Princeton Low-Dimensional Topology Workshop 2015

Date: June 15-19, 2015
Location: Room 314, Fine Hall, Washington Road, Princeton, NJ 08544
Organisers: Adam Levine, Peter Ozsváth, Sucharit Sarkar, and Steven Sivek
Primarily funded by: NSF DMS-1350037
With support from: Department of Mathematics, Princeton University

The workshop focused on recent developments in low-dimensional topology, with a particular emphasis on modern homological invariants such as Heegaard Floer homology. The workshop was open to all; the following captures most of the participants.




Schedule

• Monday, June 15
  • [09:30-10:30] Çağatay Kutluhan, University at Buffalo, A Heegaard Floer analog of algebraic torsion I will talk about joint work in progress with Gordana Matic, Jeremy Van Horn-Morris, and Andy Wand the goal of which is to define a Heegaard Floer analog of Latschev and Wendl's algebraic torsion for contact 3-manifolds. (Notes)
  • [10:30-11:00] Coffee
  • [11:00-12:00] John Baldwin, Boston College, Lagrangian concordance and contact invariants in sutured Floer theories In 2007, Honda, Kazez, and Matic defined an invariant of contact 3-manifolds with convex boundaries using sutured Heegaard Floer homology (SHF). Last year, Steven Sivek and I defined an analogous contact invariant using sutured Monopole Floer homology (SMF). In this talk, I will describe work with Sivek to prove that these two contact invariants are identified by an isomorphism relating the two sutured theories. This has several interesting consequences. First, it gives a proof of invariance for the contact invariant in SHF which does not rely on the relative Giroux correspondence between contact structures and open books (something whose proof has not yet been written down in full). Second, it gives a proof that the combinatorially computable invariants of Legendrian knots in Heegaard Floer homology can obstruct Lagrangian concordance.
  • [14:15-15:15] Yi Ni, California Institute of Technology, Fintushel--Stern knot surgery in torus bundles Suppose that X is a torus bundle over a closed surface with homologically essential fibers. Let X_K be the manifold obtained by Fintushel--Stern knot surgery on a fiber using a knot K\subset S^3. We prove that X_K has a symplectic structure if and only if K is a fibered knot.
  • [15:15-16:00] Tea
  • [16:00-17:00] Daniel Ruberman, Brandeis University, Absolutely exotic 4-manifolds We show the existence of exotic smooth structures on contractible 4-manifolds. These structures are absolute, in the sense that they do not depend on a specific marking of the boundary. This is in contrast to the phenomenon of corks, which are exotic relative to an automorphism of their boundaries. The technique is to modify a relatively exotic manifold to give an exotic one for which we have a good understanding of the automorphism group of the boundary. This is joint work with Selman Akbulut.
• Tuesday, June 16
  • [09:30-10:30] John Etnyre, Georgia Institute of Technology, Braiding and transverse knotting in Euclidean 5-space It is a well-known result of Hirsch that all 3-manifolds embed in R^5. It was recently observed by Kasuya that if a contact 3-manifold embeds in the standard contact structure on R^5 then its first Chern class must be zero. In this talk I will discuss joint work with Ryo Furukawa aimed at using braiding techniques to try to prove the converse and more generally study contact embedding in R^5. Contact embedding should be consider the higher dimensional analog of transverse knot theory. Braided embeddings give an explicit way to represent some (maybe all) smooth embeddings and should be useful in computing various invariants. In particular, if time permits I will discuss a generalization of (transverse) knot contact homology to this setting and how braided embeddings might aid in its computation.
  • [10:30-11:00] Coffee
  • [11:00-12:00] Olga Plamenevskaya, Stonybrook University, On right-veering transverse knots I will discuss some properties of right-veering braids and relation to transverse invariants.
  • [14:15-15:15] Lucas Culler, Princeton University, Integrable systems in Donaldson/Floer theory I will give examples of families of Abelian varieties that arise in 4-dimensional gauge theory (Donaldson and Seiberg-Witten theory). I will then show how a Floer homology theory satisfying certain formal properties can be used to construct such a family.
  • [15:15-16:00] Tea
  • [16:00-17:00] Dylan Thurston, Indiana University, Extremal length and degenerations of complex structures The construction of Heegaard Floer homology depends on a complex structure on a Heegaard surface, which is not combinatorial. If we want to make it more combinatorial, we may consider degenerating the complex structure. Here, we will look at one point of view on degenerating complex structures, in terms of extremal lengths. In particular, extremal lengths can characterize one complex surface with boundary embeds inside another, in a way that extends nicely to certain degenerate limits.
• Wednesday, June 17
  • [09:30-10:30] Elisenda Grigsby, Boston College, (Sutured) Khovanov homology and representation theory I'll describe some of the representation theory underlying sutured annular Khovanov homology, a variant of Khovanov homology particularly well-suited to studying links as braid closures. I'll also say a few words about potential applications. Warning: this talk contains more questions than answers. Some of what I'll discuss is joint work with Tony Licata and Stephan Wehrli, and some is joint work in progress with Tony Licata.
