On SL(2,C)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$SL(2,{\mathbb {C}})$$\end{document}-representations of torus knot groups

The aim of this paper is to study the algebraic structure of the space R(Γn,m)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R(\Gamma _{n,m})$$\end{document} of representations of the torus knot groups, Γn,m=x,y:xn=ym\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma _{n,m}=\left\langle x,y:x^{n}=y^{m}\right\rangle$$\end{document}, into the linear special group SL(2,C)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$SL(2,{\mathbb {C}})$$\end{document}.


Introduction
Knot theory is understood as the study of the equivalence classes of embeddings of the circle S 1 or the disjoint union of m copies of S 1 into S 3 , considered up to ambi- ent isotopy of S 3 .A torus knot is a knot isotopic to one that lies on the boundary (V) of an unknoted solid torus V ⊂ S 3 .A knot K ⊂ S 3 is said to be hyperbolic if S 3 ⧵ K = ℍ 3 ∕Γ , where ℍ 3 is the hyperbolic 3-space and Γ is a discrete, torsion-free subgroup of Iso + (ℍ 3 ) , isomorphic to the fundamental group 1 (S 3 ⧵ K) of the knot complement S 3 ⧵ K.
R. Riley studied the geometry of knot complements throughout representation theory in which he considered SL(2, ℂ)-representations and used the theory of Haken manifolds to prove that the complement of the figure-eight knot is homeomorphic to ℍ 3 ∕Γ , where Γ is a Kleinian subgroup of PSL(2, ℂ) , see [36] and [35].Following this line, Riley also proved that many other knots are hyperbolic and he conjectured that the knots that are neither torus knots nor satellite knots are hyperbolic knots, see [37].Riley's work prepared the setting for Thurston's geometrization conjecture.In the 80's W. Thurston in [42] showed that every knot in S 3 is either a torus knot, a satellite knot or a hyperbolic knot.
It is well known that torus knots complements do not support an unique hyperbolic geometry, because it is not possible to get faithful representations of those groups that admit a presentation of the form ⟨x, y ∶ x n = y m ⟩ into PSL(2, ℂ) with Kleinian groups as their image.Despite that, in this paper we study representations of torus knot groups into SL(2, ℂ) , and among other things, we give a simple proof of the fact that the image of any SL(2, ℂ)-representation of 1 (S 3 ⧵K n,m ) has tor- sion elements.Thus, a natural question arose, what type of topological space ℍ 3 ∕Γ is, where Γ = ( 1 (S 3 ⧵ K n,m )) .On the other hand, the complement of a torus knot S 3 ⧵ K n,m can be decomposed into pieces M 1 , ⋯ , M k such that each of them have one of the eight types of geometric structure, this decomposition is studied from an irreducible decomposition of the SL(2, ℂ)-representation affine space R(Γ n,m ) of Γ n,m ∶= 1 (S 3 ⧵ K n,m ) , in this way it is of great interest to get a complete charac- terization of these irreducible components of R(Γ n,m ) .We know that the structure of the representation variety R(Γ n,m ) and the character variety X(Γ n,m ) have been widely studied and determined in many references, for example, in [28], Muñoz and Porti give a geometric description of the character variety X(Γ n,m ) of Γ n,m into SL(2, ℂ) , GL(2, ℂ) and PGL (2, ℂ) , in [16], Liriano computes the dimension of R(Γ) , where Γ is an one-relator group with presentation ⟨ x 1 , ⋯ , x n , y | w(x 1 , ⋯ , x n ) = y k ⟩ , w(x 1 , ⋯ , x n ) is a word in the free group F(x 1 , ⋯ , x n ) and k ≥ 2 , thereby, he proves that the dimension of R(Γ n,m ) is 4, see [16,Theorem 0.4], also in [17], he provides a formula to compute the number of four-dimensional irreducible components of R(Γ n,m ) ; following this line J. Martín-Morales and A. M. Oller-Marcen, in [22], give a complete description of the character variety X(Γ n,m ) and they prove that it is pos- sible, in most cases, to recover n, m from X(Γ n,m ) .Moreover, in [23], they com- pute the total number of irreducible components of R(Γ n,m ) and their corresponding dimension, extending, in this way, the work of S. Liriano.It is appropriate to emphasize here that they do not get such number from an explicit decomposition, but instead, by using a topological result and a well known irreducible decomposition of the character variety X(Γ n,m ) .Thus, as we note, there are a lot of important results in this line.In addition, in this paper, we give a description of the matrices and the explicit changes of basis of the representation variety R(Γ n,m ) .We also yield a com- plete characterization of reducible and irreducible SL(2, ℂ)-representations of