Questions tagged [heegaard-floer-homology]

For questions about Heegaard-Floer homology (as introduced by Ozsváth-Szabó in 2003) and its uses in 3- and 4-dimensional topology.

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Why should I care about Heegaard-Floer theory?

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 ...
17 votes
0 answers
338 views

Are there two non-equivalent exotic structures on $\mathbb{R}^4$ coming from topologically slice, non-slice knots?

For a knot $K \subset S^3$, which is topologically slice but not slice (in a smooth way), there's a four manifold $\mathbb{R}^4_K$, homeomorphic but not diffeomorphic to standard euclidean $\mathbb{R}^...
Saman Habibi Esfahani's user avatar
16 votes
1 answer
742 views

Can infinitely many alternating knots have the same Alexander polynomial?

There exist many constructions of infinite families of knots with the same Alexander polynomial. However, alternating knots seem very special. While there are also many result on restricting the form ...
shestipalov's user avatar
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16 votes
1 answer
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Is Rasmussen's s-invariant of a knot an invariant of a 4-manifold?

Let $K$ be a knot in the 3-sphere $S^3$. Here we denote by $s(K)$ Rasmussen's s-invariant for $K$, and by $X_{K}(n)$ the 4-manifold obtained from the standard 4-ball $B^4$ by attaching a $2$-...
Tetsuya Abe's user avatar
16 votes
1 answer
591 views

Three-manifolds having a Reebless foliation but not a taut one

A straightforward argument reveals that a taut foliation is Reebless, and of course there are many examples of Reebless foliations that are not taut. I guess that there are many examples of three-...
Antonio Alfieri's user avatar
12 votes
1 answer
959 views

Why is Heegaard Floer Homology defined in terms of Sym$^g\Sigma_g$ instead of Pic$^g\Sigma_g$?

Recall the definition of Heegaard Floer homology: $\Sigma_g$ is a closed surface, and $\{\alpha_1,\ldots,\alpha_g\}$ and $\{\beta_1,\ldots,\beta_g\}$ are sets of attaching circles. Then Heegaard ...
John Pardon's user avatar
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12 votes
1 answer
893 views

Wanted: a nontrivial weakly inadmissible Heegaard diagram

This is a question asked by a student in my lecture. After drawing pictures for awhile, I thought it was a good one. I seek a nontrivial example of a pointed Heegaard diagram $(\Sigma,\mathbf{\alpha},\...
Jonathan Williams's user avatar
12 votes
1 answer
426 views

Is there a reasonable definition of TQFTs for n-cobordisms with connected inputs/outputs?

A question that's been on my mind for a while is whether any precise statement to the effect of "Heegaard Floer homology is a TQFT," for some reasonable definition of TQFT, can be made. Of course, a ...
Andy Manion's user avatar
  • 1,454
12 votes
0 answers
277 views

Understanding a formula in Ozsvath-Szabo

I'm a beginning graduate student reading Ozsvath-Szabo's foundational paper, Holomorphic disks and topological invariants for closed 3-manifolds. What I have trouble understanding is a formula in ...
cjackal's user avatar
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11 votes
1 answer
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Is $Sym^g$ of a Riemann Surface of genus $g$ Calabi-Yau?

The $g$-fold symmetric product of a Riemann surface of genus $g$ naturally has both a symplectic structure as well as a complex structure. Is it in fact Calabi-Yau? If so, is anything known about a ...
Steve's user avatar
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11 votes
0 answers
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The second coefficient of the Conway polynomial from Knot Floer homology

Let $\nabla_K(z)$ be the Conway polynomial and $\Delta_K(t)$ be the Alexander polynomial normalized by $\Delta_K(t)=\Delta_K(t^{-1})$ and $\Delta_K(1)=1$, These invariants are equivalent and they are ...
Tetsuya Ito's user avatar
10 votes
1 answer
1k views

Introductory article of knot Heegaard Floer Homology

I am looking for some article that gives an introduction to Heegaard Floer homology of knot. I heard that it is very useful to determine the unknotting number of a knot, but I couldn't find any ...
this_is_an_apple's user avatar
10 votes
0 answers
210 views

Kernel of "Hat to Plus" in Heegaard Floer Homology

Given a 3-manifold $M$, there is a map of Heegaard Floer groups $$f:\widehat{HF}(M) \to HF^+(M)$$ induced by the inclusion $$\widehat{CF}(M) \to CF^+(M)$$ of the respective chain complexes. Given a ...
magicker72's user avatar
10 votes
0 answers
299 views

When were bordered Heegaard Floer homology's DA bimodules invented?