  • [10:30-11:00] Coffee
  • [11:00-12:00] Jen Hom, Columbia University, Knot Floer homology and concordance We will discuss applications of knot Floer homology to concordance. In particular, we will compare the four-ball genus and concordance genus bounds given by epsilon, Upsilon(t), tau, and nu-plus, which are all concordance invariants associated to the knot Floer complex. Parts of this talk are joint work with Zhongtao Wu.
  • [14:15-15:15] Matthew Hedden, Michigan State University, Functoriality of Khovanov-Floer theories There is now a zoo of spectral sequences from Khovanov homology to various (often Floer type) homological invariants of links. In most of these situations, the Floer theory is known to possess some level of functoriality with respect to link cobordisms, and Khovanov homology is also known to have this structure. Given that the E_2 and E_infinty pages of the spectral sequences are functorial, it is natural to wonder if all the pages enjoy this structure. I'll formalize a notion of a "Khovanov-Floer theory", which is a relatively weak set of axioms placed on a mechanism for assigning filtered complexes to link diagrams. I'll then discuss how to prove that all the pages of the resulting spectral sequences are functorial, and why all the spectral sequences we known of fit into this framework (and are therefore functorial). This is joint work with John Baldwin and Andrew Lobb.
  • [15:15-16:00] Tea
  • [16:00-17:00] Aliakbar Daemi, Simons Center for Geometry and Physics, Involutions, Floer Homology, and Surgery Cube There are various exact triangles for Floer homological invariants of 3-manifolds (respectively, links) that relate the invariants of different surgeries (respectively, resolutions) of a 3-manifold (respectively, link). In some instances of these invariants, there is a group action that acts naturally on the ingredients of the theory and hence one can hope to develop an equivariant Floer thoery. In this talk, I will discuss the interaction of these group actions with the exact triangles and its applications in the special case of an invariant that is defiend in the framework of Yang-Mils gauge theory.
• Thursday, June 18
  • [09:30-10:30] Kristen Hendricks, University of California at Los Angeles, Involutive Heegaard Floer homology In joint work in progress with C. Manolescu, we use the conjugation symmetry on the Heegaard Floer complexes to define a three-manifold invariant called involutive Heegaard Floer homology, which is meant to correspond to Z_4-equivariant Seiberg-Witten Floer homology. From this we obtain two new invariants of homology cobordism, explicitly computable for surgeries on L-space knots and quasi-alternating knots, and two new concordance invariants of knots, one of which (unlike other invariants arising from Heegaard Floer homology) detects non-sliceness of the figure-eight knot.
  • [10:30-11:00] Coffee
  • [11:00-12:00] Tye Lidman, University of Texas at Austin, Knot contact homology detects torus knots Knot contact homology is an invariant of knots arising from constructions in contact topology which may in fact be a complete invariant. Using a description of knot contact homology in terms of the knot group due to Cieliebak-Ekholm-Latschev-Ng, we prove that the degree zero piece of knot contact homology detects every torus knot.
  • [14:15-15:15] Bohua Zhan, Massachusetts Institute of Technology, Combinatorial constructions of Heegaard Floer homology using bordered invariants Heegaard Floer homology, for a closed 3-manifold Y, is usually defined by counting certain holomorphic disks or curves, in a symplectic manifold coming from a Heegaard decomposition of Y. Recent developments in bordered Heegaard Floer theory makes it possible to describe the hat version, HF^(Y), in terms of purely combinatorial and algebraic constructions. This works by cutting Y along dividing surfaces into sufficiently simple pieces, then combining the bordered invariants of each resulting piece, in a way similar in spirit to a (2+1)-dimensional topological quantum field theory. I will give an overview of bordered Floer theory and then discuss my work in using it to give combinatorial constructions.
  • [15:15-16:00] Tea
  • [16:00-17:00] Zoltán Szabó, Princeton University, Algebraic methods in knot Floer homology
• Friday, June 19
  • [09:00-10:00] Robert Lipshitz, Columbia University, Remarks on equivariant Floer homology Recently, equivariant Floer homology has been used to construct a number of spectral sequences in Heegaard Floer homology and symplectic Khovanov homology. In this talk, we will give a (slightly) new construction of equivariant Floer homology and use it to tie up a number of loose ends related to these spectral sequences, addressing questions of invariance, computability, and when different sequences agree. This is joint work with Kristen Hendricks and Sucharit Sarkar.
  • [10:00-10:15] Coffee
  • [10:15-11:15] Josh Greene, Boston College, Definite surfaces and alternating links I will describe a characterization of alternating links in terms intrinsic to the link complement and use it to give new proofs of some of Tait’s conjectures.