk, I will explain an improvement on this slicing obstruction by using the 10/8 + 4 theorem of Hopkins, Lin, Shi, and Xu.
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  • Speaker: Paul Wedrich (Universität Hamburg)
  • Title: A skein relation for singular Soergel bimodules
  • Abstract: Soergel bimodules categorify Hecke algebras and lead to invariants of braids that take values in monoidal triangulated categories. In this process, the quadratic `skein relation' on Artin generators is promoted to a distinguished triangle. I will talk about an analog of this relation in the setting of singular Soergel bimodules and Rickard complexes, in which the distinguished triangle gets replaced by a longer one-sided twisted complex. Joint work with M. Hogancamp and D.E.V. Rose.
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  • Speaker: Olga Plamenevskaya (Stony Brook University)
  • Title: Unexpected Stein fillings, rational surface singularities, and line arrangements
  • Abstract: A link of an isolated complex surface singularity is a 3-manifold obtained by intersecting the surface with a small sphere centered at the singular point. The link carries a canonical contact structure, given by the complex tangencies. Milnor fibers of possible smoothings of the singular point give Stein fillings for this contact structure; deformations of the singularity give rise to Stein cobordisms between the corresponding links. This leads to many interesting questions regarding the interplay of algebraic geometry, topology, and symplectic geometry.
    After setting the stage, we will focus on the following question: do Milnor fibers and the resolution of a surface singularity yield ALL possible Stein fillings of its link? This is known to hold in some simple cases, eg for lens spaces. We show that even in the "next simplest" case, for many rational singularities, there is a plethora of "unexpected" Stein fillings that do not arise from Milnor fibers of any smoothings. To compare fillings and smoothings, we use T.de Jong-D.van Straten's description of Milnor fibers of certain rational surface singularities in terms of deformations of associated singular plane curves. On the symplectic side, we develop an analogous description of the Stein fillings via certain more general arrangements of curves. "Unexpected" arrangements then give rise to "unexpected" fillings. (Joint work with L. Starkston.)
    All the discussion will be from the topological perspective, with minimal input from algebraic geometry.
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  • Speaker: Filip Misev (Universität Regensburg)
  • Title: Families of fibred knots with the same Seifert form
  • Abstract: We will construct an infinite family of fibred, quasipositive knots having all the same Seifert matrix. The knots can be distinguished by the dilatation of their geometric monodromy. I will explain what these properties mean and why such families of knots might be of interest for the study of knot concordance.
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[K-OS] Knot Online Seminar — winter semester 2021

  • Speaker: Roland van der Veen (Rijksuniversiteit Groningen)
  • Title: Hopf algebras and 3-manifolds
  • Abstract: Why do Hopf algebras turn up so often in studying (quantum) invariants of 3-manifolds? What is their three-dimensional significance? We argue that any Hopf algebra expression can be interpreted as a marked (sutured, framed) 3-manifold and vice versa. By a Hopf algebra expression we mean any composition of (co)-products and antipodes. Composition of the Hopf algebra maps should correspond to gluing the marked 3-manifolds appropriately. For closed 3-manifolds our correspondence is inspired by the Kuperberg invariant. This is joint work in progress with Daniel Neumann and Dylan Thurston.
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  • Speaker: Richard Schwartz (Brown University)
  • Title: On the optimal paper Moebius band
  • Abstract: An old question of Halpern and Weaver asks what is the smallest aspect ratio of a Moebius band made out of paper that can be isometrically embedded in ℝ3. In my talk I will recall previous results about this, and sketch a proof of my improved lower bound for the answer. I will also explain how I reduced the whole conjecture to showing that a handful of complicated algebraic expressions are positive. Unfortunately, these expressions are too complicated for me to figure out.