This is less of a strictly mathematical question and more of a reference request. In their paper "Bordered Heegaard Floer Homology (http://arxiv.org/pdf/0810.0687.pdf)," Lipshitz-Ozsvath-Thurston (...
Andy Manion's user avatar
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8 votes
1 answer
504 views

Questions on poincare homology spheres and branched covers

I have two questions: Question 1. Suppose that $K$ is a knot in $S^3$. Let $\Sigma(K)$ be the double branched cover of $S^3$ branched along $K$. If $\Sigma(K)=\#_{i=1}^n\Sigma(2,3,5)$, then $K=\#_{i=...
user156937's user avatar
7 votes
2 answers
643 views

heegard diagram

It seems like there is an algorithm to find the Heegard diagram of a 3 manifold obtained by surgery on a link. Also someone told me I can find it in the Gompf and Stipciz's book. But I could not find ...
mark 's user avatar
  • 155
7 votes
1 answer
1k views

Wanted: differential coming from higher genus surface in Heegaard Floer homology

I am interested in studying moduli of complex surfaces which arise in computing the differential on the Heegaard Floer homology chain complex. In particular, I am interested in the generic case, when ...
John Pardon's user avatar
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7 votes
1 answer
319 views

Heegaard Floer homology of a genus two Heegaard splitting of $S^3$

This is a duplicate of a question (https://math.stackexchange.com/questions/4416204/heegaard-floer-homology-of-a-genus-two-diagram-of-s3) on stackexchange, which did not get any answer. Feel free to ...
Filippo Bianchi's user avatar
7 votes
1 answer
339 views

Admissibility in Heegaard Floer, especially with torsion Spin^c structures

I'm confused about the relationship between strong admissibility and weak admissibility for pointed diagrams in Heegaard Floer theory. For reference, here are Ozsváth-Szàbo's original definitions: ...
hfquestion's user avatar
7 votes
1 answer
437 views

In the definition of the Heegard Floer surgery exact triangle, what exactly is the correspondence between Whitney triangles and periodic domains?

I'm reading Osváth-Szabó's notes on Heegard Floer homology, in particular about the surgery exact triangle. On page 14 (numbered 42 on the document), they describe an isomorphism between the space of ...
Tom Hockenhull's user avatar
6 votes
1 answer
330 views

Untwisting Heegaard diagrams

Most Heegaard diagrams contain many rectangles, for instance from loops that circle around one of the handle disks. You can always `twist' a Heegaard diagram to get more and more rectangles (as in ...
Brian Rushton's user avatar
6 votes
1 answer
377 views

How to use a Heegaard diagram to retrieve the original 3-manifold that it represents?

(Disclaimer: I apologize that this is an introductory question for a forum like MathOverflow, but I have run out of ideas and resources to understand how this works, and I don't know where else to ask ...
Nicholas James's user avatar
6 votes
1 answer
578 views

Path of almost complex structure in the definition of Heegaard Floer homology

$\DeclareMathOperator\Sym{Sym}$In order to define Heegaard Floer Homology for a connected, closed, oriented 3 manifold, we fix a generic path of nearly symmetric almost complex structure $J_s$ over $\...
Ilknur 's user avatar
6 votes
0 answers
191 views

Is Heegaard-Floer homology the Lagrangian Floer homology of $M=\text{Sym}^g(\Sigma), L_0=\mathbb{T}_{\alpha},L_1=\mathbb{T}_{\beta}$?

