Twisted Alexander polynomials, character varieties and Reidemeister torsions of double branched covers

Topology and its Applications 204 (2016) 278–305

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Topology and its Applications www.elsevier.com/locate/topol

Twisted Alexander polynomials, character varieties and Reidemeister torsions of double branched covers Yoshikazu Yamaguchi Department of Mathematics, Akita University, 1-1 Tegata-Gakuenmachi, Akita, 010-8502, Japan

a r t i c l e

i n f o

Article history: Received 5 November 2014 Received in revised form 17 March 2016 Accepted 20 March 2016 MSC: primary 57M27, 57M05, 57M12 secondary 57M25 Keywords: Twisted Alexander polynomial Knots Metabelian representations Character varieties Reidemeister torsion Branched coverings

a b s t r a c t We give an extension of Fox’s formula of the Alexander polynomial for a double branched cover over the three-sphere. Our formula provides the Reidemeister torsion of the double branched cover along a knot for a non-trivial 1-dimensional representation. In our formula, the Reidemeister torsion is given by the product of two factors derived from the knot group. One of the factors is determined by the twisted Alexander polynomial and the other is determined by a rational function on the character variety of the knot group. As an application, we show that these products distinguish the isotopy classes of two-bridge knots up to their mirror images. © 2016 Elsevier B.V. All rights reserved.

1. Introduction This paper is intended as an attempt to classify two-bridge knots by using the twisted Alexander polynomials. For this purpose, we extend Fox’s formula of the Alexander polynomial for knots in the theory of Reidemeister torsion. Our extension establishes a connection between the special values of the twisted Alexander polynomials and the Reidemeister torsions for the double branched covers Σ2 along knots. This connection is useful to classify two-bridge knots, which is due to the following facts: 1. We have the one-to-one correspondence between the isotopy classes of two-bridge knots and the homeomorphism classes of their double branched covers, namely lens spaces. 2. We can derive the condition to classify the homeomorphism classes of lens spaces from the values of the Reidemeister torsions for the lens spaces from our extension of Fox’s formula.

E-mail address: [email protected]. http://dx.doi.org/10.1016/j.topol.2016.03.021 0166-8641/© 2016 Elsevier B.V. All rights reserved.

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We recall Fox’s formula from the viewpoint of the Reidemeister torsion, which is due to V. Turaev [18] and J. Porti [16]. Fox’s formula is expressed as |H1 (Σn ; Z)| =

n−1 

ΔK (e

2π

√

−1 n

)

=0

where Σn is the n-fold cyclic cover over S 3 branched along a knot K. This equality was extended to links and finite abelian branched covers over integral homology spheres as an equality of the Reidemeister torsions. From this viewpoint, we can regard Fox’s formula as a framework connecting two Reidemeister torsions of a cyclic branched cover and a knot exterior. In such a framework, we regard the Alexander polynomial as the Reidemeister torsion of a knot exterior, defined by the trivial representation. We deduce Fox’s formula from the Alexander polynomial of the lift  of a knot K in the cyclic branched cover Σn . We also define the Alexander polynomial of K  as the K Reidemeister torsion of the knot exterior, given by the trivial representation. As an equality of Reidemeister torsions, the following relation holds (see [18, Theorem 1.9.2]): ΔK (t) =

n−1 

ΔK (e

2π

√

−1 n

t)

=0

 where ΔK (t) is the Alexander polynomial of the knot K in Σn . Evaluating at t = 1, we can see that the Reidemeister torsions ΔK (t) and ΔK (t) for the knot exteriors turn into the orders of H1 (Σn ; Z) and H1 (S 3 ; Z). In this situation, the order of H1 (Σn ; Z) can be also regarded as the Reidemeister torsion (we refer to [18,16]). To be more precise, the order of H1 (Σn ; Z) is defined as the Reidemeister torsion by the trivial GL1 (C)-representation of π1 (Σn ). We are going to provide an extension of the framework derived from Fox’s formula by replacing representations in the Reidemeister torsions of Σ2 and knot exteriors. We require that π1 (Σ2 ) has non-trivial GL1 (C)-representations. Every GL1 (C)-representation factors through a homomorphism from H1 (Σ2 ; Z) into GL1 (C). Fox’s formula shows that every double branched cover is a rational homology sphere. Hence our main concern is double branched covers Σ2 with non-trivial H1 (Σ2 ; Z). We will replace the Alexander polynomial with the twisted Alexander polynomial as the corresponding Reidemeister torsion. Our extension consists of the following three steps: • First we need to find homomorphisms of the knot group, corresponding to a non-trivial GL1 (C)-representation of π1 (Σ2 ). We can choose irreducible metabelian representations of the knot group into SL2 (C) as the corresponding homomorphisms by the author’s previous work [14] jointly with F. Nagasato.  by the twisted • In the second step, we have to express the twisted Alexander polynomial of the knot K Alexander polynomials of K. We can use the similar formula to the Alexander polynomial, according to the previous work [5] jointly with J. Dubois.  which gives • Last, we must consider an evaluation of the twisted Alexander polynomial of the knot K the Reidemeister torsion of the double branched cover Σ2 . In the evaluation, we need an additional term to adjust the special value of the twisted Alexander polynomial. This difference is determined by the Mayer–Vietoris exact sequence for the twisted homology group of the double branched cover. We can measure the difference between the special value of the twisted Alexander polynomial and the Reidemeister torsion for the double branched cover in the Mayer–Vietoris exact sequence with the coefficient sl2 (C). The homology groups with the coefficient sl2 (C) are related to the deformation of geometric structure. We can observe the Mayer–Vietoris exact sequence in the context of the deformation of geometric structure, namely in the context of the SL2 (C)-character variety.

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The special value of the twisted Alexander polynomial gives a Reidemeister torsion defined by some cotangent vector of the character variety. However this cotangent vector does not coincide with the desired framing to give the Reidemeister torsion for Σ2 . To resolve such a difference between cotangent vectors, we measure the ratio of two cotangent vectors by a rational function on the character variety. We will carry out this modification by using the Reidemeister torsion of the Mayer–Vietoris exact sequence. As a result, we will obtain the following equality (for a precise statement, we refer to Theorem 3.19 in Section 3.5) to compute the Reidemeister torsion for Σ2 and a non-trivial GL1 (C)-representation ξ: √ 2   ΔEK ,α⊗ρ ( −1t)    | Tor(Σ2 ; Cξ )| =  lim  · |F ([ρ ])|. t→1 t2 − 1 2

Here EK is the exterior of K and ΔEK ,α⊗ρ (t) denotes the twisted Alexander polynomial of K for an irreducible metabelian SL2 (C)-representation ρ of π1 (EK ) corresponding to ξ and F denotes a rational function on the SL2 (C)-character variety. We denote by F ([ρ ]) the value of F at the conjugacy class of another irreducible metabelian representation of π1 (EK ) associated with ρ by the adjoint action on the Lie algebra sl2 (C). The metabelian representation ρ is embedded into the adjoint representation of another metabelian representation ρ . This theorem allows us to compute the Reidemeister torsion of Σ2 and a GL1 (C)-representation ξ by the knot group and SL2 (C)-representations. In the last section, we will see an application of our main theorem. Theorem 4.9 shows that we can distinguish two-bridge knots, up to mirror images, by the twisted Alexander polynomials and the rational functions F on the character varieties. We will also discuss how to compute the rational functions F on character varieties for two-bridge knots. Our computation is supported by several works on the character varieties of two-bridge knots. Especially we need the result that every two-bridge knot has Property L, which is shown in [1] by M. Boileau, S. Boyer, A. Reid and S. Wang (for details, we refer to Section 4.2). Our method also works for knots in integral homology three spheres. However we will focus on knots in S 3 for the simplicity. This paper is organized as follows. In Section 2, we review the Reidemeister torsion associated to twisted chain complexes with nontrivial homology groups and properties of character varieties, which are needed throughout this paper. We discuss twisted chain complexes and the associated Reidemeister torsion in details in Section 3. We show our main theorem which gives a connection between the twisted Alexander polynomial of a knot and the Reidemeister torsion of the double branched cover along the knot under base change conditions. Last, Section 4 reveals that all two-bridge knots satisfy the conditions required in our main theorem and we can obtain the numerical invariant which classifies two-bridge knots up to mirror images. 2. Preliminaries 2.1. Torsion for chain complexes Torsion is an invariant defined for a based chain complex. In this paper, a based chain complex means a chain complex: ∂

∂n−1

∂

∂

n 2 1 C∗ : 0 → Cn −−→ Cn−1 −−−→ · · · −−→ C1 −−→ C0 → 0

whose each chain module Ci is a vector space over a field F and equipped with a basis ci . As we shall see below, the chain complex C∗ also has a basis determined by the boundary operators ∂i and the homology classes. The torsion of a based chain complex provides a property of a chain complex in the difference between a given basis and a new one determined by the boundary operators and the homology classes.

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For each boundary operator ∂i , let Zi ⊂ Ci denote the kernel and Bi ⊂ Ci−1 the image by ∂i . The ˜i . Moreover we can chain module Ci is expressed as the direct sum of Zi and the lift of Bi , denoted by B ˜ i of homology group Hi (C∗ ): decompose the kernel Zi into the direct sum of Bi+1 and the lift H ˜i Ci = Zi ⊕ B ˜i+1 ⊕ H ˜i ⊕ B ˜i = ∂i+1 B ˜i+1 . where Bi+1 is written as ∂i+1 B ˜ i a basis of B ˜i . Choosing a basis hi of the i-th homology group Hi (C∗ ), we can take a We denote by b i i ˜ i+1 ) ∪ h ˜i ∪ b ˜ i forms a new basis of the vector space Ci . We define ˜ lift h of h in Ci . Then the set ∂i+1 (b the torsion of C∗ as the following alternating product of determinants of base change matrices: Tor(C∗ , c∗ , h∗ ) =

 i+1  ˜ i+1 ) ∪ h ˜i ∪ b ˜ i /ci (−1) ∈ F∗ = F \ {0} ∂i+1 (b

(1)

i≥0

˜ i+1 ) ∪ h ˜i ∪b ˜ i /ci ] denotes the determinant of base change matrix from ci to ∂i+1 (b ˜ i+1 ) ∪ h ˜i ∪b ˜ i. where [∂i+1 (b i i ˜ and the lift of h . The alternating Note that the right hand side is independent of the choice of bases b ∗ product in (1) is an invariant of the based chain complex (C∗ , c ) and the basis h∗ = ∪i≥0 hi . 2.2. Reidemeister torsion for CW-complexes Let W denote a finite CW-complex. We use the symbols V and ρ to denote a vector space over F and a homomorphism from π1 (W ) into GL(V ). The pair (V, ρ) is referred as a GL(V )-representation ρ of π1 (W ). We will consider the torsion of twisted chain complexes given by a CW-complex W and GL(V )-representations of its fundamental group. Definition 2.1. The twisted chain complex C∗ (W ; Vρ ) is defined as a chain complex consisting of the following twisted chain modules:  ; Z) Ci (W ; Vρ ) := V ⊗Z[π1 (W )] Ci (W  is the universal cover of W and Ci (W  ; Z) is a left Z[π1 (W )]-module with the action of π1 (W ) by where W the covering transformation. In taking the tensor product, we regard V as a right Z[π1 (W )]-module under the homomorphism ρ−1 . Namely a chain v ⊗ γc is identified with ρ(γ)−1 (v) ⊗ c in Ci (W ; Vρ ). We say that the twisted chain complex C∗ (W ; Vρ ) has coefficients Vρ and call the homology group, denoted by H∗ (W ; Vρ ), the twisted homology group. We will drop the subscript ρ for simplicity when there exists no risk of ambiguity. When we fix a basis of the vector space V , we make a basis of the twisted chain complex C∗ (W ; Vρ ). Let {ei1 , . . . , eimi } be the set of i-dimensional cells of W and set d = dimF V . Choosing a lift e˜ij of each cell and taking tensor product with a basis {v 1 , . . . , v d } of V , we have the following basis of Ci (W ; Vρ ): ci (W ; V ) = {v 1 ⊗ e˜i1 , . . . , v d ⊗ e˜i1 , . . . , v 1 ⊗ e˜imi , . . . , v d ⊗ e˜imi }. Regarding C∗ (W ; Vρ ) as a based chain complex, we define the Reidemeister torsion for W and (V, ρ) as the torsion of C∗ (W ; Vρ ) with a basis h∗ of H∗ (W ; Vρ ), i.e., Tor(W ; Vρ , h∗ ) = Tor(C∗ (W ; Vρ ), c∗ (W ; V ), h∗ ) ∈ F∗ .

