This workhsop will take place in Aussois from the 10th to the 15th of April 2022. More informations soon!
Presentation of the workshop
This workshop aims at bringing together mostly young
researchers to spend one week studying a topic in low
dimensional topology. Part of the week is dedicated to
detailed talks taking time to introduce the background of
the theory necessary to tackle more recent questions. Time
is voluntarily made available to encourage discussion that
the presentations will generate.
For this second edition, the object of study is the
relation between knot theory and four dimensional
topology. After series of talks presenting relations
between handle decomposition of four manifolds and knot
theory, we will see how the notion of sliceness for links
is related to construction of exotic pairs of
manifolds. Then we will have discussions on invariants of
knots that could measure such properties. We will focus
mostly on invariants coming from quantum topology such as
the Jones polynomial and its derivatives. It will be the
good occasion to study topological notions in dimension 4
surrounding this framework.
Chalk drawings by Jacques Darné
The full program can be downloaded here
Slice/Ribbon and knot concordance — Stavroula Makri.
A starting reference for this talk might be the first chapter of Arunima
Ray’s lecture notes at Winter Braid [Ray]. In particular the speaker should
take some time to explain the ”movie” picture, and to prove Proposition
1.3 therein. There exist many other references, including [Owen, Tei]
For knot concordance, a good starting point is the second section of
Teichner’s lectures notes [Tei]. In particular, Theorem 2.4, Proposition
2.6 and Corollary 2.11 therein should be presented. Note that a proof
of Theorem 2.11, will be a central ingredient of the talk of Fox–Milnor’s
theorem. Other possible sources are [Owen, Ray, Liv].
Handle decompositions and Kirby calculus I — Quentin Faes and
Pablo Toro Sánchez.
talk could explain what are handles and handle decompositions (see [GS,
Section 4.1]), why every smooth manifold admits handle decompositions
(see [GS, Section 4.2]), how to relate different handle decompositions
of the same manifold by handle moves (see [GS, Proposition 4.2.9] for
creation/removal of canceling pairs, [GS, Definition 4.2.10] for handle
slides, and [GS, Theorem 4.2.12] for the completeness of this set of moves),
how to visualize and represent diagrammatically handles and handle
decompositions (see [GS, Section 4.4] for a brief discussion of 1∕2-handles
and an explanation of why we don’t need to represent 3∕4-handles in
the closed case, see [GS, Section 5.4] for the dotted circle notation
for 1-handles), and how to visualize and represent diagrammatically
handle moves (see [GS, Section 5.1] for creation/removal of canceling
pairs, handle addition/subtraction, blow-ups/blow-downs). Other possible
references: [FM, Sections 9.2 & 9.3], [Sc, Sections 1.2, 1.3, & 2.5], [Ak,
Handle decompositions and Kirby calculus II — Quentin Faes and
Pablo Toro Sánchez.
This talk could explain how to represent diagrammatically handle
decompositions of disc bundles (see [GS, Example 4.1.4, Exercise 4.1.6]),
projective spaces (see [GS, Examples 4.2.4 & 4.4.2, Exercise 4.2.5]),
sphere bundles (see [GS, Section 4.6 up to Exercise 4.6.1]), plumbings (see
[GS, Example 4.6.2]), doubles (see [GS, Example 4.6.3]), bundles over
surfaces (see [GS, Example 4.6.5]), products with 3-manifolds (see [GS,
A knot which would be topologically but not smoothly slice
sliceness produces an exotic ℝ4 — Arno Wildi.
The aim of this talk is to give details of the construction sketched in
Arumina Ray’s lecture notes [Ray]. In particular, one would need to give
details about Quinn’s theorem and to explain what an exotic ℝ4 would
look like. References: [BKKPR, GS, FQ].
Fox–Milnor theorem — Manos Manouras.
The Fox–Milnor theorem asserts that the Alexander polynomial of a slice
knot factorizes as f(t)f(t-1). Elementary proof for ribbon knots. Proof
for general slice knots [Lic, Proposition 8.17].
