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...× S −→ Z, satisfying gr(x,y) + gr(y,w) = gr(x,w) for each x,y,w ∈ S. When the corresponding theory for four-manifolds is developed, this relative Z-grading can be lifted to an absolute Q-grading, see =-=[29]-=-.HOLOMORPHIC DISKS AND THREE-MANIFOLD INVARIANTS 33 but one should bear in mind that ∂ does depend on the path Js. When it is important to call attention to this dependence, we write ∂Js, see the pro...

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...n [12]. They possess many of the known or expected properties of the Seiberg-Witten Floer homologies, and have been used to define four-manifold invariants analogous to the Seiberg-Witten invariants (=-=[21]-=-, [20]). It would be a mistake, however, to think of the Ozsváth-Szabó theory as being nothing more than a convenient technical alternative to Seiberg-Witten Floer homology. The Ozsváth-Szabó invarian...

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...s four-dimensional picture can already be found in the discussion on holomorphic triangles (c.f. Section 8 of [26] and Section 9 of the present paper). These four-manifold invariants are presented in =-=[30]-=-. Although the link with Seiberg-Witten theory was our primary motivation for finding the invariants, we emphasize that the invariants studied here require no gauge theory to define and calculate, onl...

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...re for i = 2, ..., g, βi is a small exact Hamiltonian translate of γi, with µ = β2, while the β1 and γ1 are different (the γ1 here corresponds to the longitude of γ, and β1 to its meridian) Following =-=[20]-=-, the map f ∞ W,s: CFK ∞ (Y, K) −→ CFK ∞ (Y ′ , K ′ )HOLOMORPHIC DISKS AND KNOT INVARIANTS 41 induced by the cobordism is defined by f ∞ W,s ([x, i, j]) = ∑ ∑ #M(ψ) · [y, i − nw(ψ), j − nz(ψ)]. {y∈Tα...

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... and let W0 be the cobordism from −Y0 to −Y ′ 0 obtained by attaching the two-handle along γ, thought now as a curve in Y0 (rather than Y ). Then, by naturality of the maps induced by cobordisms (see =-=[11]-=-), we have a commutative diagram Z ∼ = ̂ HF(−Y0, 1 − g) ⏐ FV ↓ FW 0 −−−→ ̂ HF(−Y ′ 0, 1 − g) ∼ = Z ⏐ ↓ F V ′ ̂HF(−Y ) FW −−−→ ̂ HF(−Y ′).s18 PETER OZSVÁTH AND ZOLTÁN SZABÓ Now, FW0 induces an isomorph...

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... Subsection 3.1. The fact that FW(Θ) is a generator for ̂ HF(Y ′ ) (which we denote by Θ ′ ) follows from a direct inspection of a Heegaard triple which naturally splits into genus one summands, c.f. =-=[15]-=-. (Alternately, one could use the surgery exact sequence which in this case reads ... −−−→ ̂ HF(Y ′ ) −−−→ ̂ HF(Y ) −−−→ ̂ HF(Y ′ ) −−−→ ... to deduce that the map from ̂ HF(Y ) to ̂ HF(Y ′ ), which i...

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...ten by counting pseudo-holomorphic triangles in Sym g (Σ), as explained in Section 9 of [14]. Further invariance properties of these maps, and a generalization to other cobordisms, are established in =-=[11]-=-. In Section 4 of [17], we described the relationship between this knot filtration and the Heegaard Floer homologies of three-manifolds obtained by performing “sufficiently large” integral surgeries o...

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...ark 1.3. It is a straightforward matter to determine HF + (Y (G), t) from HF + (−Y (G), t). In the statement of the above theorem, the grading on HF + (−Y (G), t) is the absolute Q-grading defined in =-=[9]-=- and studied in [6]. Recall that when −Y (G) is an integral homology sphere, this absolute grading takes values in Z. As a qualitative remark, it is perhaps worth pointing out the following corollary ...

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...section 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 Ozsváth and Szabó’s papers =-=[16, 15, 20, 14, 19, 13]-=-. 2.1 Heegaard Floer homology Let Y be a closed, connected, oriented 3–manifold. For any Spinc –structure t on Y Ozsváth and Szabó [16] defined an Abelian group HF +(Y,t) which is an isomorphism invar...