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If we use another basis of V , then we may have the different value for C∗ (W ; Vρ ). In general, we need to require some constraint on the basis of V to remove the effect of this. However when we consider CW-complexes whose Euler characteristic is zero, the torsion Tor(C∗ (W ; Vρ ), c∗ (W ; V ), h∗ ) is not affected by the choice of basis of V . Remark 2.2. The torsion defined by another basis {u1 , . . . , ud } coincides with the product of Tor(C∗ (W ; Vρ ), c∗ (W ; V ), h∗ ) defined by {v 1 , . . . , v d } with the following factor: [{u1 , . . . , ud }/{v 1 , . . . , v d }]−χ(W ) . Remark 2.3. Choosing ρ as an SL(V )-representation, we can also remove the indeterminacy factor given by det(ρ(γ)) from the torsion Tor(W ; Vρ , h∗ ). We will focus on CW-complexes with the Euler characteristic zero and SL(V )-representations ρ of their fundamental groups. Hence the Reidemeister torsion Tor(W ; Vρ , h∗ ) is determined up to sign in this paper. It is also worth noting that the Reidemeister torsion has an invariance under conjugation of representations ρ. We often determine the Reidemeister torsion by choosing another representation in the conjugacy class of a given representation. When bases c∗ (W ; V ) and h∗ are clear from the context, we abbreviate the notation Tor(C∗ , c∗ , h∗ ) to Tor(C∗ ) and Tor(W ; Vρ , h∗ ) to Tor(W ; Vρ ). For more details on the Reidemeister torsion, we refer to Turaev’s book [19] and Milnor’s survey [12]. 2.3. Non-abelian Reidemeister torsion and Twisted Alexander polynomial We will mainly study the Reidemeister torsion of knot exteriors and the twisted chain complexes with the coefficient sl2 (C). Let EK denote the knot exterior S 3 \ N (K) where K is a knot in S 3 and N (K) is an open tubular neighborhood. From now on, we use the symbol ρ for an SL2 (C)-representation of a knot group π1 (EK ). Taking the composition with the adjoint action of SL2 (C) on the Lie algebra sl2 (C), we have a representation (sl2 (C), Ad ◦ ρ) of π1 (EK ) as follows: Ad ◦ ρ : π1 (EK ) → SL2 (C) → Aut(sl2 (C))

(2)

γ → ρ(γ) → Ad ρ(γ) : v → ρ(γ) v ρ(γ)−1 . It is called the adjoint representation of ρ. We regard the vector space sl2 (C), consisting of trace-free 2 × 2-matrices, as the right Z[π1 (EK )]-module via the action Ad ◦ ρ−1 . The following basis will be referred to as the standard basis of 3-dimensional vector space sl2 (C):

E=

0 1 0 0



,H =

1 0 0 −1



,F =

0 0 1 0

 .

(3)

Note that one can show that Ad A has eigenvalues z ±2 and 1 when the SL2 (C)-element A has the eigenvalues z ±1 . Hence Ad ◦ ρ−1 gives an SL3 (C)-representation of π1 (EK ). We will determine the torsion of the twisted chain complex C∗ (EK ; sl2 (C)) defined by the composition Ad ◦ ρ−1 . Since the twisted homology group H∗ (EK ; sl2 (C)) is non-trivial, we need to choose a basis of H∗ (EK ; sl2 (C)) in order to define the torsion of C∗ (EK ; sl2 (C)). The twisted homology group H∗ (EK ; sl2 (C)) depends on the choice of SL2 (C)-representations ρ. We follow the notation of [3,21] of SL2 (C)-representations concerning a basis of H∗ (EK ; sl2 (C)), which was introduced by J. Porti in [15]. Recall that a representation ρ is called reducible if there exists a line 0 = L ⊂ C2 invariant under the action of π1 (EK ) induced by ρ. Otherwise, the representation ρ is called irreducible. We will focus on irreducible SL2 (C)-representations of π1 (EK ).

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Definition 2.4. Let γ be a closed curve on ∂EK . An SL2 (C)-representation ρ of π1 (EK ) is γ-regular if ρ is irreducible and satisfies the following conditions: 1. dimC H1 (EK ; sl2 (C)) = 1; 2. the inclusion map from γ into EK induces the surjective homomorphism from H1 (γ; sl2 (C)) onto H1 (EK ; sl2 (C)), where H1 (γ; sl2 (C)) is the homology group of twisted chain complex for γ and the restriction of ρ on the subgroup γ ⊂ π1 (EK ); 3. if tr ρ(γ  ) = ±2 for all γ  ∈ π1 (∂EK ), then ρ(γ) = ±1. For an explicit basis of H∗ (∂EK ; sl2 (C)), see [21, Lemma 2.6.2] and [15, Proposition 3.18]. Proposition 2.5. Let γ be a closed loop on ∂EK , If ρ is γ-regular, then we can choose the following basis of the twisted homology group H∗ (EK ; sl2 (C)): ⎧ ρ 2 ⎪ ⎪ ⎨C P ⊗ T H∗ (EK ; sl2 (C)) =

C P ⊗ γ ⎪ ⎪ ⎩0 ρ

(∗ = 2) (∗ = 1)

(4)

(otherwise)

where C . . . denotes a linear span over C and we use the same notations for the lifts of chains to the  universal cover E K for simplicity. Let μ and λ denote a meridian and a corresponding preferred longitude on ∂EK . We will focus on μ-regular and λ-regular representations of π1 (EK ). We refer to [21, Section 2.6] as an exposition. Definition 2.6. We assume that ρ is γ-regular for a closed loop on ∂EK . Then we will consider the Reidemeister torsion for EK and the representation (sl2 (C), Ad ◦ ρ−1 ) with the basis as in Eq. (4) and write it simply as Tor(EK ; sl2 (C)). We also review the twisted Alexander polynomial in the context of Reidemeister torsion. We denote by α the abelianization homomorphism of π1 (EK ), i.e., α : π1 (EK ) → H1 (EK ; Z) = t , which sends μ to t. For an SL2 (C)-representation ρ of π1 (EK ), the tensor product α ⊗ Ad ◦ ρ−1 gives an action of π1 (EK ) on the vector space C(t) ⊗C sl2 (C) over the rational function field C(t). Extending the action defined by α ⊗ Ad ◦ ρ−1 to the group ring Z[π1 (EK )] linearly, we can construct the following chain complex:  C∗ (EK ; C(t) ⊗ sl2 (C)) = (C(t) ⊗ sl2 (C)) ⊗Z[π1 (EK )] C∗ (E K ; Z). If the chain complex C∗ (EK ; C(t) ⊗ sl2 (C)) is acyclic, then Tor(C∗ (EK ; C(t) ⊗ sl2 (C))) is defined as an element in C(t) \{0} up to a factor ±tk (k ∈ Z). According to the observation by P. Kirk and C. Livingston [9, Section 4], this torsion Tor(C∗ (EK ; C(t) ⊗ sl2 (C))) can be regarded as the twisted Alexander polynomial of K and Ad ◦ ρ, defined by X.-S. Lin [11] and M. Wada [20]. Definition 2.7. Suppose that the chain complex C∗ (EK ; C(t) ⊗ sl2 (C)) is acyclic. Then we call Tor(C∗ (EK ; C(t) ⊗ sl2 (C))) the twisted Alexander polynomial for K and Ad ◦ ρ−1 and denote it by ΔEK ,α⊗Ad◦ρ (t).

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Remark 2.8. Wada’s definition of the twisted Alexander polynomial allows one to easily compute ΔEK ,α⊗Ad◦ρ (t) by applying Fox differential calculus to a presentation of π1 (EK ). 2.4. Character varieties and the fixed points of an involution We review the SL2 (C)-character varieties of knot groups briefly. It is known that the twisted homology group H1 (EK ; sl2 (C)) is isomorphic to the Zariski cotangent space of the character variety of π1 (EK ) if ρ is irreducible and satisfies the conditions (1) & (3) in Definition 2.4. When we consider a base change in H1 (EK ; sl2 (C)), it is helpful to use the geometric properties and regular functions on the character variety. We start with the definition of character varieties as sets. Definition 2.9. We denote by Hom(π1 (EK ), SL2 (C)) the set of homomorphisms from a knot group π1 (EK ) into SL2 (C). Choosing an SL2 (C)-representation ρ, we call the following map χρ the character of ρ: χρ : π1 (EK ) → C γ → tr ρ(γ). The set of characters, {χρ | ρ ∈ Hom(π1 (EK ), SL2 (C))}, is denoted by X(EK ). The set X(EK ) is called the character variety because it admits the structure of an affine variety (see [2]). This affine variety can be viewed as the algebraic quotient of Hom(π1 (EK ), SL2 (C)) under conjugation. We mainly deal with irreducible SL2 (C)-representations and their characters. We will write X irr (EK ) for the subset in X(EK ), consisting of irreducible characters, which means the characters of irreducible SL2 (C)-representations of π1 (EK ). Remark 2.10. By [2, Proposition 1.5.2], two irreducible representations ρ and ρ satisfy χρ = χρ if and only if ρ is conjugate to ρ . In general, we have the following inequalities of dimensions related to the character variety X(EK ): dimC X(EK ) ≤ dimC TχZar X(EK ) ≤ dimC H 1 (EK ; sl2 (C))(= dimC H1 (EK ; sl2 (C))) ρ where TχZar X(EK ) is the Zariski tangent space at χρ (we refer to [15, Section 3.1.3]) and the last equality ρ is due to the universal coefficient theorem. The first equality holds when χρ is a smooth point and second equality holds when ρ is irreducible. Remark 2.11. According to [15, Proposition 3.5], if ρ is irreducible, then we can identify H1 (EK ; sl2 (C)) with the dual space of Zariski tangent space TχZar X(EK ). ρ The structure of X(EK ) as affine variety arises from regular functions defined as follows. Definition 2.12. We call the following evaluation Iγ the trace function of γ ∈ π1 (EK ): Iγ : X(EK ) → C χρ → χρ (γ) = tr ρ(γ).

