Knot concordance and low-dimensional manifolds
17–21 June 2019
Le Croisic

Schedule:

All lectures are going to be held at the Domain de Port-aux-Rocs (Le Croisic). (Hover for the title, click for the abstract.)

	Monday	Tuesday	Wednesday	Thursday	Friday
9:30-10:30		Lewark	Kjuchukova	Ray	M. Miller
10:30-10:50	Kick off	Coffee break	Coffee break	Coffee break	Coffee break
10:50-11:50	Kim	(11:00) Nagel	Baader	(11:00) Lecuona	Rasmussen
11:55-12:55	Stipsicz		Limouzineau		Liechti
13:00-14:30	Lunch	Lunch	Lunch	Lunch	Lunch
14:30-15:30	A. Miller	Piccirillo		Politarczyk
15:30-16:00	Coffee break	Coffee break		Coffee break
16:00-17:00	Schwartz	Zentner		Allen

Abstracts:

Samantha Allen (Dartmouth College)
Nonorientable surfaces bounded by knots: a geography problem
The 4-genus of a knot K is the minimal genus of a surface in the 4-ball whose boundary is K. Similarly, one can define the nonorientable 4-genus, γ₁(K), as the minimal first Betti number of a surface in the 4-ball whose boundary is the knot K. Current methods for computing γ₄(K) exploit the relationship between the first Betti number, b₁(F), and the normal Euler number, e(F), of nonorientable surfaces F in B⁴ such that ∂F = K. In this talk, I will introduce these definitions and consider the geography problem related to this relationship. Namely, given a knot K, what pairs (e(F), b₁(F)) are realizable for F a nonorientable surface in B⁴ bounded by K. When considering this problem, we see an interplay between classical invariants and Heegaard Floer invariants.

Sebastian Baader (University of Bern)
Four-genus and twist regions of random braids
We show that the smooth 4-genus of a closed random braid of length N is of the order sqrt(N). As an application, we derive a lower bound on the average number of twist regions in the Garside normal form of a random braid: N/3. This is heavily based on our recent collaboration with Alexandra Kjuchukova, Lukas Lewark, Filip Misev, Arunima Ray: On the average four-genus of two-bridge knots.

Min Hoon Kim (KIAS)
Freely slice good boundary links
The still open topological surgery conjecture for 4-manifolds is equivalent to the statement that all good boundary links are freely slice. In this talk, I will show that every good boundary link with a pair of derivative links on a Seifert surface satisfying a homotopically trivial plus assumption is freely slice. This subsumes all previously known methods for freely slicing good boundary links with two or more components, and provides new freely slice links. This is joint work with Jae Choon Cha and Mark Powell.

Alexandra Kjuchukova (MPIM)
Non-cyclic branched covers of S⁴
Let F be an orientable surface embedded in S⁴ locally flatly except for one point where F looks like the cone on a knot K. When S⁴ admits a connected three-fold non-cyclic cover branched along F, the signature of this cover gives rise to a ribbon obstruction for K. I'll define this obstruction and give examples where it does not vanish. (Sadly, none of these non-ribbon knots are slice; we are still looking.) Parts of this work are joint with Cahn, Geske and Shaneson.

Ana G. Lecuona (University of Glasgow)
The slope of a link: computation and applications
Together with Alex Degtyarev and Vincent Florens we continue to investigate the properties of the link invariant we introduced, the slope. In this talk we will recall how it is defined for colored links in homology spheres and we will use various examples to discuss different methods for computing it, using C-complexes and Fox calculus.

Lukas Lewark (University of Regensburg)
Concordance and Khovanov–Rozansky homologies
Khovanov–Rozansky homologies give rise to a homomorphism from the smooth concordance group to an abelian group consisting of certain indecomposable chain complexes. The concordance information contained in this homomorphism appears to be "orthogonal" to the information provided by knot Floer homology, where a similar construction exists. For example, the homomorphism sends every quasipositive (or squeezed) knot to the same chain complex as the unknot, up to a grading shift; but the images of quasi-alternating knots may be more complicated chain complexes. As an application, we find an infinitely generated free subgroup in the quotient of the concordance group by quasipositive knots. This talk is based on past and ongoing work with Andrew Lobb, and does not presume previous knowledge of Khovanov–Rozansky homologies.

Livio Liechti (University of Fribourg)
Overcommutation in groups and the Heegaard genus
We introduce the notion of overcommutation of two group elements and define a corresponding overcommutation length. We then discuss the close connection of these concepts to the existence and the Heegaard genus of knot complements (in 3-manifolds) whose fundamental groups admit representations with prescribed boundary values. This is joint work with J. Marché.

Maÿlis Limouzineau (University of Cologne)
Legendrian concordance and generating functions
There are many more Legendrian knots than smooth knots in the standard 3-dimensional contact space. To tidy them up, we can ask if or if not they admit a generating function, and how many of them do. Here we already get an invariant up to Legendrian isotopy, a complicated one however. In this talk, we will see the behaviour of this invariant with Legendrian cobordisms. For instance, they combine perfectly to construct an analogue of the smooth knot concordance group.

