� EXTRACTA MATHEMATICAE Volumen 33, Número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura EXTRACTA MATHEMATICAE Vol. 38, Num. 1 (2023), 105 – 123 doi:10.17398/2605-5686.38.1.105 Available online May 31, 2023 The character variety of one relator groups A. Cavicchioli, F. Spaggiari Dipartimento di Scienze Fisiche, Informatiche e Matematiche Universitá di Modena e Reggio Emilia, Via Campi 213/B, 41125 Modena, Italy alberto.cavicchioli@unimore.it , fulvia.spaggiari@unimore.it Received October 26, 2022 Presented by A. Turull Accepted April 23, 2023 Abstract: We consider some families of one relator groups arising as fundamental groups of 3- dimensional manifolds, and calculate their character varieties in SL(2, C). Then we give simple geometrical descriptions of such varieties, and determine the number of their irreducible compo- nents. Our paper relates to the work of Baker-Petersen, Qazaqzeh and Morales-Marcén on the character variety of certain classes of one relator groups, but we use different methods based on the concept of palindrome presentations of given groups. Key words: Finitely generated group, torus link, torus bundle, character variety, SL(2, C) repre- sentation, Kauffman bracket skein module. MSC (2020): 20C15, 57M25, 57M27. 1. Introduction Let G be a finitely presented group. A representation of G is a group homomorphism from G to SL(2,C). Two representations are said to be con- jugate if they differ by an inner automorphism of SL(2,C). A representation is reducible if it is conjugate to a representation into upper triangular matrices. Otherwise, the representation is called irreducible. The character variety of G is the set of conjugacy classes of representations of G into SL(2,C). The character variety of G is a closed algebraic subset of Cn for some n (see [8, 17]). The character variety of the fundamental group of any hyperbolic 3-man- ifold contains some topological informations about the structure of the given manifold (see [8, 25]). A general equation form for such character varieties does not exist in the literature. However, they have been calculated for many classes of (hyperbolic) 3–manifolds. Representations of two-bridge knot groups have been investigated in [3, 11, 23]. Character varieties of pretzel links and twisted Whitehead links have been determined in [27]. Recursive formulas for the character varieties of twist knots can be found in [13]. A very different method to determine the ISSN: 0213-8743 (print), 2605-5686 (online) c©The author(s) - Released under a Creative Commons Attribution License (CC BY-NC 3.0) https://doi.org/10.17398/2605-5686.38.1.105 mailto:alberto.cavicchioli@unimore.it mailto:fulvia.spaggiari@unimore.it https://publicaciones.unex.es/index.php/EM https://creativecommons.org/licenses/by-nc/3.0/ 106 a. cavicchioli, f. spaggiari character variety of twist knot groups has been proposed in [5]. The results are obtained by using special presentations of the knot groups, whose relators are palindromes (see [4]). This means that the relators read the same forwards or backwards as words in the generators. In this paper we propose a method to determine the character variety of a class of torus links which is different to that developed in [21]. Our method reduces the computations presented in the quoted paper, and permits to give an easy geometrical description of the character varieties of these torus links. Using such a description we also give simplified proofs of some algebraic results obtained in [21]. The method is then applied to the fundamental group of once-punctured torus bundles. Such manifolds can be obtained by (n+ 2, 1) Dehn filling on one boundary component of the Whitehead link (WL) exterior. Using the concept of palindrome word, we give a geometrical description of the character varieties of such torus bundles. This relates to the main result of [1], using very different techniques for computing character varieties. As a further new result, we then derive the character varieties of another family of bordered 3-manifolds, arising from (6n + 2, 2n + 1) Dehn filling on one boundary component of the WL exterior. 