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Floer theory and low-dimensional topology: Connections to contact and symplectic geometry

Ascending surfaces, complex curves, and knot Floer homology- Kyle Hayden (Columbia University)

Abstract: Ascending surfaces are a natural class of surfaces in certain symplectic 4-manifolds that behave like complex curves, but they are much more flexible. I'll present a key structural result about these surfaces and highlight two applications: one to the study of complex curves, the other to recent results of Juhasz-Miller-Zemke regarding the transverse invariant in knot Floer homology. 

 

Right-veering open books and the Upsilon invariant- Diana Hubbard (Brooklyn College, CUNY)

Abstract: Fibered knots in a three-manifold Y can be thought of as the binding of an open book decomposition for Y. A basic question to ask is how properties of the open book decomposition relate to properties of the corresponding knot. In this talk I will describe joint work with Dongtai He and Linh Truong that explores this: specifically, we can give a sufficient condition for the monodromy of an open book decomposition of a fibered knot to be right-veering from the concordance invariant Upsilon. I will discuss an application of this work to the Slice-Ribbon conjecture.

 

GRID invariants obstruct decomposable Lagrangian cobordisms- C.-M. Michael Wong (Louisiana State Unversity) 

Abstract: Ozsvath, Szabo, and Thurston defined several combinatorial invariants of Legendrian links in the 3-sphere using grid homology, which is a combinatorial version of link Floer homology. These, collectively called the GRID invariants, are known to be effective in distinguishing some Legendrian knots that have the same classical invariants. In this talk, we show that the GRID invariants provide an obstruction to the existence of decomposable Lagrangian cobordisms between Legendrian links. This obstruction is stronger than the obstructions from the Thurston-Bennequin and rotation numbers, and is closely related to a recent result by Golla and Juhasz. This is joint work with John Baldwin and Tye Lidman.

 

12 replies on “Floer theory and low-dimensional topology: Connections to contact and symplectic geometry”

Hi Diana! I really enjoyed your talk!

I have a question: you mentioned that there are right-veering maps that are not obtained by doing positive Dehn twists along a collection of sccs. Do you have a favorite example (or a favorite reference) of such maps? Thanks!

Hi Siddhi, there’s a result of Giroux that says a contact structure is Stein fillable iff the monodromy of the open book is expressable as a product of right-handed Dehn twists. For examples of right-veering maps not obtained as a product of right-handed Dehn twists, you can take the monodromy of any open book associated to a non-Stein fillable contact structure.

Hi Kyle. It was a great talk!

I have a quick question. Is there a condition for self linking number of a transverse link bounding a positive ascending surface?

Hi all, I finished watching all the talks in this section, I like them all, thanks.

Here is a question for Kyle:
Faboulus that you can do relative Boileau-Orevkov (I dare calling your excellent Theorem 1 that)! Question: If you assume that you have a rational curve in Theorm 1 (that is: no genus, but allow sigularities, just assume they are away from the two spheres), do you get to replace sigma_i by sigma_i^2 in ii)?
More genarally I would hope, that if the singularities are all of type A_{n-1}, then ii) can be promoted to sigma_i^n. (And if the singularities are a prescribed list of A_{n-1} Singularities, I would hope that you get one ii) move ”add sigma_i^n” for each singularity.)

Background: I once checked that your Theorem 1 holds in case of deformations (you get braid axes almost for free in that setting and it all becomes an Rudolph diagram argument, so what you do is the cool and hard theorem). Similarly I could check that for delta-constant deformations (which give rise to rational complex curves), one has what I ask for above; hence my question.

(And one could define arbitary notions of deformation, where one asks for rational curves and fixed sets of singularities and there should be sensible topological notions of adding certain positive braids into the braid in as an anolog of your ii)).)

I think I’ll also bother you in person at one point about this

Best Peter

Thanks, Peter! The majority of the talk is pulled from my older paper on quasipositive links and Stein surfaces (https://arxiv.org/abs/1703.10150). However, the explicit statement of Theorems 1 and 1′ from the talk don’t appear in the paper; they’re implicitly used to prove the paper’s main theorems. More precisely, they follow from Lemma 3.5 and the proof of Theorem 1.3, as applied in the proof of Theorem 1.1. (Oof, sorry!)

As for handling singular curves, please continue to bother me about it! I think that the generalization you stated should be true. It would be good to work it out, and I’ve always assumed that understanding Maciej’s paper (https://arxiv.org/abs/1101.1870) would be useful when trying to adapt this to singular curves. The one thing to keep in mind in all of this is that I’m not controlling the braid types of the ends, only their transverse isotopy type. But, by the way, if you’re willing to positively stabilize your braided ends a ton, then you can drop “positive destabilization” from the theorem statement.

