normalization by ajizai

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									       Normalization of Twisted Alexander Invariants
                                         Takahiro Kitayama∗



                                                  Abstract

           Twisted Alexander invariants of knots are well-defined up to multiplication of units.
       We get rid of this multiplicative ambiguity via a combinatorial method and define normal-
       ized twisted Alexander invariants. We can show that the invariants coincide with sign-
       determined Reidemeister torsion in a normalized setting and refine the duality theorem. As
       an application, we obtain stronger necessary conditions for a knot to be fibered than those
       previously known. Finally, we study a behavior of the highest degree of the normalized
       invariant.

       2000 Mathematics Subject Classification. Primary: 57M25, Secondary: 57M05; 57Q10

       Keywords and phrases. twisted Alexander invariant, Reidemeister torsion, fibered knot


1 Introduction
    Twisted Alexander invariants, which coincide with Reidemeister torsion ([Ki], [KL]), were
introduced for knots in the 3-sphere by Lin [L] and generally for finitely presentable groups
by Wada [Wad]. They were given a natural topological definition by using twisted homology
groups in the notable work of Kirk and Livingston [KL]. Many properties of the classical
Alexander polynomial ∆K were subsequently extended to the twisted case and it was shown
that the invariants have much information on the topological structure of a space. For example,
necessary conditions of twisted Alexander invariants for a knot to be fibered were given by
Cha [C], Goda-Morifuji [GM], Goda-Kitano-Morifuji [GKM] and Friedl-Kim [FK]. Moreover,
even sufficient conditions for a knot with genus 1 to be fibered were obtained by Friedl-Vidussi
[FV].
    It is well known that ∆K can be normalized, for instance, by considering the skein relation.
In this paper, we first obtain the corresponding result in twisted settings. The twisted Alexander
invariant ∆K,ρ associated to a linear representation ρ is well-defined up to multiplication of units
in a Laurent polynomial ring. We show that the ambiguity can be eliminated via a combinatorial
method constructed by Wada and define the normalized twisted Alexander invariant ∆K,ρ (See
Definition 4.4 and Theorem 4.5).
   ∗
    The author is supported by Research Fellowship of the Japan Society for the Promotion of Science for Young
Scientists.

                                                      1
2                                                                                  T. Kitayama


    Turaev [T2] defined sign-determined Reidemeister torsion by refining the sign ambiguity
of Reidemeister torsion for a odd-dimensional manifold and showed that the other ambiguity
depends on the choice of Euler structures. We also normalize sign-determined Reidemeister
torsion T K,ρ for a knot and define T K,ρ (t). Then we prove the equality

                                        ∆K,ρ (t) = T K,ρ (t)

(Theorem 5.7). This shows that ∆K,ρ is a simple homotopy invariant and give rise to a refined
version (Theorem 5.9) of the duality theorem for twisted Alexander invariants.
    As an application, we generalize above results for fibered knots. We can define the highest
degree and the coefficient of the highest degree term of ∆K,ρ . We show that these values are
completely determined for fibered knots (See Theorem 6.3). Finally, we obtain the following
inequality which bounds free genus g f (K) from below by using the highest degree h-deg ∆K,ρ :

                                 2 h-deg ∆K,ρ ≤ n(2g f (K) − 1).                           (1.1)

(See Theorem 6.6.)

    This paper is organized as follows. In the next section, we first review the definition of
twisted Alexander invariants for knots. We also describe how to compute them from a presen-
tation of a knot group and the duality theorem for unitary representations. In Section 3, we
review Turaev’s sign-determined Reidemeister torsion and the relation with twisted Alexander
invariants. In Section 4, we establish normalization of twisted Alexander invariants. In Section
5, we refine the correspondence with sign-determined Reidemeister torsion and the duality the-
orem for twisted Alexander invariants. Section 6 is devoted to applications. Here we generalize
the result of Cha [C], Goda-Kitano-Morifuji [GKM] and Friedl-Kim [FK] for fibered knots and
study a behavior of the highest degree and obtain (1.1).

Acknowledgement. The author would like to express his gratitude to Toshitake Kohno for
his encouragement and helpful suggestions. He also would like to thank Hiroshi Goda, Teru-
aki Kitano, Takayuki Morifuji and Yoshikazu Yamaguchi for fruitful discussions and advice
and Stefan Friedl for several stimulating comments which lead to some improvements of the
argument in this revised version.


2 Twisted Alexander invariants
   In this section, we review twisted Alexander invariants of K following [C] and [KL]. For
a given oriented knot K in S 3 , let E K := S 3 \ N(K), where N(K) denotes an open tubular
neighborhood of K and G K := π1 E K . We fix an element µ ∈ G K represented by a meridian
of E K and denote by α : G K → t be the abelianization homomorphism which maps µ to the
generator t. Let R be a Noetherian unique factorization domain and Q(R) the quotient field of
R.
    We first give a definition of a twisted homology group and a twisted cohomology group. Let
X be a connected CW-complex and X the universal covering of X. The chain complex C∗ (X) is
Normalization of twisted Alexander invariants                                                          3


a left Z[π1 X]-module via the action of π1 X as the deck transformations of X. We regard C∗ (X)
also as a right Z[π1 X]-module by defining σ · γ := γ−1 · σ, where γ ∈ π1 X and σ ∈ C∗ (X). For a
linear representation ρ : π1 X → GLn (R), R⊕n naturally has a left Z[π1 X]-module structure. We
define the twisted homology group Hi (X; R⊕n ) and the twisted cohomology group H i (X; R⊕n ) of
                                            ρ                                             ρ
ρ as follows:

                             Hi (X; R⊕n ) := Hi (C∗ (X) ⊗Z[π1 X] R⊕n ),
                                     ρ

                             H i (X; R⊕n ) := H i (HomZ[π1 X] (C∗ (X), R⊕n )).
                                      ρ

Definition 2.1. For a representation ρ : G K → GLn (R), we define ∆iK,ρ to be the order of the
i-th twisted homology group Hi (E K ; R[t, t−1 ]⊕n ), where we consider R[t, t−1 ]⊕n = R[t, t−1 ] ⊗ R⊕n .
                                                α⊗ρ
It is called the i-th twisted Alexander polynomial associated to ρ, which is well-defined up to
multiplication by a unit in R[t, t−1 ]. We furthermore define