2. Technical preliminaries We think of SL(2,C) as the 2 × 2 complex matrices of determinant 1 in the set of 2×2 complex matrices M(2,C). It is known that every matrix A ∈ M(2,C) splits as the direct sum of a scalar multiple of the identity matrix plus a trace zero matrix. In particular, we can write A = A+ + A− = αI2 + A −, with σ(A) = 2 α and σ(A−) = 0, where σ(A) denotes the trace of the matrix A and I2 denotes the 2 × 2 identity matrix. So we can write A = α + A−. For A,B ∈M(2,C), set A+ = α, B+ = β, (A− B−)+ = γ, where α, β and γ represent complex numbers or scalar diagonal matrices depending on the context. We define two families of polynomials, which naturally arise from comput- ing the n-th powers of a matrix A ∈ SL(2,C). Write A = α + A− as above, and An = fn(α) + gn(α) A −, (2.1) where σ(A) = 2 α ∈ C. The polynomial fn can be expressed in terms of gn and gn−1. the character variety of one relator groups 107 Lemma 2.1. With the above notations, we have fn(α) = αgn(α) − gn−1(α). (2.2) Proof. Since (A−)2 = α2 − 1 from [5, Lemma 2.1(3)], it follows that An = AAn−1 = (α + A−) [fn−1(α) + gn−1(α)A −] = αfn−1(α) + (α 2 − 1)gn−1(α) + [fn−1(α) + αgn−1(α)]A−. Equating this formula and (2.1) yields fn(α) = αfn−1(α) + (α 2 − 1)gn−1(α) (2.3) and gn(α) = fn−1(α) + αgn−1(α). (2.4) Multiplying (2.4) by α, we get αgn(α) = αfn−1(α) + α 2gn−1(α). Using the last expression, we can eliminate αfn−1(α) from (2.3), that is, fn(α) = αgn(α) −α2gn−1(α) + (α2 − 1)gn−1(α), which gives (2.2). Moreover, we can derive the recursive expressions of fn and gn. Lemma 2.2. The families of polynomials {fn} and {gn} are defined by the recurrence formulas gn(α) = 2 αgn−1(α) − gn−2(α) (2.5) and fn(α) = 2 αfn−1(α) − fn−2(α) (2.6) for every n ≥ 1, with the initial values g−1(α) = −1 and g0(α) = 0, f−1(α) = α and f0(α) = 1, respectively. Proof. Substituting the expression of fn−1 from (2.2) into (2.4) yields gn(α) = αgn−1(α) −gn−2(α) + αgn−1(α), 108 a. cavicchioli, f. spaggiari which gives (2.5). Multiplying by α the formula of fn−1 from (2.2), we get αfn−1(α) = α 2gn−1(α) −αgn−2(α). Using the last expression, we can eliminate α2gn−1(α) from (2.3), that is, fn(α) = 2 αfn−1(α) + αgn−2(α) −gn−1(α). By (2.5) written for n− 1, we get fn(α) = 2 αfn−1(α) + αgn−2(α) − [2 αgn−2(α) −gn−3(α)] = 2 αfn−1(α) −αgn−2(α) + gn−3(α) = 2 αfn−1(α) − [αgn−2(α) −gn−3(α)]. This implies (2.6) as the expression inside the brackets is precisely fn−2(α) by (2.2). Lemma 2.3. The following identities g2n(α) = 1 + gn−1(α) gn+1(α) and 2 gn(α) α − g2n(α) = [gn+1(α) − 1] [1 − gn−1(α)] hold. Proof. The first formula is proved by induction on n. If n = 0, 1, 2, then g20 = 1 + g−1 g1 = 0, g 2 1 = 1 + g0 g2 = 1, and g 2 2 (α) = 1 + g1 g3 = 4α 2, respectively, as g−1 = −1, g0 = 0, g1 = 1, g2(α) = 2α, and g3(α) = 4α2 − 1. Using the inductive hypothesis and (2.5), we get 1 + gn−1(α) gn+1(α) = 1 + gn−1(α) [2αgn(α) −gn−1(α)] = 1 + 2αgn(α) gn−1(α) − g2n−1(α) = 1 + 2αgn(α) gn−1(α) − 1 − gn−2(α) gn(α) = gn(α)[2αgn−1(α) − gn−2(α)] = g2n(α). For the second equality, we have [gn+1(α) − 1] [1 − gn−1(α)] = gn+1(α) − gn+1(α) gn−1(α) + gn−1(α) − 1 = gn+1(α) + 1 − g2n(α) − 1 + gn−1(α) = 2 αgn(α) − gn−1(α) − g2n(α) + gn−1(α) = 2 αgn(α) − g2n(α) the character variety of one relator groups 109 by using the first equality of the statement and the recursive formula of gn(α) in (2.5). The polynomials {gn} are related to the n-th Chebyshev polynomial of the first kind Sn(x) (see [14]), that is, gn(α) = Sn−1(2 α). Furthermore, we also have gn(α) = Fn(2 α), where Fn denotes the n-th Fibonacci polynomial (see, for example, [1, 26]). Finally, gn relates with the Hilden-Lozano-Montesinos polynomial pn (see [10]) by the formula gn+1(α) = pn(2α). Further algebraic properties of polynomials fn and gn have been described in [5, Proposition 2.3]. Through the paper we also need the following result: Lemma 2.4. Let {a,b} be a set of generators of a 2-generator group G, and let ρ be an irreducible representation of G into SL(2,C). Setting A = ρ(a) and B = ρ(b), the set B = {I2,A−,B−, (A−B−)−} is a basis for the 4-dimensional vector space M(2,C). For a proof see, for example, [12, Lemma 1.2]. Furthermore, we implicitly use the well-known fact that a representation of a group with two generators a and b is determined by the traces of these generators and of their product ab (see, for example, [9]). 3. Torus links Let C(2n) denote the rational link in Conway’s normal form (see [15, p. 24]), which is the torus link depicted in Figure 1. It is the closure of the braid σ2n1 , where σ1 is the standard generator of the braid group B2 on two strands. Equivalently, it is the closure of the braid (σ2n−1 σ2n−2 · · · σ1)2 with σ1, σ2, . . . ,σ2n−1 being the standard generators of the braid group B2n on 2n strands. Note that the torus link C(2n) is given by T(2n, 2) according to Rolfsen’s notation [24]. Theorem 3.1. The character variety of the torus link C(2n), n ≥ 1, is defined by the equation (AB − BA) gn(α) = 0. The first factor determines the character variety for abelian representations into SL(2,C), and the second factor determines the character variety for non- abelian representations of the link group Gn. 110 a. cavicchioli, f. spaggiari Figure 1: The torus link C(2n), n ≥ 1. Proof. Let Gn denote the fundamental group of the exterior of C(2n) in the oriented 3-sphere S3, i.e., Gn = π1(S3\C(2n)). The group Gn admits the finite presentation 〈a,b : (ab)n = (ba)n〉. We provide a geometric inter- pretation of the generators of Gn by representing them in Figure 1. Setting u = ab and v = b (hence a = uv−1 and b = v), we get the finite presentation 〈u,v : unv = vun〉. Sending u and v to the matrices A and B, respectively, the last relation gives AnB = BAn in SL(2,C). For n ≥ 1, we have AnB = [fn(α) + gn(α)A −] (β + B−) = βfn(α)I2 + βgn(α)A − + fn(α)B − + gn(α)A −B− and BAn = (β + B−) [fn(α) + gn(α)A −] = βfn(α)I2 + βgn(α)A − + fn(α)B − + gn(α)B −A−. Computing the difference gives AnB − BAn = (A−B− − B−A−) gn(α) hence AnB − BAn = (AB − BA) gn(α) as A−B− − B−A− = AB − BA. This produces the defining relations of the character variety of C(2n) (or Gn). The techniques used in the above proof are different from those employed by Qazaqzeh in [21, Theorem 1.2]. For a given representation ρ of the group the character variety of one relator groups 111 Gn = 〈a,b : (ab)n = (ba)n〉 into SL(2,C), the cited author denotes by tr(x) the trace of ρ(x), for any word x in the generators a and b. Then tr(a), tr(b) and tr(ab) are abbreviated by t1, t2 and t3, respectively. His result states that the defining polynomial of the character variety of Gn is given by tr ( (ab)na−1b−1 ) − tr ( (ba)n−1 ) = (t23 + t 2 2 + t 2 1 − t3t2t1 − 4) Sn−1(t3), where the first (resp. second) factor on the right side determines the character variety for abelian (resp. nonabelian) representations. Here Sk(x) is the kth Chebyshev polynomial of the first kind, defined recursively by S0(x) = 1, S1(x) = x and Sk(x) = xSk−1(x) − Sk−2(x). The proof of this formula is given by induction on n, using the trace identities and the recursive definition of the