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I would like to collect a list of applications of Heegaard-Floer theory. By applications, I don't mean things like "it can detect the unknot" or "it can detect knot genus". Algorithms for these kinds of things have been known since the '60's and '70's. Instead, I mean two kinds of things.

  1. Questions that make no reference to Heegaard-Floer theory that can be answered using Heegaard-Floer theory (and, preferably, cannot be answered in other ways).
  2. Things about knots or 3-manifolds that can be computed with Heegaard-Floer theory for which algorithms did not previously exist.

I'm asking this question here in response to a large number of talks about Heegaard-Floer theory I've attended over the years. It seems like a hot subject and lots of talented young people are working on it, but most of the talks I've attended about it addressed what seemed to me to be technical questions internal to the subject. And when I've asked the speakers this question, I never seem to get a good answer. But since it is such a hot subject, I assume there must be some killer applications.

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Ghiggini has shown that the only knot giving rise to the Poincare homology sphere by Dehn filling is the trefoil knot, as an application of Heegaard Floer homology.

Another important point is that now it is known that certain types of Heegaard Floer homology are computable, starting with Sarkar-Wang. Since it is known to be equivalent to other Floer homologies (Seiberg-Witten, Embedded Contact Homology), this allows one to compute these other invariants. The hope is to be able to get combinatorial formulae for invariants of 4-manifolds from this, which are related to Seiberg-Witten invariants. Unfortunately, sometimes the combinatorial formulae are more intricate and less well-motivated than the geometric definitions, or have the geometry suppressed.

As you noted, many talks on Heegaard Floer homology are getting quite technical, especially the bordered Floer theory. I believe that people think the added complexity is worth the investment, since it should yield new insights (and already is). But this also means that the motivation for some technical results might not be currently apparent.

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    $\begingroup$ Since you talk about surgeries, Agol, let me mention Josh Greene's work on the Berge conjecture (following earlier work of Ozsváth and Szabó, and Rasmussen, among others) and the cabling conjecture. Also, more recently he worked on Conway mutations. $\endgroup$
    – Marco Golla
    Commented Feb 17, 2012 at 22:18
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The $d$ invariant coming from the rational grading can be seen as an enhancement of the torsion linking form on a rational homology $3$-sphere. Just like the regular torsion linking form of 3-manifolds, this gives an obstruction to embedding rational homology $3$-spheres in the $4$-sphere (or any homology $4$-sphere). Similarly, this gives obstruction to $spin^c$-cobordism. Sometimes these obstructions are completely new, so it's a genuinely novel application where there were no comparable tools beforehand.

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    $\begingroup$ May I put in a word for Kim Froyshov? He used Seiberg-Witten theory, in the days before either SW Floer or Heegaard Floer theory, to define an invariant of homology 3-spheres, and gave applications to 4-manifold topology. [The Seiberg-Witten equations and four-manifolds with boundary, Math. Res. Lett. 3 (1996), no. 3, 373–390.] Ozsvath-Szabo's d-invariant was inspired by the Froyshov invariant, and is conjecturally equal to it. Of course, the big Heegaard Floer package helps one use this invariant to full advantage. $\endgroup$
    – Tim Perutz
    Commented Feb 17, 2012 at 15:18
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    $\begingroup$ Actually, I think the recent work of Kutluhan-Lee-Taubes on SW vs. Heegaard Floer implies that the d-invariant really is the Froyshov invariant. $\endgroup$
    – Tim Perutz
    Commented Feb 17, 2012 at 15:21
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It provides lots of computable invariants in contact geometry, in particular the contact invariant defined by Ozsvåth and Szabó via open books and the Giroux correspondence. For example, on Seifert fibered spaces the question of whether tight contact structures exist was completely solved by Lisca and Stipsicz, and the classification was completed in many of these cases in several other papers; and Ghiggini used it to exhibit contact structures which are strongly fillable but not Stein fillable.

Similarly, the LOSS invariant (named after Lisca-Ozsváth-Stipsicz-Szabó) and the related, easily computable Ozsváth-Szabó-Thurston grid diagram invariants of Legendrian and transverse knots were used by Ng-Ozsváth-Thurston to successfully distinguish many pairs of knots in the standard contact $S^3$, and been used to prove other properties such as the fact due to Etnyre and Vela-Vick that the complement of the binding of any open book of any contact structure has no Giroux torsion. (According to recent work of Baldwin--Vela-Vick--Vértesi, these invariants are the same for knots in the standard $S^3$.)

