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.
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.
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.
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.
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.
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!
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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?
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Abstract: In the early 90's, X.S. Lin defined a
Casson-type invariant of knots in 𝕊3 by counting
representations π1(𝕊3∖K) → 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.
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.
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
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.
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
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.
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.
Title: Non-semisimple quantum invariants of 3-Manifolds from the Kauffman bracket.
Abstract: In recent years, several constructions in
the field of quantum topology have been extended to the
non-semisimple case, producing TQFTs with remarkable new
properties. At present, all the different approaches
rely on a rather elaborate technical setup, involving
either the structure of Hopf algebras, or more abstract
categorical machineries. In this talk, we will explain
how the family of non-semisimple quantum invariants
associated with the small quantum group of
𝖘𝖑2 at odd roots of unity can
be reformulated in purely combinatorial and diagrammatic
terms, using only Temperley–Lieb categories and Kauffman
bracket polynomials. Based on joint work with
C. Blanchet and J. Murakami.
Speaker: Anthony Conway (Massachusetts Institute of Technology).
Title: Knotted surfaces with infinite cyclic knot group.
Abstract: This talk will concern embedded surfaces in
4-manifolds for which the fundamental group of the
complement is infinite cyclic. Working in the
topological category, necessary and sufficient
conditions will be given for two such surfaces to be
isotopic. This is based on joint work with Mark
Powell.
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.
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) ≤ 2g4(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 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.
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).
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, 𝕊1, or 𝕊2), 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 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.
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⊂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(𝕊3∖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 ℂℙ2 # -ℂℙ2). Joint work with Peter
Feller, see arXiv:1905.08305.
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.
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.