Chebyshev polynomials. The same elementary methods in the proof of Theorem 3.1 can be used to obtain the defining polynomial of the character variety of a class of torus knots from [20] and the characters of certain families of one relator groups from [18, 19, 22]. Namely, the authors in [18, 19] consider the group G = 〈x,y : xm = yn〉 with m and n nonzero integers, and compute the number of irreducible components of the character variety of G in SL(2,C). A defining polynomial of the SL(2,C) character variety of the torus knot of type (m, 2) has been provided by Oller-Marcén in [20]. Recurrence formulas based on (generalized) Fibonacci polynomials have been proposed in [26, Theorem 7 and Theorem 11] to derive HOMFLY poly- nomials (and hence Alexander-Conway polynomials and Jones polynomials) of torus links C(2n). Generalized Fibonacci polynomials can be related to our classes of polynomials {fn} and {gn}, as remarked above. For n = 1, gn(α) = 1, hence the equation in Theorem 3.1 reduces to AB = BA, which determines the character variety for abelian representations into SL(2,C). So in the sequel, we discuss the case C(2n+2) with n being ≥ 1. Theorem 3.1 directly gives an easy geometrical description of the character variety of such torus links. Theorem 3.2. In the complex 3-space (X,Y,Z) the character variety for nonabelian representations of the torus link C(2n + 2) consists of the union of n horizontal planes of the form Zk = 2 cos[kπ/(n + 1)], for 1 ≤ k ≤ n. Proof. We set Z = 2 α = σ(A), X = 2 β = σ(B) = σ(B−1), and Y = σ(AB−1). From the relation AB−1 = (α + A−) (β − B−) = αβ I2 + β A− − αB− − A−B−, 112 a. cavicchioli, f. spaggiari it follows that Y = σ(AB−1) = 2 αβ − 2 γ as σ(A−) = σ(B−) = 0 and σ(A−B−) = 2 γ. The roots of the second factor gn(Z/2) = 0 are given by Zk = 2 cos[kπ/(n + 1)] for any 1 ≤ k ≤ n. See [5, Proposition 2.3(9)] and [10, Proposition 1.3]. Using the Chesebro formula for gn+1(α) (see [7]), we can give a different expression for the defining equation in Theorem 3.2. Corollary 3.3. In the complex 3-space (X,Y,Z) the character variety for nonabelian representations of the torus link C(2n + 2) is defined by the equation [ (Z + √ Z2 − 4)n+1 − (Z − √ Z2 − 4)n+1 2n+1 √ Z2 − 4 ] = 0 for −2 < Z < 2 (real number). To illustrate geometrically the support of the character variety in Theorem 3.2 and Corollary 3.3 we explicitly discuss the cases n = 1, . . . , 5. If n = 1, there is one horizontal plane of the form Z = 2 cos(π/2) = 0 from Theorem 3.2. The equation of the second factor in Corollary 3.3 becomes Z = 0. If n = 2, there are two horizontal planes with equations Z = 2 cos(π/3) = 1 and Z = 2 cos(2π/3) = −1 (see Theorem 3.2). The equation of the second factor in Corollary 3.3 becomes Z2 − 1 = 0. If n = 3, there are three horizontal planes with equations Z = 2 cos(π/4) =√ 2, Z = 2 cos(π/2) = 0, and Z = 2 cos(3π/4) = − √ 2. The equation of the second factor in Corollary 3.3 becomes Z(Z2 − 2) = 0. If n = 4, there are four horizontal planes with equations Z = 2 cos(π/5) = (1 + √ 5)/2, Z = 2 cos(2π/5) = ( √ 5 − 1)/2, Z = 2 cos(3π/5) = (1 − √ 5)/2, and Z = 2 cos(4π/5) = (−1 − √ 5)/2. The equation of the second factor in Corollary 3.3 becomes Z4−3Z2 +1 = 0, which has the four roots ±(1± √ 5)/2, as requested. If n = 5, there are five horizontal planes with equations Z = 2 cos(π/6) =√ 3, Z = 2 cos(π/3) = 1, Z = 2 cos(π/2) = 0, Z = 2 cos(2π/3) = −1, and Z = 2 cos(5π/6) = − √ 3. The equation of the second factor in Corollary 3.3 becomes Z(Z2 − 1)(Z2 − 3) = 0, which has the above roots. the character variety of one relator groups 113 As remarked in [24, Example 10], the genus of the torus link C(2n + 2) is n, which precisely coincides with the number of horizontal planes in the character