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Knot Floer homology detects genus-one ﬁbred knots Paolo Ghiggini e e a CIRGET, Universit´ du Qu´bec ` Montr´ale Case Postale 8888, succursale Centre-Ville e e Montr´al (Qu´bec) H3C 3P8, Canada ghiggini@math.uqam.ca October 21, 2006 Abstract a o Ozsv´th and Szab´ conjectured that knot Floer homology detects ﬁ- bred knots. We propose a strategy to approach this conjecture based on contact topology and Gabai’s theory of sutured manifold decom- position. We implement this strategy for genus-one knots, obtaining e as a corollary that if rational surgery on a knot K gives the Poincar´ homology sphere Σ(2, 3, 5), then K is the left-handed trefoil knot. 1 Introduction Knot Floer homology is an invariant for null-homologous oriented links in a o 3-manifolds introduced independently by Ozsv´th and Szab´ in [15] and by Rasmussen in [23]. For a knot K in S 3 — the only case we will consider in this article — and any integer d the knot Floer homology group HF K(K, d) is a ﬁnitely generated graded Abelian group. Knot Floer homology can be seen as a categoriﬁcation of the Alexander polynomial in the sense that +∞ χ(HF K(K, d))T d = ∆K (T ) d=−∞ where ∆K (T ) denotes the symmetrised Alexander polynomial of K. How- ever the groups HF K(K, d), and in particular the top non trivial group, contain more information than just the Alexander polynomial, as the fol- lowing results show. 1 Theorem 1.1. ([14, Theorem 1.2]) Let K be a knot in S 3 with genus g(K). Then g(K) = max d ∈ Z : HF K(K, d) = 0 . Theorem 1.2. (See [20, Theorem 1.1]) Let K be a knot in S 3 with genus g. If K is a ﬁbred knot, then HF K(K, g) = Z. a o Ozsv´th and Szab´ formulated the following conjecture, whose evidence is supported by the computation of knot Floer homology for a large number of knots. Conjecture 1.3. (Ozsv´th–Szab´) If K is a knot in S 3 with genus g and a o HF K(K, g) = Z then K is a ﬁbred knot. In this article we propose a strategy to attack Conjecture 1.3, and we im- plement it in the case of genus-one knots. More precisely, we will prove the following result: Theorem 1.4. Let K be an oriented genus-one knot in S 3 . Then K is ﬁbred if and only if HF K(K, 1) = Z Our strategy to prove Theorem 1.4 is to deduce information about the top knot Floer homology group of non-ﬁbred knots from topological proper- ties of their complement via sutured manifold decompositions and contact structures in a way that is reminiscent of the proof of Theorem 1.1. It is well known that the only ﬁbred knots of genus one are the trefoil knots and the ﬁgure-eight knot. Therefore Theorem 1.4, together with the computation of knot Floer homology for such knots, implies the following: Corollary 1.5. Knot Floer homology detects the trefoil knots and the ﬁgure- eight knot. The following conjecture was formulated by Kirby in a remark after Problem 3.6(D) of his problem list, and by Zhang in [25]: ˆ Conjecture 1.6. (Conjecture I) If K is a knot in S 3 such that there exists a rational number r for which the 3–manifold obtained by r–surgery on K e is homeomorphic to the Poincar´ homology sphere Σ(2, 3, 5), then K is the left-handed trefoil knot. 2 ˆ Conjecture I was proved for some knots by Zhang in [25], and a major step a o toward its complete proof was made by Ozsv´th and Szab´ in [19], where ˆ they proved that a counterexample to Conjecture I must have the same knot Floer Homology groups as the left-handed trefoil knot. Corollary 1.5 provides the missing step to prove it in full generality: ˆ Corollary 1.7. Conjecture I holds. a o Corollary 1.5 has also been used by Ozsv´th and Szab´ to prove that the trefoil knot and the ﬁgure-eight knot are determined by their Dehn surgeries [22]. Shortly after a preliminary version of this paper was made accessible to the public, Yi Ni proved Conjecture 1.3 [12]. For a wide class of knots his proof follows the strategy developed in this article, then he proves gluing formulas for knot Floer homology, and uses them to reduce the remaining cases to the ones he has already considered. Acknowledgements We warmly thank Steve Boyer, Ko Honda, Joseph Maher, and Stefan Till- mann for many inspiring conversations. This work was partially supported by a CIRGET fellowship and by the Chaire de Recherche du Canada en e e alg`bre, combinatoire et informatique math´matique de l’UQAM. 2 Overview of Heegaard Floer theory a Heegaard Floer theory is a family of invariants introduced by Ozsv´th and o Szab´ in the last few years for the most common objects in low-dimensional topology. In this section we will give a brief overview of the results in Heegaard Floer theory we will need in the following, with no pretension a o of completeness. The details can be found in Ozsv´th and Szab´’s papers [17, 16, 21, 15, 20, 14]. 