(5)

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2.5. Correspondence of representations We suppose that a double branched cover Σ2 of S 3 has non-trivial first homology group H1 (Σ2 ; Z). The order of H1 (Σ2 ; Z) is equal to |ΔK (−1)|. Hence we have a finite number of abelian representations ξ from π1 (Σ2 ) into GL1 (C) = C \ {0}. Since our main concern is a relation between the Reidemeister torsions of Σ2 and EK , we seek a relation between representations of π1 (Σ2 ) and π1 (EK ). To construct ξ from a representation of π1 (EK ), we consider the following diagram:  μ ⊂E K

Σ2

μ2 ⊂ EK

S3

 where E  is the lift of μ2 . K is the 2-fold cover over EK and μ Definition 2.13. An SL2 (C)-representation is called metabelian when the image of the commutator subgroup [π1 (EK ), π1 (EK )] is an abelian subgroup in SL2 (C). By [14, Theorem 1], the following one to one correspondence holds: {ξ ⊕ ξ −1 : π1 (Σ2 ) → SL2 (C) | ξ : π1 (Σ2 ) → GL1 (C)} / conj. 1:1

←→ {ρ : π1 (EK ) → SL2 (C) | tr ρ(μ) = 0, ρ : metabelian} / conj.

(6)

Remark 2.14. By [14, Lemma 9 and Proposition 4] or [13, Proposition 1.1], every irreducible metabelian representation ρ satisfies tr ρ(μ) = 0. Every non-trivial homomorphism ξ of π1 (Σ2 ) corresponds to an irreducible metabelian representation of π1 (EK ). The number of conjugacy classes is given by 1 2 (|ΔK (−1)| − 1). To describe the relation between local systems, we review the map which gives the correspondence (6) (we refer to [14, Section 5]).  Let p denote the induced homomorphism from π1 (E  to μ2 . K ) to π1 (EK ). This homomorphism p sends μ ∗  The pull-back of ρ gives an SL2 (C)-representation p∗ (ρ) of π1 (E K ). Since this representation p (ρ) sends μ  to −1, we need a sign refinement to give an SL2 (C)-representation of π1 (Σ2 ), which is induced from the following 1-dimensional representation: √

[ · ]

−1

: π1 (EK ) → GL1 (C) √ [γ] γ → −1

where [γ] denotes an integer corresponding to the homology class under the isomorphism H1 (EK ; Z)  μ → 1 ∈ Z. √

The product of ( −1)p [ · ] and p∗ (ρ) defines an SL2 (C)-representation of π1 (Σ2 ). Here we identify π1 (Σ2 )  with the quotient group π1 (E 

where μ 

denotes the normal closure of μ . K )/ μ ∗

Definition 2.15. We say that an irreducible metabelian SL2 (C)-representation ρ corresponds to ξ if √ ( −1)p [ · ] · p∗ (ρ) induces the diagonal representation ξ ⊕ ξ −1 on π1 (Σ2 ). ∗

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We need to consider the relation between local systems of Σ2 and EK given by the SL2 (C)-representation √ ξ ⊕ ξ −1 and the GL2 (C)-representation ( −1)[ · ] ρ. From [22, Section 3 and 4], we can see that the adjoint representation defined as Eq. (2) is useful for this purpose as follows. We can find the 2-dimensional repre√ sentation ( −1)[ · ] ρ as a direct summand in the adjoint representation for another irreducible metabelian representation of π1 (EK ). Lemma 2.16. For every irreducible metabelian SL2 (C)-representation ρ of the knot group π1 (EK ), there exists another irreducible metabelian representation ρ and an SL2 (C)-matrix C satisfying the following decomposition: √

Ad ◦ ρ = (−1)[ · ] ⊕ C (

−1 )[ · ] ρ C

−1

,

where the equality holds for the ordered basis {H, E, F } of sl2 (C). Proof. Since tr ρ(μ) = 0, we can choose C ∈ SL2 (C) satisfying Cρ(μ)C

−1

=

√0 −1

√  −1 . 0

From [22, Proposition 3.1], we can decompose the adjoint representation of each irreducible metabelian representation into ψ1 ⊕ ψ2 where ψ1 is the 1-dimensional representation (−1)[ · ] and ψ2 is a 2-dimensional one. To be more precise, we can choose a representative in the conjugacy class of irreducible metabelian representations, whose adjoint representation decomposes sl2 (C) into V1 ⊕ V2 where V1 = C H and V2 = C E, F . It also follows from [22, Proposition 2.8 and the sequel of Eq. (8)] that there exists another metabelian   SL2 (C)-representation

 ρ such that each summand ψ1 and ψ2 in Ad ◦ ρ sends the meridian μ to the matrices 0 −1 and the commutator subgroup into {1} and the abelian subgroup in SL2 (C), given by −1 and −1 0 the image of ρ. 2 Remark 2.17. In Proposition 2.16, two irreducible metabelian representation ρ and ρ are usually contained in distinct conjugacy classes of SL2 (C)-representations of π1 (EK ) since they have the different image of the commutator subgroup. We refer to [22, Section 5] for explicit examples. Definition 2.18. Here and subsequently, we always use the symbol ρ to denote an √ irreducible metabelian 

0 −1 . We will denote SL2 (C)-representation of π1 (EK ) which sends the meridian μ to the matrix √ −1 0 √ by ψ1 and ψ2 the 1-dimensional representation given by (−1)[ · ] and the 2-dimensional one given by ( −1)[ · ] ρ for an irreducible metabelian SL2 (C)-representation ρ of π1 (EK ). Under the assumption in Definition 2.18, every metabelian representation ρ sends the commutator subgroup into the maximal abelian subgroup consisting of diagonal matrices in SL2 (C). We summarize the properties of the adjoint representation Ad ◦ ρ in Lemma 2.16. Remark 2.19. Suppose that ρ corresponds to ξ : π1 (Σ2 ) → GL1 (C). • The pull-back p∗ (Ad ◦ ρ ) sends μ  to the identity matrix. ∗ √  • The pull-back p∗ (Ad ◦ρ ) turns out to be an SL3 (C)-representation (−1)p [ · ] ⊕ ( −1)p [ · ] p∗ (ρ) of π1 (E K ). ∗ ∗   Since p [ · ] maps π1 (E ) onto 2Z, the pull-back p (Ad ◦ ρ ) induces the diagonal representation 1 ⊕ (ξ ⊕ K −1 ξ ) on π1 (Σ2 ). Here 1 denotes the trivial 1-dimensional representation. ∗

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• Since a preferred longitude λ is contained in the second commutator subgroup, every metabelian repre sentation sends λ to the identity matrix. Hence the restriction of p∗ (Ad ◦ ρ ) on π1 (∂ E K ) is the trivial 3-dimensional representation. We will identify μ  with μ2 in π1 (EK ) under the injection p. 3. An extension of Fox’s formula for the twisted Alexander polynomials In this section, we suppose that the homology group H1 (Σ2 ; Z) is non-trivial and consider the Reidemeister torsion of Σ2 defined by abelian representations of π1 (Σ2 ). We derive the Reidemeister torsion of Σ2 from the Mayer–Vietoris exact sequence with the coefficient sl2 (C) induced by the following decomposition: 2 1  Σ2 = E K ∪D ×S 2 2  where E K is the 2-fold cover of EK and the meridian disk ∂D × {∗} is glued with the lift of μ . We will see that the Reidemeister torsion of Σ2 is determined by the product of the Reidemeister torsions  of E K and the Mayer–Vietoris homology exact sequence in the Subsection 3.1. Proposition 3.1 gives a formula of the Reidemeister torsion for Σ2 . Subsection 3.2 provides the bases of the twisted homology groups, which are needed in our observation. In Subsections 3.3 and 3.4, we will discuss how to derive the Reidemeister  torsions of E K and the homology exact sequence from the twisted Alexander polynomial of EK and the character variety of π1 (EK ). Finally Subsection 3.5 shows our statement (Theorem 3.19) of an extension of Fox’s formula. Theorem 3.19 deduces the Reidemeister torsion for Σ2 from a knot exterior EK . This shows that the right hand side of Proposition 3.1 can be replaced with the results in the previous Subsections 3.3 and 3.4.

3.1. The equality of the Reidemeister torsions from the Mayer–Vietoris sequence ∗   When we consider the local system of E K induced by the pull-back p (Ad ◦ ρ ), we can extend this local  system to Σ2 by Lemma 2.16. We also define local systems on the boundary torus ∂ E K and the attached 2 1  solid torus D × S = Σ2 \ int(EK ) by restricting local systems of Σ2 . Note that the coefficients of these 2 1  local systems are sl2 (C). From the decomposition Σ2 = E K ∪ D × S , we have the short exact sequence of the local systems with the coefficient sl2 (C): 2 1  0 → C∗ (T2 ; sl2 (C)) → C∗ (E K ; sl2 (C)) ⊕ C∗ (D × S ; sl2 (C)) → C∗ (Σ2 ; sl2 (C)) → 0

(S )

 where T2 denotes the boundary torus of E K. We can derive the Reidemeister torsion of Σ2 with the representation 1 ⊕ (ξ ⊕ ξ −1 ) as in Remark 2.19 from the short exact sequence (S ). Proposition 3.1. Let ρ be an irreducible metabelian SL2 (C)-representation of π1 (EK ) and ρ another one as in Lemma 2.16. We suppose that the twisted chain complex C∗ (Σ2 ; Cξ ) is acyclic. Then the absolute value of Reidemeister torsion for Σ2 and ξ is expressed as | Tor(Σ2 ; ξ)|2 =

1  · | Tor(E K ; sl2 (C)) · Tor(H )| |H1 (Σ2 ; Z)|

Here H denotes the Mayer–Vietoris exact sequence of homology groups with sl2 (C)-coefficient: 2 1  · · · → Hi+1 (Σ) → Hi (T2 ) → Hi (E K ) ⊕ Hi (D × S ) → Hi (Σ2 ) → · · · ,

(H )

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 which is equipped with the bases of the twisted homology groups H∗(T2 ; sl2 (C)) and H∗ (E K ; sl2 (C)) as in Lemma 3.3. Since we consider non-acyclic chain complexes and their torsions, we must take care of the bases of the twisted homology groups. We will examine the bases of the twisted homology groups in the next Subsection 3.2. Here we need especially Lemmas 3.3 & 3.4 and Proposition 3.5. Proof of Proposition 3.1. Applying the Multiplicativity property of the Reidemeister torsion (we refer to [15, Proposition 0.11] and [21, Proposition 2.4.4]) to the short exact sequence (S ), we have the following equality of Reidemeister torsions up to a sign: 2 1  Tor(Σ2 ; sl2 (C)) Tor(T2 ; sl2 (C)) = ± Tor(E K ; sl2 (C)) Tor(D × S ; sl2 (C)) Tor(H )

(7)

2 1  Here the Reidemeister torsions for T2 , E K and D × S are defined by the non-acyclic chain complexes. By direct computation, one can show the Reidemeister torsions of T2 and D2 × S 1 are equal to ±1 for bases of  H∗ (T2 ; sl2 (C)) and H∗ (E K ; sl2 (C)) as in Lemma 3.3. By Lemma 3.4, the Reidemeister torsion Tor(Σ2 ; sl2 (C)) turns into the product:

Tor(Σ2 ; C) Tor(Σ2 ; Cξ ) Tor(Σ2 ; Cξ¯). Moreover the first factor Tor(Σ2 ; C) turns out to be the order ±|H1 (Σ2 ; Z)| by [16, Proposition 3.10] and Proposition 3.5. The last factor Tor(Σ2 ; Cξ¯) coincides with the complex conjugate of Tor(Σ2 ; Cξ ) from the definition. Combining these results, we can rewrite the equality (7) as  |H1 (Σ2 ; Z)| · | Tor(Σ2 ; Cξ )|2 = ± Tor(E K ; sl2 (C)) · Tor(H ).