torus knot groups; see also [29] and [30], and several results in order to compute, explicitly, non-abelian representations of torus knot groups.We also prove that the abelian representations in R(Γ n,m ) define an affine algebraic set that is a closed subset of the variety representation V(J n,m ) of Γ n,m , endowed with the Zariski topology.Thus, 1 3 São Paulo Journal of Mathematical Sciences (2023) 17:615-637 the non-abelian representations in R(Γ n,m ) correspond to open subsets of V(J n,m ).We denote by A(Γ n,m ) the abelian representations and by N(Γ n,m ) the non-abelian rep- resentations, then, R(Γ n,m ) = A(Γ n,m ) ∪ N(Γ n,m ) .We decompose these two subsets as a disjoint union of open sets.On the other hand, we obtain a decomposition of R(Γ n,m ) into closed subsets just by taking closures.Since, where N I (Γ n,m ) and N R (Γ n,m ) denote the set of non-abelian irreducible and reduc- ible representations, respectively, of Γ n,m , and ) ,, see [24] for more details.We also find the irreducible components of R(Γ n,m ) and some of their ideals.We prove that the closed subset A(Γ n,m ) is irreducible.Since the set of non-abelian representations in R(Γ n,m ) can be decomposed as a disjoint union of the set of reducible non-abelian representations and the set of irreducible non-abelian representations, we also study the irreducible components of each of these subsets.In the last section we use that decomposition in order to present a complete description of the character variety X(Γ n,m ) of the group of torus knots.
In this paper, the trace and transposed of the matrix A are denoted by tr(A) and A t , respectively.The following definition comes from the correspondence between SL(2, ℂ) and the group of Möbius transformations PSL(2, ℂ) .A complete classifica- tion of matrices in SL(2, ℂ) according to the value of the square of their trace can be found in [2,Theorem 4.3.4].
Definition 1 [2] Given a matrix A in SL(2, ℂ) such that A ≠ I , it is said to be a para- bolic matrix if tr 2 (A) = 4 , A is said to be an elliptic matrix if tr 2 (A) ∈ [0, 4) , the matrix is said to be a hyperbolic matrix if tr 2 (A) ∈ (4, ∞) , and A is a strictly loxo- dromic matrix if tr 2 (A) ∉ [0, ∞).
and � ∶ G → SL(2, ℂ) are said to be equivalent if there exists P ∈ SL(2, ℂ) such that, for every g ∈ G , (g) = P � (g)P −1 .Consider a finitely generated group G with finite presentation given by G = ⟨x 1 , x 2 , … , x n ∶ r 1 , r 2 , … , r m ⟩ .Then any representation ∶ G → SL(2, ℂ) is completely determined by the n-tuple ( (x 1 ), ⋯ , (x n )) ∈ SL(2, ℂ) n subject to the relations r j ( (x 1 ), ⋯ , (x n )) = I , for all j = 1, ⋯ , m .Thereby, we define the sub- set V(G) of SL(2, ℂ) n as Thus, using the natural embedding of SL(2, ℂ) into ℂ 4 , V(G) can be endowed with the structure of an affine algebraic set (the zero set of polynomials in ℂ 4n ).That is, V(G) is in bijection to the zero set in ℂ 4n of the polynomials given by the matrix entries of the group relations r j and by the determinant equal to one.Therefore, there exists a natural 1-1 correspondence between the set R(G), of representations of G on SL(2, ℂ), and the points of V(G) in which each representation ∶ G → SL(2, ℂ) is identified with the point ( (x 1 ), ⋯ , (x n )) .From this correspondence, we refer to R(G) as the space of representations of G in SL(2, ℂ) and also the algebraic variety.The space R(G) is well-defined in the following sense, for two finite sets of generators of G, the unique bijection between the corresponding spaces of representations which preserves the above identification is an isomorphism of algebraic sets, see [8], thereby, R(G) is well-defined up to isomorphism.Definition 2 A representation ∶ G → SL(2, ℂ) is called reducible if there exists a nontrivial subspace V of ℂ 2 , such that for each g ∈ G, (g)(V) ⊂ V .Otherwise, we say that is irreducible.
The character of a representation ∈ R(G) is the function ∶ G → ℂ , such that (g) = tr( (g)) , the set of all characters , ∈ R(G) , is denoted by (G) .For each g ∈ G , let g ∶ R(G) → ℂ , with g ( ) = tr( (g)) .Let T be the ring generated by all the functions g , for g ∈ G .It was proved in [8, Proposition 1.4.1] that if h 1 , ⋯ , h n are generators of G, then T is generated by the set of all functions h i 1 h i 2 ⋯h i k , where i 1 , ⋯ , i k are distinct numbers in {1, ⋯ , n} .Let g 1 , ⋯ , g s be a set of elements of G such that, g 1 , ⋯ , g s generate T. Define the map t ∶ R(G) → ℂ s , by and let X(G) = t(R(G)) .The proof of the following theorem can be found in [4, The- orem A] and [8,Corollary 1.4.5].

