Leila Ben Abdelghani

Michael Heusener

Irreducible representations of knot groups into SL(n, C)

Keywords: knot group, Alexander module, variety of representations, character variety

The aim of this article is to study the existence of certain reducible, metabelian representations of knot groups into SL(n, C) which generalise the representations studied previously by G. Burde and G. de Rham. Under specific hypotheses we prove the existence of irreducible deformations of such representations of knot groups into SL(n, C). MSC: 57M25; 57M05; 57M27

Introduction

In [START_REF] Ben Abdelghani | Deformations of metabelian representations of knot groups into SL(3, C)[END_REF], the authors studied the deformations of certain metabelian, reducible representations of knot groups into SL (3, C). In this paper we continue this study by generalizing all of the results of [START_REF] Ben Abdelghani | Deformations of metabelian representations of knot groups into SL(3, C)[END_REF] to the group SL(n, C) (see Theorem 1.1).

Let Γ be a finitely generated group. The set R n (Γ) := R(Γ, SL(n, C)) of homomorphisms of Γ in SL(n, C) is called the SL(n, C)-representation variety of Γ. It is a (not necessarily irreducible) algebraic variety. A representation ρ : Γ → SL(n, C) is called abelian (resp. metabelian) if the restriction of ρ to the first (resp. second) commutator subgroup of Γ is trivial. The representation ρ : Γ → SL(n) is called reducible if there exists a proper subspace V ⊂ C n such that ρ(Γ) preserves V . Otherwise ρ is called irreducible.

Let Γ denote the knot group of the knot K ⊂ S 3 i.e. Γ is the fundamental group of the knot complement of K in S 3 . Since the ring of complex Laurent polynomials C[t ±1 ] is a principal ideal domain, the complex Alexander module M(t) of K decomposes into a direct sum of cyclic modules. A generator of the order ideal of M(t) is called the Alexander polynomial of K . It will be denoted by ∆ K (t) ∈ C[t ±1 ], and it is unique up to multiplication by a unit c t k ∈ C[t ±1 ], c ∈ C * , k ∈ Z. For a given root α ∈ C * of ∆ K (t) we let τ α denote the (t -α)-torsion of the Alexander module. (For details see Section 2.)

The main result of this article is the following theorem which generalizes the results of [START_REF] Ben Abdelghani | Deformations of metabelian representations of knot groups into SL(3, C)[END_REF] where the case n = 3 was investigated. It also applies in the case n = 2 which was studied in [START_REF] Ben | Espace des représentations du groupe d'un noeud classique dans un groupe de Lie[END_REF] and [START_REF] Heusener | Deformations of reducible representations of 3-manifold groups into SL 2 (C)[END_REF]Theorem 1.1].

1.1 Theorem Let K be a knot in the 3-sphere S 3 . If the (t -α)-torsion τ α of the Alexander module is cyclic of the form C[t ±1 ] (t -α) n-1 , n ≥ 2, then for each λ ∈ C * such that λ n = α there exists a certain reducible metabelian representation ̺ λ of the knot group Γ into SL(n, C). Moreover, the representation ̺ λ is a smooth point of the representation variety R n (Γ), it is contained in a unique (n 2 +n-2)-dimensional component R ̺ λ of R n (Γ). Moreover, R ̺ λ contains irreducible non-metabelian representations which deform ̺ λ .

This paper is organised as follows. In Section 2 we introduce some notations and recall some facts which will be used in this article. In Section 3 we study the existence of certain reducible representations. These representations were previously studied in [START_REF] Jebali | Module d'Alexander et représentations métabéliennes[END_REF], and we treat the existence results from a more general point of view. Section 4 is devoted to the proof of Proposition 4.1, and it contains all necessary cohomological calculations. In the last section we prove that there are irreducible non-metabelian deformations of the initial reducible representation. cochain complex, the coboundary operator δ : C n (Γ; A) → C n+1 (Γ; A) is given by: δf (γ 1 , . . . , γ n+1 ) = γ 1 • f (γ 2 , . . . , γ n+1 ) + n i=1 (-1) i f (γ 1 , . . . , γ i-1 , γ i γ i+1 , . . . , γ n+1 ) + (-1) n+1 f (γ 1 , . . . , γ n ) .

The coboundaries (respectively cocycles, cohomology) of Γ with coefficients in A are denoted by B * (Γ; A) (respectively Z * (Γ; A), H * (Γ; A)). In what follows 1-cocycles and 1-coboundaries will be also called derivations and principal derivations respectively.

Let A 1 , A 2 and A 3 be Γ-modules. The cup product of two cochains u ∈ C p (Γ; A 1 ) and v ∈ C q (Γ; A 2 ) is the cochain u v ∈ C p+q (Γ; A 1 ⊗ A 2 ) defined by u v(γ 1 , . . . , γ p+q ) := u(γ 1 , . . . , γ p ) ⊗ γ 1 . . . γ p • v(γ p+1 , . . . , γ p+q ) .

(

Here A 1 ⊗A 2 is a Γ-module via the diagonal action. It is possible to combine the cup product with any Γ-invariant bilinear map A 1 ⊗ A 2 → A 3 . We are mainly interested in the product map C ⊗ C → C.

Remark

Notice that our definition of the cup product (1) differs from the convention used in [5, V.3] by the sign (-1) pq . Hence with the definition (1) the following formula holds:

δ(u v) = (-1) q δu v + u δv .
A short exact sequence

0 → A 1 i -→ A 2 p -→ A 3 → 0
of Γ-modules gives rise to a short exact sequence of cochain complexes:

0 → C * (Γ; A 1 ) i * -→ C * (Γ; A 2 ) p * -→ C * (Γ; A 3 ) → 0 .
We will make use of the corresponding long exact cohomology sequence (see [5, III. Prop. 6.1]):

0 → H 0 (Γ; A 1 ) -→ H 0 (Γ; A 2 ) -→ H 0 (Γ; A 3 ) β 0 -→ H 1 (Γ; A 1 ) -→ • • • Recall that the Bockstein homomorphism β n : H n (Γ; A 3 ) → H n+1 (Γ; A 1 ) is determined by the snake lemma: if z ∈ Z n (Γ; A 3 ) is a cocycle and if z ∈ (p * ) -1 (z) ⊂ C n (Γ; A 2 ) is any lift of z then δ 2 (z) ∈ Im(i * )
where δ 2 the coboundary operator of C * (Γ; A 2 ). It follows that any cochain z ′ ∈ C n+1 (Γ; A 3 ) such that i * (z ′ ) = δ 2 (z) is a cocycle and that its cohomology class does only depend on the cohomology class represented by z . The cocycle z ′ represents the image of the cohomology class represented by z under β n .

Remark

By abuse of notation and if no confusion can arise, we will write sometimes β n (z) for a cocycle z ∈ Z n (Γ; A 3 ) even if the map β n is only well defined on cohomology classes. This will simplify the notations.