Anyway, I know very little about deformations. Let’s chat further and I’ll make you (re-)teach me!

Thanks! I really enjoyed your talk too.

The moral answer is that the slice-Bennequin bound will be sharp. In the case of knots in the standard $(S^3,\xi)$, a transverse knot $K$ that bounds a positive ascending surface $\Sigma \subset B^4 \subset \mathbb{C}^2$ (i.e. with only positively signed critical points) will satisfy $sl(K)=-\chi(\Sigma)$. For a transverse knot $K$ in the boundary of a Stein domain $(X,J)$ such that $[K]=0 \in H_1(X)$, the self-linking number is defined relative to the class of an orientable spanning surface for $K$ in $H_2(X,K)$. In particular, if $K$ bounds a positive ascending surface $\Sigma \subset X$, then you *should* get $sl_{[\Sigma]}(K)=-\chi(\Sigma)$. I’ll have to think a bit more to give a rigorous argument, but the point is that $c_1(J)$ should now play the role that the Euler class $e(\xi)$ would usually play when calculating self-linking number using a Seifert surface in a contact 3-manifold $(Y,\xi)$.

Hi Sarah,

Thank you! That is a great question whose answer has deep connections to symplectic geometry that I may not be able to explain well. First, perhaps I was inaccurate in saying that exact Lagrangian cobordisms are “the” right notion of cobordisms. (It is indeed the most studied notion by far.) There is nothing inherently wrong with studying, for example, (possily non-exact) Lagrangian cobordisms between Legendrian links. However, there is a dichotomy: While exact Lagrangian cobordisms are rigid and heavily constrained (and hence more intimately connected to the contact geometry and smooth topology of the 3-manifold and the link), (non-exact) Lagrangian cobordisms are, in contrast, abundant in existence and rather flexible: They tend to exist whenever not obviously forbidden. So to study them, perhaps the focus would not be on existence, but on something else.

That being said, it is also very natural to want to study exact Lagrangian cobordisms in our context, since the symplectization is an exact symplectic manifold (i.e. the symplectic form is exact).

Interest in Symplectic Field Theory (e.g. the Chekanov–Eliashberg DGA) and (Heegaard, monopole, or instanton) Floer homology also plays a role in answering this question. For example, Ekholm, Honda, and Kalman proved that an exact Lagrangian cobordism from L-bot to L-top induces a DGA morphism from the DGA of L-top to that of L-bot, which makes the DGA a functor from some category of Legendrian links to the category of DGAs. The domain category obviously demands exactness of its morphisms. (As a side note, the reason that the primitive function f, briefly shown in the video, is required to be constant—rather than locally constant—is precisely to ensure that composition in this category (i.e. concatenation) makes sense.) On the Floer homology side, the three results on knot (monopole or Heegaard) Floer homology I mentioned in the video all require exactness in their statements.

Finally, I want to mention the importance of exactness of Lagrangians in Floer theories, which does not directly relate to your question but may be of interest. Recall that given a symplectic manifold M, the objects of the Fukaya category of M are the Lagrangian submanifolds of M. One issue in defining the Fukaya category (as an A-infinity category) is “bubbling”, which does not arise under certain conditions—one of which is if the Lagrangians are assumed to be exact (because of Stokes’ Theorem). (In contrast, viewing Heegaard Floer homology as a special case of Lagrangian intersection Floer homology, “bubbling” does arise in this context, but its overall contribution is zero.)

I apologize for the lengthy response. Please do not hesitate to contact me if you would like references for any of the above!

Mike

Looking back at my reply, perhaps I should have started by stating why Lagrangian cobordisms are the right notion of cobordisms for Legendrian knots. First, contact geometry and symplectic geometry are closely related: Given a contact manifold Y, the product R x Y is naturally a symplectic manifold (the symplectization). Suppose you have a Legendrian knot L in Y. Looking at the simplest case, the identity cobordism R x L in R x Y, the obvious question is how does R x L interact with the symplectic structure (which came from the contact structure of Y)? The answer is that the restriction of the symplectic form to R x L vanishes identically, which is the Lagrangian condition. In fact, this works also for cobordisms arising from Legendrian isotopies. (And in fact, these are all exact.) From this, we extrapolate the Lagrangian condition to other cobordisms.

Hi Mike,
Thanks for such a thoughtful answer, I’ve been wondering about this for a while so this helps a lot. I’ll digest it over the next few days and let you know if I have any follow up questions.
Thanks again!

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