Theorem 1 X(G) is a closed algebraic set.
We also have that the function Λ: (G) → X(G) , given by Λ( ) = t( ) = (tr( (g 1 )), ⋯ , tr( (g s ))) is an injective function.In fact, let , ∈ R(G) be two representations of G such that t( ) = t( ) , then where n 1 , ..., n s are integer numbers and n 1 ,n 2 ,...,n s are complex numbers, such that n 1 ,n 2 ,...,n s = 0 except for a finite number of them.Given that g i ( ) = g i ( ) , for i = 1, 2, ..., s , then Due to the fact that, for every g ∈ G , g ∈ T, we have, = .Thereby, we identify the points of X(G) with the correspond- ing character, and so, X(G) is called the space of characters of the group G and t( ) = , ∈ R(G) .The space X(G) does not depend, up to canonical-isomor- phism, of the generating set of the ring T, so X(G) is well-defined.See [8] for more details.A subset K of S 3 is called a knot if there is an orientation-preserving embedding ∶ S 1 → S 3 , such that K = (S 1 ) .Two knots K and K ′ are said to be equivalent is there exists an orientation-preserving homeomorphism ∶ S 3 → S 3 , such that (K) = K � .A knot K ⊂ S 3 is usually represented by a diagram in the plane or S 2 , called knot dia- gram, which is the projection of it in some plane of S 3 , such that their intersections are transverse and no more than two points of K have the same projection.
A torus knot is a knot isotopic to a simple, closed curve D that lies on the boundary (V) of an unknoted solid torus V ⊂ S 3 .Since V is a solid torus, there are two simple closed curves (a meridian) and (a longitude) on the border (V) of V such that D = m + n , where m and n are positive integer numbers.In this case, we say that the torus knot is of type (n, m) and denoted by K n,m .When m and n are relative prime numbers, then K n,m has one component, see [3] and [7].
Let K be a knot in S 3 , then the fundamental group 1 (S 3 ⧵ K) of the knot com- plement S 3 ∖K or shortly called the knot group of K has a Wirtinger presentation [3, 26] and [40] for more details.
It is well known, see [25], that the group of the torus knot K n,m has the presenta- tion Γ n,m = ⟨x, y | x n = y m ⟩ .In this way, the ring T is generated by x , y , xy .This is because xy = yx .Therefore, the character variety of Γ n,m has the form: In the last section we present a more explicit description of such algebraic variety.
Lemma 1 With the above notation.The generators x and y are torsion free.Therefore, the center Z(Γ n,m ) of the group Γ n,m of the knot K n,m is not trivial.
Proof Suppose that there is a positive integer p such that y p = 1 .Then y p ∈ ⟨r⟩ , where r = x n y −m .Because, ⟨r⟩ = {wrw −1 | w ∈ F} , F is the free group on the set {x, y} , then y p = ∏ k i=1 w i r i w −1 i .Let us denote l y (w) and l x (w) the sum of the super- scripts of the occurrences of the letters x and y, respectively, in the word w ∈ F .So, On the other hand, Then, k ∑ i=1 i = 0 .Thereby, p = 0 .In a similar way, we prove that x is torsion free.Since ⟨y m ⟩ ⊂ Z(Γ n,m ) and ⟨y m ⟩ ≠ {1} , then Z(Γ n,m ) is not trivial.

