Schedule for: 22w5164 - Interactions of gauge theory with contact and symplectic topology in dimensions 3 and 4

A buffet dinner is served daily between 5:30pm and 7:30pm in Kinnear Center 105, main floor of the Kinnear Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

The Murasugi sum operation in knot theory provides an interesting generalization of the better-known connected sum operation. The special case of plumbing is central to the Giroux correspondence, and therefore figures prominently in the study of contact structures on 3-manifolds.  Gabai's work on taut foliations and sutured manifolds showed that the Murasugi sum interacts predictably with regard to geometric features of the knot pertaining to Seifert surfaces e.g. whether they are fibered, or minimal genus.  Floer homology has provided algebraic invariants that can serve as receptors for some of the information provided by Gabai's techniques. In the context of Murasugi sums, Ni proved that the rank of a particular (extremal) knot Floer homology group is multiplicative under these operations.  We refine Ni's result to show that Murasugi sums induce a graded tensor product of the extremal knot Floer homology groups and, moreover, that in certain cases one can obtain information about other invariants derived from knot Floer homology (so-called "tau" invariants).   I'll give an overview of the Murasugi sum and some of its roles in low-dimensional topology, discuss our results, and their corresponding applications.   This is joint work with Zhechi Cheng and Sucharit Sarkar.

The talk will focus on studying Legendrian knots in the standard 3-dimensional contact Darboux ball, with an emphasis on their Lagrangian fillings. First, I will discuss some of the geometric techniques that we currently use to understand Legendrian knots and their Lagrangian fillings, including Legendrian weaves. Then, we will motivate the notion of a cluster algebra through the lens of contact and symplectic topology and present one of the new results in this area: the existence of cluster algebras associated to a large class of Legendrian knots, including all positive braids. Even if we discuss pieces of algebra, symplectic geometry, and in particular the study of Lagrangian skeleta in 4-dimensions, will be our guiding light.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

The Manolescu-Ozsvath link surgery formula is a powerful tool for computing Heegaard Floer homology. In this talk, we will describe how to interpret their link surgery formula in terms of A_infty modules over a new algebra K. We will describe how to naturally interpret this theory as an theory for bordered manifolds with torus boundary (with some aspects of the theory still in progress). A pairing theorem for gluing torus boundary components takes the form of a connected sum formula for the Manolescu-Ozsvath link surgery formula. This new bordered theory has as an antecedent and inspiration the dual-knot surgery formula of Eftekhary, Hedden and Levine. We will describe how to recover their dual-knot surgery formula as a bimodule for changing the boundary parametrization of a bordered manifold. We will also describe works in progress related to this theory.

Given n points on  a disk, we will describe how to build an A-infinity category based on the instanton Floer complex of links, and explain why it is finitely generated. This is based on work in progress with Ko Honda.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Engel manifolds are closely related to contact manifolds, but only occur in dimension 4. They are much less well understood than contact manifolds. For example, it is still unknown if "tight" Engel structures exist. A primary tool for understanding such issues for contact 3-manifolds is transverse knot theory. Every knot in a contact 3-manifold is isotopic to transverse knots, realizing infinitely many values of the associated homotopy invariant (self-linking number) when it is defined. At a 2017 AIM conference, Eliashberg suggested the analogous problem of understanding transverse (closed, oriented) surfaces in Engel manifolds, but no significant results were obtained at the time. It is easy to see that such transverse surfaces are necessarily tori with trivial normal bundles. We will discuss very recent work showing that conversely, tori with trivial normal bundles can be made transverse, analogously to knots in contact 3-manifolds. This might turn into a powerful tool for understanding Engel manifolds.

Given a link L in the solid torus, we can remove a neighborhood of L and put meridinal sutures on the resulting boundary components. It follows from earlier work with Hanselman and Watson that the resulting bordered sutured Floer homology can be described using curves in the torus. I'll discuss some properties and applications of these curves.

Please turn your cameras on - BIRS staff will take a virtual group photo.

Meet in the PDC front desk for a guided tour of The Banff Centre campus.

The Khovanov homology of a rational knot or link happens to coincide with the cohomology of its space of SU(2) representations that send meridians to traceless matrices. This coincidence is closely related to the spectral sequence from Khovanov homology to an SU(2) instanton homology defined by Kronheimer and Mrowka. Motivated by a hypothetical spectral sequence from colored sl(N) homology to a hypothetical colored SU(N) instanton homology, I'll show that the colored sl(N) homology of the Hopf link agrees with the cohomology of its space of SU(N) representations that send meridians to conjugacy classes specified by the colors. I'll also discuss work in progress for other rational knots and links. 

There has been a surge of interest over the last year in the relationship between Heegaard Floer homology and the dynamics of surface diffeomorphisms. This relationship has recently been put towards several topological ends, including a proof that Khovanov homology detects the torus knot $T(2,5)$ and the resolution of problems in Dehn surgery stemming from Kronheimer and Mrowka's work on the Property P conjecture. I will describe in detail the first of these applications. This is mostly joint work with Ying Hu and Steven Sivek.