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  • Speaker: Cristina Anghel (Université de Genève)
  • Title: Coloured Jones and coloured Alexander invariants from two Lagrangians intersected in a symmetric power of a surface
  • Abstract: The theory of quantum invariants started with the Jones polynomial and continued with the Reshetikhin–Turaev algebraic construction of invariants. In this context, the quantum group Uq(𝔰𝔩(2)) leads to the sequence of coloured Jones polynomials, and the same quantum group at roots of unity gives the coloured Alexander polynomials.
    We construct a unified topological model for these two sequences of quantum invariants. More specifically, we prove that the Nth coloured Jones and Nth coloured Alexander invariants are different specialisations of a state sum of Lagrangian intersections in configuration spaces. As a particular case, we see both Jones and Alexander polynomials from the same intersection pairing in a configuration space.
    Secondly, we present a globalised model without state sums. We show that one can read off both coloured Jones and coloured Alexander polynomials of colour N from a graded intersection between two explicit Lagrangians in a symmetric power of the punctured disk.
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  • Speaker: Gordana Matic (University of Georgia)
  • Title: Fillability of contact structures and the spectral order invariant
  • Abstract: Determining tightness and fillability of contact structures are central questions in contact topology. I will talk a bit about history, and then describe a refinement of the contact invariant of Ozsváth and Szabó in Heegaard Floer homology, the "spectral order" we defined in joint work with Cagatay Kutluhan, Jeremy Van-Horn Morris and Andy Wand. I will talk about some calculations in examples where spectral order can be used as obstruction to Stein fillability.
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  • Speaker: Allison Miller (Swarthmore College)
  • Title: Satellite operators and knot concordance
  • Abstract: The classical satellite construction associates to any pattern P in a solid torus and companion knot K in the 3-sphere a satellite knot P(K), the image of P when the solid torus is ‘tied into’ the knot K. This operation descends to a well-defined map on the set of (smooth or topological) concordance classes of knots. Many natural questions about these maps remain open: when are they surjective, injective, or bijective? How do they behave with respect to measures of 4-dimensional complexity? How do they interact with additional group or metric space structure on the concordance set?
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Exceptionnally at 15:00 CET
  • Speaker: Liam Watson (University of British Columbia)
  • Title: Immersed curve invariants in low-dimensions
  • Abstract: Various relative versions of homological invariants studied in low-dimensional topology have been recast in terms of immersed curves. These curves live in a surface that can be identified with the boundary of the object in question, and as such they provide insight into essential surfaces in the relevant setting. This story starts with Heegaard Floer homology in joint work with Jonathan Hanselman and Jake Rasmussen. I will focus on some sample applications in an attempt to show how this machinery works.

  • Speaker: Paolo Bellingeri (Université de Caen)
  • Title: Virtual Artin groups
  • Abstract: Virtual braid groups were introduced as a braid counterpart of virtual knots. From the combinatorial point of view it is interesting to remark that the virtual braid group VBn admits two surjective homomorphisms onto the symmetric group Sn. The kernels of these two homomorphisms have different meanings and applications: the first one, the virtual pure braid group VPn, coincides with the quasitriangular group QTrn considered by L Bartholdi, B Enriquez, P Etingof, E Rains in relation with Yang–Baxter equations, while the second one, usually denoted KBn, is an Artin group and it turns out to be a powerful tool to study combinatorial properties of VBn.
    Starting from the observation that the standard presentation of a virtual braid group mixes the presentations of the corresponding braid group Bn and of the corresponding symmetric group Sn together with the action of the symmetric group on its root system, we define for any Coxeter graph Γ a virtual Artin group VA[Γ] with a presentation that mixes the standard presentations of the Artin group A[Γ] and of the Coxeter group W[Γ] together with the action of W[Γ] on its root system. As in the case of VBn, we will define two surjective homomorphisms from VA[Γ] to W[Γ]: we will provide group presentations for these kernels (completely determined by root systems) and we will show several and general results on virtual Artin groups.
    This is a joint work with Luis Paris and Anne-Laure Thiel.