2

 Remark 3.2. The Reidemeister torsions Tor(E K ; sl2 (C)) and Tor(H ) depend on the choice of basis of  H∗ (E ; sl (C)). From the last equality of the proof of Proposition 3.1, it follows that the product is inK 2  dependent of the choice of basis of the twisted homology group H∗(E K ; sl2 (C)) and gives the topological invariant of Σ2 . 3.2. The bases of the twisted homology groups This subsection is devoted to compute the homology groups of the twisted chain complexes in the exact √ sequence (S ). We assume that each local system is induced from the Ad ◦ ρ = (−1)[ · ] ⊕ ( −1)[ · ] ρ. Here ρ √ and ρ are irreducible metabelian representation of π1 (EK ). Recall that the direct sum (−1)[ · ] ⊕ ( −1)[ · ] ρ is given by sl2 (C) = V1 ⊕ V2 where V1 = C H and V2 = C E, F spanned by {H} and {E, F }. To express the homology groups, we prepare some notations for cycles in the twisted chain complex C∗ (T 2 ; sl2 (C)) as in Fig. 1. We will denote lifts of chains briefly by the same symbols.

Fig. 1. The symbol m denotes a lift of the meridian μ and a lift of longitude λ is denoted by the same symbol.

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Lemma 3.3. It holds that ⎧ ⎪ sl2 (C) ⊗ T2 ⎪ ⎪ ⎪ ⎨sl (C) ⊗ μ  ⊕ sl2 (C) ⊗ λ 2 H∗ (T2 ; sl2 (C)) = ⎪ ⎪sl2 (C) ⊗ pt ⎪ ⎪ ⎩ 0 ⎧ ⎪ ⎪sl2 (C) ⊗ λ (∗ = 1) ⎨ 2 1 H∗ (D × S ; sl2 (C)) = sl2 (C) ⊗ pt (∗ = 0) ⎪ ⎪ ⎩0 otherwise.

(∗ = 2) (∗ = 1) (∗ = 0) otherwise,

Proof. We have seen in Remark 2.19 that the restriction of Ad ◦ρ on π1 (T2 ) are trivial. Thus the restriction on π1 (D2 × S 1 ) is also trivial. The chain complexes C∗ (T2 ; sl2 (C)) and C∗ (D2 × S 1 ; sl2 (C)) turn into sl2 (C) ⊗C C∗ (T2 ; C) and sl2 (C) ⊗C C∗ (D2 × S 1 ; C). 2 We also have the following decomposition of C∗ (Σ2 ; sl2 (C)) from Remark 2.19. Lemma 3.4. C∗ (Σ2 ; sl2 (C)) = C∗ (Σ2 ; C) ⊕ C∗ (Σ2 ; Cξ ) ⊕ C∗ (Σ2 ; Cξ−1 ). From the Mayer–Vietoris sequence (H ) together with Lemmas 3.3 and 3.4, the twisted homology groups  H∗ (Σ2 ; sl2 (C)) and H∗ (E K ; sl2 (C)) are expressed as follows. Proposition 3.5. Suppose that ρ is irreducible and metabelian such that Ad ◦ ρ induces the representation 1 ⊕ (ξ ⊕ ξ −1 ) of π1 (Σ2 ). If C∗ (Σ2 ; Cξ ) is acyclic, then

H∗ (Σ2 ; sl2 (C)) 

 V1

(∗ = 0, 3)

0

otherwise,

⎧ ⎪ V2 ⎪ ⎪ ⎪ ⎨sl (C) 2  H∗ (E K ; sl2 (C))  ⎪V1 ⎪ ⎪ ⎪ ⎩ 0

(∗ = 2) (∗ = 1) (∗ = 0) otherwise.

Remark 3.6. In Proposition 3.5, the basis of H∗ (Σ2 ; sl2 (C)) is given by the fundamental cycle and the base 2  point of Σ2 . Also the generators of H∗ (E K ; sl2 (C)) are given by the subset of basis of H∗ (T ; sl2 (C)), i.e.,  2 these generators are expressed as the chains T , μ  and pt tensored with bases of V2 , sl2 (C) and V1 . We also give an inverse direction of Proposition 3.5, which will be needed later. Proposition 3.7. Under the same notations of Proposition 3.5, the twisted chain complex C∗ (Σ2 ; Cξ ) is  acyclic if H∗ (E K ; sl2 (C)) is generated as follows: ⎧ 2 ⎪ ⎪V2 ⊗ T ⎪ ⎪ ⎨sl (C) ⊗ μ  2  H∗ (E K ; sl2 (C))  ⎪ V1 ⊗ pt ⎪ ⎪ ⎪ ⎩ 0

(∗ = 2) (∗ = 1) (∗ = 0) otherwise.

Proof. This follows from the Mayer–Vietoris exact sequence (H ) and Lemma 3.3.

2

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 When we choose the basis of H∗ (E K ; sl2 (C)) as in Proposition 3.7, the Reidemeister torsion Tor(H )   equals ±1. However we will need Tor(E K ; sl2 (C)) and Tor(H ) for another basis of H1 (EK ; sl2 (C)), which is given by the lift of a preferred longitude.  By an argument similar to [5, Lemma 4.3 and Eq. (4)–(8)], we can express C∗ (E K ; sl2 (C)) as the direct  sum of twisted chain complexes of EK . Let G denote the covering transformation group of E K , i.e., G = g | g 2 = 1 . Lemma 3.8. Suppose that f±1 are (1 ± g)/2 in the group ring C[G]. Then it holds that  C∗ (E K ; sl2 (C))  C∗ (EK ; sl2 (C) ⊗ C[f1 ]) ⊕ C∗ (EK ; sl2 (C) ⊗ C[f−1 ])

(8)

  where C∗ (EK ; sl2 (C) ⊗ C[f±1 ]) is defined by (sl2 (C) ⊗C C[f±1 ]) ⊗Z[π1 (EK )] C∗ (E K ; Z) via Ad ◦ ρ and the  projection π1 (EK ) → π1 (EK )/π1 (E K ) = G. Moreover we can identify C∗ (EK ; sl2 (C) ⊗ C[f1 ]) with C∗ (EK ; sl2 (C)) defined by Ad ◦ ρ and C∗ (EK ; sl2 (C) ⊗ C[f−1 ]) with the twisted chain complex of EK by (−1)[ · ] · Ad ◦ ρ .

Proof. It follows from the same way of [5, Lemma 4.3] that   C∗ (E K ; sl2 (C))  (sl2 (C) ⊗C C[G]) ⊗Z[π1 (EK )] C∗ (EK ; Z) where we take the tensor product ⊗Z[π1 (EK )] in the right hand side under the adjoint representation Ad ◦ ρ  and the projection π1 (EK ) → G = π1 (EK )/π1 (E K ). We can think of C[G] as a 2-dimensional vector space  spanned by {f1 , f−1 }. Together with C[G] = C[f1 ] ⊕ C[f−1 ], we have the decomposition of C∗ (E K ; sl2 (C)) in our statement. The element g of G acts on C[G] linearly. The vectors f±1 are the eigenvector for the eigenvalues ±1 of this  action. Therefore we can regard the chain complexes (sl2 (C) ⊗ C[f±1 ]) ⊗Z[π1 (EK )] C∗ (E K ; Z) as the twisted chain complex of EK with sl2 (C) ⊗ C the coefficient defined by the representations (Ad ⊗ ρ ) ⊗ (±1)[ · ] . This  gives the identification between (sl2 (C) ⊗ C[f±1 ]) ⊗Z[π1 (EK )] C∗ (E K ; Z) and the twisted chain complexes of  [·]  EK defined by Ad ◦ ρ and (−1) (Ad ◦ ρ ). 2 Proposition 3.9. We have the following isomorphisms:  H∗ (E K ; sl2 (C))  H∗ (EK ; sl2 (C)) ⊕ H∗ (EK ; sl2 (C) ⊗ C−1 )  H∗ (EK ; C) ⊕ H∗ (EK ; sl2 (C))⊕2

(9) (10)

where C−1 denotes the 1-dimensional representation (−1)[ · ] of π1 (EK ). Moreover H∗ (EK ; sl2 (C))  H∗ (EK ; V2 ). Here V2 is the 2-dimensional vector space for the summand ψ2 in Ad ◦ ρ . Proof. The first isomorphism follows from Lemma 3.8. The second summand in the right hand side of (9) √ is determined by the representation (−1)[ · ] (Ad ◦ ρ ) = 1 ⊕ (( −1 )[ · ] · (−1)[ · ] ρ). The SL2 (C)-representation √ (−1)[ · ] ρ is conjugate to itself ρ by [14, Proposition 3]. Thus (−1)[ · ] (Ad ◦ ρ ) is conjugate to 1 ⊕ ( −1 )[ · ] ρ = 1 ⊕ ψ2 . This conjugation between representations induces the isomorphism from twisted homology group H∗ (EK ; sl2 (C) ⊗ C−1 ) to H∗ (EK ; C) ⊕ H∗ (EK ; V2 ). √ We need to show H∗ (EK ; sl2 (C))  H∗ (EK ; V2 ) for (10). The equality Ad ◦ ρ = (−1)[ · ] ⊕ ( −1 )[ · ] ρ gives the isomorphism: H∗ (EK ; sl2 (C))  H∗ (EK ; V1 ) ⊕ H∗ (EK ; V2 )

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and the coefficient V1 is C−1 . By direct calculation, we can see that H∗ (EK ; C−1 ) = 0 which proves the proposition. 2  Proposition 3.9 shows that a basis of H∗ (E K ; sl2 (C)) can be chosen from a basis of H∗ (EK ; sl2 (C)). We focus on cycles given by the preferred longitude in C1 (EK ; sl2 (C)). This means that we assume that an irreducible metabelian SL2 (C)-representation ρ is λ-regular where λ is the preferred longitude on ∂EK (see Definition 2.4). √ To describe the basis of H∗ (EK ; sl2 (C)), we will use eigenvectors of ρ(μ) and Ad ◦ ρ (μ) = −1 ⊕ −1ρ(μ). √ The vectors √12 (E − F ) and √12 (E + F ) in V2 are eigenvectors of ρ(μ) for the eigenvalues ∓ −1.  Proposition 3.10. If ρ is λ-regular, then H∗ (E K ; sl2 (C)) is expressed as

 H∗ (E K ; sl2 (C)) 

where P ρ and Qρ are the eigenvectors

⎧ ρ 2 ρ 2 ⎪ ⎪ ⎨C P ⊗ T , Q ⊗ T

, P ⊗ λ, Q ⊗ λ C H ⊗ μ ⎪ ⎪ ⎩ H ⊗ pt C

√1 (E 2

ρ

− F ) and

√1 (E 2

ρ

(∗ = 2) (∗ = 1)

(11)

(∗ = 0)

√ + F ) for the eigenvalues ∓ −1.