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Yi Ni used Heegaard Floer Homology to prove, among many other things, that a knot admitting a lens-space surgery is fibred. I believe that no 'conventional' proof of this is known.

Ni, Yi, Knot Floer homology detects fibred knots. Invent. Math. 170 (2007), no. 3, 577–608.

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I don't think you can say that detecting the unknot and detecting knot genus have been long understood just because there were algorithms for them. You want effective algorithms, or ways of computing infinite families of examples.

Aside from that, Ozsvath and Szabo had early applications to questions about which three manifolds can be obtained from which other manifolds by what kinds of surgeries. The Ozsvath-Szabo contact invariant has a number of applications, for example to understanding symplectic fillings. There is much more, too much for me to completely follow, so that anything I say would be very incomplete.

In addition, most of the applications of Seiberg-Witten theory (e.g. the Thom Conjecture, distinguishing various non-diffeomorphic but homeomorphic smooth manifolds) can be re-proved using the Ozsvath-Szabo theory. Since the two theories are known to be equivalent, it is a matter of taste whether you want to regard these as applications of Seiberg-Witten or Ozsvath-Szabo.

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    $\begingroup$ Indeed! In a similar vein, algorithms for computing the homotopy groups of spheres have been known for ages. $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Feb 17, 2012 at 8:01
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    $\begingroup$ Yes, but computing Heegaard Floer invariants is still not very efficient. I'm not sure about knot genus calculations, but tables of isotopy classes of knots were computed up to 16 crossings using classical methods. My understanding is that the algorithms for Heegaard Floer computations are not computationally practical for that number of crossings. Thus the fact that they detect the unknot (much weaker than separating isotopy classes) adds nothing computational to the picture. $\endgroup$
    – Mitya
    Commented Feb 18, 2012 at 4:37
  • $\begingroup$ Frequently new methods of computation gives new insight. This is certainly the case for homotopy groups of spheres. $\endgroup$
    – Sean Tilson
    Commented Feb 19, 2012 at 1:56
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    $\begingroup$ An update on the state of the art in computations in Heegaard-Floer homology: Ozsvath-Szabo's recent bordered matchings package seems to extend the range of effective computation of knot Floer homology to significantly larger knots -- my understanding is that stuff on the order of 30-40 crossings is now possible. $\endgroup$
    – dvitek
    Commented Feb 12, 2018 at 6:31
  • $\begingroup$ Understanding the computational complexity of unknot recognition is a major open problem. Breakthroughs have recently been made by Lackenby, but I don't think Heegaard--Floer is involved in his arguments. A reference showing that Heegaard--Floer provides "effective algorithms" for, well, anything that can't be computed by other methods, is needed to make this answer credible. $\endgroup$
    – HJRW
    Commented Oct 9 at 13:33
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Taken from the introduction to A Tour of Bordered Floer Theory (by Lipshitz, Ozsvath, D. Thurston):
http://arxiv.org/pdf/1107.5621v1.pdf

Heegaard Floer homology, introduced in a series of papers of Zoltán Szabó and the second author, has become a useful tool in 3- and 4-dimensional topology. The Heegaard Floer invariants contain subtle topological information, allowing one to detect the genera of knots and homology classes; detect fiberedness for knots and 3-manifolds; bound the slice genus and unknotting number; prove tightness and obstruct Stein fillability of contact structures; and more. It has been useful for resolving a number of conjectures, particularly related to questions about Dehn surgery. It is either known or conjectured to be equivalent to several other gauge-theoretic or holomorphic curve invariants in low-dimensional topology, including monopole Floer homology, embedded contact homology, and the Lagrangian matching invariants of 3- and 4-manifolds. Heegaard Floer homology is known to relate to Khovanov homology, and more relations with Khovanov-Rozansky type homologies are conjectured.

(references provided in the paper for each statement above)

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Kronheimer, Mrowka, Ozsváth, and Szabó obtained a new proof of Gordon and Luecke's Knot Complement Theorem using monopole Floer homology. That's pretty good, I think. They also proved that $\mathbb{RP}^3$ can't be obtained by surgery on a nontrivial knot (I don't know if there are any other proofs now).

Kronheimer, Mrowka, Ozsváth, Szabó, Monopoles and lens space surgeries. Ann. of Math. (2) 165 (2007), no. 2, 457–546.

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