On the Heegaard Floer homology of branched double-covers

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

416 citations recorded by the source corpus.

Abstract Let L ⊂ S 3 be a link. We study the Heegaard Floer homology of the branched double-cover Σ ( L ) of S 3 , branched along L. When L is an alternating link, HF ^ of its branched double-cover has a particularly simple form, determined entirely by the determinant of the link. For the general case, we derive a spectral sequence whose E 2 term is a suitable variant of Khovanov's homology for the link L, converging to the Heegaard Floer homology of Σ ( L ) .

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

View Paper: https://www.semanticscholar.org/paper/006beaa3724fe1762b97df00a5a6534c3c09f71b

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Discrete Mathematics and Combinatorics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

On the Khovanov and knot Floer homologies of quasi-alternating links (2007, background): “Our purpose here is to calculate these invariants for the class of quasi-alternating links introduced in [19], which generalize alternating links.”
Monopoles and lens space surgeries (2003, methodology): “1(Σ(L))| = |H 1(Σ(L 0))| +|H 1(Σ(L 1)| by classical knot theory (c.f. [26]). The result now follows from Proposition 9.2, together with Theorem 2.1. (For more details, see the corresponding result in [35].) Corollary 9.6. Let K⊂ S3 be a knot for which there is a positive rational number r with the p”
On knot Floer homology and cabling (2004, background): “Another connection arises in [33], which relates Khovanov’s homology to the Floer homology of the branched double cover of a link.”
Khovanov module and the detection of unlinks (2012, background): “As shown in Ozsváth–Szabó [26, Lemma 4.18], Gromov compactness implies that aζ is a chain map.”
The Ozsváth-Szabó and Rasmussen Concordance Invariants are not Equal (2005, neutral): “(3) Strongly quasipositive knots, in particular positive knots: s(K) = 2τ (K) = 2g(K).”
Knot Floer homology and rational surgeries (2005, background): “Another interesting family is given by the branched double-covers of alternating knots in S ; cf our paper [25].”
Unoriented knot Floer homology and the unoriented four-ball genus (2015, background): “(Indeed, the same formula holds for the wider class of “quasi-alternating knots” of [23].”
Heegaard Floer homology and genus one, one‐boundary component open books (2008, background): “On the other hand, when K is quasi-alternating there is a fairly straightforward algorithm to compute the correction terms d(Σ(K), s) for all Spin structures s on Σ(K) [39].”
Link Floer homology and the Thurston norm (2006, methodology): “icities must be non-negative for the homotopy class to admit pseudo-holomorphic representatives). Moreover, this homotopy class admits a unique pseudo-holomorphic representative, compare Section 4 of [19]). This is a routine adaptation of the arguments given in that section. For the Heegaard tuple (”
A link invariant from the symplectic geometry of nilpotent slices (2004, background): “Since both presentations have the same number of strands and the same writhe, the computation above shows: (59) Proposition: Up to isomorphism of graded groups, Kh∗symp(κ) remains unchanged if we reverse the orientation of all components of κ. (C) The trefoil We now look at the left-handed trefoil k”
Localization for Involutions in Floer Cohomology (2010, result): “Together with the results of Section 4 below, and the main theorem of [20], this implies that for alternating knots the combinatorial and symplectic Khovanov homologies with Z2 coefficients have the same rank.”
Knots with unknotting number one and Heegaard Floer homology (2004, background): “Given an alternating projection of a knot K with determinant D, Goeritz form Q, an isomorphism φ : Z/DZ −→ Coker(q), and a sign ǫ = ±1, we define a function Tφ,ǫ : Z/DZ −→ Q by Tφ,ǫ(i) = −ǫ · MQ(φ(i)) − γD(i).”
Involutive Heegaard Floer homology (2015, methodology): “or example, (2 ;3;7) is not Z2-homology cobordant to any L-space. Observe that L-spaces that are Z2-homology spheres include, for example, all double branched covers over alternating knots in S3; cf. [42]. We also remark that Corollary 1.4 can be obtained using Pin(2)-equivariant Seiberg-Witten Floe”
Twisting quasi-alternating links (2007, background): “ve description of homologically thin links, and possibly a complete one. However, the recursive deﬁ- nition makes it diﬃcult to decide whether a given knot or link is quasi-alternating. Deﬁnition 1. ([11]) The set Q of quasi-alternating links is the smallest set of links such that • The unknot is in”
Khovanov homology is an unknot-detector (2010, methodology): “More generally, one can ask whether the intermediate pages of the spectral sequence, as filtered groups, are invariants of K (see [2] for the similar question concerning the spectral sequence of Ozsváth and Szabó).”