Hughes, Kim and I recently showed how to construct 3-dimensional genus-g handlebodies $H$ and $H’$ in the 4-sphere so that $H$ and $H’$ have the same boundary and are homeomorphic rel boundary, but are not smoothly isotopic rel boundary (for all g at least two). In fact, $H$ and $H’$ are not even topologically isotopic rel boundary, even when their interiors are pushed into the 5-ball. This proves a conjecture of Budney and Gabai (who recently constructed smooth 3-balls in the 4-sphere with the same boundary that are not smoothly isotopic rel boundary) for g at least two in a very strong sense.  In this talk, I’ll describe some useful facts about higher-dimensional knots that go into this construction and talk about remaining future directions. This is joint work with Mark Hughes and Seungwon Kim.

We define several equivariant concordance invariants using knot Floer homology. We show that our invariants provide a lower bound for the equivariant slice genus and use this to give a family of strongly invertible slice knots whose equivariant slice genus grows arbitrarily large, answering a question of Boyle and Issa. We also apply our formalism to several seemingly non-equivariant questions. In particular, we show that knot Floer homology can be used to detect exotic pairs of slice disks, recovering an example due to Hayden. This is joint work with Abhishek Mallick and Matthew Stoffregen.

We construct a surgery exact triangle in involutive Floer homology, and give some consequences for the homology cobordism group. This is joint work with Kristen Hendricks, Jen Hom, and Ian Zemke.

Two knots $K_0$ and $K_1$ are said to be smoothly concordant if the connected sum $K_0 \# m(K_1^r)$ bounds a disk smoothly embedded in the 4-ball. Smooth concordance is an equivalence relation, and the set C of smooth concordance classes of knots is an abelian group with connected sum as the binary operation. Satellite operations, or the process of tying a given knot $P$ along another knot $K$ to produce a third knot $P(K)$, are powerful tools for studying the algebraic structure of the concordance group. In this talk I will describe conditions on the pattern $P$ that suffice to conclude that the function $P: C \to C$ is not a homomorphism. This is joint work with Tye Lidman and Allison Miller.

Given a closed, orientable 3-manifold $Y$, it is of great interest but often a difficult problem to determine whether $Y$ may be smoothly embedded in ${\mathbb R}^4$. This is the case even for integer homology spheres, and restricting to special classes such as Seifert manifolds, the problem is open in general, with positive answers for some such manifolds and negative answers in other cases. On the other hand, under additional geometric considerations coming from symplectic geometry (such as hypersurfaces of contact type) and complex geometry (such as the boundaries of holomorphically or rationally or polynomially convex Stein domains), the problems become tractable and in certain cases a uniform answer is possible. For example, recent work, which is obtained jointly with Tom Mark, shows for Brieskorn homology spheres: no such 3-manifold admits an embedding as a hypersurface of contact type in ${\mathbb R}^4$. This implies restrictions on the topology of rationally and polynomially convex domains in ${\mathbb C}^2$. In this talk, I will provide further context and motivations for these results, and give some details of the proof.

I will present an immersed-curve approach to computing the $F[U,V]/(UV)$-version knot Floer homology of satellite knots. This is built on the immersed-curve technique in bordered Heegaard Floer homology introduced by Hanselman-Rasmussen-Watson. I will show that our method can quickly reprove several previous results on the knot Floer homology of satellite knots. This talk is based on joint work in progress with Jonathan Hanselman.

The unknotting number is the minimum number of self-crossings needed to untie a knot. This invariant seems like a simple measure of complexity for knots but turns out to be very difficult to compute. For example, there are still several 10 crossing knots for which we do not know the unknotting number. In this talk I will consider symmetric knots and the equivariant unknotting number $eu(K)$, where we require the self-crossings to occur symmetrically. I will define a suprisingly easy-to-compute lower bound on eu(K), which we call the half-linking number, when the symmetry is a point reflection across a point on the knot. I will also discuss relationships to the Arf invariant and the equivariant concordance group. This is joint work in progress with Wenzhao Chen. 

The Eliashberg-Bennequin inequality holds for surfaces with Legendrian boundary in tight contact 3-manifolds. For the tight 3-sphere, Rudolph showed it also applies to surfaces in the 4-ball. In work with Hedden, we explore the general case. Our work leads us to conjecture a 4-dimenisional characterization of tight 3-manifolds. In this talk, I’ll explain our conjecture, two related ones and discuss what we can prove so far. This is joint work with Matthew Hedden.

While visiting the University of Chicago in 1970, Tom Farrell came up with an embedding $D^{n-1} \to S^1 x D^{n-1} $ (for $n>5$) that was null in pseudo-isotopy, and conjectured it was isotopically non-trivial.  We provide a proof of this conjecture.  This result is one of several results with D. Gabai on the homotopy-type of the diffeomorphism groups of manifolds of the form $S^1 x D^{n-1},$ for all $n$. 

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.