                                     ∆K,ρ := ∆1 /∆0 ∈ Q(R)(t).
                                              K,ρ K,ρ

It is called the twisted Alexander invariant associated to ρ and well-defined up to a factor ηtl ,
where η ∈ R× and l ∈ Z.
Remark 2.2. Lin’s twisted Alexander polynomial defined in [L] coincides with ∆1 .
                                                                             K,ρ

     The homomorphisms α and α ⊗ ρ induce ring homomorphisms α : Z[G K ] → Z[t, t−1 ] and
                                                                              ˜
                              −1
Φ : Z[G K ] → Mn (R[t, t ]). For a knot diagram of K, we choose and fix a Wirtinger presentation
G K = x1 , . . . , xm | r1 , . . . , rm−1 . Let us consider the (m − 1) × m matrix AΦ whose component
                              ∂r                              ∂
is the n × n matrix Φ ∂xij ∈ Mn (R[t, t−1 ]), where ∂x j denotes Fox’s free derivative with respect
to x j . For 1 ≤ k ≤ m, let us denote by AΦ,k the (m − 1) × (m − 1) matrix obtained from AΦ by
removing the k-th column. We regard AΦ,k as an (m − 1)n × (m − 1)n matrix with coefficients in
R[t, t−1 ].
     The twisted Alexander invariants can be computed from a Wirtinger presentation as follows.
This is nothing but Wada’s construction in [Wad].
Theorem 2.3 ([HLN], [KL]). For a representation ρ : G K → GLn (R) and a Wirtinger presenta-
tion x1 , . . . , xm | r1 , . . . , rm−1 of G K , we have
                                          det AΦ,k
                              ∆K,ρ ≡                     mod ηtl     η∈R× ,l∈Z
                                       det Φ(xk − 1)
for any index k.
Remark 2.4. Wada shows in [Wad] that the twisted Alexander invariant is well-defined up to
a factor ηtln . He also shows that in case that ρ is a unimodular representation, the twisted
Alexander invariant is well-defined up to a factor ±tln if n is odd and up to only tln if n is even.
   It is also known that the twisted Alexander invariants have the following duality. We extend
complex conjugation to C(t) by taking t → t−1 .
Theorem 2.5 ([Ki], [KL]). Given a representation ρ : G K → U(n) (resp. O(n)), we have

                                ∆K,ρ (t) ≡ ∆K,ρ (t)    mod ηtl   η∈R× ,l∈Z .
4                                                                                                         T. Kitayama


3 Sign-determined Reidemeister torsion
   In this section, we review the definition of Turaev’s sign-determined Reidemeister torsion.
See [T1], [T2] for more details. For two bases u and v of an n-dimensional vector space over a
field F, [u/v] denotes the determinant of the base change matrix from v to u.
                           ∂n                       ∂1
    Let C∗ = (0 → Cn − Cn−1 → · · · − C0 → 0) be a chain complex of finite dimensional
                           →                →
vector spaces over F. For given bases bi of Im ∂i+1 and hi of Hi (C∗ ), we can choose a basis
bi ∪ hi ∪ bi−1 of Ci as follows. First, we choose a lift hi of hi in Ci+1 and obtain a basis bi ∪ hi of
     ˜ ˜                                                 ˜                                        ˜
Ker ∂i . Consider the exact sequence

                                  0 → Im ∂i+1 → Ker ∂i → Hi (C∗ ) → 0.

Then we choose a lift bi−1 of bi−1 in Ci and obtain a basis (bi ∪ hi ) ∪ bi−1 of Ci . Consider the
                      ˜                                           ˜      ˜
exact sequence
                               0 → Ker ∂i → Ci → Im ∂i → 0.

Definition 3.1. For given bases c = (ci ) of C∗ and h = (hi ) of H∗ (C∗ ), we choose a basis b = (bi )
of Im ∂∗ and define
                                                          n
                     Tor(C∗ , c, h) := (−1)|C∗ |                                                 ∈ F ×,
                                                                                           i+1
                                                               [bi ∪ hi ∪ bi−1 /ci ](−1)
                                                                     ˜    ˜
                                                         i=0

where
                                            n       j                 j
                                 |C∗ | :=       (        dim Ci )(         dim Hi (C∗ )).
                                            j=0 i=0                  i=0


Remark 3.2. It can be easily checked that Tor(C∗ , c, h) does not depend on the choices of b, bi
                                                                                              ˜
    ˜
and hi .
   Now let us apply the above algebraic torsion to the geometric situations. Let X be a con-
nected finite CW-complex. By a homology orientation of X we mean an orientation of the
homology group H∗ (X; R) =     H (X; R) as a real vector space.
                              i i

Definition 3.3. For a representation ρ : π1 X → GLn (F) such that the twisted homology group
        ⊕n
H∗ (X; Fρ ) vanishes and a homology orientation o, we define the sign-determined Reidemeister
torsion T ρ (X, o) of ρ and o as follows. We choose a lift ei of each cell ei in X and bases h of
                                                           ˜
H∗ (X; R) which is positively oriented with respect to o and f1 , . . . , fn of F ⊕n . Then,

                                T ρ (X, o) := τn Tor(C∗ (X) ⊗ρ F ⊕n , c) ∈ F × ,
                                               0                      ˜

where

                    τ0 := sgn Tor(C∗ (X; R), c, h),
                     c := e1 , . . . , edimC∗ ,
                     c := e1 ⊗ f1 , . . . , e1 ⊗ fn , . . . , edimC∗ ⊗ f1 , . . . , edimC∗ ⊗ fn .
                     ˜     ˜                ˜                 ˜                     ˜
Normalization of twisted Alexander invariants                                                         5


Remark 3.4. It is known that T ρ (X, o) does not depend on the choice of ei , h and f1 , . . . , fn and
                                                                         ˜
is well-defined as a simple homotopy invariant up to multiplication of an element in Im(det ◦ρ).
See [T1].
    Here let us consider the knot exterior E K . In this case, we can equip E K with its canonical
homology orientation ωK as follows. We have H∗ (E K ; R) = H0 (E K ; R) ⊕ t and define ωK :=
[ [pt], t ], where [pt] is the homology class of a point.

Definition 3.5. For a representation ρ : G K → GLn (F) such that the twisted homology group
H∗ (X; F(t)⊕n ) vanishes, the sign-determined Reidemeister torsion T K,ρ (t) of ρ is defined by
              α⊗ρ
T α⊗ρ (E K , ωK ). Here we consider α ⊗ ρ : G K → GLn (F[t, t−1 ]) → GLn (F(t)).

