# On knot Floer homology and lens space surgeries

@article{Ozsvath2003OnKF, title={On knot Floer homology and lens space surgeries}, author={Peter S. Ozsvath and Zolt{\'a}n Imre Szab{\'o}}, journal={Topology}, year={2003}, volume={44}, pages={1281-1300} }

Abstract In an earlier paper, we used the absolute grading on Heegaard Floer homology HF + to give restrictions on knots in S 3 which admit lens space surgeries. The aim of the present article is to exhibit stronger restrictions on such knots, arising from knot Floer homology. One consequence is that the non-zero coefficients of the Alexander polynomial of such a knot are ± 1 . This information can in turn be used to prove that certain lens spaces are not obtained as integral surgeries on knots…

## 386 Citations

On Floer homology and the Berge conjecture on knots admitting lens space surgeries

- Mathematics
- 2007

We complete the first step in a two-part program proposed by Baker, Grigsby, and the author to prove that Berge’s construction of knots in the three-sphere which admit lens space surgeries is…

Heegaard Floer homology and knots determined by their complements

- Mathematics
- 2015

In this paper we investigate the question of when different surgeries on a knot can produce identical manifolds. We show that given a knot in a homology sphere, unless the knot is quite special,…

A note on the knot Floer homology of fibered knots

- MathematicsAlgebraic & Geometric Topology
- 2018

We prove that the knot Floer homology of a fibered knot is nontrivial in its next-to-top Alexander grading. Immediate applications include new proofs of Krcatovich's result that knots with $L$-space…

The mapping cone formula in Heegaard Floer homology and Dehn surgery on knots in S3

- Mathematics
- 2017

We write down an explicit formula for the $+$ version of the Heegaard Floer homology (as an absolutely graded vector space over an arbitrary field) of the results of Dehn surgery on a knot $K$ in…

Heegaard Floer genus bounds for Dehn surgeries on knots

- Mathematics
- 2012

We provide a new obstruction for a rational homology 3-sphere to arise by Dehn surgery on a given knot in the 3-sphere. The obstruction takes the form of an inequality involving the genus of the…

AN INTRODUCTION TO KNOT FLOER HOMOLOGY

- Mathematics
- 2014

This is a survey article about knot Floer homology. We present three constructions of this invariant: the original one using holomorphic disks, a combinatorial description using grid dia- grams, and…

Finite knot surgeries and Heegaard Floer homology

- Mathematics
- 2012

It is well known that any 3‐manifold can be obtained by Dehn surgery on a link, but not which ones can be obtained from a knot or which knots can produce them. We investigate these two questions for…

Knots which admit a surgery with simple knot Floer homology groups

- Mathematics
- 2010

The importance of studying knots inside rational homology spheres which have simple knot Floer homology came up in the study of the Berge conjecture using techniques from Heegaard Floer homology by…

On the geography and botany of knot Floer homology

- Mathematics
- 2014

This paper explores two questions: (1) Which bigraded groups arise as the knot Floer homology of a knot in the three-sphere? (2) Given a knot, how many distinct knots share its Floer homology?…

The reduced knot Floer complex

- Mathematics
- 2015

We define a “reduced” version of the knot Floer complex CFK − (K) CFK − ( K ) , and show that it behaves well under connected sums and retains enough information to compute Heegaard Floer d…

## References

SHOWING 1-10 OF 41 REFERENCES

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…

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…

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…

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…

Alternating knots satisfy strong property P

- Mathematics
- 1999

Abstract. Suppose a manifold is produced by finite Dehn surgery on a non-torus alternating knot for which Seifert's algorithm produces a checkerboard surface. By demonstrating that it contains an…

Spherical space forms and Dehn filling

- Mathematics
- 1996

Abstract This paper concerns those Dehn fillings on a torally bounded 3-manifold which yield manifolds with a finite fundamental group. The focus will be on those torally bounded 3-manifolds which…

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…

Torsion invariants of $Spin^c$-structures on 3-manifolds

- Mathematics
- 1997

Recently there has been a surge of interest in the Seiberg-Witten invariants of 3-manifolds, see [3], [4], [7]. The Seiberg-Witten invariant of a closed oriented 3-manifold M is a function SW from…

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…

The geometries of 3-manifolds

- Mathematics
- 1983

The theory of 3-manifolds has been revolutionised in the last few years by work of Thurston [66-70]. He has shown that geometry has an important role to play in the theory in addition to the use of…