2.1 Heegaard Floer homology Let Y be a closed, connected, oriented 3–manifold. For any Spinc –structure t on Y Ozsv´th and Szab´ [17] deﬁned an Abelian group HF + (Y, t) which is a o 3 an isomorphism invariant of the pair (Y, t). When c1 (t) is not a torsion ele- ment in H 2 (Y ) the group HF + (Y, t) is ﬁnitely generated; see [16, Theorem 5.2 and Theorem 5.11]. If there is a distinguished surface Σ in Y which is clear from the context, we will use the shortened notation HF + (Y, d) = HF + (Y, t). t ∈ Spinc (Y ) c1 (t), [Σ] = 2d If d = 0 HF + (Y, d) is ﬁnitely generated because HF + (Y, t) = 0 only for ﬁnitely many Spinc –structures. Heegaard Floer homology is symmetric; in fact there is a natural isomorphism HF + (Y, d) ∼ HF + (Y, −d) for any 3– = manifolds Y and any integer d: see [16, Theorem 2.4]. There is an adjunction inequality relating HF + to the minimal genus of embedded surfaces which can be stated as follows: Theorem 2.1. [16, Theorem 1.6] If Σ has genus g, then HF + (Y, d) = 0 for all d > g − 1. Any connected oriented cobordism X from Y1 to Y2 induces a homomor- phism FX : HF + (Y1 , t1 ) → HF + (Y2 , t2 ) which splits as a sum of homomorphisms indexed by the Spinc –structures on X extending t1 and t2 . When X is obtained by a single 2–handle addition FX ﬁts into a surgery exact triangle as follows: Theorem 2.2. ([16, Theorem 9.12]) Let (K, λ) be an oriented framed knot in Y , and let µ be a meridian of K. Denote by Yλ (K) the manifold obtained from Y by surgery on K with framing λ, and by X the cobordism induced by the surgery. If we choose a surface Σ ⊂ Y \ K to partition the Spinc - structures, then the following triangle FX HF + (Y, d) / HF + (Yλ (K), d) hRRR RRR kkk RRR kkk RRR kkkkk R ukkk HF + (Yλ+µ (K), d) is exact for any d ∈ Z. 4 2.2 a o The Ozsv´th–Szab´ contact invariant A contact structure ξ on a 3–manifold Y determines a Spinc –structure tξ on Y such that c1 (tξ ) = c1 (ξ). To any contact manifold (Y, ξ) we can associate an element c+ (ξ) ∈ HF + (−Y, tξ )/ ± 1 which is an isotopy invariant of ξ, see [20]. In the following we will always abuse the notation and consider c+ (ξ) as an element of HF + (−Y, tξ ), although it is, strictly speaking, deﬁned only up to sign. This abuse does not lead to mistakes as long as we do not use the additive structure on HF + (−Y, tξ ). The proof of the following lemma is contained in the proof of [14, Theorem 1.2]; see also [10, Theorem 2.1] for a similar result in the setting of monopole Floer homology. Lemma 2.3. Let Y be a closed, connected oriented 3–manifold with b1 (Y ) = 1, and let ξ be a weakly symplectically ﬁllable contact structure on Y such that c1 (ξ) is non trivial in H 2 (Y ; R). Then c+ (ξ) is a primitive element of HF + (−Y, tξ )/ ± 1. Given a contact manifold (Y, ξ) and a Legendrian knot K ⊂ Y there is an op- eration called contact (+1)–surgery which produces a new contact manifold (Y ′ , ξ ′ ); see [2] and [3]. Y ′ is obtained by surgery along K with coeﬃcient +1 with respect to the framing induced on K by ξ, and ξ ′ coincides with ξ out- a o side a neighbourhood of K. The Ozsv´th–Szab´ contact invariant behaves well with respect to contact (+1)–surgeries: Lemma 2.4. ([11, Theorem 2.3]; see also [20]) Suppose (Y ′ , ξ ′ ) is obtained from (Y, ξ) by a contact (+1)–surgery, and let −X be the cobordism induced by the surgery with the opposite orientation. Then + F−X (c+ (ξ)) = c+ (ξ ′ ). 2.3 Knot Floer homology Knot Floer homology is a family of ﬁnite dimensional graded Abelian groups HF K(K, d) indexed by d ∈ Z attached to any oriented knot K in S 3 ; see a o Ozsv´th and Szab´ [15]. If K is a knot we denote the 0–surgery on K by YK . The knot Floer homology of K is related to the Heegaard Floer homology of YK by the following: 5 Proposition 2.5. ([15, Corollary 4.5] and [14, Theorem 1.2]). Let K be a knot of genus g > 1. Then HF K(K, g) = HF + (YK , g − 1). u Another property of knot Floer homology we will need later is a K¨nneth- like formula for connected sums. Proposition 2.6. ([15, Corollary 7.2]) Let K1 and K2 be knots in S 3 , and denote by K1 #K2 their connected sum performed. If HF K(K1 , d) is a free Abelian group for every d (or if we work with coeﬃcients in a ﬁeld), then HF K(K1 #K2 , d) = HF K(K1 , d1 ) ⊗ HF K(K1 , d1 ). d1 +d2 =d 3 Taut foliations and Heegaard Floer homology 3.1 Controlled perturbation of taut foliations Eliashberg and Thurston in [4] introduced a new technique to construct symplectically ﬁllable contact structures by perturbing taut foliations. In this section we show how to control the perturbation in the neighbourhood of some closed curves. We need to introduce some terminology about con- foliations, following Eliashberg and Thurston [4]. Deﬁnition 3.1. A confoliation on an oriented 3–manifold is a tangent plane ﬁeld η deﬁned by the kernel of a 1–form α such that α ∧ dα ≥ 0. Given a confoliation η on M we deﬁne its contact part H(η) as H(η) = {x ∈ M : α ∧ dα(x) > 0}. Lemma 3.2. Let Σ be a compact leaf with trivial germinal holonomy (See [1, Section 2.3] for the deﬁnition). in