    In the later section, we generalize the following theorem.

Theorem 3.6 ([Ki], [KL]). For a representation ρ : G K → GLn (F) such that the twisted homol-
ogy group H∗ (X; F(t)⊕n ) vanishes, we have
                     α⊗ρ


                                ∆K,ρ (t) ≡ T K,ρ (t)   mod ηtl    η∈F × ,l∈Z .



4 Construction
    Now we establish one of our main results. We get rid of the multiplicative ambiguity of
twisted Alexander invariants via a combinatorial method. For f (t) = p(t)/q(t) ∈ Q(R)(t) (p, q ∈
R[t, t−1 ]), we define

                   deg f := deg p − deg q,
                 h-deg f := (the highest degree of p) − (the highest degree of q),
                 l-deg f := (the lowest degree of p) − (the lowest degree of q),
                              (the coefficient of the highest degree term of p)
                    c( f ) :=                                                 .
                              (the coefficient of the highest degree term of q)
    We make use of a combinatorial group theoretical approach constructed by Wada in [Wad].

Definition 4.1. Given a finite presentable group G = x1 , . . . , xm | r1 , . . . , rn , the operations of
the following types for any word w in x1 , . . . , xm , are called the strong Tietze transformations:

  Ia. To replace one of the relators ri by its inverse ri−1 .

  Ib. To replace one of the relators ri by its conjugate wri w−1 .

  Ic. To replace one of the relators ri by ri r j for any j      i.

  II. To add a new generator y and a new relator yw−1 . (Namely, the resulting presentation is
       x1 , . . . , xm , y | r1 , . . . , rn , yw−1 .)

If a presentation is transformable to another by a finite sequence of operations of above types
and their inverse operations, we say that the two presentations are strongly Tietze equivalent.
6                                                                                            T. Kitayama


Remark 4.2. The deficiency of G does not change via the strong Tietze transformations.
   Wada shows the following lemma.
Lemma 4.3 ([Wad]). All the Wirtinger presentations of a given link in S 3 are strongly Tietze
equivalent to each other.
    Let ϕ : Z[G K ] → Z be the augmentation homomorphism. (Namely, ϕ(γ) = 1 for any element
γ of G K .) For a fixed presentation x1 , . . . , xm | r1 , . . . , rm−1 of G K , we denote Aϕ,k and Aα,k by
                                                                                                     ˜
    ∂ri              ∂ri
 ϕ ∂x j      and α ∂x j
                   ˜         as in Section 2.
          j k              j k
    We eliminate the ambiguity of ηtl in Definition 2.1 as follows.
Definition 4.4. Given a representation ρ : G K → GLn (R), we choose a presentation
 x1 , . . . , xm | r1 , . . . , rm−1 of G K which is strongly Tietze equivalent to a Wirtinger presenta-
tion and an index 1 ≤ k ≤ m such that h-deg α(xk ) 0. Then we define the normalized twisted
Alexander invariant associated to ρ as
                                              δn     det AΦ,k            1    1
                                     ∆K,ρ := n d                ∈ Q(R)( 2 )(t 2 ),
                                            ( t ) det Φ(xk − 1)
where
                      := det ρ(µ),
                    δ := sgn(h-deg α(xk ) det Aϕ,k ),
                          1
                    d := (h-deg det Aα,k + l-deg det Aα,k − h-deg α(xk )).
                                        ˜               ˜
                          2
Theorem 4.5. ∆K,ρ is an invariant of a linear representation ρ.
Proof. From Lemma 4.3, we have to check (i) the independence of the choice of k and (ii) the
invariance for each operation of Definition 4.1.
    We assume that we can choose another index k also satisfying the condition h-deg α(xk )
0. We set
                      δ := sgn(h-deg α(xk ) det Aϕ,k ),
                           1
                      d := (h-deg det Aα,k + l-deg det Aα,k − h-deg α(xk )).
                                        ˜               ˜
                           2
    Since                               m
                                              ∂ri
                                                   (x j − 1) = ri − 1,
                                        j=1
                                              ∂x j
we have
                                                           ∂ri
                 det AΦ,k det Φ(xk − 1) = det . . . , Φ        Φ(xk − 1), . . . ,
                                                          ∂xk
                                                                                     
                                              
                                              
                                                               ∂ri                   
                                                                                      
                                                                                      
                                        = det . . . , −
                                              
                                              
                                                            Φ      Φ(x j − 1), . . .  ,
                                                                                      
                                                                                      
                                                                                      
                                                         j k
                                                               ∂x j
                                                              ∂ri
                                            = det . . . , −Φ        Φ(xk − 1), . . . ,
                                                              ∂xk
                                            = (−1)n(k−k ) det AΦ,k det Φ(xk − 1).
Normalization of twisted Alexander invariants                                                   7


Similarly, we obtain

                         det Aα,k det α(xk − 1) = (−1)k−k det Aα,k det α(xk − 1).
                              ˜       ˜                        ˜       ˜

Hence d = d. Moreover, by dividing this equality by (t − 1) and taking t → 1, we can see that

                          h-deg α(xk ) det Aϕ,k = (−1)k−k h-deg α(xk ) det Aϕ,k .

Hence δ = (−1)k−k δ. This concludes the proof of (i).
   Next, we consider the strong Tietze transformations. Since

                                           ∂(ri−1 )        ∂ri
                                                    = −ri      ,
                                             ∂x j         ∂x j
                                        ∂(wri w−1 )     ∂ri
                                                    =w       ,
                                           ∂x j        ∂x j
                                           ∂(ri rl ) ∂ri         ∂rl
                                                    =       + ri      ,
                                             ∂x j     ∂x j       ∂x j

the changes of each value by the transformation Ia, Ib and Ic are as follows. By the transfor-
mation Ia, det AΦ,k → (−1)n det AΦ,k , δ → −δ and d does not change. By the transformation Ib,
det AΦ,k → ( tn )deg α(w) det AΦ,k , δ does not change and d → d + deg α(w). By the transformation
Ic and II, it is easy to see that all the values do not change. This concludes the proof of (ii).
    From the construction, the following lemma holds.
Lemma 4.6. (i)For a representation ρ : G K → GLn (R),
                                                                    1        l
                                 ∆K,ρ (t) ≡ ∆K,ρ (t) mod            2   , ηt 2   η∈R× ,l∈Z .