variety of C(2n + 2), i.e., the degree of the polynomial gn+1(Z/2). Since the character varieties of Gn and Gm have different number of irre- ducible components if n 6= m, we derive the following well-known result (see [21, Corollary 1.3]). Corollary 3.4. The groups Gn and Gm are isomorphic if and only if n = m. Note that Corollary 3.4 also follows from the theory of Seifert manifolds since the torus link complement C(2n) is a Seifert fiber space with one excep- tional fiber. Let M be an oriented compact 3-manifold. Then the Kauffman bracket skein module K(M) of M is defined to be the quotient of the module freely generated by equivalence classes of framed links in M over Z[t,t−1] by the smallest submodule containing Kauffman relations (see [2] for more details). The topological meaning of this module has been explained in [2] for t = −1. More precisely, setting t = −1 and tensoring such a module with C produces a natural algebra structure, denoted K−1(M), over C. Furthermore, this algebra is canonically isomorphic to the coordinate ring of the character variety of π1(M) after factoring it by its nilradical (see [2, Theorem 10]). Then Theorem 3.2 allows to give a simplified proof of Theorem 1.4 from [21]. Theorem 3.5. Let M denote the exterior of C(2n + 2), n ≥ 1, in the oriented 3-sphere, K(M) the Kauffman bracket skein module of M, and N the (t + 1)-torsion submodule of K(M). Then the quotient K(M)/N is a free module over Z[t,t−1] with a basis B = {xiyjzk : i,j ≥ 0, 0 ≤ k ≤ n}, where x, y, and z represent the conjugacy classes of uv−1, v, and u in the presentation 〈u,v : unv = vun〉 of π1(M), respectively. Proof. By Theorem 3.2 the coordinate ring of the character variety of π1(M) admits the basis B (over C) indicated in the statement. In fact, the horizontal planes Z = 2 cos[kπ/(n + 1)], 1 ≤ k ≤ n, plus the neutral element for k = 0, give n + 1 conjugacy classes of the statement. By [21] the quotient of K−1(M) over its nilradical is isomorphic (over C) to K−1(M). Hence B is linearly independent (over C) in K−1(M). Then it is a basis for K(M)/N. For a description of K(M), when M is the exterior of a 2-bridge link, we refer to [16]. 114 a. cavicchioli, f. spaggiari 4. Once-punctured torus bundles Let us consider the once-punctured torus bundles with tunnel number one, that is, the once-punctured torus bundles that arise from filling one boundary component of the Whitehead link (WL) exterior. See Figure 2. Figure 2: A planar projection of the Whitehead link. The character varieties of such manifolds have been determined in [1]. Using the concept of palindrome word, we compute the defining polynomials of these character varieties with different techniques with respect to [1]. Up to homeomorphism, the monodromy of the once-punctured torus bundle Mn = (T × I)/Qn is Qn = τc1 τn+2c2 , where c1 and c2 are curves forming a basis for the fiber T (a torus) and τc means a right-handed Dehn twist about the curve c. Here I = [0, 1]. The manifold Mn can be obtained by (n + 2, 1) Dehn filling on one boundary component of the WL exterior, and it is the exterior of a certain genus one fibered knot in the lens space L(n + 2, 1). It is known that Mn is hyperbolic if and only if |n| > 2, contains an essential torus (i.e., is toroidal) if and only if |n| = 2, and is a Seifert fiber space if and only if |n| ≤ 1. See, for example, [1, Lemma 2.8]. By [1, Lemma 2.5], the fundamental group π1(Mn) is isomorphic to Γn = 〈a,b : a−n = b−1 ab2 ab−1 〉. (4.1) We provide a geometric interpretation of the generators of Γn by repre- senting them in Figure 2. We choose meridians µ0, µ1 and longitudes λ0, λ1 on the oriented components K0, K1 of WL, respectively, (see Figure 2) such the character variety of one relator groups 115 