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Knot Floer Homology Detects Genus-One Fibred Knots

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Knot Floer Homology Detects Genus-One Fibred Knots Knot Floer homology detects genus-one fibred knots Paolo Ghiggini CIRGET, Universit´edu Qu´ebec `aMontr´eal Case Postale 8888, succursale Centre-Ville Montr´eal (Qu´ebec) H3C 3P8, Canada [email protected] July 6, 2019 Abstract Ozsv´ath and Szab´oconjectured that knot Floer homology detects fi- bred knots. We propose a strategy to approach this conjecture based on Gabai’s theory of sutured manifold decomposition and contact topol- ogy. We implement this strategy for genus-one knots, obtaining as a corollary that if rational surgery on a knot K gives the Poincar´e homology sphere Σ(2, 3, 5), then K is the left-handed trefoil knot. 1 Introduction Knot Floer homology was introduced independently by Ozsv´ath and Szab´o in [14] and by Rasmussen in [22]. For any knot K in S3 and any integer d the knot Floer homology group HF\ K(K, d) is a finitely generated graded Abelian group. arXiv:math/0603445v3 [math.GT] 5 Dec 2006 Knot Floer homology can be seen as a categorification of the Alexander polynomial in the sense that i χ(HF\ K(K, d))T =∆K (T ) X where ∆K(T ) denotes the symmetrised Alexander polynomial. However the groups HF\ K(K, d), and in particular the top non trivial group, contain more information than just the Alexander polynomial, as the following results show. 1 Theorem 1.1. ([13, Theorem 1.2]) Let g(K) denote the genus of K. Then g(K) = max d ∈ Z : HF\ K(K, d) 6= 0 . Theorem 1.2. ([19, Theorem 1.1]) Let g be the genus of K. If K is a fibred knot, then HF\ K(K, g)= Z. Ozsv´ath and Szab´oformulated the following conjecture, whose evidence is supported by the computation of knot Floer homology for a large number of knots. Conjecture 1.3. (Ozsv´ath–Szab´o) If K is a knot in S3 with genus g and 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 S3. 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 properties of their complement via sutured manifold decomposition and contact struc- tures 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 [23]. Conjecture 1.6. (Conjecture Iˆ) If K is a knot in S3 such that there exists a rational number r for which the 3–manifold obtained by r–surgery on K is homeomorphic to the Poincar´ehomology sphere Σ(2, 3, 5), then K is the left-handed trefoil knot. 2 Conjecture Iˆ was proved for some knots by Zhang in [23], and a major step toward its complete proof was made by Ozsv´ath and Szab´oin [18], where they proved that a counterexample to Conjecture Iˆ 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. Corollary 1.7. Conjecture Iˆ holds. Corollary 1.5 has also been used by Ozsv´ath and Szab´oto prove that the trefoil knot and the figure-eight knot are determined by their Dehn surgeries [12]. Acknowledgements We warmly thank Steve Boyer, Ko Honda, Joseph Maher, and Stefan Till- mann for many inspiring conversations. 2 Overview of Heegaard Floer theory Heegaard Floer theory is a family of invariants introduced by Ozsv´ath and Szab´oin 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 of completeness. The details can be found in Ozsv´ath and Szab´o’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 Ozsv´ath and Szab´o[16] defined an Abelian group HF +(Y, t) which is an isomorphism invariant of the pair (Y, t). When c1(t) is not a torsion element in H2(Y ) the group HF +(Y, t) is finitely generated; see [15, Theo- rem 5.2 and Theorem 5.11]. If there is a distinguished surfaceΣin Y which is clear from the context, we use the shortened notation HF +(Y, d)= HF +(Y, t). t ∈ SpinMc(Y ) hc1(t), [Σ]i = 2d 3 This notation makes sense because HF +(Y, t) 6= 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 [15, 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. [15, Theorem 1.6] If Σ has genus g, then HF +(Y, d) = 0 for all d > g. 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 fit into a surgery exact triangle as follows. Theorem 2.2. ([15, Theorem 9.12]) Let (K, λ) be an oriented framed knot in Y , and let µ 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 kk RRR kkk RRR kkk RRR kkk R ukkk + HF (Yλ+µ(K), d) is exact for any d ∈ Z. 2.2 The Ozsv´ath–Szab´ocontact invariant c A contact structure ξ on a 3–manifold Y determines a Spin –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 [19]. 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 4 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 [13, Corol- lary 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 2 + that c1(ξ) is non trivial in H (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 operation called contact (+1)–surgery which produces a new contact mani- fold (Y ′,ξ′); see [2] and [3]. The Ozsv´ath–Szab´ocontact invariant behaves well with respect to contact (+1)–surgeries. Lemma 2.4. ([11, Theorem 2.3]; see also [19]) Suppose (Y ′,ξ′) is obtained from (Y,ξ) by a contact (+1)–surgery. Let −X be the cobordism induced by the surgery with opposite orientation. Then + + + ′ F−X (c (ξ)) = c (ξ ). 2.3 Knot Floer homology Knot Floer homology is a family of finitely dimensional graded Abelian groups HF\ K(K, d) indexed by d ∈ Z attached to any oriented knot K in 3 S ; see Ozsv´ath and Szab´o[14]. 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 Proposition 2.5. ([14, Corollary 4.5] and [13, Corollary 1.2]). Let K be a knot of genus g> 1. Then + HF\ K(K, g)= HF (YK, g − 1). Another property of knot Floer homology we will need later is a K¨unneth- like formula for connected sums. 3 Proposition 2.6. ([14, Corollary 7.2]) Let K1 and K2 be knots in S , and \ denote by K1#K2 their connected sum. 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+Md2=d 5 The definition of Knot Floer homology can be extended to links. To any link L in S3 with |L| components, Ozsv´ath and Szab´oassociate a null- homologous knot κ(L) in #|L|−1(S2 × S1).
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