a taut smooth foliation F on a 3– manifold M , and let γ be a non-separating closed curve in Σ. Then we can modify F in a neighbourhood of γ so that we obtain a new taut smooth foliation with non trivial linear holonomy along γ. 6 Proof. The holonomy of Σ determines the germ of F along Σ (see [1, Theo- rem 3.1.6]), therefore Σ has a neighbourhood N = Σ × [−1, 1] such that F|N is the product foliation. Pick γ ′ ⊂ Σ = Σ × {0} such that it intersects γ in a unique point, and call Σ′ = Σ \ γ ′ . The boundary of Σ′ has two components ′ ′ ′ γ+ and γ− . For every point x ∈ γ ′ denote by x± the points in γ± which correspond to x. Choose a diﬀeomorphism f : [−1, 1] → [−1, 1] such that 1. f (x) = x if x ∈ [−1, −1 + ǫ] ∪ [1 − ǫ, 1] for some small ǫ; 2. f (0) = 0; 3. f ′ (0) = 1, ′ ′ then re-glue γ− × [−1, 1] to γ+ × [−1, 1] identifying (x− , t) to (x+ , f (t)). Lemma 3.3. Let (M, F) be a foliated manifold, and let γ be a curve with non-trivial linear holonomy contained in a leaf Σ. Then F can be approxi- mated in the C 0 –topology by confoliations such that γ is contained in their contact parts and is a Legendrian curve with twisted number zero with respect to the framing induced by Σ. Proof. The proof of this Lemma is an easy computation on the explicit contact form constructed in [4, Proposition 2.6.1], however we provide the details to the beneﬁt of the reader. We identify a neighbourhood N of γ with S 1 × [−1, 1]× [−1, 1] with coordinates (x, y, z) so that N ∩ Σ = {z = 0}, γ = S 1 × {0} × {0}, and the tangent distribution T F|N is the kernel of a 1– form dz + v(x, z)dx, see [4, Proposition 1.1.5]. a(x, 0) = 0 for all x ∈ [−1, 1] because {z = 0} is contained in a leaf of F. If F has non trivial linear holonomy along γ we can also assume that ∂v > C for a positive constant ∂z C. Take a smooth monotone function h : R → [0, 1] which is equal to 1 near 0, positive on [0, 1), and vanishes on [1, +∞). Then for any ǫ > 0 the 1– form αǫ = dz + v(x, z)dx + ǫh(y 2 + z 2 )dy deﬁnes a confoliation and is a contact form in a smaller neighbourhood of γ. In fact (αǫ ∧ dαǫ ) = (ǫh(y 2 + z 2 ) ∂v − 2ǫzh′ (y 2 + z 2 )v(x, z))dx ∧ dy ∧ dz ≥ 0 because ∂v > 0, h′ ≤ 0 and ∂z ∂z zv(x, z) ≥ 0, moreover (αǫ ∧ dαǫ )|γ = ǫ ∂v dx ∧ dy ∧ dz > 0. The curve γ is ∂z ∂ Legendrian because αǫ |γ = dz + ǫdy and T γ is generated by ∂x . Since the kernel of αǫ never becomes tangent to Σ along γ for ǫ > 0, γ has twisting number zero with respect to the faming induced by Σ. 7 For any subset A ⊂ M we deﬁne its saturation A as the set of all points in M which can be connected to A by a curve tangent to η. Deﬁnition 3.4. A confoliation η on M is called transitive if H(η) = M . Lemma 3.5. Let (M, F) be a smooth taut foliated manifold, let Σ be a compact leaf of F with trivial germinal holonomy, and let γ ⊂ Σ be a closed non-separating curve. Then F can be approximated in the C 0 –topology by contact structures such that γ is a Legendrian curve with twisting number zero with respect to Σ. Proof. First we apply Lemma 3.2 to create non-trivial linear holonomy along γ, so that we can apply Lemma 3.3 to perturb F to a contact structure in a neighbourhood of γ, and γ becomes a Legendrian curve with twisting number zero. In this way we obtain a confoliation η. The approximation of η by contact structures is done in two steps. First η is C 0 –approximated by a transitive confoliation η [4, Proposition 2.7.1], then η is C 1 –approximated by a contact structure ξ [4, Proposition 2.8.1]. The ﬁrst step is done by perturbing the confoliation in arbitrarily small neighbourhoods of curves contained in M \ H(η), then we can assume that a contact neighbourhood V of γ is not touched in the ﬁrst step. Let h : M → [0, 1] be a smooth function supported in V such that h ≡ 1 in a smaller neighbourhood of γ. Since η and ξ can be made arbitrarily close in the C 1 –topology of the space of tangent 2–plane ﬁelds, we can choose deﬁning forms α and β such that the 1–form β + h(α − β) is C 1 –close to β, therefore it is a contact form because the contact condition is open in the C 1 –topology. The contact structure deﬁned by this form coincides with η near γ and is C 0 –close to F. Proposition 3.6. ([4, Corollary 3.2.5] and [5, Corollary 1.4]) Let F be a taut foliation on M . Then any contact structure ξ which is suﬃciently close to F as a plane ﬁeld in the C 0 topology is weakly symplectically ﬁllable. 