(ii)If ρ is trivial (i.e., Φ = α),
                               ˜

                                               − t− 2 ) = (t 2 − t− 2 )∆K,ρ (t),
                                           1        1       1           1
                                      K (t 2


where    K (z)   is the Conway polynomial of K.
Proof. Since (i) is clear from Theorem 2.3 and the definition, we prove (ii). We set

                                          f (t) = (t 2 − t− 2 )∆K,ρ (t).
                                                        1       1




It is easy to see that
                                         f (t) ≡ ∆K (t)         mod ±t .
Moreover, we can check that

                                                        f (1) = 1,
                                           h-deg f + l-deg f = 0,

which establishes the formula.
8                                                                                           T. Kitayama


5 Relation to sign-determined Reidemeister torsion
    In this section, we generalize Theorem 2.5 and Theorem 3.6. Here we only consider the
case that R is a field F.
    First, we also normalize sign-determined Reidemeister torsion as twisted Alexander invari-
ants.

Definition 5.1. For a representation ρ : G K → GLn (F) such that the twisted homology group
H∗ (E K ; F(t)⊕n ) vanishes, we define T K,ρ (t) as follows. We choose a lift ei in E K of each cell ei ,
              α⊗ρ                                                              ˜
bases h of H∗ (E K ; R) which is positively oriented with respect to ωK and f1 , . . . , fn of F(t)⊕n .
Then
                                      τn
                        T K,ρ (t) := n0 d Tor(C∗ (E K ) ⊗α⊗ρ F(t)⊕n , c) ∈ F(t)× ,
                                                                      ˜
                                    ( t)
where

                 := det ρ(µ),
             τ0 := sgn Tor(C∗ (E K ; R), c, h),
                    1
             d := (h-deg Tor(C∗ (E K ) ⊗α Q(t), c0 ) + l-deg Tor(C∗ (E K ) ⊗α Q(t), c0 )),
                                                             ˜                              ˜
                    2
               c := e1 , . . . , edimC∗ ,
             c0 := e1 ⊗ 1, . . . , edimC∗ ⊗ 1 ,
             ˜       ˜               ˜
               c := e1 ⊗ f1 , . . . , e1 ⊗ fn , . . . , edimC∗ ⊗ f1 , . . . , edimC∗ ⊗ fn .
               ˜     ˜                ˜                 ˜                     ˜

Remark 5.2. We can also define normalized Reidemeister torsion for a link by a similar method.
    One can prove the following lemma by a similar way as in the non-normalized case. As a
reference, see [T1].

Lemma 5.3. T K,ρ is invariant under homology orientation preserving simple homotopy equiva-
lence.

Remark 5.4. From the result of Waldhausen [Wal], the Whitehead group Wh(G K ) is trivial for
a knot group G K in general. Therefore homotopy equivalence between finite CW-complexes
whose fundamental groups are knot groups is simple homotopy equivalence.
    Let F be a field with (possibly trivial) involution f → f¯. We extend the involution to F(t)
by taking t → t−1 . We equip F(t)⊕n with the standard hermitian inner product (·, ·) defined by

                                             (v, w) := t vw,
                                                          ¯

where v, w ∈ F(t)⊕n and t v is the transpose of v. For a representation ρ : G K → GLn (F), we
define a representation ρ† : G K → GLn (F) by

                                           ρ† (γ) := ρ(γ−1 )∗ ,

where γ ∈ G K and A∗ := t A for a matrix A.
   We can also refine the duality theorem for sign-determined Reidemeister torsion as follows.
Normalization of twisted Alexander invariants                                                                 9


Theorem 5.5. If the twisted homology group H∗ (E K ; F(t)⊕n ) vanishes for a representation
                                                               α⊗ρ
ρ : G K → GLn (F), then so does H∗ (E K ; F(t)⊕n † ) and we have
                                              α⊗ρ


                                        T K,ρ† (t) = (−1)n T K,ρ (t).

    The proof is based on the following observation. Let (E K , {ei }) denote the PL manifold E K
with the dual cell structure and choose a lift ei which is the dual of ei . In the remainder of this
                                               ˜                        ˜
section, for abbreviation, we write

            Cq := Cq (E K ) ⊗α Q(t),                              Cρ,q := Cq (E K ) ⊗α⊗ρ F(t)⊕n ,
            Cq := Cq (∂E K ) ⊗α Q(t),                             Cρ,q := Cq (∂E K ) ⊗α⊗ρ F(t)⊕n ,
           Cq := Cq (E K , ∂E K ) ⊗α Q(t),                        Cρ,q := Cq (E K , ∂E K ) ⊗α⊗ρ F(t)⊕n ,
            Dq := Cq (E K ) ⊗α Q(t),                              Dρ,q := Cq (E K ) ⊗α⊗ρ† F(t)⊕n ,
            Bq := Im(∂ : Cq+1 → Cq ),                             Bρ,q := Im(∂ : Cρ,q+1 → Cρ,q ),
            Bq := Im(∂ : Cq+1 → Cq ),                             Bρ,q := Im(∂ : Cρ,q+1 → Cρ,q ).

   Note that since direct computation gives

                                        H∗ (∂E K ; F(t)⊕n ) = 0
                                                       α⊗ρ                                                 (5.1)

(See, for example, [KL, Subsection 3.3.].), we have
                                              i
                            dim Bρ,i =             (−1)i− j dim Cρ, j
                                             j=0
                                              i
                                                                                                           (5.2)
                                        =          (−1)i− j n dim C j = n dim Bi
                                             j=0


Similarly, if H∗ (E K ; F(t)⊕n ) = 0, then from (5.1) and the long exact sequence of the pair
                                 α⊗ρ
(E K , ∂E K ), H∗ (E K , ∂E K ; F(t)⊕n ) = 0 and so
                                     α⊗ρ

                                            dim Bρ,i = n dim Bi .                                          (5.3)

   The well known inner product

                            [·, ·] : Cq (E K ) × C3−q (E K , ∂E K ) → Z[G K ]

(See, for example, [M, Lemma 2.].) defined by

                                   [˜ i , e j ] :=
                                    e ˜                       (˜ i , e j · γ−1 )γ,
                                                               e ˜
                                                      γ∈G K

where (·, ·) denote the intersection number, induces an inner product

                                       ·, · : Dρ,q × Cρ,3−q → C(t)
10                                                                                                      T. Kitayama


defined by
                                     ei ⊗ v, e j ⊗ w := (v, [˜ i , e j ] · w),
                                     ˜       ˜               e ˜
where v, w ∈ C(t)⊕n . We see at once that this is well-defined. Thus