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  • Speaker: Dror Bar-Natan (University of Toronto)
  • Title: Yarn-ball knots — A modest light conversation on how knots should be measured
  • Abstract: Let there be scones! Our view of knot theory is biased in favour of pancakes. Technically, if K is a 3D knot that fits in volume V (assuming fixed-width yarn), then its projection to 2D will have about V4/3 crossings. You'd expect genuinely 3D quantities associated with K to be computable straight from a 3D presentation of K. Yet we can hardly ever circumvent this V4/3>>V "projection fee". Exceptions include linking numbers (as we shall prove), possibly include the hyperbolic volume, and likely include finite type invariants (as we shall discuss in detail). But knot polynomials and knot homologies seem to always pay the fee. Can we exempt them? Joint with Itai Bar-Natan, Iva Halacheva, and Nancy Scherich. All the material for this talk, including up-to-date title and abstract, is available here.
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  • Dror made a small mistake towards the end of the talk, which changes some of the numbers throughout the talk but does not change anything qualitative (namely, for finite type, 3D seems to beat 2D, with the same algorithms). A correction will be posted here soon.

  • Speaker: Renaud Detcherry (Université de Bourgogne)
  • Title: A quantum obstruction to purely cosmetic surgeries
  • Abstract: The cosmetic surgery conjecture asks whether it is possible that two Dehn-surgeries on the same non-trivial knot in 𝕊3 give the same oriented 3-manifolds. We will present new obstructions for a knot to admit purely cosmetic surgeries, using Witten–Reshetikhin–Turaev invariants. In particular, we will show that if a knot K admits purely cosmetic surgeries, then the slopes of the surgery are ±1/5k unless the Jones polynomial of K is 1 at the fifth root of unity.
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  • Speaker: Patrick Orson (Max Planck Institute for Mathematics)
  • Title: Mapping class group of simply-connected 4-manifolds
  • Abstract: We study the mapping class group of a compact simply-connected 4-manifold; that is the set of self-diffeomorphisms (or self-homeomorphisms, in the topological category), up to isotopy. For a manifold with nonempty boundary, one assumes the self-automorphisms fix the boundary pointwise. In both the smooth and topological categories, we provide sufficient conditions for two automorphisms to be pseudoisotopic. Pseudoisotopy is weaker than isotopy, but in the topological category we are able to use this theorem to compute the mapping class group in many cases. We use our theorem to prove new topological unknotting results for embedded 2-spheres in many classes of 4-manifold. This is joint work with Mark Powell.

  • Speaker: Benjamin Ruppik (Max Planck Institute for Mathematics)
  • Title: Handlebodies, trivial tangles and group trisections for knotted surfaces
  • Abstract: This is reporting on a project started at the Summer Trisectors Workshop 2020 with Sarah Blackwell, Rob Kirby, Michael Klug, and Vincent Longo. We give an algorithmic proof based on Stallings folding of the following folklore fact: An epimorphism from a surface group to a free group is geometrically realized by a handlebody bounding a surface. Similarly, special epimorphisms from free groups to free groups are geometrically realized by a trivial tangle in a handlebody. Pairs of these epimorphisms determine bridge splittings of links in a Heegaard splitting of a 3-manifold [extending work of Stallings and Jaco]. Triples of epimorphisms with pairwise compatibility describe bridge trisections of surfaces in 4-manifolds [extending Abrams–Gay–Kirby]. Consequently, isotopy classes of smoothly embedded surfaces in 4-manifolds are in bijection with group trisections of knotted surface groups up to an equivalence relation.
  • Slides.