Is Heegaard Floer homology the Lagrangian Floer homology of $M=\text{Sym}^g(\Sigma), L_0=\mathbb{T}_{\alpha},L_1=\mathbb{T}_{\beta}$? I am interested in the relationship between the theories ...
contingent's user avatar
5 votes
3 answers
1k views

Is there a version of Seiberg-Witten-Floer or Heegard-Floer homology for 3-manifolds with boundary?

Recently, the Seiberg-Witten-Floer homology created by Kronheimer and Mrowka has important applications in Taubes' proofs of Weinstein conjecture and Arnold Chord Conjecture. Also, Cagatay Kutluhan, ...
Cheuk Hwang's user avatar
5 votes
3 answers
1k views

Heegaard Floer Homology of double branched cover

The question is the following: Let $K\subset S^{3}$ be a knot, consider the double branched cover $Y$ of $S^{3}$ over $K$. We know $Y$ has a unique spin structure $\mathfrak{s}$, now the question is: ...
user44651's user avatar
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5 votes
2 answers
747 views

Heegard diagrams for three-manifolds

I have a basic question about the Heegaard diagrams involved in providing a framework for calculation of Floer-Homology of three-manifolds. Typically such diagrams look like Figure 1 and Figure 2 here ...
user267839's user avatar
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5 votes
1 answer
490 views

What is the Heegaard Floer Homology of a connect sum of $S^2 \times S^1$s?

There are many conjecturally-equivalent three-manifold Floer homologies, of which my understanding is the most-computable is Heegaard Floer homology. What is the (Heegaard) Floer homology of a ...
Theo Johnson-Freyd's user avatar
5 votes
1 answer
224 views

Upsilon of an alternating knot

I have a couple of questions about how Oz-Stip-Sz computes the upsilon function invariant of an alternating knot in their upsilon ($\Upsilon$) paper here.1 This is theorem 1.14 (on the bottom of page ...
Anon's user avatar
  • 51
5 votes
1 answer
339 views

Computation of \tau invariant

I am trying to understand the following inequality, $$0 \leq \tau (K_{+}) - \tau(K_{-}) \leq 1$$ from the following paper by Livingston. \ https://arxiv.org/pdf/math/0311036.pdf . At page 737 , he ...
Monkey.D.Luffy's user avatar
5 votes
1 answer
686 views

Sarkar's Maslov index formula

I have difficulty understanding Sarkar's maslov index formula in symmetric products from http://arxiv.org/abs/math/0609673. If $D$ is an $n$-sided region with corner points $p_1,\ldots, p_n$ then it ...
Reza Rezazadegan's user avatar
5 votes
1 answer
239 views

Does $H_*(A^-_0(K))=\mathbb{F}[U]$ imply that $K$ is an L-space knot?

Let $K$ be a knot in the three-sphere. Let $A_s^-(K)$ be the Alexander filtrations of the knot Floer complex $CFK^{\infty}$. Would $A_0^-(K)$ has homology $\mathbb{F}[U]$ imply that K is an L-space ...
Yajing Liu's user avatar
5 votes
0 answers
126 views

Do knot/link Floer homology detect variations of link genus?

$\newcommand{\wHFK}{\widehat{\mathrm{HFK}}}\newcommand{\wHFL}{\widehat{\mathrm{HFL}}}$Ni has shown that the knot Floer homology $\wHFK$ of an oriented link $L$ (in $S^3$ or more generally homology 3-...
Lukas Lewark's user avatar
4 votes
1 answer
256 views

On Ozsváth and Szabó's branched covering description of holomorphic disks in symmetric products

On page 25 of Holomorphic Disks and Topological Invariants for 3-manifolds (https://arxiv.org/pdf/math/0101206.pdf), the following lemma appears. Given any holomorphic disk $u \in M(x,y)$, there is a ...
Knot All Knots's user avatar
4 votes
1 answer
138 views

Topological type of complement of Heegaard curves in Heegaard surface $(\Sigma - \alpha - \beta)$