that [µi,λi] = 1, for i = 0, 1, and λi ∼ 0 in S3\Ki. Then we have µ0 = a−1, µ1 = a 2ba−1, λ0 = xab −1a−2 and λ1 = az, where x and z are represented in Figure 2. The Wirtinger presentation of the group π(WL) = π1(S3\WL) has generators a, b, x, y and z and relations ya−1 = aba−1, z = ab−1a−1ba−1, yx = a2ba−1y and xz = a−1x. Then we obtain the relation xz = a−1b−1ab2 after doing the appropriate elimination. Eliminating x = b−1ab2, y = ab and z = ab−1a−1ba−1 yields a finite presentation for π(WL) with generators a and b and relation b−1ab2ab−1a−1ba−1 = a−1b−1ab2. (4.2) A presentation for Γn can be obtained from that of π(WL) by adding the surgery relation µ −(n+2) 0 λ0 = 1 (4.3) where µ0 = a −1 and λ0 = xab −1a−2 = b−1ab2ab−1a−2. Substituting these formulas into (4.3) gives an+2b−1ab2ab−1a−2 = 1 hence anb−1ab2ab−1 = 1 which is equivalent to the relation in (4.1). Now (4.2) is a consequence of the relation in (4.1), so it can be dropped. In fact, we have the following sequences of Tietze transformations: (b−1ab2ab−1)a−1ba−1 = a−1b−1ab2, a−na−1ba−1 = a−1b−1ab2, a−n = b−1ab2ab−1, which is the relation of Γn. Theorem 4.1. For every n ∈ Z, let Mn be the once-punctured torus bundle of tunnel number one, and Γn = π1(Mn). In the complex plane (X,Z), the defining equation of the character variety of Γn is given by [gn+1(Z/2) − 1] [X2 − 1 + gn−1(Z/2)] = 0. In the hyperbolic case |n| > 2, the character variety for nonabelian represen- tations of Γn (or Mn) consists of the hyperelliptic curve given by X2 + gn−1(Z/2) − 1 = 0 116 a. cavicchioli, f. spaggiari and a finite number of horizontal lines (counted with their multiplicities) of the form Z = Zk, where Zk is a root of the equation gn+1(Z/2) − 1 = 0. Proof. From the relation in (4.1), or equivalently ba−nb = ab2a, sending a and b to the matrices A and B, respectively, gives the relation in SL(2,C) B A−n B = AB2 A which is palindrome in the left and right sides. Set A = α + A− and B = β + B−. As a direct application of the Cayley-Hamilton theorem, the formula A−n = fn(α) − gn(α) A− holds. By direct calculations on palindrome words, it follows B A−n B = q0 I2 + q1 A − + q2 B − where q0 = (2 β 2 − 1) fn(α) − 2 β γ gn(α) , q1 = −gn(α) , q2 = 2 β fn(α) − 2 γ gn(α) , with A+ = α, B+ = β and (A−B−)+ = γ, i.e., σ(A) = 2 α, σ(B) = 2 β and σ(A−B−) = 2 γ. As above, by direct computations on palindromes, we have AB2 A = q ′ 0 I2 + q ′ 1 A − + q ′ 2 B − where q ′ 0 = (2 α 2 − 1) (2 β2 − 1) + 4 αβ γ , q ′ 1 = 2 α (2 β 2 − 1) + 4 β γ , q ′ 2 = 2 β . Equating qi = q ′ i, i = 0, 1, 2, gives the defining polynomials of the character variety for Γn (or Mn). From q2 = q ′ 2 we derive an expression of γ in terms of α and β. So the representation (up to conjugacy) is only determined by the traces σ(A) = 2α and σ(B) = 2β. Substituting the cited expression of γ into q1 = q ′ 1 yields the defining equation of the character variety. In fact, q0 = q ′ 0 the character variety of one relator groups 117 is a consequence of the other equations. Thus the character variety of Γn has equation g2n(α) + 2 α (2 β 2 − 1) gn(α) + 4 β2 [fn(α) − 1] = 0. We can express fn(α) in terms of gn(α) and gn−1(α). Multiply out gives the equation g2n(α) + 2 α (4 β 2 − 1) gn(α) − 4 β2 [gn−1(α) + 1] = 0. Set Z = 2 α ∈ C and X = 2 β ∈ C. Then we get g2n(Z/2) + Z (X 2 − 1) gn(Z/2) − X2 [gn−1(Z/2) + 1] = 0 or, equivalently, g2n(Z/2) + [gn(Z/2) Z − gn−1(Z/2) − 1] X 2 − gn(Z/2) Z = 0 hence g2n(Z/2) + [gn+1(Z/2) − 1] X 2 − gn(Z/2) Z = 0. By Lemma 2.3, the defining equation of the character variety of Γn is given by the first formula in the statement. The last sentence of the theorem follows from [5, Proposition 2.3]. Since gn(Z/2) = Fn(Z), Theorem 4.1 relates to Theorem 5.1 from Baker and Petersen [1] in the