The Alexander module Given a knot K ⊂ S 3 , we let X = S 3 \V (K) denote its complement where V (K) is a tubular neighborhood of K . Let Γ = π 1 (X) denote the fundamental group of X and h : Γ → Z, h(γ) = lk(γ, K), the canonical projection. Recall also that a knot complement X is aspherical (see [7, 3.F]). In what follows we will identify the cohomology of the knot complement and of the knot group Γ. Note that there is a short exact splitting sequence

1 → Γ ′ → Γ → t | -→ 1
where Γ ′ = [Γ, Γ] denote the commutator subgroup of Γ and where the surjection is given by γ → t h(γ) . Hence Γ is isomorphic to the semi-direct product Γ ′ ⋊ Z. Note that Γ ′ is the fundamental group of the infinite cyclic covering X ∞ of X . The abelian group Γ ′ /Γ ′′ ∼ = H 1 (X ∞ , Z) turns into a Z[t ±1 ]-module via the action of the group of covering transformations which is isomorphic to t | -. The Z[t ±1 ]-module H 1 (X ∞ , Z) is a finitely generated torsion module called the Alexander module of K . Note that there are isomorphisms of Z[t ±1 ]-modules

H * (Γ; Z[t ±1 ]) ∼ = H * (X; Z[t ±1 ]) ∼ = H * (X ∞ , Z)
where Γ acts on

Z[t ±1 ] via γ p(t) = t h(γ)
C ⊗ Γ ′ /Γ ′′ ∼ = H 1 (Γ; C[t ±1 ]) of the Alexander module. As C[t ±1 ] is a principal ideal domain, the Alexander module H 1 (Γ; C[t ±1 ]) decomposes into a direct sum of cyclic modules of the form C[t ±1 ]/(t -α) k , α ∈ C * \ {1} i.e. there exist α 1 , . . . α s ∈ C * such that H 1 (Γ; C[t ±1 ]) ∼ = τ α 1 ⊕ • • • ⊕ τ αs where τ α j = nα j i j =1 C[t ±1 ] (t -α j ) r i j denotes the (t -α j )-torsion of H 1 (Γ; C[t ±1 ]). A generator of the order ideal of H 1 (X ∞ , C) is called the Alexander polynomial ∆ K (t) ∈ C[t ±1 ] of K i.e. ∆ K (t) is the product ∆ K (t) = s j=1 nα j i j =1 (t -α j ) r j i .
Notice that the Alexander polynomial is symmetric and is well defined up to multiplication by a unit [START_REF] Burde | Knots[END_REF]), and hence the (t -1)-torsion of the Alexander module is trivial.

c t k of C[t ±1 ], c ∈ C * , k ∈ Z. Moreover, ∆ K (1) = ±1 = 0 (see
For completeness we will state the next lemma which shows that the cohomology groups H * (Γ; C[t ±1 ]/(t -α) k ) are determined by the Alexander module H 1 (Γ; C[t ±1 ]). Recall that the action of Γ on C[t ±1 ]/(t -α) k is induced by γ p(t) = t h(γ) p(t).

Lemma

Let K ⊂ S 3 be a knot and Γ its group. Let α ∈ C * and let

τ α = nα i=1 C[t ±1 ] (t -α) r i denote the (t -α)-torsion of the Alexander module H 1 (Γ; C[t ±1 ]). Then if α = 1 we have that τ 1 is trivial and H q (Γ; C[t ±1 ]/(t -1) k ) ∼ = C for q = 0, 1 0 for q ≥ 2.
Moreover, for α = 1 we have:

H q (Γ; C[t ±1 ]/(t-α) k ) ∼ = 0 for q = 0 and q ≥ 3, nα i=1 C[t ±1 ] (t -α) min(k,r i ) for q = 1, 2. In particular, H 1 (Γ; C[t ±1 ]/(t -α) k ) = 0 if and only H 1 (Γ; C[t ±1 ]) has non- trivial (t -α)-torsion i.e if ∆ K (α) = 0.
Proof. Let M be a C[t ±1 ]-module, then by the extension of scalars [START_REF] Brown | Cohomology of groups[END_REF]III.3] we have an isomorphism

H q (Γ; M) ∼ = H q (Hom C[t ±1 ] (C * (X ∞ , C), M). Since C[t ±1
] is a principal ideal domain, we can apply the universal coefficient theorem and obtain [START_REF] Burde | Knots[END_REF]Prop. 8.16]) so we can apply the above isomorphisms to the modules C[t ±1 ]/(t -α) k with α = 1 or α = 1. Notice also that for λ = α the multiplication by (t -λ) induces an isomorphism of C[t ±1 ]/(t -α) k . 2

H q (Γ; M) ∼ = Ext 1 C[t ±1 ] (H q-1 (X ∞ , C), M) ⊕ Hom C[t ±1 ] (H q (X ∞ , C), M). Now H 0 (X ∞ , C) ∼ = C ∼ = C[t ±1 ]/(t -1) and H k (X ∞ , C) = 0 for k ≥ 2 (see
Representation variety. Let Γ be a finitely generated group. The set of all homomorphisms of Γ into SL(n) has the structure of an affine algebraic set (see [START_REF] Lubotzky | Varieties of representations of finitely generated groups[END_REF] for details). In what follows this affine algebraic set will be denoted by R(Γ, SL(n)) or simply by R n (Γ). Let ρ : Γ → SL(n) be a representation. The Lie algebra sl(n) turns into a Γ-module via Ad •ρ. This module will be simply denoted by sl(n

) ρ . A 1-cocycle or derivation d ∈ Z 1 (Γ; sl(n) ρ ) is a map d : Γ → sl(n) satisfying d(γ 1 γ 2 ) = d(γ 1 ) + Ad •ρ(γ 1 )(d(γ 2 )) , ∀ γ 1 , γ 2 ∈ Γ .
It was observed by André Weil [START_REF] Weil | Remarks on the cohomology of groups[END_REF] that there is a natural inclusion of the Zariski tangent space T Zar ρ (R n (Γ)) ֒→ Z 1 (Γ; sl(n) ρ ). Informally speaking, given a smooth curve ρ ǫ of representations through ρ 0 = ρ one gets a 1cocycle d : Γ → sl(n) by defining

d(γ) := d ρ ǫ (γ) d ǫ ǫ=0 ρ(γ) -1 , ∀γ ∈ Γ .
It is easy to see that the tangent space to the orbit by conjugation corresponds to the space of 1-coboundaries

B 1 (Γ; sl(n) ρ ). Here, b : Γ → sl(n) is a coboundary if there exists x ∈ sl(n) such that b(γ) = Ad •ρ(γ)(x) -x.
A detailed account can be found in [START_REF] Lubotzky | Varieties of representations of finitely generated groups[END_REF].

For the convenience of the reader, we state the following result which is implicitly contained in [START_REF] Ben Abdelghani | Deformations of metabelian representations of knot groups into SL(3, C)[END_REF][START_REF] Heusener | Deformations of reducible representations of 3-manifold groups into SL 2 (C)[END_REF][START_REF] Heusener | Deformations of reducible representations of 3-manifold groups into PSL 2 (C)[END_REF]. A detailed proof of the following streamlined version can be found in [START_REF] Heusener | Deformations of reducible representations of knot groups into SL(n, C)[END_REF]:

2.4 Proposition Let M be an orientable, irreducible 3-manifold with infinite fundamental group π 1 (M) and incompressible tours boundary, and let ρ :

π 1 (M) → SL(n) be a representation. If dim H 1 (M; sl(n) ρ ) = n -1 then ρ is a smooth point of the SL(n)- representation variety R n (π 1 (M)). More precisely, ρ is contained in a unique component of dimension n 2 + n -2 -dim H 0 (π 1 (M); sl(n) ρ ).

Reducible metabelian representations

Recall that every nonzero complex number α ∈ C * determines an action of a knot group Γ on the complex numbers given by γ x = α h(γ) x for γ ∈ Γ and x ∈ C. The resulting Γ-module will be denoted by

C α . Notice that C α is isomorphic to C[t ±1 ]/(t -α).
It is easy to see that a map Γ → GL(2, C) given by

γ → 1 z 1 (γ) 0 1 α h(γ) 0 0 1 = α h(γ) z 1 (γ) 0 1 (2)
is a representation if and only if the map z 1 : Γ → C α is a derivation i.e.

δz 1 (γ 1 , γ 2 ) = α h(γ 1 ) z 1 (γ 2 ) -z 1 (γ 1 γ 2 ) + z 1 (γ 1 ) = 0 for all γ 1 , γ 2 ∈ Γ.
The representation given by ( 2) is non-abelian if and only if α = 1 and the cocycle z is not a coboundary. Hence it follows from Lemma 2.3 that such a reducible non abelian representation exists if and only if α is a root of the Alexander polynomial. These representations were first studied independently by G. Burde [START_REF] Burde | Darstellungen von Knotengruppen[END_REF] and G. de Rham [START_REF] De | Introduction aux polynômes d'un noeud[END_REF].