Proof. We will prove the cycles as in Eq. (11) give bases in the right hand side of Eq. (10). We consider  the images under the isomorphism Φ from C∗ (E K ; sl2 (C)) to the direct sum of C∗ (EK ; sl2 (C) ⊗ C[f±1 ]). This isomorphism is given by Φ(x ⊗ c) = (x ⊗ 1) ⊗ c = x ⊗ (f1 + f−1 ) ⊗ c. The image P ρ ⊗ T2 under Φ is expressed as Φ(P ρ ⊗ T2 ) = P ρ ⊗ f1 ⊗ (1 + μ)T 2 + P ρ ⊗ f−1 ⊗ (1 + μ)T 2 = 2P ρ ⊗ f1 ⊗ T 2 . We also have the equality that Φ(Qρ ⊗ T2 ) = 2Qρ ⊗ f−1 ⊗ T 2 . It follows from the λ-regularity of ρ that these images of P ρ ⊗ T2 and Qρ ⊗ T2 give generators of the direct sum of H2 (EK ; sl2 (C)). Similarly we can show the images of P ρ ⊗ λ and Qρ ⊗ λ turn into 2P ρ ⊗ λ and 2Qρ ⊗ λ which give non-trivial homology classes. The image Φ(H ⊗ μ ) turns out to be 2H ⊗ f−1 ⊗ μ since H is also an eigenvector for the eigenvalue −1 of Ad ◦ ρ (μ). The cycle H ⊗ f−1 ⊗ μ gives a generator of H1 (EK ; C) in Eq. (10). Last it follows from  the isomorphism in Proposition 3.9 that H ⊗ pt gives a generator of H0 (EK ; C)  H0 (E K ; sl2 (C)). 2 3.3. Torsions of cyclic covers over knot exteriors  The purpose of this subsection is to compute the torsion Tor(E K ; sl2 (C)) using the twisted Alexander polynomial of EK as follows. Proposition 3.11. If ρ is λ-regular, then  Tor(E K ; sl2 (C)) = ± lim

t→1

ΔE  (t) K ,α⊗Ad◦ρ t2 − 1

 where we choose the basis of H∗ (E K ; sl2 (C)) as in (11). The rest of this subsection is devoted to proving Proposition 3.11. To prove the above equality, we apply the method developed in [21,4], which is based on the following idea.

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 The Reidemeister torsion for the non-acyclic chain complex C∗ (E K ; sl2 (C)) coincides with the torsion    of the quotient C∗ (EK ; sl2 (C))/C∗ where C∗ is a subchain complex which is isomorphic to the twisted  homology group H∗ (E K ; sl2 (C)). When each chain complex in this decomposition:     C∗ (E K ; sl2 (C))  C∗ ⊕ C∗ (EK ; sl2 (C))/C∗

also defines chain complexes in the coefficient C(t) ⊗ sl2 (C), we can express the torsion of the quotient   complex as a quotient of the twisted Alexander polynomial for E K by the polynomial corresponding to C∗   with the coefficient C(t) ⊗ sl2 (C). Then the torsion of C∗ (EK ; sl2 (C))/C∗ , which coincides with the torsion  of C∗ (E K ; sl2 (C)), is recovered by evaluating this rational function at t = 1.  To do this, we need to choose a suitable subchain complex C∗ in C∗ (E K ; sl2 (C)). The subchain complex  C∗ is generated by the basis in Proposition 3.10. Definition 3.12. Suppose that an irreducible metabelian representation ρ is λ-regular. We define the sub chain complex C∗ of C∗ (E K ; sl2 (C)) as 0 → C2 → C1 → C0 → 0 C2 = C P ρ ⊗ T2 , Qρ ⊗ T2 ,

C1 = C H ⊗ μ , P ρ ⊗ λ, Qρ ⊗ λ ,

C0 = C H ⊗ pt .

Note that the restriction of boundary operators are 0-homomorphism and the homology group H∗(C∗ )    coincides with H∗ (E K ; sl2 (C)). We denote by C∗ the chain complex defined as the quotient of C∗ (EK ; sl2 (C))   by C∗ . It follows that C∗ is acyclic from the induced homology long exact sequence from the short exact sequence:   0 → C∗ → C∗ (E K ; sl2 (C)) → C∗ → 0.

(12)

We can also define the following subchain complex with the variable t in the twisted chain complex  for E K. Definition 3.13. Under the assumption that ρ is λ-regular, we denote by C∗ (t) the subchain complex of  C∗ (E K ; C(t) ⊗ sl2 (C)) defined by C2 (t) = C(t) 1 ⊗ P ρ ⊗ T2 , 1 ⊗ Qρ ⊗ T2 , C1 (t) = C(t) 1 ⊗ H ⊗ μ , 1 ⊗ P ρ ⊗ λ, 1 ⊗ Qρ ⊗ λ , C0 (t) = C(t) 1 ⊗ H ⊗ pt and the boundary operators are given by ∂

∂

2 2 0 → C2 (t) −−→ C1 (t) −−→ C0 (t) → 0 ⎛ ⎞ 0 0   0 ⎠ , ∂1 = t2 − 1 0 0 . ∂2 = ⎝ t2 − 1 0 t2 − 1

  We also denote by C∗ (t) the quotient C∗ (E K ; C(t) ⊗ sl2 (C)) by C∗ (t).

 In the definition of C∗ (t), we need a closed loop on ∂ E K whose homology class is trivial, i.e., which ∗ is included in ker p (α). This is a reason to choose cycles given by the longitude, corresponding to the  generators of H1 (E K ; sl2 (C)) as in the isomorphism (11).

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  We can recover the torsion of C∗ (E K ; sl2 (C)) by substituting t = 1 into the torsion of C∗ (t), as discussed   below. Moreover the torsion of C∗ (t) is determined by that of C∗ (EK ; C(t) ⊗ sl2 (C)). According to the definition of the twisted Alexander polynomial by Wada [20] and the interpretation as the Reidemeister  torsion by Kirk and Livingston [9], we regard the Reidemeister torsion of the chain complex C∗ (E K ; C(t) ⊗   sl2 (C)) as the twisted Alexander polynomial for EK and the pull-backs of Ad ◦ ρ and α. This viewpoint  gives a computation method of Tor(E K ; sl2 (C)) by an evaluation of the twisted Alexander polynomial. The proof of Proposition 3.11 will be divide into two parts. First we need to prove the torsion of C∗ (t) is 2 given by ΔE  (t)/(t − 1). In the second step, evaluating this rational function at t = 1 we see the K ,α⊗Ad◦ρ  torsion Tor(C∗ (t)) turns out to be Tor(C∗ ) = ± Tor(E K ; sl2 (C)). The first step is divided into Lemmas 3.14

and 3.15. Lemma 3.14. The subchain complex C∗ (t) is an acyclic chain complex. The torsion of this chain complex is equal to ±(t2 − 1). Proof. This lemma follows from the construction. 2 ∗   Lemma 3.15. If ρ is λ-regular, then the twisted chain complex C∗ (E K ; C(t) ⊗ sl2 (C)) defined by p (Ad ◦ ρ )  is acyclic. Moreover the torsion of C∗ (E K ; C(t) ⊗ sl2 (C)) is given by the twisted Alexander polynomial ΔE  (t). K ,α⊗Ad◦ρ

 Proof. By [5, Lemma 4.3], we have the following decomposition of C∗ (E K ; C(t) ⊗ sl2 (C)):   C∗ (E K ; C(t) ⊗ sl2 (C))  (C(t) ⊗ sl2 (C) ⊗ C[G]) ⊗Z[π1 (EK )] C∗ (EK ; Z)  = (C(t) ⊗ sl2 (C) ⊗ C[f1 ]) ⊗Z[π1 (EK )] C∗ (E K ; Z)  ⊕ (C(t) ⊗ sl2 (C) ⊗ C[f−1 ]) ⊗Z[π1 (EK )] C∗ (E K ; Z)

(13)

where we take tensor product under the representations α, Ad ◦ ρ and the projection π1 (EK ) → G. The right hand side of Eq. (13) is isomorphic to the direct sum: C∗ (EK ; C(t) ⊗ sl2 (C)) ⊕ C∗ (EK ; C(t) ⊗ (sl2 (C) ⊗ C−1 )). Since we assume that ρ is λ-regular, the acyclicity of C∗ (EK ; C(t) ⊗ sl2 (C)) follows from [21, Proposition 3.1.1]. The second summand C∗ (EK ; C(t) ⊗ (sl2 (C) ⊗ C−1 )) is defined by α ⊗ (−1)[ · ] (Ad ◦ ρ ). We have √ seen that the representation (−1)[ · ] (Ad ◦ ρ ) is conjugate to 1 ⊕ ( −1 )[ · ] ρ = 1 ⊕ ψ2 in the proof of Proposition 3.9. This conjugation of representation induces the following isomorphism between the homology groups: H∗ (EK ; C(t) ⊗ (sl2 (C) ⊗ C−1 ))  H∗ (EK ; C(t)) ⊕ H∗ (EK ; C(t) ⊗ V2 ). It is known that H∗ (EK ; C(t)) is trivial and Proposition 3.9 shows that H∗ (EK ; C(t) ⊗ V2 ) is isomorphic to H∗ (EK ; C(t) ⊗ sl2 (C)), which is trivial.  It remains to prove that the torsion of C∗ (E K ; C(t) ⊗ sl2 (C)) coincides with the twisted Alexander polynomial. This follows from the result of [9, Section 4]. 2  We need to show the evaluation of Tor(C∗ (t)) gives the torsion Tor(E K ; sl2 (C)). This is an application of [21, Proposition 3.3.1] to our situation. Proof of Proposition 3.11. Applying [21, Proposition 3.3.1] to our situation, we have

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 Tor(E K ; C(t) ⊗ sl2 (C)) = ± Tor(C∗ ). t→1 Tor(C∗ (t)) lim

(14)

On the left hand side of (14), it follows from the Multiplicativity property of the Reidemeister torsion to   0 → C  (t) → C∗ (E K ; C(t) ⊗ sl2 (C)) → C∗ (t) → 0

and Lemmas 3.14 and 3.15 that  ΔE  (t) Tor(E K ; C(t) ⊗ sl2 (C)) K ,α⊗Ad◦ρ = .  2 Tor(C∗ (t)) t −1  On the right hand side of (14), we can see Tor(C∗ ) = ± Tor(E K ; sl2 (C)). This is follows from the equalities     Tor(C∗ ) Tor(C∗ ) = ± Tor(E K ; sl2 (C)) Tor(H (C∗ , C∗ (EK ; sl2 (C)), C∗ ))

by the Multiplicativity property of the Reidemeister torsion to the short exact sequence (12) and Tor(C∗ ) = ±1,

  Tor(H (C∗ , C∗ (E K ; sl2 (C)), C∗ )) = ±1

 by the construction of C∗ and the choice of a basis of H∗ (E K ; sl2 (C)).