sense that we obtain a similar hyperelliptic curve. More precisely, these authors prove that if |n| > 2, then there is a unique canonical component of the SL(2,C) character variety of Mn, and it is birational to the hyperelliptic curve given by w2 = −ĥn(y) ̂̀n(y) in the complex plane (w,y), where the polynomials ĥn and ̂̀n are specific factors of Fibonacci polynomials. If n is not congruent to 2 (mod 4), this is the only component of the SL(2; C) character variety which contains the characters of an irreducible representa- tion. If n ≡ 2 (mod 4), there is an additional component which is isomorphic to C. If n is not equal to −2, all the components consisting of characters of reducible representations are isomorphic to affine conics (including lines) and consist of characters of abelian representations. However, the methods used by the cited authors (based on the invariant theory) are similar to those developed by Qazaqzeh in [21] for the class of torus links. To illustrate geometrically the support of the character variety in Theorem 4.1 we explicitly discuss the hyperbolic cases n = 3, . . . , 6. 118 a. cavicchioli, f. spaggiari If n = 3, the equation X2 + g2(Z/2) − 1 = 0 becomes X2 + Z − 1 = 0 as g2(α) = 2α = Z. Furthermore, the equation g4(Z/2) − 1 = 0 becomes Z3 − 2Z − 1 = (Z + 1) (Z2 −Z − 1) = 0 as g4(α) = 8α 3 − 4α = Z3 − 2Z. Then, in the complex plane (X,Z), the character variety of Γ3 (or M3) consists of the parabola Z = 1 −X2 and the union of three horizontal lines with equations Z = −1 and Z = (1 ± √ 5)/2. If n = 4, the equation X2 + g3(Z/2) − 1 = 0 becomes X2 + Z2 − 2 = 0 as g3(α) = 4α 2−1 = Z2−1. Furthermore, the equation g5(Z/2)−1 = 0 becomes Z4 − 3Z2 = Z2(Z2 − 3) as g5(α) = 16α4 − 12α2 + 1 = Z4 − 3Z2 + 1. So, in the complex plane (X,Z), the character variety of Γ4 (or M4) consists of the ellipse X2 + Z2 = 2 and the union of four (counted with their multiplicities) horizontal lines with equations Z = 0 (counted twice) and Z = ± √ 3. If n = 5, the equation X2 + g4(Z/2) − 1 = 0 becomes X2 +Z3−2Z−1 = 0 or, equivalently, X2 + (Z + 1) (Z2 −Z−1) = 0. The equation g6(Z/2)−1 = 0 becomes Z5 − 4Z3 + 3Z − 1 = 0 as g6(α) = 32 α 5 − 32 α3 + 6 α = Z5 − 4 Z3 + 3 Z. Thus, in the complex plane (X,Z), the character variety of Γ5 (or M5) consists of the elliptic cubic (in fact, the Newton divergent parabola) of equation X2 = −Z3 + 2Z + 1 and the union of five horizontal lines with equations of the form Z = Zk, where Zk is a root of Z5 − 4Z3 + 3Z − 1 = (Z2 + Z − 1) (Z3 −Z2 − 2Z + 1) = 0. From the first factor we get Z1,2 = (−1 ± √ 5)/2. The equation Z3 −Z2 − 2Z + 1 = 0 becomes x3 + px + q = 0 with p = −7 3 and q = 7 27 by using the transformation Z = x+ 1 3 . Since ∆ = q2 4 + p3 27 = − 49 108 < 0, there are three real roots x1 = 2a, x2 = −a−b √ 3 and x3 = −a + b √ 3, where a + ib = 3 √ w and w = −q 2 + i √ ∆. If n = 6, the equation X2 + g5(Z/2) − 1 = 0 becomes X2 + Z4 − 3Z2 = 0. The equation g7(Z/2) − 1 = 0 becomes Z6 − 5Z4 + 6Z2 − 2 = (Z2 − 1) (Z4 − 4Z2 + 2) = 0 as g7(α) = 64 α 6 − 80 α4 + 24 α2 − 1 = Z6 − 5Z4 + 6Z2 − 1. the character variety of one relator groups 119 Thus, in the complex plane (X,Z), the character variety of Γ6 consists of the hyperelliptic quartic X2 = −Z4 + 3Z2 and the union of six horizontal lines with equations of the form Z = Zk, where Zk takes on the values ±1 and ± √ 2 ± √ 2. 