Knot Floer homology detects genus-one fibred knots by sdfgsg234

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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. It is well known that the trefoil knots and the figure-
eight knot are the only three fibred knots with genus one. By Theorem 1.4 if
HF K(K, 1) = Z then K is either one of the trefoil knots or the figure-eight
knot. These knots are alternating, therefore their knot Floer homology is
determined by their Alexander polynomial and signature by [13, Theorem
1.3]. Then the statement follows from the fact that the trefoil knots and the
figure-eight knot have different Alexander polynomial, and the right-handed
and the left-handed trefoil knot have different signature.

Proof of Corollary 1.7. The proof is immediate from Corollary 1.5 and from
                                                                          e
[19, Theorem 1.6] asserting that, if surgery on a knot K gives the Poincar´
homology sphere, then the knot Floer homology of K is isomorphic to the
knot Floer homology of the left-handed trefoil knot.


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[25] X. Zhang. On property I for knots in S 3 . Trans. Amer. Math. Soc.,
     339(2):643–657, 1993.




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