2

  Remark 3.16. The acyclic chain complexes C∗ (E K ; C(t) ⊗ sl2 (C)) and C∗ (t) correspond to the non-acyclic    complexes C∗ (E K ; sl2 (C)) and C∗ in evaluating at t = 1. However the acyclic chain complex C∗ (t) corre2 sponds to the acyclic one. It makes sense to evaluate ΔE  (t)/(t − 1) in terms of torsions of acyclic K ,α⊗Ad◦ρ chain complexes.

3.4. Torsion of the Mayer–Vietoris homology exact sequence We will express the torsion of the Mayer–Vietoris exact sequence (H ) as the special value of a rational function on the SL2 (C)-character variety. In Proposition 3.18, it is expressed as I2 − 4 Tor(H ) = λ2 Iμ − 4



dIμ dIλ

2    

. χ=χρ

This is due to the identification between the twisted homology group H1 (EK ; sl2 (C)) and the cotangent space of the SL2 (C)-character variety. The ratio of two cotangent vectors, namely two 1-forms, is expressed as the rational function on the character variety. We will show that the special value of the rational function at an irreducible metabelian character gives the torsion of the exact sequence (H ). To observe the torsion of Mayer–Vietoris exact sequence (H ), we set the bases of each homology groups in the sequence. We assume that H∗ (Σ2 ; Cξ ) = 0. The coefficient vector space sl2 (C) has the standard basis {E, H, F }. However we set a basis of sl2 (C) as {H, P ρ , Qρ } where P ρ and Qρ are defined as √12 (E ∓ F ). Lemma 3.17. Suppose that sl2 (C) is given by the basis {H, P ρ , Qρ } and ρ is λ-regular. If we choose the  bases of H∗ (T2 ; sl2 (C)), H∗ (D2 ×S 1 ; sl2 (C)) as in Lemma 3.3 and the basis of H∗ (E K ; sl2 (C)) as in Eq. (11), then the torsion Tor(H ) is given by the determinant of the base change matrix (a 2 × 2-matrix) T = ({i∗ (P ρ ⊗ μ ), i∗ (Qρ ⊗ μ )}/{P ρ ⊗ λ, Qρ ⊗ λ}) 2   in H1 (E K ; sl2 (C)), where i∗ denotes the induced homomorphism from H1 (T ; sl2 (C)) to H1 (EK ; sl2 (C)) by  2  the inclusion T → EK .

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Proof. It follows from the definition of torsion for an acyclic complex that the torsion Tor(H ) is equal to the determinant of the isomorphism: 2 1  H1 (T2 ; sl2 (C)) → H1 (E K ; sl2 (C)) ⊕ H1 (D × S ; sl2 (C)),

⎛

⎞ 1 00 ⎜0 ⎟ ⎟ whose representation matrix is ⎜ ⎝ 0 T O ⎠. Here 13 denotes the 3 × 3 identity matrix. 2 13 13 We need to observe where vectors P ρ ⊗ λ and Qρ ⊗ λ live in the decomposition of the twisted homology  group H1 (E K ; sl2 (C)) as in the proof of Proposition 3.9:  H1 (E K ; sl2 (C))  H1 (EK ; sl2 (C)) ⊕ H1 (EK ; sl2 (C) ⊗ C−1 )  H1 (EK ; sl2 (C)) ⊕ H1 (EK ; C) ⊕ H∗ (EK ; V2 ⊗ C−1 ). As we have seen in the proof of Proposition 3.10, the vector P ρ ⊗ λ is contained in the first summand H1 (EK ; sl2 (C)) and Qρ ⊗λ is contained in H∗ (EK ; V2 ⊗C−1 )  H1 (EK ; sl2 (C)). To compute the determinant of the base change matrix T , it is enough to consider the ratio between two vector P ρ ⊗ μ  (resp. Qρ ⊗ μ ) and ρ ρ P ⊗ λ (resp. Q ⊗ λ) in H∗ (EK ; sl2 (C)). By Lemma 3.17, we can give the torsion Tor(H ) by the special value of the rational function on the character variety. Proposition 3.18. Under the assumption of Lemma 3.17, we can express the torsion Tor(H ) for the Mayer– Vietoris sequence (H ) as I2 − 4 Tor(H ) = λ2 Iμ − 4



dIμ dIλ

2    

. χ=χρ

Proof. We have the isomorphism induced by the conjugation between (−1)[ · ] ρ and ρ: isom.

H1 (EK ; V2 ⊗ C−1 ) −−−−→ H1 (EK ; V2 )( H1 (EK ; sl2 (C))) where we can identify H1 (EK ; V2 ) with H1 (EK ; sl2 (C)) by Proposition 3.9. Under this isomorphism, the vector Qρ ⊗ μ (resp. Qρ ⊗λ) in H1 (EK ; V2 ⊗C−1 ) is sent to the vector P ρ ⊗ μ (resp. P ρ ⊗λ) in H1 (EK ; sl2 (C)). Therefore the determinant of the base change matrix T in Lemma 3.17 turns into the square of determinant of base change matrix, given by [P ρ ⊗ μ /P ρ ⊗ λ]2 . It follows from [15, Proposition 4.7] that I2 − 4 [P ⊗ μ /P ⊗ λ] = λ2 Iμ − 4 ρ

ρ

2



dIμ dIλ

2    

2

(15)

χ=χρ .

3.5. The statement of an extension of Fox’s formula for double branched covers From the results in the previous two Subsections 3.3 & 3.4, we can rewrite the right hand side of the equality of the Reidemeister torsions in Proposition 3.1. The right hand side becomes a quantity that is determined by the knot exterior EK and π1 (EK ). Theorem 3.19. We suppose that ρ is λ-regular and H∗ (Σ2 ; Cξ ) = 0. If we choose the bases of H∗ (T2 ; sl2 (C)),  H∗ (D2 × S 1 ; sl2 (C)) as in Lemma 3.3 and the basis of H∗ (E K ; sl2 (C)) as in the isomorphism (11), then we have the following equality:

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Y. Yamaguchi / Topology and its Applications 204 (2016) 278–305

 √ 

2  2

2  dI I Δ ( −1 t) − 4  E ,α⊗ρ μ  K 2 λ | Tor(Σ2 ; ξ)| =  lim · 2  2−1 t→1  t I − 4 dI λ  μ  χ=χ

ρ

   .   

(16)

Proof of Theorem 3.19. We start with the equality of Proposition 3.1: | Tor(Σ2 ; Cξ )|2 =

1  · | Tor(E K ; sl2 (C)) · Tor(H )|. |H1 (Σ2 ; Z)|

By Propositions 3.11 & 3.18, we can rewrite the above equality as follows:    2 

ΔE   (t) Iλ2 − 4 dIμ  1 K ,α⊗Ad◦ρ 2  · lim · 2 | Tor(Σ2 ; Cξ )| =  |H1 (Σ2 ; Z)|  t→1 t2 − 1 Iμ − 4 dIλ  χ=χ

ρ

   .   

(17)

It follows from [5, Theorem 4.1] that ΔE  (t) = ΔEK ,α⊗Ad◦ρ (t)ΔEK ,α⊗Ad◦ρ (−t). K ,α⊗Ad◦ρ By [22, Theorem 4.2 and Theorem 4.4], we can express ΔEK ,α⊗Ad◦ρ (t) as √ ΔEK ,α⊗ρ ( −1 t) . ΔEK ,α⊗Ad◦ρ (t) = ±ΔK (−t) t+1 The right hand side o (17) turns into     I 2 − 4 dI 2   1  μ  λ ·  lim ΔK (t)ΔK (−t)P (t)P (−t) · 2  |H1 (Σ2 ; Z)|  t→1 Iμ − 4 dIλ 

χ=χρ

      

√ where P (t) = ΔEK ,α⊗ρ ( −1 t)/(t2 − 1). Fox’s formula shows that ΔK (1)ΔK (−1) = ±|H1 (Σ2 ; Z)|. By [6] and [22, Proposition 4.1], P (t) is an even function on the variable t. Hence we obtain the desired equality:   √  

2  2

2   dI I Δ ( −1 t) − 4  EK ,α⊗ρ μ  λ . 2 | Tor(Σ2 ; Cξ )|2 =  lim ·  2  2 t −1 Iμ − 4 dIλ    t→1 χ=χρ

Remark 3.20. The assumption that C∗ (Σ2 ; Cξ ) is acyclic is equivalent to the condition for an irreducible metabelian representation ρ to be μ -regular from Propositions 3.5, 3.7 & 3.9. Actually, this is equivalent to the condition that ρ is μ-regular since the base change formula (15) between μ  and μ is equal to 4. Remark 3.21. The assumption of Lemma 3.17 requires that ρ is λ-regular. It is a sufficient condition for each factor in the right hand side of Eq. (16) to be well-defined as a complex number. Remark 3.22. The left hand side of Eq. (16) is well-defined as a topological invariant of Σ2 when H∗ (Σ2 ; Cξ ) = 0. Theorem 3.19 shows that the product in the right hand side of (16) is independent of the choice of basis of H∗ (EK ; sl2 (C)), even though each factor depends on such a choice. Eq. (16) gives us a method to compute a topological invariant of Σ2 from π1 (EK ). 4. Application to two-bridge knots We discuss the twisted Alexander polynomial for irreducible metabelian representations and the rational  2 I 2 −4 dI function Iλ2 −4 dIμλ on the character varieties for two-bridge knots. Subsection 4.1 deals with the nonμ  hyperbolic two-bridge knots and we will discuss the case of hyperbolic two-bridge knots in Subsection 4.2.

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4.1. Non-hyperbolic two-bridge knots From [7] it is known that we have no satellite knots in two-bridge knots. Hence every non-hyperbolic two-bridge knot is a torus knot of type (2, q) where q is an odd integer. We give the explicit forms for the  2 I 2 −4 dI Reidemeister torsion Tor(EK ; sl2 (C)) and the rational function Iλ2 −4 dIμλ on the character variety for μ 

(2, q)-torus knot. For simplicity, we assume that q is a positive odd integer. We start with the description of character varieties of torus knots, according to the lecture note of D. Johnson [8]. Here we adopt the following presentation of the knot group for (p, q)-torus knot K (see Fig. 2): π1 (EK ) = x, y | xp = y q .