We extend these considerations to a map Γ → GL(3, C). It follows easily that

γ →   α h(γ) z 1 (γ) z 2 (γ) 0 1 h(γ) 0 0 1   (3)
is a representation if and only if δz 1 = 0 and δz 2 + z 1 h = 0 i.e.

δz 1 (γ 1 , γ 2 ) = 0 for all γ 1 , γ 2 ∈ Γ, δz 2 (γ 1 , γ 2 ) + z 1 (γ 1 )h(γ 2 ) = 0 for all γ 1 , γ 2 ∈ Γ.
It was proved in [1, Theorem 3.2] that the 2-cocycle z 1 h represents a nontrivial cohomology class in H 2 (Γ; C α ) provided that z 1 is not a coboundary and that the (t -α)-torsion of the Alexander module is semi-simple i.e.

τ α = C[t ±1 ]/(t -α) ⊕ • • • ⊕ C[t ±1 ]/(t -α).
Hence if we suppose that z 1 is not a coboundary then it is clear that a non-abelian representation Γ → GL(3, C) given by (3) can only exist if the (t -α)-torsion τ α of the Alexander module has a direct summand of the form

C[t ±1 ]/(t -α) s , s ≥ 2.
Representations Γ → GL(n, C) of this type were studied in [START_REF] Jebali | Module d'Alexander et représentations métabéliennes[END_REF] where it was shown that the whole structure of the (t -α)-torsion of the Alexander module can be recovered. Note that every metabelian representation of Γ factors through the metabelian group Γ ′ /Γ ′′ ⋊ Z.

Let α ∈ C * be a non-zero complex number and n ∈ Z, n > 1. In what follows we consider the cyclic

C[t ±1 ]-module C[t ±1 ]/(t -α) n-1 and the semi-direct product C[t ±1 ] (t -α) n-1 ⋊ Z
where the multiplication is given by (p

1 , n 1 )(p 2 , n 2 ) = (p 1 + t n 1 p 2 , n 1 + n 2 ).
Let I n ∈ SL(n) and N n ∈ GL(n) denote the identity matrix and the upper triangular Jordan normal form of a nilpotent matrix of degree n respectively.

For later use we note the following lemma which follows easily from the Jordan normal form theorem:

3.1 Lemma Let α ∈ C * be a nonzero complex number and let C n be the C[t ±1 ]-module with the action of t k given by

t k a = α k a J k n ( 4
)
where a ∈ C n and

J n = I n + N n .Then the C[t ±1 ]-module C n is isomorphic to C[t ±1 ]/(t -α) n .
There is a direct method to construct a reducible metabelian representations of

C[t ±1 ]/(t -α) n-1 ⋊ Z into GL(n, C) (see [4, Proposition 3.13]). A direct calculation gives that (a, 0) → 1 a 0 I n-1 , (0, 1) → α 0 0 J -1 n-1 defines a faithful representation C[t ±1 ]/(t -α) n-1 ⋊ Z → GL(n, C).
Therefore, we obtain a reducible, metabelian, non-abelian representation

̺ : Γ → GL(n, C) if the Alexander module H 1 (X ∞ , C) has a direct summand of the form C[t ±1 ] (t -α) s with s ≥ n -1 ≥ 1: ̺ : Γ ∼ = Γ ′ ⋊ Z → Γ ′ /Γ ′′ ⋊ Z → (C ⊗ Γ ′ /Γ ′′ ) ⋊ Z → C[t ±1 ] (t -α) s ⋊ Z → C[t ±1 ] (t -α) n-1 ⋊ Z → GL(n, C) given by ̺(γ) = 1 z(γ) 0 I n-1 α h(γ) 0 0 J -h(γ) n-1 . (5)
It is easy to see that a map ̺ : Γ → GL(n) given by ( 5) is a homomorphism if and only if z :

Γ → C n-1 is a cocycle i.e. for all γ 1 , γ 2 ∈ Γ we have z(γ 1 γ 2 ) = z(γ 1 ) + α h(γ 1 ) z(γ 2 )J h(γ 1 ) n-1 . (6)
For a better description of the cocycle z, we introduce the following notations: for m, k ∈ Z, k ≥ 0, we define

h k (γ) := h(γ) k where m k := m(m-1)•••(m-k+1) k! if k > 0 1 if k = 0. ( 7
)
It follows directly from the properties of the binomial coefficients that for each k ∈ Z, k ≥ 0, the cochains h k ∈ C 1 (Γ; C) are defined and verify:

δh k + k-1 i=1 h i h k-i = 0. ( 8
)
3.2 Lemma Let z : Γ → C n-1 be a map verifying [START_REF] Burde | Darstellungen von Knotengruppen[END_REF] and let zk : Γ → C α , z = (z 1 , . . . , zn-1 ), denote the components of z. Then the cochains zk ,

1 ≤ k ≤ n -1, satisfy δz k + k-1 i=1 h i zk-i = 0 .
In particular z1 : Γ → C α is a cocycle.

Proof. Note that h 0 ≡ 1, h 1 = h, J m n-1 = (I n-1 + N n-1 ) m = i≥0 m i N i n-1 and (x 1 , . . . , x n-1 )J m n-1 = (x ′ 1 , x ′ 2 , . . . , x ′ n-1 )
where

x ′ k = k-1 i=0 m i x k-i = x k + k-1 i=1 m i x k-i .
It follows from this formula that z(γ

1 γ 2 ) = z(γ 1 ) + α h(γ 1 ) z(γ 2 )J h(γ 1 ) n-1 holds if and only if for k = 1, . . . , n -1 we have zk (γ 1 γ 2 ) = zk (γ 1 ) + α h(γ 1 ) zk (γ 2 ) + k-1 i=1 h i (γ 1 ) α h(γ 1 ) zk-i (γ 2 ) .
In other words 0 = δz k + k-1 i=1 h i zk-i holds.

2

From now on we will suppose that for α ∈ C * \ {1} the (t -α)-torsion of the Alexander module is cyclic of the form

τ α = C[t ±1 ] (t -α) n-1 ,
where n ≥ 2 .

This is equivalent to the fact that α is a root of the Alexander polynomial ∆ K (t) of multiplicity n -1 and that dim H 1 (Γ; C α ) = 1 (see Lemma 2.3).

Let us recall also that by Lemma 2.3, the following dimension formulas hold:

dim H q (Γ; C) = 1 for q = 0, 1; 0 for q ≥ 2, (9)
and dim H q (Γ; C α ±1 ) = 1 for q = 1, 2; 0 for q = 1, 2.

(10)

Remark

Notice that by Blanchfield-duality the (t -α -1 )-torsion of the Alexander module H 1 (Γ; C[t ±1 ]) is also of the form

τ α -1 = C[t ±1 ]/(t -α -1 ) n-1 .
More precisely, the Alexander polynomial ∆ K (t) is symmetric and hence α -1 is also a root of ∆ K (t) of multiplicity n -1 and dim

H 1 (Γ; C α -1 ) = 1.
Let ̺ : Γ → GL(n) be a representation given by ( 5) i.e. for all γ ∈ Γ we have

̺(γ) = 1 z(γ) 0 I n-1 α h(γ) 0 0 J -h(γ) n-1
.

We will say that ̺ can be upgraded to a representation into GL(n + 1, C) if there is a cochain zn : Γ → C α such that the map Γ → GL(n + 1, C) given by

γ → 1 (z(γ), zn (γ)) 0 I n α h(γ) 0 0 J -h(γ) n is a representation. 3.4 Lemma Suppose that the (t -α)-torsion of the Alexander module is cyclic of the form τ α = C[t ±1 ] (t -α) n-1 , n ≥ 2 and let ̺ : Γ → GL(n, C)
be a representation given by [START_REF] Brown | Cohomology of groups[END_REF].