5. Cusped manifolds from Dehn fillings For every n ≥ 0, let Nn be the one-cusped 3-manifold obtained by per- forming a (6n+ 2, 2n+ 1) Dehn filling on one boundary component of the WL exterior, leaving the other component open. See Figure 3. It is known that Nn is hyperbolic for every n ≥ 1. Among all fillings of one cusp of the Whitehead exterior we focus on the (6n+ 2, 2n+ 1) fillings since their fundamental group has a simple palindrome presentation. See (5.1) below. However, the proposed techniques for comput- ing character varieties of such manifolds can also be applied in the general case. By [6, Proposition 4.1] the fundamental group π1(Nn) is isomorphic to Λn = 〈a, b : aba = (b3 a−3)2n b3 〉 = 〈a, b : aba = b3 (a−3 b3)2n 〉. (5.1) We provide a geometric interpretation of the generators of Λn in Figure 3. Figure 3: Another planar projection of the Whitehead link. Theorem 5.1. For every n ≥ 0, let Nn be the one-cusped 3-manifold obtained by (6n + 2, 2n + 1) Dehn filling on one boundary component of the 120 a. cavicchioli, f. spaggiari WL exterior, and let Λn = π1(Nn). In the complex 3-space (X,Y,Z), the character variety of the group Λn (or Nn) is determined by the equations Y + (Z2 − 1)g2n(δ) = 0 , (X2 − 1)g2n+1(δ) − 1 = 0 , where δ is given by 2δ = X3Z3 − 2XZ3 − 2X3Z −X2Y Z2 + X2Y + Y Z2 + 5XZ −Y. Proof. From the relation in (5.1), sending a and b to the matrices A and B, respectively, gives the relation in SL(2,C) AB A = (B3 A−3)2n B3, which is palindrome in the left and right sides. By direct computations on palindromes, we obtain AB A = q̄0 I2 + q̄1 A − + q̄2 B − where q̄0 = (2α 2 − 1)β + 2αγ , q̄1 = 2αβ + 2γ , q̄2 = 1 , with A+ = α, B+ = β and (A−B−)+ = γ, as usual. Define L = B3A−3. Then L = δ + L−, where σ(L) = 2δ. We get L = p0I2 + p1A − + p2B − + p3A −B− where p0 = (4α 3 − 3α)(4β3 − 3β) − 2(4α2 − 1)(4β2 − 1)γ , p1 = −(4α2 − 1)(4β3 − 3β) , p2 = (4α 3 − 3α)(4β2 − 1) , p3 = (4α 2 − 1)(4β2 − 1) . Since σ(A−) = σ(B−) = 0 and σ(A−B−) = 2γ, we obtain δ = p0 + γp3. (5.2) the character variety of one relator groups 121 It follows that L− = −γp3I2 + p1A− + p2B− + p3A−B− (5.3) and L−B− = (β2 − 1)p2I2 + (β2 − 1)p3A− −γp3B− + p1A−B−. (5.4) Using (5.3) and (5.4) we obtain (B3 A−3)2n B3 = L2nB3 = [f2n(δ) + g2n(δ) L −] [4β3 − 3β + (4β2 − 1)B−] = q̄ ′ 0I2 + q̄ ′ 1A − + q̄ ′ 2B − where q̄ ′ 0 = (4β 3 − 3β)f2n(δ) + [(4β2 − 1)(β2 − 1)p2 − (4β3 − 3β)γp3]g2n(δ) , q̄ ′ 1 = −(4α 2 − 1)g2n(δ) , q̄ ′ 2 = (4β 2 − 1)f2n(δ) + [(4β3 − 3β)p2 − (4β2 − 1)γp3]g2n(δ) . By (2.2) and (5.2) and using the above expressions of p2 and p3 in terms of α and β, the polynomial q̄ ′ 2 becomes q̄ ′ 2 = (4β 2 − 1)[δg2n(δ) −g2n−1(δ)] + (4β2 − 1)δg2n(δ) = 2δ(4β2 − 1)g2n(δ) − (4β2 − 1)g2n−1(δ) = (4β2 − 1) g2n+1(δ). By Lemma 2.4, equating q̄i = q̄ ′ i, i = 0, 1, 2, gives the equations of the character variety of the group Λn. We see that q̄0 = q̄ ′ 0 is a consequence of the other two equations. We set Z = 2α = σ(A), X = 2β = σ(B), and Y = σ(AB) = 2αβ + 2γ. Solving α, β and γ as functions of X, Y and Z and substituting into p0 and p3, equation (5.2) becomes the formula of 2δ given in the statement. Expressing q̄1 = q̄ ′ 1 and q̄2 = q̄ ′ 2 in terms of X, Y and Z yields the first two equations in the statement of the theorem. Acknowledgements Work performed under the auspices of the scientific group G.N.S.A.G.A. of the C.N.R (National Research Council) of Italy and partially supported by the MUR (Ministero dell’Universitá e della Ricerca) of Italy within the project Strutture Geometriche, Combinatoria e loro Applicazioni, and by a research grant FAR 2022 of the University of Modena and Reggio Emilia. The authors would like to thank the anonymous referee for his/her useful comments and suggestions, which improved the final version of the paper. 122 a. cavicchioli, f. spaggiari References [1] K.L. Baker, K.L. Petersen, Character varieties of once-punctured torus bundles with tunnel number one, Internat. J. 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Knot Theory Ramifications 25 (2) (2016), 1650007, 16 pp. Introduction Technical preliminaries Torus links Once-punctured torus bundles Cusped manifolds from Dehn fillings