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  • Speaker: Maggie Miller (Stanford University)
  • Title: Exotic Brunnian surfaces
  • Abstract: For any integer n>1, I will show how to construct pairs {F,G} of n-component surface links in the 4-ball that are exotic (i.e. topologically but not smoothly equivalent) yet Brunnian (trivial if any single component is deleted). This means that corresponding components of F and G are smoothly equivalent, and in fact every proper sublink of F and G are smoothly equivalent, yet somehow F and G as wholes fail to be smoothly equivalent -- suggesting a very subtle exotic behavior that cannot be localized to any component of F or G. This is joint work with K. Hayden, A. Kjuchukova, S. Krishna, M. Powell, and N. Sunukjian.
  • Notes.
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  • Speaker: Daniel Cristofaro-Gardiner (University of Maryland)
  • Title: The Simplicity Conjecture
  • Abstract: In the 60s and 70s, there was a flurry of activity concerning the question of whether or not various subgroups of homeomorphism groups of manifolds are simple, with beautiful contributions by Fathi, Kirby, Mather, Thurston, and many others. A funnily stubborn case that remained open was the case of area-preserving homeomorphisms of surfaces. For example, for balls of dimension at least 3, the relevant group was shown to be simple by work of Fathi from the 1970s, but the answer in the two-dimensional case was not known. I will explain recent joint work proving that the group of compactly supported area preserving homeomorphisms of the two-disc is in fact not a simple group, which answers the "Simplicity Conjecture" in the affirmative. Our proof uses a new tool for studying area-preserving surface homeomorphisms, called periodic Floer homology (PFH) spectral invariants; these recover the classical Calabi invariant of monotone twists. I will also briefly mention a generalization of our result to compact surfaces of any genus.
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  • Speaker: Sebastian Baader (Universität Bern)
  • Title: Signature spectrum of positive braids
  • Abstract: We derive a lower bound for all Levine–Tristram signatures of positive braid links, linear in terms of the first Betti number. As a consequence, we obtain upper and lower bounds on the ratio of fixed pairs of Levine–Tristram signature invariants, valid uniformly on all positive braid monoids.
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[K-OS] Knot Online Seminar — spring semester 2021

  • Speaker: Christian Blanchet (Université de Paris)
  • Title: Heisenberg homology on surface configurations.
  • Abstract: Motivated by the famous Lawrence–Krammer–Bigelow representations of the classical braid groups, we define and study homology of unordered configuration spaces of a surface with one boundary component, with a local system which is based on Heisenberg extension of the homology group. We show that for any positive genus this Heisenberg extension is the quotient of the braid group of the surface which forces centrality of the classical generators.
    We compute the homology groups and study the action of mapping classes. We obtain a representation of the Chillingworth subgroup, a representation of a central extension of the Torelli subgroup and a twisted representation of the whole Mapping Class Group. We give explicit computation in the case of 2 points configurations.
    This is joint work with Martin Palmer and Awais Shaukat.
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  • Speaker: Vincent Colin (Université de Nantes)
  • Title: Variations on Khovanov homology.
  • Abstract: We use a generalization of Heegaard–Floer homology to higher dimension to propose variants of Khovanov homology for links in 𝕊3 and braids in fibration over the circle. Along the way, several algebras of geometric origin arise. This is joint work in progress with Ko Honda and Yin Tian.
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  • 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.
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  • 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.
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  • 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.
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  • 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.
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  • 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.
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  • 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.
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  • 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.
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  • 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.
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  • 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.
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  • 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.
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  • 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.
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  • 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.
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  • 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.
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  • 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.
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  • 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.
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  • 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.
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  • 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.
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  • 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.
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  • 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.
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[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.
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  • 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.
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  • 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).
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  • 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.
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  • 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).
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  • 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.
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  • 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.
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  • 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.
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  • 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.
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  • 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.
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  • 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)
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  • 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.
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  • 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.
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  • 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.
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[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.
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  • 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.
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  • 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.
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  • 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.
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  • 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.
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  • 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.
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  • 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.
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  • 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.
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  • 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.
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  • 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.
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  • 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.
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  • 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.
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  • 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.
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Credit for the jingle: Daniel Maszkowicz.

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