Then ̺ cannot be upgraded to a representation into GL(n + 1, C) unless z1 : Γ → C α is a coboundary.

Proof. By Lemma 3.1 the C[t ±1 ]-module C n-1 with the action given by t a = α a J n-1 is isomorphic to C[t ±1 ]/(t -α) n-1 . Hence it follows from the universal coefficient theorem that for l ≥ n -1 we have:

H 1 (Γ; C[t ±1 ]/(t -α) l ) ∼ = Hom C[t ±1 ] H 1 (Γ; C[t ±1 ]), C[t ±1 ]/(t -α) l ∼ = Hom C[t ±1 ] C[t ±1 ]/(t -α) n-1 , C[t ±1 ]/(t -α) l ∼ = (t -α) l-n+1 C[t ±1 ]/(t -α) l ∼ = C[t ±1 ]/(t -α) n-1 . Hence if l > n -1 then every cocycle z : Γ → C[t ±1 ]/(t -α) l , given by z(γ) = (z 1 (γ), . . . , zl (γ)
) is cohomologous to a cocycle for which the first l -n + 1 components vanish. This proves the conclusion of the lemma. 2

Notice that the unipotent matrices J n and J -1 n are similar: a direct calculation shows that P n J n P -1 n = J -1 n where P n = (p ij ), p ij = (-1) j j i . The matrix P n is upper triangular with ±1 in the diagonal and P 2 n is the identity matrix, and therefore P n = P -1 n .

Hence ̺ is conjugate to a representation ̺ : Γ → GL(n, C) given by

̺(γ) = α h(γ) z(γ) 0 J h(γ) n-1 =        α h(γ) z 1 (γ) z 2 (γ) . . . z n-1 (γ) 0 1 h 1 (γ) . . . h n-2 (γ) . . . . . . . . . . . . . . . . . . . . . 1 h 1 (γ) 0 . . . . . . 0 1        (11)
where z = (z 1 , . . . , z n-1 ) :

Γ → C n-1 satisfies z(γ 1 γ 2 ) = α h(γ 1 ) z(γ 2 ) + z(γ 1 )J h(γ 2 ) n-1 . It follows directly that z(γ) = z(γ)P n-1 J h(γ)
n-1 and in particular z 1 = -z 1 . The same argument as in the proof of Lemma 3.2 shows that the cochains z k : Γ → C α verify:

δz k + k-1 i=1 z i h k-i = 0 for k = 1, . . . , n -1.
Therefore, the representation ̺ : Γ → GL(n, C) can be upgraded into a representation Γ → GL(n + 1, C) if and only if n-1 i=1 z i h n-i is a coboundary.

Hence we obtain the following:

3.5 Proposition Suppose that the (t -α)-torsion of the Alexander module is cyclic of the form

τ α = C[t ±1 ] (t -α) n-1 , n ≥ 2. Let ̺, ̺ : Γ → GL(n, C
) be the representations given by ( 5) and (11) respectively where z1 = -z 1 : Γ → C α is a non-principal derivation. Then the representations ̺ and ̺ can not be upgraded to representations Γ → GL(n + 1, C) i.e. the cocycles

n-1 i=1 h i zn-i and n-1 i=1 z i h n-i represent nontrivial cohomology classes in H 2 (Γ; C α ).
Proof. The proposition follows from Lemma 3.4 and the above considerations. 2

Cohomological computations

We suppose throughout this section that K ⊂ S 3 is a knot and that the (t-α)-torsion of its Alexander module is cyclic of the form τ α = C[t, t -1 ] (tα) n-1 , n ≥ 2, where α ∈ C * is a nonzero complex number. Let ̺ : Γ → GL(n) be a representation given by [START_REF] Heusener | Deformations of reducible representations of 3-manifold groups into PSL 2 (C)[END_REF] where z 1 : Γ → C α is a non-principal derivation:

̺(γ) = α h(γ) z(γ) 0 J h(γ) n-1 =        α h(γ) z 1 (γ) z 2 (γ) . . . z n-1 (γ) 0 1 h 1 (γ) . . . h n-2 (γ) . . . . . . . . . . . . . . . . . . . . . 1 h 1 (γ) 0 . . . . . . 0 1       
.

We choose an n-th root λ of α = λ n and we define a reducible metabelian representation ̺ λ : Γ → SL(n) by

̺ λ (γ) = λ -h(γ) ̺(γ) (12)
The aim of the following sections is to calculate the cohomological groups of Γ with coefficients in the Lie algebra sl(n) Ad •̺ λ . Notice that the action of Γ via Ad •̺ and Ad •̺ λ preserve sl(n) and coincide since the center of GL(n) is the kernel of Ad : GL(n) → Aut(gl(n)). Hence we have the following isomorphisms of Γ-modules:

sl(n) Ad •̺ λ ∼ = sl(n) Ad •̺ and gl(n) Ad •̺ = sl(n) Ad •̺ ⊕ C I n ( 13
)
where Γ acts trivially on the center CI n of gl(n). We will prove the following result:

4.1 Proposition Let K ⊂ S 3 be a knot and suppose that the (t -α)torsion of the Alexander module of K is of the form

τ α = C[t ±1 ] (t -α) n-1 .
Then for the representation

̺ λ : Γ → SL(n) we have H 0 (Γ; sl(n) Ad •̺ λ ) = 0 and dim H 1 (Γ; sl(n) Ad •̺ λ ) = dim H 2 (Γ; sl(n) Ad •̺ λ ) = n -1 .
Notice that Propositions 4.1 and 2.4 will proof the first part of Theorem 1.1. The proof of Proposition 4.1 will occupy the rest of this section. Throughout this section we will consider gl(n) as a Γ-module via Ad •̺ and for simplicity we will write gl(n) for gl(n) Ad •̺ . It follows form Equation (13) that

H * (Γ; gl(n)) ∼ = H * (Γ; sl(n)) ⊕ H * (Γ; C) .
In order to compute the cohomological groups H * (Γ, gl(n)) and describe the cocycles, we will construct and use an adequate filtration of the coefficient algebra gl(n).

The setup

Let (E 1 , . . . , E n ) denote the canonical basis of the space of column vectors. Hence

E j i := E i t E j , 1 ≤ i, j ≤ n, form the canonical basis of gl(n). Note that for A ∈ GL(n), Ad A (E j i ) = (AE i )( t E j A -1
). The Lie algebra gl(n) turns into a Γ-module via Ad •̺ i.e. for all γ ∈ Γ we have

γ • E j i = (̺(γ)E i )( t E j ̺(γ -1 )) . Explicitly we have γ • E 1 1 =      α h(γ) 0 . . . 0      α -h(γ) , z 1 (γ -1 ), . . . , z n-1 (γ -1 ) = E 1 1 + α h(γ) z 1 (γ -1 )E 2 1 + • • • + α h(γ) z n-1 (γ -1 )E n 1 ; (14)
for 1 < k ≤ n:

γ • E k 1 = α h(γ) E k 1 + α h(γ) h 1 (γ -1 )E k+1 1 + • • • + α h(γ) h n-k (γ -1 )E n 1 ; (15) γ • E 1 k =            z k-1 (γ) h k-2 (γ) . . . h 1 (γ) 1 0 . . .            α -h(γ) , z 1 (γ -1 ), . . . , z n-1 (γ -1 ) (16)
and for 1 < i, j ≤ n:

γ • E j i =            z i-1 (γ) h i-2 (γ) . . . h 1 (γ) 1 0 . . .            0, . . . , 0, 1, h 1 (γ -1 ), . . . , h n-j (γ -1 ) . ( 17
)
For a given family (X i ) i∈I , X i ∈ gl(n), we let X i |i ∈ I ⊂ gl(n) denote the subspace of gl(n) generated by the family.