3.2 An estimate on the rank of HF + (Y ) coming from taut foliations Let η be a ﬁeld of tangent planes in the 3–manifold Y , and let S be an em- bedded compact surface with non empty boundary such that ∂S is tangent to η. 8 Deﬁnition 3.7. Let v be the positively oriented unit tangent vector ﬁeld of ∂S. We deﬁne the relative Euler class e(η, S) ∈ H 2 (S, ∂S) of η on S as the obstruction to extending v to a nowhere vanishing section of η|S , i. e. if v is a generic extension of v to a section of η|S , then e(η, S) = P D[˜−1 (0)]. ˜ v Although properly speaking e(η, S) is an element of H 2 (S, ∂S), we can iden- tify it with an integer number via the isomorphism H 2 (S, ∂S) = Z. If η is the ﬁeld of the tangent planes of the leaves of a foliation F we write e(F, S) for e(η, S). If η is a contact structure e(η, S) is the sum of the rotation numbers of the components of ∂S computed with respect to S. Theorem 3.8. Let Y be a 3–manifold with H2 (Y ) ∼ Z, and let Σ be a genus = minimising closed surface representing a generator of H2 (Y ). Call Σ+ and Σ− the two components of ∂(Y \ Σ). Suppose that Σ has genus g(Σ) > 1 and that Y admits two smooth taut foliations F1 and F2 such that Σ is a compact leaf for both, and the holonomy of both foliations on Σ has the same Taylor series as the identity. If there exists a properly embedded surface S ⊂ Y \ Σ with boundary ∂S = α+ ∪ α− such that 1. α+ ⊂ Σ+ and α− ⊂ Σ− are non separating curves, and 2. e(F1 , S) = e(F2 , S) (see Figure 1), then rk HF + (Y, g − 1) > 1. Requiring that the holonomy of Σ has the same Taylor series as the identity is not as strong a restriction as it seems, because foliations constructed by sutured manifold theory have this property; see the induction hypothesis in the proof of [6, Theorem 5.1]. The strategy of the proof is to view F1 and F2 as taut foliations on −Y and to approximate them by contact structures ξ1 and ξ2 on −Y so that c(ξi ) ∈ HF + (Y, g − 1), then to construct a new 3–manifold Yφ together with + a 4–dimensional cobordism W from −Y to −Yφ so that F−W (c+ (ξ1 )) and + F−W (c+ (ξ2 )) are linearly independent in HF + (Yφ ). This implies that c+ (ξ1 ) and c+ (ξ2 ) are linearly independent in HF + (Y, g − 1). The entire subsection is devoted to the proof of Theorem 3.8. We choose a diﬀeomorphism φ : Σ+ → Σ− such that φ(α+ ) = α− , and we form the new 3–manifold Yφ by cutting Y along Σ and re-gluing Σ+ to Σ− after acting by φ. The diﬀeomorphism φ exists because α+ and α− are non separating. 9 α+ S α− Figure 1: A somewhat misleading picture of S ⊂ Y \ Σ It is well known that every element of the mapping class group of a closed surface of genus g > 1 can be written as a product of positive powers of pos- itive Dehn twists around non separating curves; see [5, Footnote 3]. In order to construct the 4–dimensional cobordism W from Y to Yφ we decompose φ as a product φ = τc1 . . . τck where each τci is a positive Dehn twist around a non-separating curve ci ⊂ Σ. Then we identify a tubular neighbourhood N of Σ with Σ × [−1, 1], we choose distinct points t1 , . . . , tk in (−1, 1) and see ci as a curve in Σ × {ti }. The surface Σ × {ti } induces a framing on ci , and Yφ is obtained by (−1)–surgery on the link C = c1 ∪ . . . ∪ ck ⊂ Y , where the surgery coeﬃcient of ci is computed with respect to that framing. Equivalently, −Yφ is obtained by (+1)–surgery on the same link C seen as a link in −Y . We denote by W the smooth 4–dimensional cobordism obtained by adding 2–handles to −Y along the curves ci with framing +1, and by −W the same cobordism with opposite orientation, so that −W is obtained by adding 2–handles to Y along the curves ci with framing −1. Lemma 3.9. The map + F−W : HF + (Y, g − 1) → HF + (Yφ , g − 1) induced by the cobordism −W is an isomorphism. Proof. The proof is similar to the proof of [18, Lemma 5.4]. By the com- + position formula [21, Theorem 3.4] the map F−W is a composition of maps 10 induced by elementary cobordisms Wi each of which is obtained from a single 2–handle addition corresponding to the surgery along the curve ci , therefore we can assume without loss of generality that W = W1 . In this case Yφ is + obtained from Y by (−1)–surgery along c1 , therefore by Theorem 2.2 F−W ﬁts into the exact triangle F−W HF + (Y, g − 1) / HF + (Yφ , g − 1) hRRR RRR lll RRR lll RRR lllll R ulll HF + (Y0 , g − 1) where Y0 is obtained by 0–surgery on c1 . Since c1 bounds a disc in Y0 there is a surface Σ′ ⊂ Y0 with χ(Σ′ ) > χ(Σ) = 2g −2. Such surface violates the adjunction inequality (Theorem 2.1), there- fore HF + (Y0 , g − 1) = 0. This implies that F−W is an isomorphism. We can assume that the tubular neighbourhood N of Σ is foliated as a product in both foliations. In fact Y is diﬀeomorphic to (Y \Σ)∪(Σ×[−1, 1]) where Σ+ is identiﬁed with Σ×{−1}, and Σ− is identiﬁed with Σ×{1}. Then we can extend the foliations Fi |Y \Σ to foliations on (Y \ Σ) ∪ (Σ × [−1, 1]) which are product foliations on Σ × [−1, 1]. We call the resulting foliated manifolds (Y, F1 ) and (Y, F2 ) again. This operation does not destroy the smoothness of F1 and F2 because their holonomies along Σ have the same Taylor series as the identity. We see S as a surface in Y \N , so that α+ is identiﬁed to a curve in Σ×{−1}, and α− is identiﬁed to a curve in Σ × {1}. By Lemma 3.5 we can control the perturbations of F1 and F2 so that α+ , α− , and ci for all i become Legendrian curves with twisting number zero for both ξ1 and ξ2 , where the