The Jones polynomial of ribbon links — Emmanuel Graaf.
The idea is to study the behaviour of (this time) the Jones polynomial
for ribbon knots following [Eis]. The first property (Theorem 1) says that
for an n-component ribbon link, the nullity (multiplicity of the zero at
q = i) of the Jones polynomial is exactly n - 1. The definition of the
Jones polynomials in terms of skein relations is expected (Theorem 2.2),
without any proof. A basic algebraic study (Sec. 2, involving usual notions
of Seifert surfaces/matrix, signature...) should be conducted leading to the
upper bound (Lemma 1) of the Theorem 1 (for any link). For the lower
bound part, the notion of band diagram and ribbon link could be recalled
from the Talk 1 of the workshop. The strategy of proof for Prop. 4.1 (lower
bound for ribbon links) has to be presented: applying Kauffman skein
relations to ribbon band moves. The consequence of Theorem 1 could be
outlined: first the notion of surface invariants of finite type w.r.t. band
crossing changes (Sec. 5), second the statement of Theorem 2.
Trisection I: existence and examples — Anthony Saint-Cricq.
Definition (unbalanced), stabilization moves, proof of existence via
Morse functions [LCM, Section 2.1], diagrams, relation between trisection
diagrams and Kirby diagrams (both directions). [GK, Gay]
Trisection II: Uniqueness — Delphine Moussard.
”Two trisections of a given smooth 4–manifold can be made isotopic
by some number of stabilizations” [GK]. Proof via Morse functions and
homotopy between Morse functions.
Rohlin theorem, Rohlin invariant — Daniel Galvin.
The aim of this talk is to introduce the audience to the intersection form on
4 manifolds. A good occasion for that is to give a proof of Rohlin theorem.
Reference: Saveliev, ”Lecture on the topology of 3-manifolds”,[Sav]. Then
saying a word on the Rohlin invariant might make no harm. If time
permits, say a word on the surgery formula for Rohlin invariant.
Freedman theorem — Yohan Mandin.
The aim of this talk is to give some insight on the proof of the theorem
ΔK = 1 implies K is topologically slice. The main reference might be
[BKKPR, Section 21.6.3], see also [FQ] and [GT, Appendix]. It seems
that part of the argument relies on a delicate Theorem (Disc embedding
theorem), and we clearly do not have time to provide a proof of this
technical important point. The speaker should rather explain how the
theorem of Freedman is related to this point, and say what is possible to
say in the given time.
The RGB link construction and candidates for exotic spheres —
We will outline a construction of Piccirillo and Manolescu that produces
many candidates for exotic smooth structures on S4. We will explain how
pairs of knots K,K′ with homeomorphic 0-surgeries but with K slice, K′
non slice may be used to produce an exotic sphere.
Then we will discuss the RGB link construction that produces many
pairs of knots (K,K′) with diffeomorphic traces, and how to extract good
candidates from this construction.
Khovanov and Lee’s homologies: — Léo Schelstraete.
We will sketch the construction of Khovanov’s link homology Kh(L) that
categorifies the Jones polynomial. We will also present a variation of
Khovanov’s homology Kh′(L), introduced by Lee, which is related to
Khovanov’s by a spectral sequence and we will show that for a knot K,
one has simply has Kh′(K) = ℚ ⊗ ℚ as vector space. The information in
Lee’s homology is purely contained in the gradings; we will discuss how
to compute gradings and bases for Lee’s homology, and how to sometimes
compute Lee’s homology from Khovanov homology using the spectral
References: [Ras10] [Lee02]
Rasmussen’s invariant and the Conway knot — Hyeonhee Jin.
We will explain how the grading information contained in Lee’s homology
gives rise to a concordance invariant (now known as Rasmussen’ s
invariant). We will explain how this invariant can be compared to other
classical concordance invariants such as knot signatures.