Remark

A first consequence of these calculations is that if c ∈ C 1 (Γ; C) is a cochain, then for 2 ≤ i ≤ n and 1 ≤ j ≤ n we have:

δ gl (cE j i ) = (δc)E j i + (h 1 c)E j i-1 + • • • + (h i-2 c)E j 2 + (z i-1 c)E j 1 + x where x : Γ×Γ → E l k | 1 ≤ k ≤ i, j < l ≤ n is a 2-cochain.
Here δ gl and δ denote the coboundary operators of C 1 (Γ; gl(n)) and C 1 (Γ; C) respectively.

In what follows we will also make use of the following Γ-modules: for

0 ≤ i ≤ n -1, we define C(i) = E l k | 1 ≤ k ≤ n, n -i ≤ l ≤ n . We have C(i) =                     0 • • • 0 c 1,n-i • • • c 1,n 0 • • • 0 c 2,n-i • • • c 2,n . . . . . . . . . . . . . . . 0 • • • 0 c n-1,n-i • • • c n-1,n 0 • • • 0 c n,n-i • • • c n,n        : c i,j ∈ C              (18)
and gl(n

) = C(n-1) ⊃ C(n-2) ⊃ • • • ⊃ C(0) = E n 1 , . . . , E n n ⊃ C(-1) = 0. We will denote by X + C(i) ∈ C(k)/C(i) the class represented by X ∈ C(k), 0 ≤ i < k ≤ n -1.

Cohomology with coefficients in C(i)

The aim of this subsection is to prove that for 0 ≤ i ≤ n -2 the cohomology groups H * (Γ; C(i)) vanish (see Proposition (4.7)). First we will prove this for i = 0 and in order to conclude we will apply the isomorphism C(0) ∼ = C(i)/C(i -1) (see Lemma 4.5). Finally Lemma 4.6 permits us to compute a certain Bockstein operator.

Lemma

The vector space E n 1 is a submodule of C(0) and thus of gl(n) = C(n -1) and we have

H 0 (Γ; E n 1 ) = 0, dim H 1 (Γ; E n 1 ) = dim H 2 (Γ; E n 1 ) = 1.
More precisely, the cocycles

z 1 E n 1 ∈ Z 1 (Γ; E n 1 ) and n-1 i=1 z i h n-i E n 1 ∈ Z 2 (Γ; E n 1 )
represent generators of H 1 (Γ; E n 1 ) and H 2 (Γ; E n 1 ) respectively.

Proof. The isomorphism E n 1 ∼ = C α and Lemma 2.3 imply the dimension formulas. The form of the generating cocycles follows from the isomorphism E n 1 ∼ = C α and Proposition 3.5. 2

Lemma

The Γ-module C(0)/ E n 1 is isomorphic to C[t ±1 ]/(t -1) n-1 . In particular, we obtain:

1. for q = 0, 1 dim H q Γ; C(0)/ E n 1 = 1 and H 2 Γ; C(0)/ E n 1 = 0,

the vector E n

2 represents a generator of H 0 (Γ; C(0)/ E n 1 ) and the cochain v1 : Γ → C(0) given by

v1 (γ) = h 1 (γ)E n n + h 2 (γ)E n n-1 + • • • + h n-2 (γ)E n 2 represents a generator of H 1 (Γ; C(0)/ E n 1 ).
Proof. First notice that C(0)/ E n 1 is a (n -1)-dimensional vector space. More precisely, a basis of this space is represented by the elements

E n n , E n n-1 , . . . , E n 2 .
It follows from (17) that the action of Γ on C(0)/ E n 1 factors through h : Γ → Z. More precisely, we have for all γ ∈ Γ such that h(γ) = 1 and for all 0

≤ l ≤ n -1 γ • E n n-l = E n n-l + E n n-l-1
Here we used the fact that if h(γ) = 1 then h i (γ) = 0 for all 2 ≤ i ≤ n -1.

On the other hand 1 = (t -1) 0 , (t -1), . . . , (t -1) n-2 represents a basis of C[t ±1 ]/(t -1) n-1 and we have for all γ ∈ Γ such that

h(γ) = 1: γ • (t -1) l = (t -1) l + (t -1) l+1 + p where p ∈ (t -1) n-1 C[t ±1 ] and 0 ≤ l ≤ n -2. Hence the bijection ϕ : {(t -1) l | 0 ≤ l ≤ n -2} → {E n n-l | 0 ≤ l ≤ n -2}
given by ϕ

: (t -1) l → E n n-l , 0 ≤ l ≤ n -2, induces an isomorphism of Γ-modules ϕ : C[t ±1 ]/(t -1) n-1 ∼ = --→ C(0)/ E n 1 .
Now, the first assertion follows from Lemma 2.3.

Moreover, it follows from the above considerations that E n 2 represents a generator of H 0 (Γ; C(0)/ E n 1 ). To prove the second assertion consider the following short exact sequence

0 → C[t ±1 ]/(t -1) n-2 (t-1)• ---→ C[t ±1 ]/(t -1) n-1 → C → 0
which gives the following long exact sequence in cohomology:

0 → H 0 (Γ; C[t ±1 ]/(t -1) n-2 ) ∼ = -→ H 0 (Γ; C[t ±1 ]/(t -1) n-1 ) → H 0 (Γ; C) β 0 -→ H 1 (Γ; C[t ±1 ]/(t -1) n-2 ) → H 1 (Γ; C[t ±1 ]/(t -1) n-1 ) ∼ = -→ H 1 (Γ; C)→H 2 (Γ; C[t ±1 ]/(t -1) n-2 ) = 0 .
The isomorphisms and the vanishing of

H 2 (Γ; C[t ±1 ]/(t -1) n-2 ) follow di- rectly from Lemma 2.3.
Hence the Bockstein operator β 0 is an isomorphism: the element e 0 = 1 ∈ C[t ±1 ]/(t -1) n-1 projects onto a generator of H 0 (Γ; C) and if δ n-1 denotes the coboundary operator of C * (Γ; C[t ±1 ]/(t -1) n-1 ) we obtain:

δ n-1 (e 0 )(γ) = (γ -1) • e 0 = h 1 (γ)e 1 + h 2 (γ)e 2 + • • • + h n-2 (γ)e n-1 = (t -1) • h 1 (γ)e 0 + h 2 (γ)e 1 + • • • + h n-2 (γ)e n-2 .
Hence the cocycle γ → h 1 (γ)e 0 + h 2 (γ)e 1 + • • • + h n-2 (γ)e n-2 represents a generator of H 1 (Γ; C[t ±1 ]/(t -1) n-2 ). To conclude, recall that the isomorphism

C[t ±1 ]/(t -1) n-1 ∼ = C(0)/ E n 1 is induced by the map ϕ : e l → E n n-l , 0 ≤ l ≤ n -2. 2 4.5 Lemma For i ∈ Z, 0 ≤ i ≤ n -3, the Γ-module C(i + 1)/C(i) is isomorphic to C(0).
Proof. It follows from (17) that, for all i ∈ Z, 0 ≤ i ≤ n -2, the bijection

φ : {E n-(i+1) n-j + C(i) | 0 ≤ j ≤ n -1} → {E n n-j | 0 ≤ j ≤ n -1}
given by φ(E

n-(i+1) n-j + C(i)) = E n n-j induces an isomrphism of Γ-modules φ : C(i + 1)/C(i) → C(0). 2
Let us recall the definition of the cochains h i ∈ C 1 (Γ; C), given by h i (γ) = h(γ) i (see Equation ( 7)). Recall also that for 1 ≤ i ≤ n -1 the cochains h i ∈ C 1 (Γ; C) verify Equation [START_REF] Davis | Lecture notes in algebraic topology[END_REF]:

