# Holomorphic disks and knot invariants

@article{Ozsvath2002HolomorphicDA, title={Holomorphic disks and knot invariants}, author={Peter S. Ozsvath and Zolt{\'a}n Imre Szab{\'o}}, journal={Advances in Mathematics}, year={2002}, volume={186}, pages={58-116} }

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 basic properties of these invariants, including an Euler characteristic calculation, and a description of the behavior under connected sums. Then, we establish a relationship with HF + for surgeries along the knot. Applications include calculation of HF + of three-manifolds obtained by surgeries on some… Expand

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#### References

SHOWING 1-10 OF 32 REFERENCES

Heegaard Floer homology and alternating knots.

- Mathematics
- 2003

In an earlier paper, we introduced a knot invariant for a null-homologous knot K in an oriented three-manifold Y , which is closely related to the Heegaard Floer homology of Y . In this paper we… 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

Holomorphic triangles and invariants for smooth four-manifolds

- Mathematics
- 2001

Abstract In earlier articles, the authors introduced invariants for closed, oriented three-manifolds, defined using a variant of Lagrangian Floer homology in the symmetric products of Riemann… Expand

Knot Floer homology, genus bounds, and mutation

- Mathematics
- 2003

Abstract In an earlier paper, we introduced a collection of graded Abelian groups HFK (Y,K) associated to knots in a three-manifold. The aim of the present paper is to investigate these groups for… Expand

Holomorphic disks and three-manifold invariants: Properties and applications

- Mathematics
- 2004

In [27], we introduced Floer homology theories HF − (Y,s), HF ∞ (Y,s), HF + (Y,t), � HF(Y,s),and HFred(Y,s) associated to closed, oriented three-manifolds Y equipped with a Spin c structures s ∈ Spin… Expand

Holomorphic disks and three-manifold invariants: Properties and applications

- Mathematics
- 2001

In [27], we introduced Floer homology theories HF - (Y,s), HF∞(Y,s), HF + (Y, t), HF(Y,s),and HF red (Y, s) associated to closed, oriented three-manifolds Y equipped with a Spiny structures s ∈ Spin… 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 topological invariants for closed three-manifolds

- Mathematics
- 2004

The aim of this article is to introduce certain topological invariants for closed, oriented three-manifolds Y , equipped with a Spin c structure. Given a Heegaard splitting of Y = U0 ∪Σ U1, these… Expand

Heegaard Floer homologies and contact structures

- Mathematics
- 2002

Given a contact structure on a closed, oriented three-manifold $Y$, we describe an invariant which takes values in the three-manifold's Floer homology $\HFa$. This invariant vanishes for overtwisted… Expand

Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary

- Mathematics
- 2001

Abstract In Ozsvath and Szabo (Holomorphic triangles and invariants for smooth four-manifolds, math. SG/0110169, 2001), we introduced absolute gradings on the three-manifold invariants developed in… Expand