Toroidal knot representations
In this section we prove some important properties about representations of torus knots groups.These results are necessaries for the understanding of the last section.

Lemma 2 Let us consider the infinite sequence of polynomials in
Let A be a matrix in SL(2, ℂ) and = tr(A).Then, for each positive integer k > 1, Proof We prove this by induction on the number k.If k = 2, then Thus, the lemma holds, when k = 2. Now suppose that We have that matrices, consisting of the image of the generators, and embed SL(2, ℂ) in ℂ 4 by using the entries of the matrices.Then R(Γ n,m ) is in bijection with the zero set in ℂ 8 of the polynomials given by the matrix entries of the group relation x n = y m and by the determinant equal to one.That is, A n = B m , det A = 1 and det B = 1 .On the other hand, by Lemma 2, we have that: From Theorem 3, the ideal J n,m corresponding to R(Γ n,m ), can be generated by Abelian representations in R(Γ n,m ) also define affine algebraic sets, more- over they correspond to a closed subset of V(J n,m ), with the Zariski topology.Thus, the non-abelian representations in R(Γ n,m ) correspond to an open subset of V(J n,m ).We denote the abelian representations by A(Γ n,m ) and the non-abelian ones by N(Γ n,m ) .Then, R(Γ n,m ) = A(Γ n,m ) ∪ N(Γ n,m ) .We study these two subsets in a more detailed way.We can obtain a decomposition of R(Γ n,m ) into closed subsets just by taking closures, but the unions are no longer disjoint.That is,