δh i + i-1 j=1 h j h i-j = 0.
4.6 Lemma Let δ gl denote the coboundary operator of C * (Γ; gl(n)). Then for all 0 ≤ k ≤ n -2 there exists a cochain

x k-1 ∈ C 2 (Γ; C(k -1)) such that δ gl n i=2 h n-i+1 E n-k i = n-1 i=1 z i h n-i E n-k 1 + x k-1
Proof. Equation ( 17) and Remark 4.2 imply that

δ gl (h n-i+1 E n-k i ) = z i-1 h n-i+1 E n-k 1 + i-1 l=2 h i-l h n-i+1 E n-k l + δh n-i+1 E n-k i + x i,k-1 where x i,k-1 ∈ C 2 (Γ; C(k -1)
) and δ is the boundary operator of C * (Γ; C). Therefore,

δ gl ( n i=2 h n-i+1 E n-k i ) = n i=2 z i-1 h n-i+1 E n-k 1 + n i=2 i-1 l=2 h i-l h n-i+1 E n-k l + n i=2 δh n-i+1 E n-k i + x k-1 .
where

x k-1 = n i=2 x i,k-1 ∈ C 2 (Γ; C(k - 1

)). A direct calculation gives that

n i=2 i-1 l=2 h i-l h n-i+1 E n-k l = n-1 l=2 n i=l+1 h i-l h n-i+1 E n-k l = n-1 l=2 n-l i=1 h i h n-l+1-i E n-k l . Thus δ gl (h n-i+1 E n-k i ) = n-1 i=1 z i h n-i E n-k 1 + δh 1 E n-k n + n-2 i=1 δh n-i + n-i-1 l=1 h l h n-i-l E n-k i + x k-1 .
Now δh 1 = 0 and by [START_REF] Davis | Lecture notes in algebraic topology[END_REF] we have δh n-i + n-i l=1 h l h n-i+1-l = 0. Hence we obtain the claimed formula. 2

Proposition

For all i ∈ Z, 0 ≤ i ≤ n -2 and q ≥ 0 we have

H q (Γ; C(i)) = 0.
This sequence induces a long exact sequence in cohomology

0 → H 0 (Γ; C(i 0 )) → H 0 (Γ; C(i 0 + 1)) → H 0 (Γ; C(i 0 + 1)/C(i 0 )) → H 1 (Γ; C(i 0 )) → H 1 (Γ; C(i 0 + 1)) → H 1 (Γ; C(i 0 + 1)/C(i 0 )) → H 2 (Γ; C(i 0 )) → H 2 (Γ; C(i 0 + 1)) → H 2 (Γ; C(i 0 + 1)/C(i 0 )) → 0 .
Using the hypothesis, we conclude that the groups H q (Γ; C(i 0 + 1)) and H q (Γ; C(i 0 + 1)/C(i 0 )) are isomorphic for q = 0, 1, 2. By Lemma 4.5, we obtain H q (Γ; C(i 0 + 1)) ∼ = H q (Γ; C(0)) = 0 for q = 0, 1, 2. 2

Cohomology with coefficients in gl(n)

In this subsection we will prove Proposition 4.1.

Proof of Proposition 4.1. In order to compute the dimensions of the cohomology groups H * (Γ; gl(n)), we consider the short exact sequence

0 → C(n -2) C(n -1) = gl(n) ։ gl(n)/C(n -2) → 0 . (21)
The sequence (21) gives rise to the following long exact cohomology sequence:

0 → H 0 (Γ; gl(n)) → H 0 (Γ; gl(n)/C(n -2)) → H 1 (Γ; C(n -2)) → H 1 (Γ; gl(n)) → H 1 (Γ; gl(n)/C(n -2)) → H 2 (Γ; C(n -2)) → H 2 (Γ; gl(n)) → H 2 (Γ; gl(n)/C(n -2)) → 0 .
As H q (Γ; C(n -2)) = 0 we conclude that

H q (Γ; gl(n)) ∼ = H q (Γ; gl(n)/C(n -2)) .
It remains to understand the quotient gl(n)/C(n -2).

Clearly the vectors E 1 n , . . . , E 1 1 represent a basis of gl(n)/C(n -2) and there exists a Γ-module M such that the following sequence

0 → E 1 1 + C(n -2) gl(n)/C(n -2) ։ M → 0 (22)
is exact. Now the sequence (22) induces the following exact cohomology sequence:

0 → H 0 (Γ; E 1 1 + C(n -2) ) → H 0 (Γ; gl(n)/C(n -2)) → H 0 (Γ; M) → H 1 (Γ; E 1 1 + C(n -2) ) → H 1 (Γ; gl(n)/C(n -2)) → H 1 (Γ; M) → H 2 Γ; E 1 1 + C(n -2) → H 2 (Γ; gl(n)/C(n -2)) → H 2 (Γ; M) → 0 . (23)
Observe that the action of Γ on E 1 1 + C(n -2) is trivial. Therefore, E 1 1 + C(n -2) and C are isomorphic Γ-modules. By Lemma 2.3 we obtain dim H q (Γ; E 1 1 + C(n -2) ) = 1 for q = 0, 1 and H 2 (Γ; E 1 1 + C(n -2) ) = 0. To complete the proof we will make use of Lemma 4.8, which states that the Γ-module M is isomorphic to

C[t ±1 ]/(t -α -1 ) n-1 . Recall that Lemma 2.3 implies that H 0 (Γ; C[t ±1 ]/(t -α -1 ) n-1 ) = 0 and dim H q (Γ; C[t ±1 ]/(t -α -1 ) n-1 ) = n -1, for q = 1, 2.
Therefore, sequence (23) gives:

H q (Γ; gl(n)) ∼ = H q (Γ; gl(n)/C(n -2)) ∼ = H 0 (Γ; C) for q = 0; H 2 (Γ; M) for q = 2
and the short exact sequence:

0 → H 1 (Γ; C) H 1 (Γ; gl(n)/C(n -2)) ∼ = H 1 (Γ; gl(n)) ։ H 1 (Γ; M) → 0 . 2 4.8 Lemma The Γ-module M is isomorphic to C[t ±1 ]/(t -α -1 ) n-1 . Con- sequently H 0 (Γ; M) = 0, dim H q (Γ; M) = n -1, q = 0, 1.
Proof of Lemma 4.8.

The proof is similar to the proof of Lemma 4.4. As a C-vector space the dimension of M is n -1 and a basis is given by E

1 n , . . . , E 1 2
where

E 1 i = E 1 i +C(n-2) ∈ M is the class represented by E 1 i , 2 ≤ i ≤ n. In order to prove that M is isomorphic to C[t ±1 ]/(t -α -1 ) n-1 observe that by (16) γ • E 1 k = α -h(γ) E 1 k + h 1 (γ)E 1 k-1 + • • • + h k-2 (γ)E 1 2 + X k where X k ∈ E 1 1 + C(n -2)
. Therefore, the action of Γ on M factors through h : Γ → Z. More precisely, we have for all γ ∈ Γ such that h(γ) = 1

γ • E 1 k = α -1 (E 1 k + E 1 k-1 ) .
On the other hand e l = α(t -α -1 ) l , 0 ≤ l ≤ n -2, represents a basis of

C[t ±1 ]/(t -α -1
) n-1 and we have for all γ ∈ Γ such that h(γ) = 1:

γ • e l = α -1 (e l + e l+1 ) + p where p ∈ (t -α -1 ) n-1 C[t ±1 ].
Hence the bijection ψ :

{e l | 0 ≤ l ≤ n -2} → {E 1 k | 2 ≤ k ≤ n} given by ϕ : e l → E 1 n-l , 0 ≤ l ≤ n -2, induces an isomorphism of Γ-modules ψ : C[t ±1 ]/(t -α -1 ) n-1 ∼ =
-→ M . Finally, the dimension equations follow from Lemma 2.3 and Remark 3.3.

2

We obtain immediately that under the hypotheses of Proposition 4.1 the representation ̺ λ is a smooth point of the representation variety R n (Γ). This proves the first part of Theorem 1.1.

