Holomorphic disks and genus bounds

By Peter S. Ozsv'ath, Zolt'an Szab'o. 2003

541 citations recorded by the source corpus.

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 genus of a knot. Thisleads to new proofs of certain results previously obtained using Seiberg-Witten monopole Floer homology (in collaboration with Kronheimer and Mrowka). It also leads to a purely Morse-theoretic interpretation of the genus of a knot. The method of proof shows that the canonical element of Heegaard Floer homology associated to a weakly symplectically fillable contact structure is non-trivial. In particular, for certain three-manifolds, Heegaard Floer homology gives obstructions to the existence of taut foliations.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/46c61258ace3d7a6aed8039d51b9bcf86981d7ad

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Citations in the corpus

Knot Floer Homology of (1, 1)-Knots (2003, methodology): “Throughout this paper, we use the same notation as in [ 22 ] and omit the explanation here.”
Knot Floer homology and rational surgeries (2005, background): “1, ĤFK(K, s) = 0 for all s 6= 0, and hence, according to [21], K is the unknot.”
On the geography and botany of knot Floer homology (2014, background): “The first, proved by Ozsváth and Szabó is that if Ĥ FK (K ) ∼= F then K must be the unknot [74].”
NOTIONS OF POSITIVITY AND THE OZSVÁTH–SZABÓ CONCORDANCE INVARIANT (2005, background): “In another direction, Ozsváth and Szabó have proved several remarkable theorems about the relationship between their Floer homology and the topology of lens spaces see [22, 25].”
Concordance homomorphisms from knot Floer homology (2014, background): “In fact, the above inequality is sharp [26], but that is not of importance to the present applications.”
On the spectral sequence from Khovanov homology to Heegaard Floer homology (2008, background): “2, recall that c(ξL) = 0 implies that ξL is not strongly symplectically fillable [16].”
Knot Floer homology detects genus-one fibred knots (2006, background): “ection we will give a brief overview of the results in Heegaard Floer theory we will need in the following, with no pretension of completeness. The details can be found in Ozsvath and Szabo´’s papers [19, 18, 15, 17, 22, 16]. 2.1 Heegaard Floer homology Let Y be a closed, connected, oriented 3–manif”
$C^0$ Approximations of foliations (2015, methodology): “In [36], Ozsvath-Szabo use the Eliashberg-Thurston existence theorem to show that L-spaces do not admit transversely orientable, taut, C(2) foliations.”
On Floer homology and the Berge conjecture on knots admitting lens space surgeries (2007, unknown): “2 of [26]) Suppose K ⊂ S(3) satisfies rk(ĤFK(S(3), K)) = 1.”
Topologically slice knots of smooth concordance order two (2012, background): “The width determines the Thurston norm of the knot complement [28, 35].”
Knot polynomials and knot homologies (2005, background): “5 [26] Let g(K) denote the Seifert genus of K.”
Lectures on open book decompositions and contact structures (2004, methodology): “Moreover, in [34] it is shown that the Thurston norm is determined by Heegaard Floer Homology.”
Link Floer homology detects the Thurston norm (2006, background): “In [13], Ozsváth and Szabó defined link Floer homology for oriented links in rational homology 3–spheres.”
Few remarks about symplectic filling (2003, methodology): “This and other applications of the results of this article in the Heegaard Floer homology theory are discussed in the paper of Peter Ozsv´ath and Zolt´an Szab´o, see [ 27 ].”