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...21] and [25], this construction is extended to give an invariant, knot Floer homology ̂HFK, for null-homologous knots in a closed, oriented three-manifold. This construction is further generalized in =-=[23]-=- to the case of oriented links. The definition of all these invariants involves counts of holomorphic disks in the symmetric product of a Riemann surface, which makes them rather challenging to calcul...

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...O’s and X’s by {Oi} n i=1 and {Xi} n i=1 , and we denote the two sets by O and X, respectively. (We use here the notation from [MOST07]; the Oi correspond to the “white dots” of [MOS06] and the wi of =-=[OS05b]-=-, while the Xi to the “black dots” of [MOS06] and the zi of [OS05b].) Given a planar grid diagram G, we place it in a standard position on the plane by placing the bottom left corner at the origin, an...

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... = 2, thus c(S, t0) = −2g(S). So we get that ⊕ SFH(Y (S)) = SFH(Y (K), s), s∈Spin c (Y (K)): 〈 c1(s,t0),[S] 〉=−2g(S) which in turn is isomorphic to ̂HFK(Y, K, [S], −g(S)) ≈ ̂HFK(Y, K, [S], g(S)), see =-=[7]-=-. Note that we get ̂HFK(Y, K, [S], g(S)) if we decompose along −S instead of S. □FLOER HOMOLOGY AND SURFACE DECOMPOSITIONS 23 Corollary 8.3. Let K be a knot in a rational homology 3-sphere Y whose Se...

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... component of ∂M has at least two sutures. We can reduce this case to the case treated above using Proposition [Ju08b, Proposition 9.2]. □ Remark. Link Floer homology of a link L ⊂ S 3 was defined in =-=[OS08a]-=-. It agrees with the sutured Floer homology of the sutured manifold S 3 (L) introduced in Example 2.3. In [OS08b] it is shown that if L has no trivial components then the link Floer homology of L dete...

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...er result of the author. Our argument uses the techniques due to Ozsváth and Szabó, Hedden and the author. 57R58, 57M27; 57R30 1 Introduction Link Floer homology was introduced by Ozsváth and Szabó =-=[13]-=-, as a multifiltered theory for links in rational homology 3–spheres. This theory generalizes an earlier invariant for knots, the knot Floer homology (see Ozsváth and Szabó [11] and Rasmussen [15])....

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...tinguish homeomorphic four-manifolds with different smooth structures. In a different direction, there also exist Heegaard Floer invariants for null-homologous knots and links in three-manifolds (see =-=[17, 24, 22]-=-), which have applications to knot theory. One feature shared by the Donaldson, Seiberg-Witten, and Heegaard Floer invariants is that their original definitions are based on counting solutions to some...

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...e-sphere, this invariant is a bigraded Abelian group, whose graded Euler characteristic is the Alexander polynomial. Moreover, in this case, knot Floer homology detects the genus of the knot [21]. In =-=[25]-=-, the constructions from knot Floer homology are generalized to the case of links in S3 . For an ℓ-component link, this gives a multi-graded Abelian group, with one grading for each component of the l...

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...duct of moduli spaces M(Aleft) ×B M(Aright), where B = Sym nw (D) × Sym nw (D) and the fibre product is taken with respect to the above evaluation maps. This is a consequence of a gluing theorem (see =-=[18]-=- Theorem 5.1 for the proof in a very closely related situation and [3] for a discussion of gluing in a general context). Finally, we will prove that (eva right , evb right ) : M(Aa right ) × M(Ab righ...

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... of papers, Ozsváth and Szabó defined invariants for a wide variety of topological and geometric objects in low dimensions, including threeand four- manifolds, knots and links, and contact structures =-=[23, 24, 26, 32, 27]-=-. These invariants proved to be quite powerful, with striking applications to questions in Dehn surgery, contact and symplectic geometry, knot concordance, and questions about unknotting numbers (to n...