twisting number is computed with respect to the framing induced by Σ. ′ ′ This implies that we can construct contact structures ξ1 and ξ2 on −Yφ by (+1)–contact surgery on ξ1 and ξ2 . ′ Lemma 3.10. c(ξi ) = 0 for i = 1, 2. Proof. By hypothesis b1 (Y ) = 1, and c1 (ξ1 ) and c1 (ξ2 ) are non torsion be- cause c1 (ξi ), [Σ] = c1 (Fi ), [Σ] = χ(Σ) < 0, therefore Lemma 2.3 applies and gives c+ (ξi ) = 0 for i = 1, 2. Moreover + + c+ (ξ1 ) = F−W (c+ (ξ1 )) and c+ (ξ1 ) = F−W (c+ (ξ1 )) by Lemma 2.4 and by ′ ′ 11 ′ the composition formula [21, Theorem 3.4] because ξi is obtained form ξi by a sequence of contact (+1)–surgeries, therefore from Lemma 3.9 it follows ′ that c+ (ξi ) = 0 for i = 1, 2. Lemma 3.11. Let Nφ be the subset of Yφ diﬀeomorphic to Σ × [−1, 1] ob- tained from N ⊂ Y by performing (−1)–surgery on C = c1 ∪ . . . ∪ cn , and call S ′ the annulus in Nφ bounded by the curves α+ and α− . If we deﬁne S = S ∪ S ′ , then ′ ′ c1 (ξ1 ), [S] = c1 (ξ2 ), [S] . Proof. Because α+ and α− are Legendrian curves with twisting number 0 ′ ′ ′ ′ with respect to both ξ1 and ξ2 , and ξ1 and ξ2 are both tight by Lemma 3.10, from the Thurston–Bennequin inequality we obtain ′ ′ e(S ′ , ξ1 ) = e(S ′ , ξ2 ) = 0. ′ In the complements of N and Nφ we have ξi |−(Y \N ) = ξi |−(Yφ \Nφ ) , therefore ′ ) = e(S, ξ ). Since ξ and ξ are C 0 –close to F and F and ∂S = e(S, ξi i 1 2 1 2 α+ ∪ α− is tangent to both ξi and Fi , we have e(S, ξi ) = e(S, Fi ) for i = 1, 2. From the additivity property of the relative Euler class we obtain c1 (ξi ), [S] = e(S, ξi ) + e(S ′ , ξi ) = e(S, Fi ), ′ ′ ′ ′ ′ therefore c1 (ξ1 ), [S] = c1 (ξ2 ), [S] . Lemma 3.11 implies that the the Spinc –structures sξ1 and sξ2 induced by ′ ′ ξ1 and ξ2 are not isomorphic, therefore c+ (ξ1 ) and c+ (ξ2 ) are linearly in- ′ ′ ′ ′ dependent because c+ (ξ1 ) ∈ HF + (Y, sξ1 ), c+ (ξ2 ) ∈ HF + (Y, sξ2 ), and they ′ ′ ′ ′ are both non zero by Lemma 3.10. This implies that c+ (ξ1 ) and c+ (ξ2 ) + + are linearly independent too because c+ (ξi ) = F−W (c+ (ξi )) and F−W is an ′ isomorphism by Lemma 3.9, therefore it proves Theorem 3.8. 4 Applications of Theorem 3.8 4.1 sutured manifolds In order to apply Theorem 3.8 we need a way to construct taut foliations in 3–manifolds. This is provided by Gabai’s sutured manifold theory. 12 Deﬁnition 4.1. ([6, Deﬁnition 2.6]) A sutured manifold (M, γ) is a com- pact oriented 3–manifold M together with a set γ ⊂ ∂M of pairwise disjoint annuli A(γ) and tori T (γ). Each component of A(γ) is a tubular neigh- bourhood of an oriented simple closed curve called suture. Finally every component of ∂M \ γ is oriented, and its orientation must be coherent with the orientation of the sutures. We deﬁne R+ (γ) as the subset of ∂M \ γ where the orientation agrees with the orientation induced by M on ∂M , and R− (γ) as the subset of ∂M \ γ where the two orientations disagree. We deﬁne also R(γ) = R+ (γ) ∪ R− (γ). Deﬁnition 4.2. ([6, Deﬁnition 2.2] Let S be a compact oriented surface S = n Si with all Si connected. We deﬁne the norm of S to be i=1 x(S) = −χ(Si ). i:χ(Si )<0 Deﬁnition 4.3. ([6, Deﬁnition 2.4]) Let S be a properly embedded oriented surface in the compact oriented manifold M . We say that S is norm min- imising in H2 (M, γ) if ∂S ⊂ γ, S is incompressible, and its norm x(S) is minimal in the homology class of S in H2 (M, γ). If S realises the minimal norm in its homology class and all its connected components have negative Euler characteristic, then it is incompressible. Deﬁnition 4.4. ([6, Deﬁnition 2.10]) A sutured manifold (M, γ) is taut if R(γ) is norm minimising in H2 (M, γ). We will give the following deﬁnition only in the simpler case when no com- ponent of γ is a torus, because this is the case we are interested in. Deﬁnition 4.5. ([6, Deﬁnition 3.1] and [8, Correction 0.3]) Let (M, γ) be a sutured manifold with T (γ) = ∅, and S be a properly embedded oriented surface in M such that 1. no component of S is a disc with boundary in R(γ) 2. no component of ∂S bounds a disc in R(γ) 3. for every component λ of ∂S ∩ γ one of the following holds: (a) γ is a non-separating properly embedded arc in γ, or 13 (b) λ is a simple closed curve isotopic to a suture in A(γ). Then S deﬁnes a sutured manifold decomposition S (M, γ) (M ′ , γ ′ ) where M ′ = M \ S and γ ′ = (γ ∩ M ′ ) ∪ ν(S+ ∩ R− (γ)) ∪ ν(S− ∩ R+ (γ)), R+ (γ ′ ) = ((R+ (γ) ∩ M ′ ) ∪ S+ ) \ int(γ ′ ), R− (γ ′ ) = ((R− (γ) ∩ M ′ ) ∪ S− ) \ int(γ ′ ), where S+ and S− are the portions of ∂M ′ corresponding to S where the normal vector to S points respectively out of or into ∂M ′ . A taut sutured manifold decomposition is a sutured manifold decomposition S (M, γ) (M ′ , γ ′ ) such that both (M, γ) and (M ′ , γ ′ ) are taut sutured manifolds. Deﬁnition 4.6. ([6, Deﬁnition 4.1]) A sutured manifold hierarchy is a se- quence of taut sutured manifold decompositions S1 Sn (M0 , γ0 ) (M1 , γ1 ) ... (Mn , γn ) where (Mn , γn ) = (R × [0, 1], ∂R × [0, 1]) for some surface with boundary R. The main results in sutured manifold theory are that for any taut sutured manifold (M, γ) there is a sutured manifold hierarchy starting from (M, γ) [6, Theorem 4.2], and that we can construct a taut foliation on (M, γ) from a sutured manifold hierarchy starting from (M, γ) such that R(γ) is union of leaves [6, Theorem 5.1]. Thus sutured manifold theory translates the prob- lem about the existence of taut foliations into a ﬁnite set of combinatorial data. The particular result we will use in our applications is the following. Proposition 4.7. Let M be a closed, connected, irreducible, orientable 3– manifold, and let Σ be a genus minimising connected surface representing a non-trivial class in H2 (M ; Q). Denote by (M1 , γ1 ) the taut sutured manifold where M1 = M \ Σ and γ1 = ∅. If g(Σ) > 1 and there is a properly embedded surface S in M1 yielding a taut sutured manifold decomposition, then M admits a smooth taut foliation F such that: 1. Σ is a closed leaf, 14 Figure 2: Cut-and-paste surgery 2. if f is a representative of the germ of the holonomy map around a closed curve δ ⊂ Σ, then dn f 1, n = 1, (0) = dtn 0, n > 1, 3. e(F, S) = χ(S). Proof. By [6, Theorem 4.2] M1 admits a taut sutured manifold hierarchy S Sn (M1 , γ1 ) (M2 , γ2 ) ... (Mn , γn ), then take the foliation F1 on M1 constructed from that hierarchy using the construction in [6, Theorem 5.1]. In particular F1 is smooth because g(Σ) > 1 and ∂M1 is union of leaves. We obtain F by gluing the two components of ∂M1 together. The smoothness of F along Σ and part (2) come from the Induction Hy- pothesis (iii) in the proof of [6, Theorem 5.1]. Part (3) is a consequence of Case 2 of the construction of the foliation in the proof of [6, Theorem 5.1]: in fact F has a leaf which coincides with S outside a small neighbourhood of Σ, and spirals toward Σ inside that neighbourhood. Remark 4.8. A properly embedded surface S in M1 = M \ Σ gives a taut sutured manifold decomposition S (M1 , γ1 ) (M2 , γ2 ) 15 if for translates Σ′ and Σ′ of the boundary components of ∂M \ Σ the + − surfaces S +Σ′ and S +Σ′ obtained by cut-and-paste surgery (see Figure 2) + − are norm minimising in H2 (M1 , ∂S). In fact S + Σ′ and S + Σ′ are isotopic + − to R+ (γ2 ) and to R− (γ2 ) respectively, and being norm minimising in M1 clearly implies being norm-minimising in the smaller manifold M2 = M1 \ S. 4.2 The Alexander polynomial and the topology of knot com- plements In this section we state and prove a folklore result about the Alexander polynomial for the beneﬁt of the reader. Deﬁnition 4.9. A 3–manifold M is a homology product if ∂M = Σ+ ∪ Σ− and the maps (ι± )∗ : H1 (Σ± , Z) → H1 (M, Z) induced by the inclusions ι± : Σ± → M are isomorphisms. Let K be a knot in S 3 , and let Σ be a connected Seifert surface of K. Denote by YK the 3–manifold obtained by 0–surgery on K, and by Σ the surface in YK obtained by capping Σ oﬀ with a meridian disc of the solid torus of the surgery. Denote MΣ = YK \ Σ and ∂MΣ = Σ+ ∪ Σ− . b b Lemma 4.10. If the Alexander polynomial of K is monic, and Σ is a Seifert surface of K such that its genus g is equal to the degree of the Alexander polynomial ∆K , then MΣ is a homology product. b Proof. Consider MΣ = S 3 \ (Σ × [−ǫ, ǫ]) and call Σ± = Σ × {±ǫ}. It is easy to see that the maps H1 (Σ± , Z) → H1 (MΣ , Z) are isomorphisms if and only b if the maps H1 (Σ± , Z) → H1 (MΣ , Z) are isomorphisms. Let a1 , . . . a2g be curves in Σ representing a basis of H1 (Σ). Each curve ai gives rise to two curves: a+ in Σ+ and a− in Σ− . By [24, Lemma 8.C.14] i i there are curves α1 , . . . , α2g in MΣ representing a basis of H1 (MΣ ) such that lk(ai , αj ) = δij , where the linking numbers are computed in S 3 . In H1 (MΣ ) we write (ι+ )∗ (ai ) = a+ = cij αj , therefore cij = lk(aj , a+ ), so i i the map (ι+ )∗ : H1 (Σ+ , Z) → H1 (M+ , Z) is represented in the bases (ai ) and αi ) by the transpose of the Seifert matrix of Σ. By [24, Corollary 8.C.5] the determinant of (ι+ )∗ is equal to the coeﬃcient in degree g of the Alexander polynomial ∆K . (Observe the diﬀerent convention about the Alexander polynomial used in [24]). 16 4.3 Some preliminary results Let M be a homology product with torus boundary T+ ∪ T− , and take two simple closed curves α+ and β+ in T+ . There are simple closed curves α− and β− in T− such that both α+ ∪ −α− and β+ ∪ −β− bound a surface in M . We may assume also that α+ and β+ , as well as α− and β− , intersect transversally in a unique point. Let µ be a properly embedded arcs with one point on T− and one on T+ , and oriented from T− to T+ . We give T+ the orientation induced by the orientation of M following the usual outward normal convention, and we give T− the opposite one. + Denote by Sn (α) the set of the connected and properly embedded surfaces in M which are bounded by α+ ∪−α− and which intersect the arc µ transver- − sally in exactly n