                                                  Dρ,q         (Cρ,3−q )∗ .                                   (5.4)

The differential ∂q of Dρ,q corresponds with (−1)q ∂∗ of (Cρ,3−q )∗ under this isomorphism. Sim-
                                                   3−q
ilarly, we have
                                         Dq (C3−q )∗ .                                     (5.5)

Lemma 5.6. For any representation ρ : G K → GLn (F),

                                Hq (E K ; F(t)⊕n † )
                                              α⊗ρ
                                                                 H3−q (E K ; F(t)⊕n )∗ .
                                                                                 α⊗ρ


Proof. From (5.4) and the universal coefficient theorem, we can see that

                             Hq (E K ; F(t)⊕n † )
                                           α⊗ρ
                                                          H3−q (E K , ∂E K ; F(t)⊕n )∗ .
                                                                                 α⊗ρ


From (5.1) and the long exact sequence of the pair (E K , ∂E K ),

                               H∗ (E K ; F(t)⊕n )
                                             α⊗ρ              H∗ (E K , ∂E K ; F(t)⊕n ).
                                                                                   α⊗ρ


This completes the proof.

     Now we prove the theorem.

Proof of Theorem 5.5. Lemma 5.6 gives the first assertion. We use the notation of Definition
5.1. We choose an orthonormal basis f1 , . . . , fn of F(t)⊕n with respect to the hermitian product
(·, ·) defined above. Let c , c , c0 , c0 , c and c be induced bases of C∗ (∂E K ), C∗ (E K , ∂E K ), C∗ ,
                                           ˜     ˜
                         ˜      ˜
C∗ , Cρ,∗ and Cρ,∗ by c, c0 and c. We set

                    c∗ := e1 , . . . , edimC∗ ,
                    c∗ := e1 ⊗ 1, . . . , edimC∗ ⊗ 1 ,
                    ˜0
                    c∗ := e1 ⊗ f1 , . . . , e1 ⊗ fn , . . . , edimC∗ ⊗ f1 , . . . , edimC∗ ⊗ fn .
                    ˜     ˜                 ˜                 ˜                     ˜

     From (5.4) and the duality for algebraic torsion ([T2, Theorem 1.9]),

                          Tor(Dρ,∗ , c∗ ) = (−1)
                                     ˜                    i   dim Bρ,i−1 dim Bρ,i
                                                                                    Tor(Cρ,∗ , c ).
                                                                                               ˜

     On the other hand, from the exact sequence

                                       0 → Cρ,∗ → Cρ,∗ → Cρ,∗ → 0

and the multiplicativity for algebraic torsion ([T2, Theorem 1.5]),

                   Tor(Cρ,∗ , c) = (−1)
                              ˜              i   dim Bρ,i−1 dim Bρ,i
                                                                       Tor(Cρ,∗ , c ) Tor(Cρ,∗ , c ).
                                                                                  ˜              ˜
Normalization of twisted Alexander invariants                                                                      11


   Therefore, we obtain
                                         i (dim Bρ,i−1 +dim Bρ,i−1 ) dim Bρ,i
              Tor(Cρ,∗ , c) = (−1)
                         ˜                                                      Tor(Cρ,∗ , c )Tor(Dρ,∗ , c∗ ).
                                                                                           ˜             ˜       (5.6)

Similarly,
                                            i (dim Bi−1 +dim Bi−1 ) dim Bi
                 Tor(C∗ , c0 ) = (−1)
                          ˜                                                     Tor(C∗ , c0 )Tor(D∗ , c∗ ).
                                                                                         ˜            ˜0         (5.7)
   We set
                                 1
                            d := (h-deg Tor(C∗ , c0 ) + l-deg Tor(C∗ , c0 )),
                                                  ˜                    ˜
                                 2
                                 1
                            d∗ := (h-deg Tor(D∗ , c∗ ) + l-deg Tor(D∗ , c∗ )).
                                                  ˜0                    ˜0
                                 2
From (5.7), we have
                                                     d = d − d∗ .                                                (5.8)
   From Lemma 4.6(ii) and Theorem 5.7,

                                  lim τ0 (t 2 − t− 2 ) Tor(C∗ , c0 ) = −
                                             1         1
                                                                ˜                     K (0)
                                  t→1
                                                                                = −1.

Similarly,
                                      lim τ∗ (t 2 − t− 2 ) Tor(D∗ , c∗ ) = −1,
                                                 1         1
                                           0                        ˜0
                                      t→1

where
                                       τ∗ := sgn Tor(C∗ (E K ; R), c∗ , h).
                                        0

By multiply (5.7) by (t 2 − t− 2 ) and taking t → 1, we obtain
                        1         1




                                                     i (dim Bi−1 +dim Bi−1 ) dim Bi
                                  τ0 = −(−1)                                          τ0 τ∗ ,
                                                                                          0                      (5.9)

where
                                             τ0 := lim Tor(C∗ , c0 ).
                                                                ˜
                                                       t→1

   From (5.2), (5.3), (5.6), (5.8) and (5.9),
                                  τn
                    T K,ρ (t) =     0
                                         Tor(Cρ,∗ , c)
                                                    ˜
                                ( tn )d
                                         (τ0 )n                   (τ∗ )n
                              = (−1)  n
                                            n )d
                                                 Tor(Cρ,∗ , c ) · n d∗ Tor(Dρ,∗ , c∗ ).
                                                            ˜       0
                                                                                  ˜
                                        ( t                      ( t)
   Direct computation gives
                                                 Tor(C∗ , c0 ) = τ0 td .
                                                          ˜
(See, for example, [KL, Subsection 3.3.].) Since the normalized invariants do not change via
conjugation of representations, we can assume ρ(µ) and ρ(λ) are diagonal matrices. This de-
duces
                                 Tor(Cρ,∗ , c ) = (τ0 )n ( tn )d .
                                            ˜
12                                                                                                T. Kitayama