As a second part of the talk, we will review some new techniques of
Piccirillo. We will review how knots with diffeomorphic knot traces can
be used to obstruct sliceness, and will sketch the RGB link construction
that produces examples of pairs of such knots. Finally, we will apply this
technique to the case of the Conway knot, and deduce the non sliceness of
the Conway from the non sliceness of another knot with the same trace,
for which the s invariant can obstruct sliceness.
Broken surface diagrams and Roseman moves for knotted surfaces — Jacques Darné.
In this talk the speaker should define what a knotted surface in dim. 4
means. Then the language of broken surface diagrams must be presented:
namely how to represent an embedding of a surface in a 4-manifold using
these objects [CS, CKS]. The theorem of Roseman [R] will be presented,
at least displaying the moves on broken surfaces diagrams under which
the knotted surface is unchanged. They are an analogue of Reidemeister
moves for knot diagrams. We expect the speaker to explain this invariance
for some moves, at least giving a flavour [R, CS]. Why Roseman set of
moves is sufficient should be mentioned and maybe outlined [R], as well as
the redundancy in the moves [Y, K]. As an illustration, the speaker will
sketch the proof (only a few cases) that tricolorability is an invariant of
knotted surfaces, by showing that it remains unchanged under Roseman
moves [D]. The construction of this talk could be helped by [AM].
Cut-diagrams and Milnor-type invariants for knotted surfaces —
In this talk, we study another category of diagrams associated with
knotted surfaces, called cut-diagrams [AMY]. They can be seen as a
generalization of Gauss diagrams for knots. The speaker will explain the
construction of cut diagram from a broken surface diagram, then the
general definition of cut diagrams of [AMY]. A translation of Roseman
moves in terms of cut-diagrams moves will also be presented. The question
of the injectivity of the map from knotted surfaces to cut diagrams
will be discussed (Tanaka’s result that the peripheral system is not a
complete invariant [T], unlike in the knot case [W], should be mentioned).
If time allows, the speaker will define Milnor-like invariants [Mi] associated
with cut diagrams, and explain why it is a concordance invariant and/or
why it vanishes on knotted spheres [AMY]. The speaker could also show
how tricolorability of knotted surfaces extends to cut-diagrams. The
construction of this talk could be helped by [AM].
Exotic surfaces and Khovanov homology — Laura Marino.
The aim of this talk is to explain the strategy of Hayden and Sundberg
[HS] to construct exotic pairs of surfaces in B4, these surfaces are
isotopic through ambient homeomorphisms but not through ambient
diffeomorphism. The main tool is to evaluate their corresponding classes
in Khovanov homology against a well-chosen element.
- Giulio Belleti
- Léo Bénard
- Boris Colombari
- Marco De Renzi
- Renaud Detecherry
- Jacques Darné
- Quentin Faes
- Daniel Galvin
- Emmanuel Graff
- Hyenhee Jin
- Corentin Lunel
- Stavroula Makri
- Yohann Mandin
- Manos Manouras
- Laura Marino
- Jules Martel
- Gregor Masbaum
- Jean-Baptiste Meilhan
- Delphine Moussard
- Louis-Hadrien Robert
- Anthony Saint-Criq
- Leo Schelstraete
- Pablo Toro-Sanchez
- Paula Truöl
- Arno Wildi
- Bangwin Wang
- Hao Xu
The full program can be downloaded here
Monday, April 11th
- 09:30-10:30: Makri
- 11:00-12:00: Sánchez
- 14:30-15:30: Faes
- 16:00-17:00: Wildi
Tuesday, April 12th
- 09:30-10:30: Manouras
- 11:00-12:00: Graff
- 14:30-15:30: Saint-Cricq
- 16:00-17:00: Moussard
Wednesday, April 13th
- 09:30-10:30: Galvin
- 11:00-12:00: Mandin
Thursday, April 14th
- 09:30-10:30: Truöl
- 11:00-12:00: Schelstraete
- 14:30-15:30: Jin
- 16:00-17:00: Marino
Friday, April 15th
- 09:30-10:30: Darné
- 11:00-12:00: Colombari