positive points and in no negative points, and by Sn (α) the set of the surfaces with the same property bounded by −α+ ∪ α− . Let + − Sn (β) and Sn (β) be the same for the curves β+ and β− . Let κ+ (α) be the n + minimal genus of the surfaces in Sn (α) and deﬁne κ− (α), κ+ (β), and κ− (β) n n n in analogous ways. Lemma 4.11. Let M be a homology product with toric boundary. Then the sequences {κ+ (α)}, {κ− (α)}, {κ+ (β)}, and {κ− (β)} deﬁned above are non n n n n increasing. Proof. We prove the lemma only for {κ+ (α)} because the other cases are n similar. Let Sn be a surface in Sn (α) with genus g(Sn ) = κ+ (α), and call + + + n + + Sn+1 the surface in Sn+1 (α) constructed by cut-and-paste surgery between + + Sn and T+ . By deﬁnition g(Sn+1 ) ≥ κ+ (α), and g(Sn+1 ) = g(Sn ) = + n+1 + κ+ (α) because T+ is a torus, therefore κ+ (α) ≥ κ+ (α). n n n+1 Lemma 4.12. Let M be an irreducible 3–manifold with toric boundary. If M is a homology product but is not homeomorphic to a product, then for all n ≥ 0 either κ+ (α) = 0 and κ− (α) = 0, or κ+ (β) = 0 and κ− (β) = 0. n n n n + − Proof. Assume that there are annuli Aα ∈ Sn (α)∪Sn (α) and Aβ ∈ Sn (β)∪+ Sn− (β). If we make A and A transverse their intersection consists of one α β segment from α+ ∩ β+ to α− ∩ β− and a number of homotopically trivial closed curves. By standard arguments in three-dimensional topology we can isotope Aα and Aβ in order to get rid of the circles because M is irreducible, therefore we can assume that Aα ∩ Aβ consists only of the segment. The 17 boundary of M \ (Aα ∪ Aβ ) is homeomorphic to S 2 , therefore M \ (Aα ∪ Aβ ) is homeomorphic to T+ \ (α+ ∪ β+ )) × [0, 1] ∼ D 3 because M is irreducible. = This proves that M is homeomorphic to T+ × [0, 1]. 4.4 Application to genus-one knots The goal of this section is to prove Theorem 1.4 and its corollaries. We will ﬁrst relate HF K(K, 1) to a Heegaard Floer homology group of a related manifold, then we will construct two taut foliations in that manifold and apply Theorem 3.8. We resume the notation of the subsection 4.2: let K be a genus-one knot in S 3 , and let YK be the the 3–manifold obtained as 0–surgery on K. Let T be a minimal genus Seifert surface for K and let T be the torus in YK obtained by capping T oﬀ with a meridian disc of the solid torus of the surgery. Denote MT = YK \ T and ∂MT = T+ ∪ T− , where T+ is given b b the orientation induced by the orientation of MT by the outward normal b convention, and T− is given the opposite one. Lemma 4.13. Let K0 and K be knots in S 3 , and denote by M the 3– manifold obtained by gluing S 3 \ ν(K0 ) to S 3 \ ν(K) via an orientation- reversing diﬀeomorphism f : ∂(S 3 \ ν(K0 )) → ∂(S 3 \ ν(K)) mapping the meridian of K0 to the meridian of K and the longitude of K0 to the longitude of K. Then M is diﬀeomorphic to the 3–manifold YK0 #K . Proof. Glue a solid torus S1 = S 1 × [−ǫ, ǫ] × [0, 1] to S 3 \ ν(K0 ) ⊔ S 3 \ ν(K) so that S 1 × [−ǫ, ǫ] × {0} is glued to a neighbourhood of a meridian of K0 and S 1 × [−ǫ, ǫ] × {1} is glued to a neighbourhood of a meridian of K. The resulting manifold M ′ is diﬀeomorphic to S 3 \ ν(K0 #K), so that if we glue a solid torus S2 = S 1 × D 2 to M ′ = S 3 \ ν(K0 #K) mapping the meridian of S2 to the longitude of K0 #K we obtain YK0 #K . Now we look at the gluing in the inverse order. First we glue S1 to S2 so that S 1 × {−ǫ, ǫ} × [0, 1] ⊂ ∂S1 is glued to disjoint neighbourhoods of two parallel longitudes of S2 , therefore the resulting manifold is diﬀeomorphic to T 2 × [0, 1]. Then we glue S1 ∪ S2 = T 2 × [0, 1] to S 3 \ ν(K0 ) ⊔ S 3 \ ν(K) and we obtain M . Proof of Theorem 1.4. Let K be a non ﬁbred genus-one knot. In this proof we will assume that the symmetrised Alexander polynomial ∆k (T ) of K has 18 degree one and is monic. If this is not the case then rk HF K(K, 1) = 1 because the coeﬃcient of ∆K (T ) in degree one is equal to χ(HF K(K, 1)). In order to estimate the rank of HF K(K, 1) we cannot apply Theorem 3.8 directly; we have to increase the genus of K artiﬁcially ﬁrst. Let K0 be any ﬁbred knot with genus one, say the ﬁgure-eight knot. By [7, Theorem 3] u K is ﬁbred if and only if K#K0 is ﬁbred. Also, the K¨nneth formula for connected sums (Proposition 2.6) gives HF K(K#K0 , 2) ∼ HF K(K, 1) ⊗ HF K(K0 , 1) ∼ HF K(K, 1), = = because HF K(K0 , d) is a free Abelian group for all d by [13, Theorem 1.3] because K0 is an alternating knot, and HF K(K0 , 1) ∼ Z because K0 is = ﬁbred. Therefore from Proposition 2.5 we have HF K(K, 1) ∼ HF + (YK#K0 , 1), = so we have reduced ourselves to proving that, under our hypothesis on the Alexander polynomial of K, if K is not ﬁbred then rkHF + (YK#K0 , 1) > 1. Let T and T0 be genus-one Seifert surfaces for K and K0 respectively, and call T and T0 the corresponding capped-oﬀ surfaces in YK and YK0 . Let µ be a properly embedded curve in MT joining T+ and T− which closes to the b core of the surgery torus in YK , then MT \ν(µ) is