Thus
                                             (τ0 )n
                                                    Tor(Cρ,∗ , c ) = 1.
                                                               ˜
                                            ( tn )d
     It can be easily seen that
                                       (τ∗ )n
                                         0
                                               Tor(Dρ,∗ , c∗ ) = T K,ρ† (t).
                                                          ˜
                                      ( tn )d∗
This proves the theorem.
     In the normalized setting, Theorem 3.6 also holds.
Theorem 5.7. For a representation ρ : G K → GLn (F) such that the twisted homology group
H∗ (E K ; F(t)⊕n ) vanishes, we have
              α⊗ρ

                                               ∆K,ρ (t) = T K,ρ (t).
Proof. We choose a Wirtinger presentation G K = x1 , . . . , xm | r1 , . . . , rm−1 and take the CW-
complex W corresponding with the presentation. Namely, W has one vertex, m edges and
(m − 1) 2-cells attached by the relations r1 , . . . , rm−1 . Let the words x1 , . . . , xm and r1 , . . . , rm−1
also denote the cells. It is easy to see that the exterior E K collapses to W. This implies that W
is simple homotopy equivalent to E K from Remark 5.4. Thus we can compute the normalized
torsion T K,ρ as that of W from Lemma 5.3.
    C∗ (W; R) is
                                      m−1                 m
                                                  ∂2                     ∂1
                             0 →                 →
                                            Rr j −                 →
                                                               Rxi − Rpt → 0,
                                      j=1               i=1

where
                                                ∂1 = 0,
                                                              ∂r j
                                                ∂2 = ϕ               .
                                                              ∂xi
Let c0 = pt, c1 = x1 , . . . , xm and c2 = r1 , . . . , rm−1 . We choose b1 = ∂c2 and h0 = [pt],
h1 = [xk ] (1 ≤ k ≤ m). Then
                                                         [b1 ∪ h1 /c1 ]
                                                                ˜
                                    τ0 = sgn(−1)|C∗ (W;R)|
                                                       [h0 /c0 ][b1 /c2 ]
                                                         ˜        ˜
                                                               
                                                   
                                                   
                                                   
                                                              0
                                                   
                                                   
                                                             .
                                                              .
                                                                
                                                   
                                                   
                                                             .
                                                                
                                                                
                                                   
                                                               
                                                   
                                                   
                                                   
                                                             0
                                                                
                                                                
                                                                
                                                   
                                                   ϕ           
                                                              1
                                                      ∂r j
                                                   
                                       = − sgn det 
                                                               
                                                                
                                                                
                                                   
                                                   
                                                   
                                                      ∂xi       
                                                                
                                                   
                                                   
                                                             0
                                                                
                                                                
                                                   
                                                   
                                                             .
                                                   
                                                   
                                                             .
                                                              .
                                                                
                                                   
                                                               
                                                                
                                                              0
                                       = (−1)k+m+1 δ.
Normalization of twisted Alexander invariants                                                                            13


    We define an involution ¯ : Z[G K ] → Z[G K ] by extending the inverse operation γ → γ−1 of
                            ·
G K linearly. We can choose lifts pt, xi and r j such that C∗ (W) ⊗α⊗ρ F(t)⊕n is
                                      ˜      ˜
                                           ∂2
                                           ˜                                    ∂1
                                                                                ˜
    0 →                                      →
                             F(t)(˜ j ⊗ fl ) −
                                  r                                            →
                                                               F(t)( xi ⊗ fl ) −
                                                                     ˜                            F(t)( pt ⊗ fl ) → 0,
            1≤ j≤m−1,1≤l≤n                       1≤i≤m,1≤l≤n                           1≤l≤n

where

                                       ∂1 ( xi ⊗ fl ) = pt ⊗ Φ( xi − 1) fl
                                       ˜ ˜                      ˜
                                                                    
                                                         m
                                                                    ∂r j 
                                                                    
                                       ∂2 (˜ j ⊗ fl ) =
                                       ˜ r                  xi ⊗ Φ   fl .
                                                            ˜       
                                                                    
                                                                    ∂x 
                                                        i=1             i


Let c0 = pt ⊗ f1 , . . . , pt ⊗ fn , c1 = x1 ⊗ f1 , . . . , x1 ⊗ fn , . . . , xm ⊗ f1 , . . . , xm ⊗ fn and c2 =
                                                         ˜           ˜          ˜               ˜
 r1 ⊗ f1 , . . . , r1 ⊗ fn , . . . , rm−1 ⊗ f1 , . . . , rm−1 ⊗ fn . We choose b0 = ∂ xk ⊗ f1 , . . . , xk ⊗ fn and
 ˜                 ˜                 ˜                   ˜                            ˜                 ˜
                                                                         ⊕n
b1 = ∂c2 . Since the twisted homology group H∗ (W; F(t)α⊗ρ ) vanishes, |C∗ (W) ⊗α⊗ρ F(t)⊕n | = 0
and so
                                                                       [b1 ∪ b0 /c1 ]
                                                                               ˜
                     Tor(C∗ (W) ⊗α⊗ρ F(t)⊕n , c0 , c1 , c2 ) =
                                              ˜ ˜ ˜
                                                                      [b0 /c0 ][b1 /c2 ]
                                                                                 ˜
                                                                                             
                                                                          
                                                                          
                                                                                           0
                                                                          
                                                                          
                                                                                           .
                                                                                            .
                                                                                              
                                                                          
                                                                          
                                                                          
                                                                                           .
                                                                                              
                                                                                              
                                                                                              
                                                                          
                                                                                             
                                                                          
                                                                          
                                                                                           0
                                                                                              
                                                                                              
                                                                          
                                                                          
                                                                                             
                                                                                              
                                                                                              
                                                                      det  Φ ∂xij          I
                                                                                  ∂r
                                                                          
                                                                          
                                                                                             
                                                                                              
                                                                                              
                                                                          
                                                                          
                                                                                             
                                                                                              
                                                                                              
                                                                          
                                                                          
                                                                                           0
                                                                                              
                                                                          
                                                                                             
                                                                          
                                                                          
                                                                                           .
                                                                                            .
                                                                                              
                                                                          
                                                                          
                                                                                           .
                                                                                              