Fig. 2. The diagram of (2, q)-torus knot K (q > 0) and generators x and y of π1 (EK ).

For any (2, q)-torus knot K, the irreducible components X irr (EK ) consists of (q − 1)/2 components X1,b = {χ ∈ X irr (EK ) | χ(y) = 2 cos πb q } with odd integer b satisfying 0 < b < q. Each components has the local coordinate Iμ . From now on, the symbol K denotes (2, q)-torus knot in this subsection. Every character of irreducible metabelian representation is contained in the subset defined by Iμ = 0 from the correspondence (6). Remark 4.1. It follows from (|ΔK (−1)| −1)/2 = (q −1)/2 that each component X1,b has only one irreducible metabelian representation as the origin under the local coordinate Iμ . √ We can compute directly the rational function ΔEK ,α⊗ρ ( −1t)/(t2 −1) in Theorem 3.19 and its evaluation at t = 1. Lemma 4.2. We suppose that the character of an irreducible metabelian representation ρ is contained in X1,b . √ The square of limt→1 ΔEK ,α⊗ρ ( −1t)/(t2 − 1) in Theorem 3.19 is expressed as

√ 2 2

ΔEK ,α⊗ρ ( −1 t) q = t→1 t2 − 1 4 sin2 (jπ/q) lim

where j = (q − b)/2. Proof. It follows from [10, Theorem 4.2] that ΔEK ,α⊗ρ (t) = (t2 + 1)



(t2 + ζq )(t2 + ζq− )

1≤≤(q−1)/2 =j

for an irreducible metabelian representation ρ whose character is contained in the component X1,b . It holds that

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298

√ ΔEK ,α⊗ρ ( −1 t) =− lim t→1 t2 − 1

From the relation that

(q−1)/2 (t − ζq )(t − ζq− ) =1



(1 − ζq )(1 − ζq− ).

1≤≤(q−1)/2 =j

= 1 + t + · · · + tq−1 , we can rewrite the above equality as

√ −q −q ΔEK ,α⊗ρ ( −1 t) . = = 2 j −j t→1 t2 − 1 4 sin (jπ/q) (1 − ζq )(1 − ζq )

2

lim

We proceed to compute the rational function irr

2 Iλ −4 2 −4 Iμ 



dIμ dIλ

2 . We will see that this function is constant on

the whole X (EK ). Lemma 4.3. For (2, q)-torus knot K, we have Iλ2 − 4 Iμ2 − 4



dIμ dIλ

2 =

1 q2

on all components of X irr (EK ). Proof. It is known that the preferred longitude λ is represented by μ2q x−2 in the torus knot group π1 (EK ). ±1 be the We can deduce the relation between the regular functions Iλ and Iμ on the character variety. Let M eigenvalues of the matrix corresponding to μ  under an irreducible representation. It follows from [8] or [10, Proposition 3.7] that χ(x) = 2 cos π/2. The element x2 is sent to −1 by every irreducible  representation.  ! 1 q −q ±1 2    Hence the function Iλ is given by −(M + M ). Since M is given by Iμ ± I − 4 , we can express 2

μ 

the function Iλ as 1 Iλ = − q 2



! Iμ +

Iμ2

! q  q 2 . − 4 + Iμ − Iμ − 4

The derivative of Iλ by Iμ is given by −q 1 dIλ =! · q dIμ Iμ2 − 4 2 =!

−q Iμ2

−4

 Iμ +

! ! q  q Iμ2 − 4 − Iμ − Iμ2 − 4

−q ). q − M (M

Squaring both sides, we have the equality:

dIλ dIμ

2 = =

q2 q + M −q )2 − 4) · ((M −4

Iμ2

q2 · (Iλ2 − 4), Iμ2 − 4

which gives the desired relation. 2 We recall the Reidemeister torsion for Σ2 . It is known that the double branched cover along (2, q)-torus knot is the lens space of type (q, 1). We denote by γ a generator of π1 (Σ2 ) = π1 (L(q, 1)), i.e., π1 (Σ2 ) = γ | γ q = 1 .

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299

Proposition 4.4. √Let K be (2, q)-torus knot and ξ a non-trivial homomorphism from π√ 1 (Σ2 ) to GL1 (C) sending γ to e2π −1/q . We denote by ξ j the homomorphism of π1 (Σ2 ) sending γ to e2π −1j/q . Then the Reidemeister torsion for Σ2 and ξ j ⊕ ξ −j is expressed as Tor(Σ2 ; Cξj ⊕ Cξ−j ) = 

1 2

2 .

4 sin (jπ/q)

Proof. For the direct sum representation, the torsion Tor(Σ2 ; Cξj ⊕ Cξ−j ) turns into the product: Tor(Σ2 ; Cξj ) · Tor(Σ2 ; Cξ−j ). We can see that ξ −j is the same as ξ j since the image by ξ is contained in the unit circle of C. Hence the Reidemeister torsion for ξ −j is the complex conjugate of Tor(Σ2 ; Cξj ). It is known that the Reidemeister torsion Tor(Σ2 ; Cξj ) is given by (ζqj ) (1 − ζqj )2 where ζq = e2π

√ −1/q

and ∈ Z. Therefore Tor(Σ2 ; Cξj ⊕ Cξ−j ) turns out to be 1 (1 −

ζqj )2 (1

− ζq−j )2

which completes the proof. 2 Remark 4.5. For precise presentations of π1 (Σ2 ) and ξ, we refer to the correspondences (20) and (21). The conjugacy class [ρ] of metabelian representations in X1,b corresponds to [ρk ] for k = (q − b)/2 in the correspondence (20). We can check our formula in Theorem 3.19. We choose an irreducible metabelian representation ρ whose character is contained in the component X1,b , where 1 < b < q and b is odd. The right hand side of (16) turns into the product √ 2 2 2

2

Iλ − 4 dIμ q ΔEK ,α⊗ρ ( −1 t) 1 lim · 2 = t→1 t2 − 1 Iμ − 4 dIλ q2 4 sin2 (jπ/q) 2

1 = 4 sin2 (jπ/q)



where j = (q − b)/2. This value agrees with the Reidemeister torsion Tor(Σ2 ; Cξj ⊕ Cξ−j ) in Proposition 4.4. 4.2. Hyperbolic two-bridge knots We will show that every irreducible metabelian representation is λ-regular for hyperbolic tow–bridge knot  groups. This guarantees that the torsion Tor(E K ; sl2 (C)) is non-zero and Tor(H ) does not diverge. We will derive the λ-regularity from the Property L of hyperbolic two-bridge knots. We recall Property L for knots, following [1] by M. Boileau, S. Boyer, A. Reid and S. Wang. Definition 4.6. Let K be a knot in S 3 . We denote by K(0) the manifold obtained from by a longitudinal surgery on K. We say that K has Property L if the SL2 (C)-character variety of the manifold K(0) contains only finitely many characters of irreducible representations.

300

Y. Yamaguchi / Topology and its Applications 204 (2016) 278–305

It follows from [1, Proposition 1.10] that any hyperbolic two-bridge knot K has Property L. This property gives us the λ-regularity of an irreducible metabelian representation. Proposition 4.7. Suppose that ρ is an irreducible metabelian SL2 (C)-representation of π1 (EK ). Then ρ is λ-regular if and only if dimC X(K(0)) = 0 at the character of ρ and its character is a smooth point of X(K(0)). Note that every metabelian SL2 (C)-representation sends the longitude λ to 1. This means that all metabelian representations induce a homomorphism from the fundamental group π1(K(0)) = π1 (EK )/ λ

into SL2 (C). We use the symbol ρ¯ for the induced representation of π1 (K(0)) by ρ. Corollary 4.8. If K is a hyperbolic two-bridge knot, then all irreducible metabelian SL2 (C)-representations of π1 (EK ) are λ-regular. Proof. Since K has Property L, the character variety X(K(0)) consists of only points. It follows from Hilbert’s Nullstellensatz that each component of the character variety X(K(0)) is defined by the maximal ideal (X1 − a1 , X2 − a2 , . . . , ). By direct calculation, we can check that every conjugacy class of irreducible metabelian representations satisfies the conditions to be λ-regular in Proposition 4.7. 2 Proof of Proposition 4.7. We assume that an SL2 (C)-representation ρ is irreducible and metabelian. First the induced homomorphism of π1 (EK )/ λ

is also irreducible since it has the same image as ρ. From the construction of K(0) = EK ∪ D2 × S 1 , in which the meridian circle ∂D2 × {∗} is glued along the longitude λ, we have the Mayer–Vietoris exact sequence of homology group with the coefficient sl2 (C): · · · → Hi+1 (K(0)) → Hi (∂EK ) → Hi (EK ) ⊕ Hi (D2 × S 1 ) → Hi (K(0)) → · · · Since both representations of π1 (EK ) and π1 (K(0)) are irreducible, in particular they are non-abelian. Hence it follows that H0 (EK ; sl2 (C)) = 0,

H0 (K(0); sl2 (C))  H3 (K(0); sl2 (C)) = 0.

Since Ad◦ρ(μ) has the eigenvalues ±1 and the multiplicity of −1 is 2, we choose P ρ ∈ sl2 (C) as an eigenvector of the eigenvalue 1. We can express the twisted homology groups H∗(∂EK ; sl2 (C)) and H∗ (D2 × S 1 ; sl2 (C)) as ⎧ ρ 2 ⎪ (∗ = 2) ⎪ ⎨C P ⊗ T H∗ (∂EK ; sl2 (C))  C P ρ ⊗ μ, P ρ ⊗ λ (∗ = 1) ⎪ ⎪ ⎩ P ρ ⊗ pt (∗ = 0) 

H∗ (D2 × S 1 ; sl2 (C)) 

C

C P

ρ

⊗ μ

(∗ = 1)

C P

ρ

⊗ pt

(∗ = 0)

Now we start with if part. The assumption on X(K(0)) implies 0 = dimC X(K(0)) = dimC TχZar X(K(0)) = dimC H1 (K(0); sl2 (C)) ρ ¯ where TχZar X(K(0)) denotes the Zariski tangent space at the character χρ¯. The second equality follows ρ ¯ from that the character χρ¯ is a smooth point and the third equality from that ρ¯ is irreducible. We have also H2 (K(0); sl2 (C)) = 0 by Poincaré duality and the universal coefficient theorem. Substituting these results in the Mayer–Vietoris exact sequence, we have