Decomposition of R(0 n,m ) as irreducible closed subsets
This section is devoted to find the irreducible components of R(Γ n,m ) and some of their ideals.We prove that the closed subset A(Γ n,m ) is irreducible.Then, since the set of non-abelian representations in R(Γ n,m ) can be decomposed as a disjoint union of the set of reducible non-abelian representations and the set of irreducible non-abelian representations, in this section we also study the irreducible components of each of these subsets.First we have that A(Γ n,m ) ≅ R((Γ n,m ) ab ).Moreover, if gcd(n, m) = 1 , then (Γ n,m ) ab is a cyclic group isomorphic to ℤ , see [26] Proposition 4.2.for an alge- braic proof.Thus, Any one of the structures in ( 1) is isomorphic to SL(2, ℂ) as it will be proved in the following proposition.Proof Let X, Y be two matrices in SL(2, ℂ).Assume that (X m , X n ) = (Y m , Y n ), that is, X m = Y m and X n = Y n .By Bezout's identity there exist integer numbers r and s such that 1 = rm + sn.Thus, X = X rm+sn = X rm X sn = Y rm Y sn = Y rm+sn = Y.So, f is an injective function.On the other hand, since gcd(n, m) = 1 and hence ⟨x, y ∶ x n = y m , 1 = [x, y]⟩ is a cyclic group, it follows that for each pair of matrices A, B in SL(2, ℂ) such that AB = BA and A n = B m , there exists X ∈ SL(2, ℂ) such that A = X m and B = X n .Therefore, the function f is onto.
Let n and m be two positive integers greater than or equal to 2. The following lemma will be useful to prove the non existence of non-abelian representations of Γ n,m in SL(2, ℂ) such that the images under it of x and y in SL(2, ℂ) are both para- bolic matrices.
Since the matrices 1 1 0 1 and a 0 a commute, then A and B commute, which contradicts the fact that is a non-abelian representation.
Proof We reason by contradiction, let us assume that there exists a non-abelian representation ∶ Γ n,m → SL(2, ℂ) such that A ∶= (x) is a parabolic matrix.From Theorem 5, the matrix B ∶= (y) is not parabolic.Then there exist non singular matrices P and Q such that and see [2].
Since A n = B m , then 1 n 0 1 and u m 0 0 u −m are conjugates, and hence . By factoring we have that (u 2m − 1) 2 = 0 and u 2m = 1 .From this, it follows that B 2m = I , and hence A 2n = I , which is a contradiction.
Similarly we can prove that there does not exist a non-abelian representation of Γ n,m in SL(2, ℂ) such that B ∶= (y) is a parabolic matrix.
Corollary 1 Let ∶ Γ n,m → SL(2, ℂ) be a non-abelian representation of Γ n,m in SL(2, ℂ).Then, (x) and (y) are not parabolic matrices.Proof From Theorem 5, (x) and (y) both are not parabolic matrices.Thus, from Theorem 6 the result holds. .We have that , then is a representation.On the other hand, AB ≠ BA , thus is non abelian.
As we can see, representations of groups of toroidal knots are not faithful.
and B = (y) , then A and B have finite order.
The following notation will be useful in order to simplify the proof and certain definitions given in the rest of this paper.For c ∈ ℂ, with c ≠ 0, we denote by

Lemma 5
Let A be a matrix in SL(2, ℂ) and n be a positive integer such that A n = ±I.Then there exists a non singular matrix P ∈ SL(2, ℂ) and a complex num- ber u ∈ ℂ such that A = PD(u)P −1 and u n = ±1.
Proof If A = I or A = −I, the result holds.Let us assume that A ≠ I and A ≠ −I.It is well known that the characteristic polynomial of the matrix A is ch A (x) = x 2 − tr(A)x + det(A).Thus, according to the number of different eigenval- ues of A, we have the following cases: 2 , for some u ∈ ℂ.Then, tr(A) = 2u, det(A) = u 2 = 1 , and hence u −1 = u.Given that A ≠ I and A ≠ −I, its minimal polynomial must be equal to ch A (x) and, from the canonical Jordan form, the matrix A is similar to a matrix in Jordan canonical form e. there exists a matrix P ∈ SL(2, ℂ) such that A = PJ(u)P −1 .We have that and, by induction, that J(u ).Thus, u n = ±1 and u −n (u + u 3 + u 5 + ⋯ + u 2n−1 ) = 0. Then u + u 3 + u 5 + ⋯ + u 2n−1 = 0 which contradicts u 2 = 1 and u ≠ 0. Therefore, this case is not possible. (2) São Paulo Journal of Mathematical Sciences (2023) 17:615-637 Case 2: If ch A (x) = (x − r 1 )(x − r 2 ), for some r 1 ≠ r 2 in ℂ.Then, tr(A) = r 1 + r 2 and det(A) = r 1 r 2 = 1.Thus, A is a diagonalizable matrix, and hence the matrix A is similar to a matrix in Jordan canonical form D(u), with u = r 1 and u −1 = r 2 i.e. there exists a matrix P ∈ SL(2, ℂ) such that A = PD(u)P −1 .We have, that and D(u) n = u n 0 0 u −n .Therefore, u n = ±1.