                                                                                             
                                                                                            0
                                                                  =
                                                                        det Φ(xk − 1)
                                                                                                   ∂ri
                                                                                     det t Φ       ∂x j
                                                                  = (−1)n(k+m)                            .
                                                                                     det t Φ(xk − 1)
Similarly, we have
                                                                                           ∂ri
                                                                              det α
                                                                                  ˜        ∂x j
                       Tor(C∗ (W) ⊗α Q(t), c0 , c1 , c2 ) = (−1)(k+m)
                                           ˜ ˜ ˜                                                    .
                                                                              det α(xk − 1)
                                                                                  ˜
Hence d = −d and so
                                            T K,ρ (t) = (−1)n ∆K,ρ† (t),
where we consider the trivial involution on F. From Theorem 5.5, we obtain the desired for-
mula.
   From the above theorems and the following lemma, we have the duality theorem for nor-
malized twisted Alexander invariants.
14                                                                                              T. Kitayama


Lemma 5.8. If H∗ (E K ; F(t)⊕n ) does not vanish, then we have
                            α⊕ρ

                                        ∆K,ρ (t) = ∆K,ρ† (t) = 0.
Proof. If H∗ (E K ; F(t)⊕n ) does not vanish, then neither does H∗ (E K ; F(t)⊕n † ) from Lemma 5.6.
                        α⊗ρ                                                   α⊗ρ
Since
                                 2
                                      dim Hq (E K ; F(t)⊕n ) = nχ(E K )
                                                        α⊕ρ
                                q=0

                                                              = 0,

from the assumption and (5.1), we have H1 (E K ; F(t)⊕n )
                                                     α⊗ρ                 0 and so ∆K,ρ (t) = 0. Similarly, we
obtain ∆K,ρ† (t) = 0, which proves the lemma.
Theorem 5.9. Given a representation ρ : G K → GLn (F), we have

                                        ∆K,ρ† (t) = (−1)n ∆K,ρ (t).
    For a unitary representation ρ, the difference between the highest coefficient of ∆K,ρ (t) and
the lowest coefficient of it is not clear from Theorem 2.5 because of the ambiguity. However,
this difference is strictly determined from the following corollary.
Corollary 5.10. For a representation ρ : G K → U(n) or O(n), we have

                                        ∆K,ρ (t) = (−1)n ∆K,ρ (t).
Example 5.11. Let K be the (p, q) torus knot (p, q > 1 and (p, q) = 1). It is well known that the
knot group has a presentation
                                     G K = x, y | x p y−q ,
where h-deg α(x) = q and h-deg α(y) = p. The 2-dimensional complex W corresponding with
this presentation is K(G K , 1). Therefore we can use this presentation for the computation via
Lemma 5.3, Remark 5.4 and Theorem 5.7.
    From the result of Klassen [Kl], all the irreducible S U(2)-representations up to conjugation
are given as follows:
                 ρa,b,s : G K → S U(2) :
                                cos aπ + i sin aπ        0
                          x→         p          p
                                                                   ,
                                       0          cos p − i sin aπ
                                                      aπ
                                                                 p
                                cos bπ + i sin bπ cos πs       sin bπ sin πs
                          y→         q          q                   q
                                                                                  ,
                                    − sin bπ sin πs
                                           q
                                                         cos bπ − i sin bπ cos πs
                                                              q          q

where a, b ∈ N, 1 ≤ a ≤ p − 1, 1 ≤ b ≤ q − 1, a ≡ b mod 2 and 0 < s < 1. The normalized
twisted Alexander invariants of the torus knot for these representations are as follows:
                                                    pq              pq
                                               (t 2 − (−1)a t− 2 )2
                      ∆K,ρa,b,s (t) = p                                          .
                                     (t − 2 cos bπ + t−p )(tq − 2 cos aπ + t−q )
                                                 q                     p
Normalization of twisted Alexander invariants                                                  15


6 Applications
  Now we consider applications of the normalized invariants. First we generalize the result of
Goda-Kitano-Morifuji and Friedl-Kim. We denote by g(K) the genus of K.
  Their results are as follows.

Theorem 6.1 ([GKM]). For a fibered knot K and a unimodular representation ρ : G K →
S L2n (F), c(∆K,ρ ) is well-defined and is 1.

Theorem 6.2 ([C],[FK]). For a fibered knot K and a representation ρ : G K → GLn (R), ∆1 is
                                                                                     K,ρ
a monic polynomial and deg ∆K,ρ = n(2g(K) − 1), where “monic” means that the highest and
lowest coefficients of a polynomial are units.

   In the normalized setting, we have the following theorem.

Theorem 6.3. For a fibered knot K and a representation ρ : G K → GLn (R),

                             deg ∆K,ρ = 2 h-deg ∆K,ρ = n(2g(K) − 1),
                                                  n g(K)− 1
                              c(∆K,ρ ) = c(     K)        2.




Proof. The equality deg ∆K,ρ = n(2g(K) − 1) can be obtained from Theorem 6.2. Since we have
∆K,i◦ρ = ∆K,ρ , where i is the natural inclusion GLn (R) → GLn (Q(R)), we can assume R is a field
F.
    Let ψ denote the automorphism of a surface group induced by the monodromy map. We can
take the following presentation of the knot group by using the fibered structure:

                       x1 , . . . , x2g , h | ri := hxi h−1 ψ∗ (xi )−1 , 1 ≤ i ≤ 2g(K)

where α(xi ) = 1 for all i and α(h) = t. It is easy to see that the corresponding CW-complex
is homotopy equivalent to the exterior E K . Thus we can compute the invariant by using the
presentation as in Example 5.11.
    Since
                                         
                                   ∂ri h − ∂xi i i = j
                                                ∂ψ∗ (x )
                                         
                                         
                                       =  ∂ψ∗ (xi )         ,
                                   ∂x j − ∂x j
                                                        i j

we have

                   det Aα,2g(K)+1 = t2g(K) + · · · + 1,
                        ˜
                                                                               ∂ψ∗ (xi )
                     det AΦ,2g+1 =     2g(K) 2ng(K)
                                            t         + · · · + (−1)n det(Φ(             )),
                                                                                ∂x j
                   det Φ(h − 1) = tn + · · · + (−1)n .

From the classical theorem of Neuwirth which states that the degree of the Alexander polyno-
mial of a fibered knot equals the twice genus, we can determine that the lowest degree term of
16                                                                                                           T. Kitayama


the first equality is 1. Since

                                                                          − t− 2 )
                                                                      1        1
                                           δ = sgn c(       K)   K (t 2
                                                                                     t=1
                                              = c(    K)
                                                     1
                                           d = g(K) − ,
                                                     2

h-deg ∆K,ρ = n(g(K) − 1 ) and c(∆K,ρ ) = c(
                      2                                    K)
                                                             n 2g(K)−1
                                                                       .
     Next we study a behavior of the highest degree of a normalized invariant.