Y. Yamaguchi / Topology and its Applications 204 (2016) 278–305

H2 (EK ; sl2 (C)) = C P ρ ⊗ T 2 ,

301

H1 (EK ; sl2 (C)) = C P ρ ⊗ λ ,

(18)

which means ρ is λ-regular. If ρ is λ-regular, then we can express the twisted homology group H∗ (EK ; sl2 (C)) as Eq. (18). We also obtain that H∗ (K(0); sl2 (C)) = 0 from the Mayer–Vietoris exact sequence. The following inequality of dimensions implies that dimC X(K(0)) = 0 and the character χρ¯ is a smooth point: 0 ≤ dimC X(K(0)) ≤ dimC TχZar X(K(0)) = dimC H1 (K(0); sl2 (C)). 2 ρ ¯ For every two-bridge knot, each factor in the right hand side of Eq. (16) is defined as a non-zero number since H∗ (Σ2 ; Cξ ) = 0. It is known that the isotopy types of two-bridge knots correspond to the topological types of double branched covers, i.e., lens spaces. The set of Reidemeister torsions for lens spaces and ξ k ⊕ξ −k distinguishes topological types of lens spaces. To be more precise, the Reidemeister torsion distinguishes the PL-isomorphism types of lens space. However there is no difference between the PL-category and the topological category for three-dimensional manifolds. Therefore we can say that the Reidemeister torsion is a topological invariant. Together with Theorem 3.19, we can conclude the following theorem regarding the set of rights hand side in Eq. (16) as an invariant of two-bridge knots. Theorem 4.9. Let K be a two-bridge knot. For every irreducible metabelian SL2 (C)-representation of π1 (EK ), we have √ • ΔEK ,α⊗ρ ( −1t)/(t2 −1) turns into a Laurent polynomial and gives a non-zero complex number at t = 1;  2 I 2 −4 dI • the rational function Iλ2 −4 dIμλ gives a non-zero complex number at the character of ρ. μ 

Moreover by the set of the following products as in Eq. (16):  √ 

2 2

2   Iλ − 4 dIμ   lim ΔEK ,α⊗ρ ( −1 t)   t→1 t2 − 1 Iμ2 − 4 dIλ  

χ=χρ

      

for (|ΔK (−1)| − 1)/2 characters of irreducible metabelian representations, we can distinguish the isotopy classes of two-bridge knots up to mirror images. Proof. Without loss of generality, we can assume that two-bridge knots K and K  have the same knot determinant |ΔK (−1)| = |ΔK  (−1)| = p. This means that the corresponding double branched covers are L(p, q) and L(p, q  ). The Reidemeister torsion of L(p, q) with ξ ⊕ ξ −1 for some GL1 (C)-representation of π1 (L(p, q)) is expressed as  −1 (ζ − 1)(ζ r − 1)(ζ −1 − 1)(ζ −r − 1) where ζ is a p-th root of unity and r satisfies that qr ≡ 1 mod p. Since we suppose that the set of the Reidemeister torsions for L(p, q) coincides with that for L(p, q  ). Hence we can find the Reidemeister torsion for L(p, q  ) satisfies that 



(ζ − 1)(ζ r − 1)(ζ −1 − 1)(ζ −r − 1) = (ζ k − 1)(ζ kr − 1)(ζ −k − 1)(ζ −kr − 1) Hence we have the following equivalent relations from the same argument in [19, Proof of Theorem 10.1] and Franz Independent Lemma: 1 ≡ ±k, r ≡ ±kr

or 1 ≡ ±kr , r ≡ ±k

(mod p).

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302

This condition derives that q  ≡ ±q or q  ≡ ±q ±1 . When q  is congruent to q ±1 , two lens spaces are orientation preserving homeomorphic which means that K and K  are isotopic. In the other case, two lens spaces are orientation reversing homeomorphic and K is isomorphic to the mirror image of K  . 2 Last, we discuss how to compute the twisted Alexander polynomial and the rational function on the character variety for a hyperbolic two-bridge knot exterior. To compute them, we need explicit forms for irreducible metabelian representations from a hyperbolic two-bridge knot group into SL2 (C). It is useful for this purpose to apply Riley’s method to make non-abelian SL2 (C)-representations of two-bridge knot groups. Let K be a hyperbolic two-bridge knot. We start with the following presentation of π1(EK ) obtained from a Wirtinger presentation: π1 (EK ) = x, y | wx = yw

(19)

where x and y are meridians and w is a word in x and y as in Fig. 3.

Fig. 3. The generators x and y for a two-bridge knot K.

By the method of Riley [17] and [14, Theorem 3], we can express the set of conjugacy classes of irreducible metabelian SL2 (C)-representation of π1 (EK ) as follows. Let p denote |ΔK (−1)|. For all conjugacy classes of irreducible metabelian SL2 (C)-representations, we can choose the following representatives: {[ρk ] | k = 1, . . . , (p − 1)/2} where ρk is an irreducible SL2 (C)-representation given by the correspondence:

√ √  −1 −√−1 , ρk : x → 0 − −1 where uk = (ekπ tation ξ k :

√ −1/p

− e−kπ

y →

√ √ −1 − −1uk

√0 − −1

 (20)

√ −1/p 2

) . Moreover the representative ρk corresponds to the GL1 (C)-represen-

ξ k : π1 (Σ2 ) = γ | γ p = 1  γ → e2kπ

√ −1/p

∈ GL1 (C)

(21)

where γ is a lift of xy −1 in π1 (EK ) into π1 (Σ2 ). Remark on computation of the twisted Alexander polynomial We simply denote by ρ an irreducible metabelian representation. We follow the definition and computation of the twisted Alexander polynomial along Wada’s way. The twisted Alexander polynomial for EK and ρ is expressed as  d  det α ⊗ ρ dx wxw−1 y −1 ΔEK ,α⊗ρ (t) = det(t ρ(y) − 1) d where dx denotes the Fox differential by x. d d The Fox differential dx wxw−1 y −1 turns into w+(1 −wxw−1 ) dx w. The numerator is a Laurent polynomial whose coefficients are polynomials in C[uk ]. The denominator det(tρ(t) − 1) turns into t2 + 1.

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303

Some more details on the rational function of a character variety To express the rational function  2 2 Iλ −4 dIμ , we need to find which representative gives another metabelian ρ whose adjoint represen2 I −4 dIλ μ 

√ −1 )[ · ] ρ.

tation is conjugate to (−1)[ · ] ⊕ (

Proposition 4.10. We assume that ρ denotes ρk given by (20). Then we can choose another metabelian representation ρ in Theorem 4.9 as ρk in (20), where k is an integer in {1, . . . , (p − 1)/2} satisfying 2k  ≡ k or 2k  ≡ −k mod p. √

Proof. We compare the traces of the images of xy −1 by Ad ◦ ρ and (−1)[ · ] ⊕ ( −1 )[ · ] ρ. Under the correspondence 20, the traces of ρ(xy −1 ) and ρ (xy −1 ) are equal to uk +2 and uk +2. If an SL2 (C)-element A has the eigenvalues ζ ±1 , then the adjoint action has the eigenvalues ζ ±2 and 1. Hence the trace of Ad ◦ ρ (xy −1 ) is expressed as   tr Ad ◦ ρ (xy −1 ) = 1 + (uk + 2)2 − 2 = (uk + 2)2 − 1. √

Since the homology class of xy −1 is trivial, the trace of the corresponding matrix by (−1)[ · ] ⊕ ( expressed as

−1 )[ · ] ρ

is

√   −1 −1 tr (−1)[xy ] ⊕ ( −1)[xy ] ρ(xy −1 ) = 1 + uk + 2. Since these traces are same, we have the equality: 

(e2k π

√ −1/p



+ e−2k π

√ −1/p 2

) = (ekπ

√ −1/p

+ e−kπ

√ −1/p 2

)

We can rewrite this as cos which means that 2k = k or 2k  = p − k.

kπ 2k  π = ± cos p p

2

−wx−2σ in the presentation (19) where ← − is the word of reverse order w w It is known that λ is expressed as ← of w and σ is an integer given by the sum of exponents of x and y in w. In Riley’s construction, non-abelian representations are given by the correspondence: x →

√  s 1/√s , 0 1/ s

√



√ s 0√ √ y → −u s 1/ s

(22)

and the equality: W1,1 + (1 − s)W1,2 = 0

(23)

where Wi,j is the (i, j)-entry of the matrix corresponding to w. The left hand side of Eq. (23) is a polynomial in s + 1/s and u. Hence we can regard the trace function Iλ as a function on s + 1/s, in particular, a function on Iμ = Iμ2 . Let ρ√s,u be the SL2 (C)-representation defined by the correspondence in Eq. (22). From Remark 2.19, the trace function Iλ gives 2 at (s, u) = (−1, uk ). More precisely, the behavior of Iλ is expressed as follows.

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304

Proposition 4.11. We can express the trace function Iλ as a function defined by the equality that Iλ − 2 = −Iμ2 · H(Iμ )

H(0) = 0

and

on Iμ near (s, u) = (−1, uk ) and the value of the rational function Iλ2 − 4 Iμ2 − 4



dIμ dIλ

2    

= χ=χρ

2 Iλ −4 2 −4 Iμ 



dIμ dIλ

2

at the character of ρ as

1 . H(0)

Proof. The character of ρ corresponds to the pair (−1, uk ). The function Iλ − 2 has a zero at ((−1, uk )). Hence Iλ − 2 is written locally as a function c · Iμk · H(Iμ ) where c is a constant, k is a positive integer  2 I 2 −4 dI and H(0) = 0. Theorem 4.9 guarantees that the rational function Iλ2 −4 dIμλ is well-defined as a non-zero μ  complex number, which deduces k = 2.  2 I 2 −4 dI We can rewrite the rational function Iλ2 −4 dIμλ as μ 

Iλ2 − 4 Iμ2 − 4



dIμ dIλ

2

2

2

dIμ Iλ2 − 4 Iμ2 − 4 dIμ = 2 Iμ − 4 Iμ2 − 4 dIμ dIλ 2

I 2 − 4 dIμ = 4 λ2 Iμ − 4 dIλ

since Iμ is Iμ2 − 2. Substituting Iλ = −Iμ2 · H(Iμ ), we can deduce the proposition. 2  μ ) where H(−2)  Remark 4.12. If we choose a parameter as Iμ (= s+1/s), then we have Iλ = −(Iμ +2)H(I = 0.  2 2 Iλ −4 dIμ  The value of 2 is given by 1/H(−2). Iμ −4

dIλ

In the case that K is the figure eight knot, the Alexander polynomial ΔK (t) is t2 − 3t + 1. We have the two conjugacy classes of irreducible metabelian representations of π1 (EK ) since |ΔK (−1)| = 5. For every irreducible metabelian representation ρ, the twisted Alexander polynomial ΔEK ,α⊗ρ (t) is given by t2 + 1 (we refer to [22, Section 5.2]). From Example 1 in [15, Section 4.4], we can see that Iλ − 2 = −Iμ2 (−Iμ2 + 5). Therefore for every metabelian representation, the product in Theorem 4.9 is given by

−t2 + 1 t→1 t2 − 1 lim

2

1 1 = . −02 + 5 5

This coincides with the Reidemeister torsion of Σ2 = L(5, 3) for ξ k ⊕ ξ −k

1 4 sin(2kπ/5) sin(kπ/5)

2 =

1 5

for k = 1, 2. Acknowledgements The author wishes to express his thanks to Professor Kunio Murasugi for drawing the author’s attention to the classification of two-bridge knots by the twisted Alexander polynomial. He also would like to thank the referee for his/her careful reading and helpful comments to improve the manuscript. This research was supported by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists and Grant-in-Aid for JSPS Fellows Grant Number 10J07961.

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