Proposition

Let K be a knot in the 3-sphere S 3 . If the (t-α)-torsion τ α of the Alexander module is cyclic of the form C[t, t -1 ] (t -α) n-1 , n ≥ 2, then the representation ̺ λ is a smooth point of the representation variety

R n (Γ); it is contained in a unique (n 2 + 2n -2)-dimensional component R ̺ λ of R n (Γ).
Proof. By Proposition 2.4 and Proposition 4.1, the representation

̺ λ is contained in a unique component R ̺ λ of dimension (n 2 + n -2). Moreover, dim Z 1 (Γ; sl(n)) = dim H 1 (Γ; sl(n)) + dim B 1 (Γ; sl(n)) = (n -1) + (n 2 -1) = n 2 + n -2 .
Hence the representation ̺ λ is a smooth point of R n (Γ) which is contained in an unique (n

2 + n -2)-dimensional component R ̺ λ . 2
For a later use, we describe more precisely the derivations v k : Γ → sl(n), 1 ≤ k ≤ n -1, which represent a basis of H 1 (Γ; sl(n)).

Corollary There exists cochains

z - 1 , • • • , z - n-1 ∈ C 1 (Γ; C α -1 ) such that δz - k + k-1 i=1 h i z - k-i = 0 for k = 1, . . . , n -1 and z - 1 : Γ → C -1
α is a non-principal derivation. Moreover, there exist cochains g k : Γ → C and

x k : Γ → C(n -2), 1 ≤ k ≤ n -1, such that the cochains v k : Γ → sl(n) given by v k = g k E 1 1 + z - k E 1 2 + • • • + z - 1 E 1 k+1 + x k
are cocycles and represent a basis of H 1 (Γ; sl(n)).

Proof. Recall that the vector space M admits as a basis the family E and that it is isomorphic to C[t ±1 ]/(t -α -1 ) n-1 . Moreover it is easily seen that M is isomorphic to the Γ-module of column vectors C n-1 where the action is given by

t k a = α -k J k n-1 a. Hence a cochain z -: Γ → M with coordinates z -= t (z - n-1 , • • • , z - 1 ) is a cocycle in Z 1 (Γ; M) if and only if for all γ 1 , γ 2 ∈ Γ z -(γ 1 γ 2 ) = z -(γ 1 ) + α -h(γ 1 ) J h(γ 1 ) n-1 z -(γ 2 ).
It follows, as in the proof of Lemma 3.2, that this is equivalent to

z - k (γ 1 γ 2 ) = z - k (γ 1 ) + α -h(γ 1 ) z - k (γ 2 ) + k-1 i=1 h i (γ 1 )α -h(γ 1 ) z - k-i (γ 2 ).
In other words, for 1

≤ k ≤ n -1, 0 = δz - k + k-1 i=1 h i ⌣ z - k-i . By Remark 3.3, if z - 1 ∈ Z 1 (Γ; C α -1 ) is a non-principal derivation, there exist cochains z - k : Γ → C α -1 , 2 ≤ k ≤ n -1, such that 0 = δz - k + k-1 i=1 h i ⌣ z - k-i .
Consequently, as dim H 1 (Γ; M) = n -1, the cochains

z - k = z - k E 1 2 + • • • + z - 1 E 1 k+1 , 1 ≤ k ≤ n -1,
represent a basis of H 1 (Γ; M). The proof is completed by noticing that the projection H 1 (Γ; gl(n)) → H 1 (Γ; M) restricts to an isomorphism between H 1 (Γ; sl(n)) and H 1 (Γ; M). 2

Irreducible SL(n) representations

This section will be devoted to the proof of the last part of Theorem 1.1. At first, we proved that the representation ̺ λ is a smooth point of R n (Γ) which is contained in a unique (n 2 + n -2)-dimensional component R ̺ λ . Then, to prove the existence of irreducible representations in that component, we will make use of Corollary 4.10 and Burnside's theorem on matrix algebras.

Proof of the last part of Theorem 1.1. To prove that the component R ̺ λ contains irreducible non metabelian representations, we will generalize the argument given in [START_REF] Ben Abdelghani | Deformations of metabelian representations of knot groups into SL(3, C)[END_REF] for n = 3.

Let Γ = S 1 , . . . , S n | W 1 , . . . , W n-1 be a Wirtinger presentation of the knot group. Modulo conjugation of the representation ̺ λ , we can assume that z 1 (S 1 ) = . . . = z n-1 (S 1 ) = 0. This conjugation corresponds to adding a coboundary to the cochains z i , 1 ≤ i ≤ n -1. We will also assume that the second Wirtinger generator S 2 verifies z 1 (S 2 ) = b 1 = 0 = z 1 (S 1 ). This is always possible since z 1 is not a coboundary. Hence

̺ λ (S 1 ) = α -1/n α 0 0 J n-1 and ̺ λ (S 2 ) = α -1/n α b 0 J n-1 where b = (b 1 , . . . , b n-1 ) with b 1 ∈ C * and b i = z i (S 2 ) ∈ C for 2 ≤ i ≤ n-1. Let v n-1 ∈ Z 1 (Γ; sl(n))
be a cocycle such that:

v n-1 = g n-1 E 1 1 + z - 1 E 1 n + z - 2 E 1 n-1 + . . . + z - n-1 E 1 2 + x n-1
given by Corollary 4.10. Up to adding a coboundary to the cocycle z - 1 we assume that z - 1 (S 1 ) = 0. Notice that, by Lemma 5.5 of [START_REF] Ben Abdelghani | Deformations of metabelian representations of knot groups into SL(3, C)[END_REF], z - 1 (S 2 ) = 0. Let ρ t be a deformation of ̺ λ with leading term v n-1 :

ρ t = I n + t v n-1 + o(t) ̺ λ , where lim t→0 o(t) t = 0 .
We may apply the following lemma (whose proof is completely analogous to that of Lemma 5.3 in [START_REF] Ben Abdelghani | Deformations of metabelian representations of knot groups into SL(3, C)[END_REF]) to this deformation for A(t) = ρ t (S 1 ). for all sufficiently small t.

Therefore, we may suppose that a n1 (t) = 0, and since

a n1 (t) = tλ n-1 z - 1 (S 1 ) + δc(S 1 ) + o(t) , for c ∈ C, it follows that a ′ n1 (0) = λ n-1 (z - 1 (S 1 ) + (α -1 -1)c) = 0
and hence c = 0. For B(t) = ρ t (S 2 ), we obtain b ′ n1 (0) = λ n-1 z - 1 (S 2 ) = 0. Hence, we can apply the following technical lemma (whose proof will be postponed to the end of this section).

Lemma

Let A(t) = (a ij (t)) 1≤i,j≤n and B(t) = (b ij (t)) 1≤i,j≤n be matrices depending analytically on t such that

A(t) = a 11 (t) 0 0 A 11 (t) , A(0) = ̺ λ (S 1 ) = α -1/n α 0 0 J n-1 and B(0) = ̺ λ (S 2 ) = α -1/n α b 0 J n-1 .
If the first derivative b ′ n1 (0) = 0 then for sufficiently small t, t = 0, the matrices A(t) and B(t) generate the full matrix algebra M(n, C).