Allison Miller (Rice University)
New lower bounds on the 4-genera of knots
A knot is slice if it bounds a embedded disc in the 4-ball. There are (at least) two natural generalizations of sliceness: one might weaken either `disc' to `small genus surface' or `the 4-ball' to `any 4-manifold that is simple in some sense.' In this talk, I'll discuss joint work with Jae Choon Cha and Mark Powell that gives new evidence that these two approaches measure very different things. Our tools include Casson–Gordon style representations of knot groups, L² signatures of 3-manifolds, and the notion of a minimal generating set for a module.

Maggie Miller (Princeton University)
Concordance of light bulbs
I will describe a concordance version of Gabai's 4-dimensional light bulb theorem. The light bulb theorem says that two 2-spheres R and R' that are homotopically embedded in a 4-manifold and mutually intersect another 2-sphere G (with trivial normal bundle) in one point are isotopic (modulo a condition on 2-torsion in the fundamental group of the 4-manifold). I'll show that if we relax the condition on R' so that R' intersects G in many points, then we may still conclude that R and R' are concordant (modulo the same 2-torsion obstruction). The proof is constructive, based on Gabai’s construction in the proof of the light bulb theorem. This will entail explicitly constructing a 3-manifold handle by handle in a movie of 5-dimensional space.

Matthias Nagel (University of Oxford)
Slice disks in stabilized 4-balls
We consider knots K which bound (nullhomotopic) slice disks in a stabilized 4-ball, that is in D⁴ # nS²×S^². We explain how to construct examples of such disks, and discuss bounds on the minimal number n of stabilizations needed. Then we compare this minimal number to the 4-genus of K. This is joint work with Anthony Conway.

Lisa Piccirillo (University of Texas at Austin)
Exotic Mazur manifolds and property R
The simplest compact contractible 4-manifolds, other than the 4-ball, are Mazur manifolds (from a handle theoretic perspective). We produce the first pairs of Mazur manifolds that are homeomorphic but not diffeomorphic. Our diffeomorphism obstruction comes from a new repurposing of Heegaard Floer concordance invariants as smooth 4-manifold invariants. As a corollary, we produce integer homology 3-spheres admitting multiple distinct S¹×S² surgeries, which gives counterexamples to a generalization of Property R, resolving a question from Problem 1.16 in Kirby's list. This is joint work in progress with Kyle Hayden and Tom Mark.

Wojciech Politarczyk (Warsaw University)
Cabling formula for twisted Blanchfield forms
Many interesting examples of knots are obtained with the aid of the satellite construction. Therefore it is crucial to understand how knot invariants behave under this construction. In this talk we will discuss a cabling formula for twisted Blanchfield forms and associated twisted signatures. As an example we will re-derive classical cabling formulas of Litherland for Casson–Gordon signatures. This is a joint work with Antony Conway and Maciej Borodzik.

Jake Rasmussen (Cambridge University)
The SL₂(R) character variety and a refined Lin invariant
I'll describe an invariant of knots in S³ defined by counting elliptic representations of the fundamental group which limit to parabolics. The first such invariant was defined by X. S. Lin using the SU(2) character variety. I'll explain some parallels between this invariant and constructions in Seiberg–Witten theory/Heegaard Floer homology, and make some conjectures about its behavior. This is joint work with Nathan Dunfield.

Arunima Ray (MPIM)
Geometrically transverse spheres in 4-manifolds
The disc embedding theorem for simply connected 4-manifolds was proved by Freedman in 1982 and forms the basis for his proofs of the h-cobordism theorem, the Poincaré conjecture, the exactness of the surgery sequence, and the classification of simply connected manifolds, all in the topological category and dimension four. The disc embedding theorem for more general 4-manifolds is proved in the book of Freedman and Quinn. However, the geometrically transverse spheres claimed in the outcome of the theorem are not constructed. We close this gap by constructing the desired transverse spheres. We also outline where and why such transverse spheres are necessary. This is a joint project with Mark Powell and Peter Teichner.

Hannah Schwartz (Bryn Mawr College)
"Exotic" spheres in four-manifolds
Throughout this talk, we will compare the notions of topological isotopy, smooth isotopy, and smooth equivalence (via an ambient diffeomorphism preserving homology) between homotopic 2-spheres smoothly embedded in a 4-manifold. In particular, we will construct pairs of spheres that are 1) both topologically isotopic and smoothly equivalent, but not smoothly isotopic, and 2) smoothly equivalent but not topologically isotopic. Indeed, the examples satisfying condition 2) show that Gabai's recent "4D Light Bulb Theorem" does not hold without the 2-torsion hypothesis.

András Stipsicz (Rényi Institute of Mathematics)
Connected knot homology of covering involution
Adapting ideas of Hendricks–Manolescu and Hendricks–Hom–Lidman, we define a new set of knot invariants, and compute them in some particular cases, providing some novel independence results in the knot concordance group. This is a joint project with Antonio Alfieri and Sungkyung Kang.

Raphael Zentner (Regensburg University)
SU(2)-cyclic surgeries and surgery obstructions
We obtain an infinite class of graph manifolds of which we can prove that none of it can be obtained by surgery on a knot in S³. The result is obtained by understanding when a surgery on a knot is SU(2)-cyclic for various classes of knots, combined with some results from hyperbolic geometry due to Gordon and Luecke. In particular, no gauge theory is involved in the argument. This is joint work with Steven Sivek.

Acknowledgments: We acknowledge support from Défimaths and CNRS.