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.
Title: Non-rational, non-cuspidal plane curves
via Heegaard Floer homology.
Abstract: Let C be a complex curve in
ℂℙ^{2}. 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
ℂℙ^{2}, for which "genus cannot be
traded for double points". This is a joint project with
Beibei Liu and Ian Zemke.
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.
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.
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.
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
𝕊^{1}×𝕊^{3}, 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.
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 BV_{2} 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 BV_{n}(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 BV_{n}(H) and some other
cases. This is a joint work with Julio Aroca.
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
𝖘𝖑_{2} 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.
04/02/2021
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.
Abstract: A knot in 𝕊^{3} 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.
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.
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) ≤ 2g_{4}(K)
– 1 + n
for all n-stranded braids β with closure a
knot K.
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.
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 LB_{n}.
Since LB_{n} contains a copy of the braid group B_{n} as a subgroup, a natural approach to look for linear representations is to extend known representations of the braid group B_{n}. 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 LB_{n} we quotient its group algebra by the ideal generated by (σ_{i} + 1)(σ_{i} – 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 LH_{n}. 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.
Title: Twisted L^{2} 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 L^{2} 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 L^{2} 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).
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.
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, 𝕊^{1}, or 𝕊^{2}), while emphasizing the
techniques that keep these proofs straightforward (and
perhaps even gauge-theory free).
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.
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.
Title: The mapping class group of connect sums of 𝕊^{2} × 𝕊^{1}.
Abstract: Let M_{n} denote the connect sum of n copies of 𝕊^{2} × 𝕊^{1}.
Laudenbach showed that the mapping class group Mod(M_{n}) is an extension
of the group Out(F_{n}) by (ℤ/2)^{n}, where the latter group is the "sphere
twist" subgroup of Mod(M_{n}). 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.
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.
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.
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)
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.
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.
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.
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.
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.
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.
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)
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.
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.
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).
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.
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.
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⊂𝕊^{3}. 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 π_{1}(𝕊^{3}∖L). 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.
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 ℂℙ^{2} # -ℂℙ^{2}). Joint work with Peter
Feller, see arXiv:1905.08305.
A technical problem occured during the recording. Only the first 23 minutes of the talk are available.
09/04/2020
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 SL_{2}(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.
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.