homeomorphic to S 3 \ν(T ). b We can divide ∂(MT \ ν(µ)) in two pieces: ∂h (MT \ ν(µ)) = ∂MT \ ν(µ) b b b called the horizontal boundary, and ∂v (MT \ ν(µ)) = ∂ν(µ) \ ∂MT called the b b vertical boundary. By Lemma 4.10 MT is a homology product because ∆K (T ) has degree one b and is monic. Moreover by [9, Corollary 8.3] it is irreducible, therefore we can assume without loss of generality that κ+ (α) = 0 and κ− (α) = 0 for any n n n ≥ 0 by Lemma 4.12. Lemma 4.11 implies that the sequences {κ+ (α)} and n {κ− (α)} are eventually constant. Fix from now on in this section a positive n integer m such that κ+ (α) = κ+ (α) and κ− (α) = κ− (α) for all i ≥ 0, m+i m m+i m + + − − + and choose surfaces Sm ∈ Sm (α) and Sm ∈ Sm (α) such that Sm has genus κm+ (α) and S − has genus κ− (α). m m By Lemma 4.13 the 3–manifold YK#K0 is diﬀeomorphic to the union of S 3 \ ν(K) and S 3 \ ν(K0 ) along their boundary via an identiﬁcation ∂(S 3 \ ν(K)) → ∂(S 3 \ ν(K)) mapping meridian to meridian and longitude to longitude. In YK#K0 the Seifert surfaces T and T0 are glued together to give a closed surface Σ with g(Σ) = 2 which minimises the genus in its homology class. Call MΣ = YK#K0 \ Σ. 19 If we denote by µ0 a segment in MT0 which closes to the core of the surgery b torus in YK0 , we can see MΣ as (MT \ ν(µ)) ∪ (MT0 \ ν(µ0 )) glued together b b along their vertical boundary components. + − From Sm and Sm we can construct surfaces S+ and S− in MΣ by gluing a ± copy of T0 to each one of the m components of Sm ∩ ∂v (MT \ ν(µ)). From an b abstract point of view S+ and S+ are obtained by performing a connected + sum with a copy of T0 at each of the m intersection points between Sm or Sm and µ, therefore g(S± ) = κ± (α) + m. − m Consider the taut sutured manifold (MΣ , γ) where γ = ∅. We claim that b S+ (MΣ , γ) (MΣ \ S+ , γ+ ) b S− (MΣ , γ) (MΣ \ S− , γ− ) are taut sutured manifold decompositions. By Remark 4.8 this is equivalent to proving that the surfaces S+ + Σ+ , S+ + Σ− , S− + Σ+ , and S− + Σ− obtained by cut-and-paste surgery between S± and Σ± are norm minimising in H2 (MΣ , α+ ∪ α− ). We recall that T+ and Σ+ are oriented by the outward normal convention, while T− and Σ− are oriented by the inward normal convention. For this reason µ ∩ T+ and µ ∩ T− consist both of one single positive point. We will consider only S+ + Σ+ , the remaining cases being similar due to the above consideration. Let S ⊂ MΣ be a surface with ∂ S = α− ∪ α+ in the same relative homology class as S+ + Σ+ and norm minimising in H2 (MΣ , α− ∪ α+ ). We can see S as the union of two (possibly discon- nected) properly embedded surfaces with boundary S ⊂ MT \ ν(µ) and b S0 ⊂ MT0 \ ν(µ0 ), then χ(S) = χ(S) + χ(S0 ). We can easily modify S with- b out increasing its genus so that it intersects ∂v (MT \ν(µ)) and ∂v (MT0 \ν(µ0 )) b b in homotopically non trivial curves. The number of connected components of ∂S0 = ∂S ∩ ∂v (MT \ ν(µ)) counted with sign is m + 1. b Since χ(S) + χ(S0 ) = χ(S) = 2 − 2g(S), and S is norm minimising, both S and S0 minimise the norm in their relative homology classes. MT0 \ ν(µ0 ) is b a product T0 × [0, 1], therefore χ(S0 ) is equal to the negative of the number of components of ∂S0 counted with sign, i. e. χ(S0 ) = −(m + 1). We can modify S0 without changing χ(S0 ) so that it consists of some boundary parallel annuli and m + 1 parallel copies of T0 , then we push the boundary parallel annuli into MT \ ν(µ), so that we have a new surface S ′ ⊂ MT \ ν(µ) b b 20 whose intersection with ∂v (MT \ ν(µ)) consists of exactly m + 1 positively b oriented non trivial closed curves. If we glue discs to these curves we obtain + + a surface Sm+1 ∈ Sm+1 such that + #(m+1) g(S) = g(Sm+1 #T0 ) = κ+ + m + 1. m+1 Since #m + + g(S+ + Σ+ ) = g(S+ ) + 1 = g(Sm #T0 ) + 1 = g(Sm ) + m + 1 = κ+ + m + 1 m and κ+ = κ+ , we conclude that g(S+ + Σ+ ) = g(S), then g(S+ + Σ+ ) is m+1 m norm minimising in its relative homology class. By Proposition 4.7 the taut sutured manifold decompositions b S+ (MΣ , γ = ∅) (MΣ \ S+ , γ+ ) b S− (MΣ , γ = ∅) (MΣ \ S− , γ− ) provide taut smooth foliations F+ and F− such that Σ is a closed leaf for both so that, in particular, c1 (F+ ), [Σ] = c1 (F− ), [Σ] = χ(Σ) = −2. Moreover e(F+ , S+ ) = χ(S+ ) and e(F− , S− ) = χ(S− ). + − Take any surface R ∈ S0 (α), then −R ∈ S0 (α), therefore [S+ ] = [R] + m[Σ] and [S− ] = −[R] + m[Σ] as relative homology classes in H2 (MΣ , α+ ∪ α− ), therefore e(F+ , S+ ) = e(F+ , R) + mχ(Σ) = e(F+ , R) − 2m and e(F− , S− ) = e(F− , −R) + mχ(Σ) = −e(F− , R) − 2m. This implies + e(F+ , R) = χ(S+ ) + 2m = χ(Sm ) = −2κ+ (α) m (1) and − e(F− , R) = −χ(S− ) − 2m = −χ(Sm ) = 2κ− (α). m (2) ± Recall that χ(Sm ) = −2κ± (α) because Sm and Sm have 2 boundary com- m + − ponents each. Equations 1 and 2 imply that e(F+ , R) = e(F− , R) because κ± (α) > 0, so we can apply Theorem 3.8. m 21 Proof of Corollary 1.5. 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