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...m(K,s) m,s (cf. [21], [25]). The topological significance of this invariant is illustrated by the result that g(K) = max{s ∈ Z ∣ ∣̂HFK∗(K,s) ̸= 0}, where here g(K) denotes the Seifert genus of K (cf. =-=[20]-=-), and also the fact that ̂HFK∗(K,g(K)) has rank one if and only if K is fibered ([8] in the case where g(K) = 1 and [16] in general). The invariant is defined as a version of Lagrangian Floer homolog...

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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...

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...rollary 1.3 in [20], where it is seen as a corollary to a more general result which constrains the structure of the “knot Floer homology” of K, c.f. [21] and [30]. Indeed combining results from [20], =-=[26]-=- and [23], we get that for such a knot, the Seifert genus, the four-ball genus, and the degree of the Alexander polynomial all agree. Moreover, hyperbolic knots with L-space surgeries all provide infi...

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... ξ) are linearly independent elements in homology HF(−Y,sξ). But Y is an L-space, so HF(−Y,sξ) has rank 1, a contradiction. (X) = 0 for any On the other hand, the fact that Y is an L-space implies =-=[24]-=- that b + 2 symplectic filling X of a contact structure on Y . By the argument in [9], it follows that b1(X) = 0. Now, observe that the space −Y can be represented as the boundary of the plumbing show...

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... the three-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...

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...racterization of L-spaces, though L-spaces do come with some rigid geometric restrictions. Most notably, Ozsváth and Szabó prove that if Y is an Lspace then Y contains no co-orientable taut foliation =-=[33]-=-. In [1], we use this result to produce an infinite family of hyperbolic 3-manifolds with no co-orientable taut foliations, extending the first known set of such 3-manifolds, which was discovered by R...

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...≡k (p)HF +(K0, si) ⊕FW 1 ,si −−−−−→ HF +(S3) FW2 ,k −−−−→ HF +(K−p, t ′ k ) −−−−→ FW2,k = ∑ i≡k (p) FW2,ti. Let hi be the rank of FW1,si : HF + (K0, si) → HF + (S 3 ). Combining results from [10] and =-=[13]-=- gives Proposition 2.1. Suppose K−p is a lens space. Then p ≥ 2g(K) − 1, and hi is nonzero if and only if −g(K) + 1 ≤ i ≤ g(K) − 1. Moreover, for −p/2 ≤ k ≤ p/2 rank coker FW2,k = rank HF + (K0, sk) =...

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...d by the open book (S1,r, φ) weakly symplectically fillable by a filling with b+2 (W ) > 0 whenever φ is a pseudo-Anosov diffeomorphism whose fractional Dehn twist coefficients are all at least 1? In =-=[18]-=-, Ozsváth and Szabó show that if (M, ξ) is weakly symplectically fillable by such a filling, and b1(M) = 0 (in which case this weak filling may be perturbed to a strong filling [17]), then there exi...

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...sis that p/q surgery on K agrees with that of the unknot forces the knot Floer homology of K to agree with that of the unknot. Combining this with the fact that knot Floer homology detects the unknot =-=[7]-=-, the Dehn surgery characterization of the unknot follows. In a beautiful recent paper, Ghiggini [2] shows that Heegaard Floer homology also detects the trefoil and the figure eight knot. Appealing to...