Hence for sufficiently small t = 0 we obtain that A(t) = ρ t (S 1 ) and B(t) = ρ t (S 2 ) generate M(n, C). By Burnside's matrix theorem, such a representation ρ t is irreducible

To conclude the proof of Theorem 1.1, we will prove that all irreducible representations sufficiently close to ̺ λ are non-metabelian. In order to do so, we will make use of the following result of H. Boden and S. Friedel [4, Theorem 1.2]: for every irreducible metabelian representation ρ : Γ → SL(n) we have tr ρ(S 1 ) = 0. Now, we have tr ̺ λ (S 1 ) = λ -1 (λ n +n-1) and we claim that λ n +n-1 = 0. Notice that α = λ n is a root of the Alexander polynomial ∆ K (t) and λ n + n -1 = 0 would imply that 1 -n is a root of ∆ K (t). This would imply that t + n -1 divides ∆ K (t) and hence n divides ∆ K (1) = ±1 which is impossible since n ≥ 2. Therefore, tr(ρ(S 1 )) = 0 for all irreducible representations sufficiently close to ̺ λ . This proves Theorem 1.1. 2

Remark

Let ρ λ : Γ → SL(n) be the diagonal representation given by ρ λ (µ) = diag(λ n-1 , λ -1 I n-1 ) where µ is a meridian of K . The orbit O(ρ λ ) of ρ λ under the action of conjugation of SL(n) is contained in the closure O(̺ λ ). Hence ̺ λ and ρ λ project to the same point χ λ of the variety of characters

X n (Γ) = R n (Γ) SL(n).
It would be natural to study the local picture of the variety of characters X n (Γ) = R n (Γ) SL(n) at χ λ as done in [11, § 8]. Unfortunately, there are much more technical difficulties since in this case the quadratic cone Q(ρ λ ) coincides with the Zariski tangent space Z 1 (Γ; sl(n) ρ λ ). Therefore the third obstruction has to be considered.

Proof of lemma 5.2. The proof follows exactly the proof of Proposition 5.4 in [START_REF] Ben Abdelghani | Deformations of metabelian representations of knot groups into SL(3, C)[END_REF]. We denote by A t ⊂ gl(n) the algebra generated by A(t) and B(t).

For any matrix A we let P A (X) denote its characteristic polynomial. We have P A 11 (0) = (λ -1 -X) n-1 and a 11 (0) = λ n-1 . Since α = λ n = 1 we obtain P A 11 (0) (a 11 (0)) = 0. It follows that P A 11 (t) (a 11 (t)) = 0 for small t and hence In the next step we will prove that

A t      1 0 . . . 0     
= C n and (1, 0, . . . , 0)A t = C n , for small t ∈ C n .

It follows from this that A t contains all rank one matrices since a rank one matrix can be written as v ⊗ w where v is a column vector and w is a row vector. Note also that A(v ⊗ w) = (Av) ⊗ w and (v ⊗ w)A = v ⊗ (wA). Since each matrix is the sum of rank one matrices the proposition follows. Now consider the vectors (1, 0, . . . , 0)A(0), (1, 0, . . . , 0)B(0), . . . , (1, 0, . . . , 0)B(0) n-1 .

Then for 1 ≤ k ≤ n -1 :

(1, 0, . . . , 0)B(0

) k = λ -k (α k , b k-1 j=0 α k-1-j J j )
and the dimension D of the vector space (1, 0, . . . , 0)A(0), (1, 0, . . . , 0)B(0), . . . Here, J = J n-1 = I n-1 + N n-1 where N n-1 ∈ GL(n -1, C) is the upper triangular Jordan normal form of a nilpotent matrix of degree n -1. Then a direct calculation gives that dim b, bJ, . . . , bJ n-2 = dim b, bN, . . . , bN n-2 = n -1 , as b 1 = 0.

Thus dim (1, 0, . . . , 0)A(0), (1, 0, . . . , 0)B(0), . . . (1, 0, . . . , 0)B(0) n-1 = n and the vectors (1, 0, . . . , 0)A(0), (1, 0, . . . , 0)B(0), . . . , (1, 0, . . . , 0)B(0) n-1 form a basis of the space of row vectors. This proves that (1, 0, . . . , 0)A t is the space of row vectors for sufficiently small t.

In the final step consider the n column vectors for the first column of B(t); then

a 1 (t) = a 11 (t) 0 , a i+2 (t) = A i (t) b 11 (t) b(t) , 0 ≤ i ≤ n -2.
Define the function f (t) := det(a 1 (t), . . . , a n (t)) and g(t) by: f (t) = a 11 (t)g(t), where g(t) = det b(t), A 11 (t)b(t), . . . , A n-2 11 (t)b(t) . Now, for k ≥ 0 the k -th derivative g (k) (t) of g(t) is given by:

1 n

 1 , . . . , E 1 2

1 P

 1 A 11 (t) (a 11 (t)) P A 11 (t) (A(t)0, . . . , 0) ∈ C[A(t)] ⊂ A t .

  (1, 0, . . . , 0)B(0) n-1 is equal toD = dim (α, 0), (α, b), (α 2 , αb + bJ), . . . , (α n-1 , b k-1 j=0 α k-1-j J j ) = dim (α, 0), (0, b), (0, bJ), • • • (0, bJ n-2 ) .

a 1 (

 1

s 1 ,

 1 ...,s n-1 c s 1 ,...,s n-1 det b (s 1 ) (t), (A 11 (t)b(t)) (s 2 ) , . . . ,(A n-2 11 (t)b(t)) (s n-1 )wherec s 1 ,...,s n-1 = k s 1 ,...,s n-1 = k! s 1 !...s n-1 ! if s 1 + • • • + s n-1 = k; 0 othewise. As b(0) = 0 one have, for 0 ≤ k ≤ n -2, g (k) (0) = 0 and consequently f (k) (0) = 0 for all 0 ≤ k ≤ n -2. Now, for k = n -1, we have g (n-1) (0) (n -1)! = det b ′ (0), (A 11 (t)b(t)) ′ (0), . . . , (A n-2 11 (t)b(t)) ′ (0) = det b ′ (0), A 11 (0)b ′ (0), . . . , A n-2 11 (0)b ′ (0) = det b ′ (0), (λ -1 J)b ′ (0), . . . , (λ -1 J) n-2 b ′ (0) = det b ′ (0), λ -1 Nb ′ (0), . . . , λ -(n-2) N n-2 b ′ (0) = 0 since b ′ n1 = 0.

Acknowledgements. Both authors are pleased to acknowledge the support by the French-Tunisian CMCU project n o 12G/1502. Moreover, the first author was supported by the project Erasmus mundus E-GOV-TN. The second author acknowledges support from the ANR projects SGT and Mod-Group.

Proof. For q ≥ 3 we have H q (Γ; C(i)) = 0 since the knot exterior X has the homotopy type of a 2-dimensional complex. We start by proving the result for i = 0. Consider the short exact sequence

As the

]/(t -α) are isomorphic, the sequence (19) gives us a long exact sequence in cohomology:

Here, for q = 0, 1, we denoted by β q 0 : H q (Γ; C(0)/ E n 1 ) → H q+1 (Γ; E n 1 ) the Bockstein homomorphism. By Lemma 4.4, E n 2 represents a generator of H 0 (Γ; C(0)/ E n 1 ), so

By Lemma 4.3 z 1 E n 1 is a generator of H 1 (Γ; E n 1 ), and by Lemma 4.4 dim H 0 (Γ; C(0)/ E n 1 ) = 1 = dim H 1 (Γ; E n 1 ), thus β 0 0 is an isomorphism. Consequently H 0 (Γ; C(0)) = 0 as H 0 (Γ; E n 1 ) = 0 by Lemma 4.3. Now by Lemma 4.4, the cochain v1 : Γ → C(0) given by

represents a generator of H 1 (Γ; C(0)/ E n 1 ) and by Lemma 4.6

Moreover, by Proposition 3.5 the cocycle

. Thus β 1 0 is an isomorphism and H q (Γ; C(0)) = 0 for q = 1, 2. Now suppose that H q (Γ; C(i 0 )) = 0 for 0 ≤ i 0 ≤ n -3, q = 0, 1, 2 and consider the following short exact sequence of Γ-modules: 0 → C(i 0 ) C(i 0 + 1) ։ C(i 0 + 1)/C(i 0 ) → 0 .

(20)

Thus, f (n-1) (0) = a 11 (0)g (n-1) (0) = 0 and f (t) = 0 for sufficiently small t, t = 0. 2