Definition 6.4. A Seifert surface for a knot K is said to be canonical if it is obtained from a
diagram of K by applying the Seifert algorithm. The minimum genus over all canonical Seifert
surfaces is called the canonical genus and denoted by gc (K). A Seifert surface S is said to be
free if π1 (S 3 \ S ) is a free group. This condition is equivalent to that S 3 \ N(S ) is a handlebody,
where N(S ) is an open regular neighborhood of S . The minimum genus over all free Seifert
surfaces is called the free genus and denoted by g f (K).

Remark 6.5. Since every canonical Seifert surfaces is free, we have the following fundamental
inequality:
                                  g(K) ≤ g f (K) ≤ gc (K).
     We obtain an estimate of free genus from below via the highest degree of the invariants.

Theorem 6.6. For a representation ρ : G K → GLn (R), the following inequality holds:

                                           2 h-deg ∆K,ρ ≤ n(2g f (K) − 1).

Proof. We choose a free Seifert surface S with genus g f (K) and take a bicollar S × [−1, 1] of S
such that S × 0 = S . Let ι± : S → S 3 \ S be the embeddings whose images are S × {±1}. Picking
generator sets {a1 , . . . , a2g f (K) } of π1 S and {x1 , . . . , x2g f (K) } of π1 (S 3 \ S ) and setting ui := (ι+ )∗ (ai )
and vi := (ι− )∗ (ai ) for all i, we have a presentation

                               x1 , . . . , x2g f (K) , h | ri := hui h−1 v−1 , 1 ≤ i ≤ 2g f (K)
                                                                           i

of G K where α(xi ) = 1 for all i and α(h) = t.
    Collapsing surfaces S × ∗ and the handlebody S 3 \ (S × [−1, 1]) to bouquets, we can realize
the 2-dimensional complex corresponding with this presentation as a deformation retract of E K .
Therefore we can compute the invariant by using the presentation as in Example 5.11.
    Since
                                         ∂ri    ∂ui ∂vi
                                             =h     −     ,
                                        ∂x j    ∂x j ∂x j
we have

                                   h-deg ∆K,ρ = h-deg det AΦ,2g f (K)+1 − nd − n
                                              ≤ 2ng f (K) − nd − n.
Normalization of twisted Alexander invariants                                                                        17


The proof is completed by showing that d = g f (K) − 1 .               2
     Let V be the Seifert matrix with respect to the basis [a1 ], . . . , [a2g f (K) ] ∈ H1 (S ; Z) and
[a1 ]∗ , . . . , [a2g f (K) ]∗ ∈ H1 (S 3 \ S ; Z) the dual basis, i. e. lk([ai ], [a j ]∗ ) = δi, j . We denote by A± the
matrices representing (ι± )∗ : H1 (S ; Z) → H1 (S 3 \S ; Z) with respect to the bases [a1 ], . . . , [a2g f (K) ]
and [x1 ], . . . , [x2g f (K) ] and by P the base change matrix of H1 (S 3 \ S ; Z) from [x1 ], . . . , [x2g f (K) ]
to [a1 ]∗ , . . . , [a2g f (K) ]∗ .
     It is well known that the matrices representing (ι+ )∗ and (ι− )∗ : H1 (S ; Z) → H1 (S 3 \ S ; Z)
with respect to the bases [a1 ], . . . , [a2g f (K) ] and [a1 ]∗ , . . . , [a2g f (K) ]∗ are V and t V. Hence

                                         det Aα,2g f (K)+1 = det(tt A+ − t A− )
                                              ˜
                                                           = det(tA+ − A− )
                                                           = det(tPV − Pt V)
                                                           = ± det(tV − t V),

and d = g f (K) −     1
                      2
                          as required.

Example 6.7. Let K be the knot 11n 73 illustrated in Figure 1. The normalized Alexander poly-
nomial of K is t2 − 2t + 3 − 2t−1 + t−2 .




                                                       Figure 1

    The Wirtinger presentation of the diagram in Figure 1 consists of 11 generators and 10
relations:

                                     −1 −1                                         −1 −1
                              x5 x1 x5 x2 ,                                x11 x2 x11 x3 ,
                                     −1 −1                                         −1 −1
                              x9 x4 x9 x3 ,                                 x7 x5 x7 x4 ,
                                     −1 −1                                         −1 −1
                              x1 x5 x1 x6 ,                                 x8 x7 x8 x6 ,
                                     −1 −1                                         −1 −1
                              x5 x8 x5 x7 ,                                x10 x9 x10 x8 ,
                                      −1 −1                                        −1 −1
                              x4 x10 x4 x9 ,                               x2 x10 x2 x11 .
18                                                                                   T. Kitayama


Let ρ : G K → S L2 (F2 ) be a nonabelian representation over F2 defined as follows:
                                              
                                          1 0
                                          
                                              
                                                
                                              
                                           1 1 , if i = 4, 8
                                          
                                          
                                          
                                          
                                                
                                          
                                          
                                              
                                          0 1
                                          
                                              
                                 ρ(xi ) =     
                                                
                                          1 0 , if i = 7, 9 .
                                          
                                          
                                                
                                          
                                          
                                          1 1
                                          
                                          
                                               
                                          
                                              
                                                
                                                
                                          
                                           0 1 , otherwise
                                              

      From them, We have the following:

                                    ∆K,ρ (t) = t5 + t + t−1 + t−5 .

Since deg ∆K,ρ 2(deg ∆K − 1), K is not fibered.
   Moreover, from Proposition 6.6,

                                       10 ≤ 2(2g f (K) − 1).

Therefore
                                             g f (K) ≥ 3.
On the other hand, we obtain a canonical Seifert surface with genus 3 by applying the Seifert
algorithm to the diagram in Figure 1. Thus

                                        g f (K) ≤ gc (K) ≤ 3.

By these inequalities we conclude

                                        g f (K) = gc (K) = 3.

Remark 6.8. From the result of Friedl and Kim [FK],

                                    deg ∆K,ρ ≤ n(2g(K) − 1).

Therefore g(K) also equals 3 in the above example.


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Normalization of twisted Alexander invariants                                              19


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      Graduate School of Mathematical Sciences, the University of Tokyo,
      3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, Japan
      E-mail address: kitayama@ms.u-tokyo.ac.jp

								
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