Suppose $(\Sigma, \alpha, \beta)$ is a genus-$g$ Heegaard diagram for a closed, oriented $3$-manifold $Y$, i.e. $\Sigma$ is an orientable genus-$g$ surface, and $(\alpha_1, \dots, \alpha_g)$ and $(\...
Matija Sreckovic's user avatar
4 votes
0 answers
643 views

maslov index of a holomorphic disk

I am studying some introductory papers on heegard floer homology and I do not understand the meaning of the Maslov index of a holomorphic disk. I could not find any definition in any of the papers. I ...
mark 's user avatar
  • 155
3 votes
2 answers
292 views

base point in Heegaard Floer homology

It is stated in Mcduff's overview paper "FLOER THEORY AND LOW DIMENSIONAL TOPOLOGY", end of page 9, that the Heegaard Floer homology without considering a base point depends only on the homology of ...
HuiRong's user avatar
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3 votes
1 answer
94 views

Extending curves on a surface to a basis for its first homology satisfying intersection criteria

The title suggests a broader scope of inquiry, but my question mostly pertains to the following example: Let $(Y, \mathcal{Z}, \phi)$ be a bordered 3-manifold with Heegaard diagram $\mathcal{H}$ of ...
contingent's user avatar
3 votes
1 answer
192 views

Algebraic variations of the full knot Floer complex

In Hom's paper (arXiv link), p.20, Section 3.3 ends with "There are other algebraic modifications one may consider, such as setting $U^n = 0$ or $UV = 0$", referring to the knot Floer ...
horned-sphere's user avatar
3 votes
1 answer
437 views

Does knot Floer homology detect knot genus in rational homology spheres?

My question is the following: Does knot Floer homology detect the genus of null-homologous knot in rational homology spheres? If the answer is yes, I would like to have a reference for the ...
christian's user avatar
  • 481
3 votes
1 answer
372 views

Decorations in Szabo's combinatorial spectral sequence

Szabo in http://arxiv.org/abs/1010.4252 gives a combinatorial candidate for what an explicit calculation of the spectral sequence of branched double covers should yield. In other words he gives a ...
Reza Rezazadegan's user avatar
3 votes
0 answers
181 views

Definition of the dual spider number and the formula for the first chern class of the triangle

In the process of trying to understand various maps in Heegaard Floer homology I got stuck on the definition of the dual spider number, which, it seems to me, has a combinatorial definition directly ...
shestipalov's user avatar
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2 votes
0 answers
126 views

Examples of counting holomorphic curves in cylindrical reformulation of Heegaard Floer

In 2005, Robert Lipshitz reformulated Heegaard Floer in a "cylindrical setting" by counting holomorphic curves in $\Sigma \times [0,1] \times \mathbb{R}$ where $\Sigma$ is a Heegaard surface ...
semper-lux's user avatar
2 votes
0 answers
103 views

Ozsvath-Szabo orientation convention for Seifert fibred spaces

I am confused by the orientation convention that Ozsvath and Szabo use in On Heegaard Floer homology and Seifert Fibered Surgeries and would appreciate if someone clarifies this for me. On page 15 ...
shestipalov's user avatar
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2 votes
1 answer
352 views

Heegard Floer homology [closed]

I am new to Heegard Floer. So far I understand that different HF groups are invariants of a three manifold. But I do not understand what these groups actually measure. I mean it seems to me that they ...
kaave's user avatar
  • 21
1 vote
0 answers
79 views

Resolving a mismatch in indexing conventions of knot/link Floer homologies

I have trouble matching the indexing conventions for Ozsvath-Szabo's knot Floer homology with link Floer homology. Say we have a knot $K$ in a 3-sphere. Then we can consider the filtered chain ...
cjackal's user avatar
  • 355
1 vote
0 answers
353 views

Nice proof of the Reidemeister-Singer’s theorem?

Is there a nice proof (preferably with pictures) of the Reidemeister-Singer theorem? I'd prefer some classical methods, perhaps in a book or lecture notes? I want to learn how things are done.
Jake B.'s user avatar
  • 1,423
0 votes
1 answer
438 views

On the proof of Robert Lipshitz's formula on Maslov index.

Hello. I am a beginning graduate student who wants to study Heegaard Floer Homologies. I am now reading the paper https://arxiv.org/abs/1301.4919 Errata to 'A cylindrical reformulation of Heegaard ...
HFH Fan's user avatar