# An algorithm for computing some Heegaard Floer homologies

@article{Sarkar2006AnAF, title={An algorithm for computing some Heegaard Floer homologies}, author={Sucharit Sarkar and Jiajun Wang}, journal={Annals of Mathematics}, year={2006}, volume={171}, pages={1213-1236} }

In this paper, we give an algorithm to compute the hat version of Heegaard Floer homology of a closed oriented three-manifold. This method also allows us to compute the filtration coming from a null-homologous link in a three-manifold.

#### Figures from this paper

#### 130 Citations

A Combinatorial Description of the U2=0 Version of Heegaard Floer Homology

- Mathematics
- 2008

We show that every 3–manifold admits a Heegaard diagram in which
a truncated version of Heegaard Floer homology (when the holomorpic disks pass
through the basepoints at most once) can be computed… Expand

Combinatorial Heegaard Floer homology and nice Heegaard diagrams

- Mathematics
- 2009

Abstract We consider a stabilized version of HF of a 3-manifold Y (i.e. the U = 0 variant of Heegaard Floer homology for closed 3-manifolds). We give a combinatorial algorithm for constructing this… Expand

Notes on bordered Floer homology

- Mathematics
- 2012

This is a survey of bordered Heegaard Floer homology, an extension of the Heegaard Floer invariant HF-hat to 3-manifolds with boundary. Emphasis is placed on how bordered Heegaard Floer homology can… Expand

Introduction to the basics of Heegaard Floer homology

- Mathematics
- 2010

This paper provides an introduction to the basics of Heegaard Floer homology with some emphasis on the hat theory and to the contact geometric invariants in the theory. The exposition is designed to… Expand

Combinatorial Heegaard Floer homology and sign assignments

- Mathematics
- 2013

We provide an intergral lift of the combinatorial definition of Heegaard Floer homology for nice diagrams, and show that the proof of independence using convenient diagrams adapts to this setting.

Dehn Twists in Heegaard Floer Homology

- Mathematics
- 2009

We derive a new exact sequence in the hat-version of Heegaard Floer homology. As a consequence we see a functorial connection between the invariant of Legendrian knots and the contact element. As an… Expand

Algebraic torsion via Heegaard Floer homology

- Mathematics
- 2015

We outline Hutchings's prescription that produces an ECH analog of Latschev and Wendl's algebraic $k$-torsion in the context of $ech$, a variant of ECH used in a proof of the isomorphism between… Expand

HEEGAARD FLOER HOMOLOGY AND FIBRED 3-MANIFOLDS By YI NI Dedicated to the memory of

- 2009

Given a closed 3-manifold Y , we show that the Heegaard Floer homology determines whether Y fibres over the circle with a fibre of negative Euler characteristic. This is an analogue of an earlier… Expand

HEEGAARD FLOER HOMOLOGY AND FIBRED 3-MANIFOLDS By YI NI Dedicated to the memory of

- 2009

Given a closed 3-manifold Y , we show that the Heegaard Floer homology determines whether Y fibres over the circle with a fibre of negative Euler characteristic. This is an analogue of an earlier… Expand

Combinatorial proofs in bordered Heegaard Floer homology

- Mathematics
- 2016

Using bordered Floer theory, we give a combinatorial construction and proof of invariance for the hat version of Heegaard Floer homology. As a part of the proof, we also establish combinatorially the… Expand

#### References

SHOWING 1-10 OF 19 REFERENCES

Heegaard Diagrams and Holomorphic Disks

- Mathematics
- 2004

A three-manifold equipped with a Heegaard diagram can be used to set up a Floer homology theory whose differential counts pseudo-holomorphic disks in the $g$-fold symmetric product of the Heegaard… Expand

Heegaard diagrams and Floer homology

- Mathematics
- 2006

We review the construction of Heegaard�Floer homology for closed three-manifolds
and also for knots and links in the three-sphere. We also discuss three applications of this
invariant to knot theory:… Expand

A combinatorial description of knot Floer homology

- Mathematics
- 2006

Given a grid presentation of a knot (or link) K in the three-sphere, we describe a Heegaard diagram for the knot complement in which the Heegaard surface is a torus and all elementary domains are… Expand

Combinatorial cobordism maps in hat Heegaard Floer theory

- Mathematics
- 2006

In a previous article, Sarkar and Wang [15] gave a combinatorial description of the hat version of Heegaard Floer homology for three-manifolds. Given a cobordism between two connected… Expand

Link Floer homology and the Thurston norm

- Mathematics
- 2006

We show that link Floer homology detects the Thurston norm of a link complement. As an application, we show that the Thurston polytope of an alternating link is dual to the Newton polytope of its… Expand

Holomorphic disks and knot invariants

- Mathematics
- 2002

Abstract We define a Floer-homology invariant for knots in an oriented three-manifold, closely related to the Heegaard Floer homologies for three-manifolds defined in an earlier paper. We set up… Expand

Knot Floer homology and the four-ball genus

- Mathematics
- 2003

We use the knot filtration on the Heegaard Floer complex to define an integer invariant tau(K) for knots. Like the classical signature, this invariant gives a homomorphism from the knot concordance… Expand

Holomorphic disks and topological invariants for closed three-manifolds

- Mathematics
- 2001

The aim of this article is to introduce certain topological invariants for closed, oriented three-manifolds Y, equipped with a Spiny structure. Given a Heegaard splitting of Y = U 0o U Σ U 1 , these… Expand

Holomorphic disks and genus bounds

- Mathematics
- 2004

We prove that, like the Seiberg-Witten monopole homology, the Heegaard Floer homology for a three-manifold determines its Thurston norm. As a consequence, we show that knot Floer homology detects the… Expand

Floer homology and knot complements

- Mathematics
- 2003

We use the Ozsvath-Szabo theory of Floer homology to define an invariant of knot complements in three-manifolds. This invariant takes the form of a filtered chain complex, which we call CF_r. It… Expand