Lemma 6
Let u ∈ ℂ be a complex number such that u ≠ 0, u 2 ≠ 1 and let Proof First, we prove (1).We have that det u .Let us assume that a order to prove Equation (3), we consider the matrix P defined by for each y ∈ ℂ − {0}.First we observe that bc + (a − u) 2 ≠ 0 and, hence, P ∈ SL(2, ℂ).In fact, by contradiction, let us assume that bc + (a − u) 2 = 0 i.e.
which implies (au−1)(a−u)+bcu u = 0 and from this, (au − 1)(a − u) Now, by expanding the indicated operations on the left hand, we have that Let us assume that a = u.
follows that a 1 u − bc = 1 i.e. −bc = 0. Now, we can prove directly the equations in (4) by expanding their right sides and then simplifying.

Finally, if P =
x y z w , then any solution of the equation system AP − PD(u) = 0, det P = 1, is equal to one of the forms given in Lemma 6, depending on a ≠ u or a = u.

If a = u then
Proof The proof of this lemma is similar to the one of Lemma 6.
Then is reducible and a non-abelian representation if and only if there exist a matrix M ∈ SL(2, ℂ) and complex numbers u, v ∈ ℂ such that u n = ±1 and v m = ±1 and Let us assume that is reducible and non-abelian.Since is reducible, ) and u, v, a 12 , b 12 ∈ ℂ, with u, v ≠ 0. Now, given that is non-abelian, by Lemma 4 we have that ( (x)) n = ±I and ( (y)) m = ±I, and hence, by Lemma 5, u is a n-th root of unity and v is a m-th root of unity or u n = −1 and v m = −1 .Then, using (4) and (5), and Thus, taking w = Where u is a n-th root of unity and v is a m-th root of unity, or u n = −1 and v m = −1, with u 2 ≠ 1 and v 2 ≠ 1.
We have that is irreducible if and only if On the other hand, Thus, is a non abelian irreducible representation if and only if tz tz Then, is irreducible if and only if there exist matrices P, Q ∈ SL(2, ℂ) and complex numbers u, v ∈ ℂ such that u n = ±1 and v m = ±1, u 2 ≠ 1, v 2 ≠ 1, (x) = PD(u)P −1 , (y) = QD(v)Q −1 and the matrices P and Q satisfy one of the following properties: ) be a representation of Γ n,m in SL(2, ℂ).Let us assume that is irreducible and non-abelian.Since is non-abelian, by Lemma 4, we have that ( (x)) n = ±I and ( (y)) m = and hence, by Lemma 5, there exist P, Q ∈ SL(2, ℂ) and u, v ∈ ℂ, such that and with u a n-th root of unity and v a m-th root of unity or u n = −1 and v m = −1 .More- over, the matrices P and Q satisfy (3) or (4).For each pair P = (p ij ) and Q = (q ij ) of these matrices we define where

3
São Paulo Journal of Mathematical Sciences (2023) 17:615-637 Thus, each column of P is linearly independent with each column of Q if and only if R(P, Q) ≠ 0. Given that is irreducible, it follows that each column of P is linearly independent with each column of Q.On the other hand, by Lemma 6, we have that a 12 a 21 = 0 if and only if (a 11 − u)(1 − ua 11 ) = 0. Similarly, b 12 b 21 = 0 if and only if (b 11 − v)(1 − vb 11 ) = 0. Thus, taking into account the above and according to the different possibilities that result for a 12 , a 21 , b 12 and b 21 , we can compute R(P, Q) for all possible matrices P and Q satisfying (3) or (4).Then, the only possibilities for P and Q in order to be an irreducible representation correspond to one of the descriptions given in ( 1)-( 9), in the statement of the theorem.Reciprocally, suppose that (x) = A = PD(u)P −1 and (y) = B = QD(v)Q −1 , where u is a n-th root of unity and v is a m-th root of unity or u n = −1 and v m = −1 .Also, P and Q are matrices in SL(2, ℂ) that satisfy any of the properties (1) to (9).Then, R(P, Q) ≠ 0, and hence, each column of P is linearly independent with each column of Q.Thus, is irreducible.
It is clear that the non-abelian irreducible representations form an open subset of V(J n,m ).Furthermore, we should note that each of the properties (1)-( 9) defines a subset of V(J n,m ) which is a principal open set of some closed subset of V(J n,m ), with the Zariski topology.Thus, by Theorem 9 we have a decomposition of the set of irreducible non-abelian representations as an union of open principal sets.

Decomposition of X(0 n,m ) as irreducible closed subsets
In this section, we use the results developed in the previous one in order to get a complete decomposition of the character variety X(Γ n,m ).
We have that g ( ) = tr( (g)), for each ∈ R(Γ n,m ) and g ∈ Γ n,m .Then, Now, since R(Γ n,m ) = A(Γ n,m ) ∪ N(Γ n,m ), we can decompose the character variety X(Γ n,m ) as follows: where and Proposition 10 Let n, m be two positive integers such that n, m > 1 and gcd(n, m) = 1.Then, X A (Γ n,m ) is isomorphic to the algebraic set of all triplets such that t = tr(X) and X ∈ SL(2, ℂ).Each k (x) is the polynomial defined in (2).
Proof From Proposition 4, we have that the function f ∶ SL(2, ℂ) → A(Γ n,m ), defined by is a bijective morphism of algebraic sets.Then, On the other hand, by Lemma 2 it follows that such that (u, v) ∈ U (n,m) .That is, u is a n-th root of unity and v is a m-th root of unity, or u n = −1 and v m = −1; with u 2 ≠ 1 and v 2 ≠ 1, in both cases.

1 3 SãoTheorem 3
Paulo Journal of Mathematical Sciences (2023) 17:615-637 Therefore, the lemma holds.For each pair of positive integers m and n, with n, m > 1, the set R(Γ n,m ) of representations over SL(2, ℂ) can be endowed with the structure of an affine alge- braic set.Indeed, there exists a bijective function from R(Γ n,m ) to the algebraic set where A = (a ij ) , B = (b ij ) are matrices in SL(2, ℂ) and I = ( ij ) is the identity matrix.Moreover, = tr(A) and = tr(B) are polynomials in the diagonal entries of the matrices A and B, respectively.

Proposition 4
Let n, m be two positive integers such that n, m > 1 and gcd(n, m) = 1.Then the map f ∶ SL(2, ℂ) → A(Γ n,m ), defined by is a bijective morphism of algebraic sets.Thereby, A(Γ n,m ) is an irreducible affine algebraic variety of dimension 3.

− 1 = 1 . 5
B m−1 = B −1 B m and B −1 = d − b −c a , then It follows that Thus, we obtain that c = 0, −b + ak = −b + dk and hence Now, we can prove by induction in m that Thus, a m = 1 and k = ma m−1 b .Given that det(B) = 1 , it follows that a 2 = 1 , that is, a ∈ {1, −1}.Therefore, if m is odd, then a = 1 and b = k m .Besides, if m is even, then either a = 1 or a = −1 , so b = k m or b = − k m , respectively.In any case, Theorem If ∶ Γ n,m → SL(2, ℂ) is a non-abelian representation and A = (x) , B = (y) , then A and B both are not parabolic matrices.

Theorem 7 .
There exist non-abelian representations of Γ n,m in SL(2, ℂ) , for each n, m ≥ 2. Proof We consider three cases Case 1: n = 2 and m = 2 .Let us consider defined by (x) = A = We have that A 2 = B 2 , then is a representation.On the other hand, AB ≠ BA , thus is non abelian.Case 2: n = 2 and m = 3 or n = 3 and m = 2. Let us consider defined by (x) = A = 0 − 1 1 0 and (y) = B = 0 − 1 1 1

Case 3 :∶
n and m are positive integers n, m ≥ 3. Let n = e 2p n i and m = e 2q m i be two primitive roots of the unity, with n ≠ ±1 and m ≠ ±1 .Let us take Then A n and B m ∈ SL(2, ℂ) .Now, taking P = ⟨x, y ∶ x n = y m ⟩ → SL(2, ℂ) , defined by we obtain that is a representation.Now, if and only if, 2 n = 1 or 2 m = 1 .Therefore is a non abelian representation.