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Citation: Planat, M.; Amaral, M.M.; Fang, F.; Chester, D.; Aschheim R.; Irwin, K. Character Varieties and Algebraic Surfaces for the Topology of Quantum Computing. Symmetry 2022, 14, 915. https://doi.org/ 10.3390/sym14050915 Academic Editors: Stefan Heusler and Ivan Arraut Received: 19 April 2022 Accepted: 28 April 2022 Published: 30 April 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). symmetry S S Article Character Varieties and Algebraic Surfaces for the Topology of Quantum Computing Michel Planat 1,*,† , Marcelo M. Amaral 2,† , Fang Fang 2,† , David Chester 2,† , Raymond Aschheim 2,† and Klee Irwin 2,† 1 Institut FEMTO-ST CNRS UMR 6174, Université de Bourgogne-Franche-Comté, F-25044 Besançon, France 2 Quantum Gravity Research, Los Angeles, CA 90290, USA; marcelo@quantumgravityresearch.org (M.M.A.); fang@quantumgravityresearch.org (F.F.); davidc@quantumgravityresearch.org (D.C.); raymond@quantumgravityresearch.org (R.A.); klee@quantumgravityresearch.org (K.I.) * Correspondence: michel.planat@femto-st.fr † These authors contributed equally to this work. Abstract: It is shown that the representation theory of some finitely presented groups thanks to their SL2(C) character variety is related to algebraic surfaces. We make use of the Enriques–Kodaira classification of algebraic surfaces and related topological tools to make such surfaces explicit. We study the connection of SL2(C) character varieties to topological quantum computing (TQC) as an alternative to the concept of anyons. The Hopf link H, whose character variety is a Del Pezzo surface fH (the trace of the commutator), is the kernel of our view of TQC. Qutrit and two-qubit magic state computing, derived from the trefoil knot in our previous work, may be seen as TQC from the Hopf link. The character variety of some two-generator Bianchi groups, as well as that of the fundamental group for the singular fibers Ẽ6 and D̃4 contain fH . A surface birationally equivalent to a K3 surface is another compound of their character varieties. Keywords: SL2(C) character varieties; algebraic surfaces; magic state quantum computing; topologi- cal quantum computing; aperiodicity 1. Introduction Let M be a three- or a four-manifold. An important invariant of M is the fundamental group π1(M). It classifies the equivalence classes under the homotopy of the loops con- tained in M. If M is the complement of a link or a knot embedded in the three-dimensional space, π1(M) is called a knot group. The Wirtinger representation explicitly describes the knot group with generators and relations based on a diagram of the knot. In [1], the authors introduced a technique for describing all representations of a finitely presented group Γ in the group SL2(C). Representations of Γ in SL2(C) are ho- momorphisms ρ : Γ → SL2(C). The character of a representation ρ is a map κρ : Γ → C defined by κρ(γ) = tr(ρ(γ)), γ ∈ Γ. The set of characters of representations Γ in SL2(C) is R(Γ) = Hom(Γ, SL2(C)), which is a complex affine algebraic set. The set of characters is defined to be X(Γ) = { κρ|ρ ∈ R(Γ) } . Given a manifold M with fundamental group Γ = π1(M), one refers to the affine algebraic set X̃(π1(M)) as the character variety of M. The character varieties of some three- manifolds of the Bianchi type were investigated in [2] and more generally in [3]. In the latter reference, a Sage software was developed to make the character variety explicit [4]. In the present paper, one finds that such character varieties decompose into algebraic surfaces that can be recognized through the Enriques–Kodaira classification [5]. Such surfaces are candidates for a new type of topological quantum computing different from anyons [6–8]. Related ideas are in [9,10]. A previous work of our group [11,12] proposed to relate the fundamental group of some three-manifolds to quantum computing, but did not employ the representation theory. Symmetry 2022, 14, 915. https://doi.org/10.3390/sym14050915 https://www.mdpi.com/journal/symmetry Symmetry 2022, 14, 915 2 of 10 In Section 2, we introduce our mathematical concepts about algebraic surfaces, SL2(C) character varieties, and magic state quantum computing. The Hopf link H and the three- dimensional surface fH(x, y, z) = xyz− x2 − y2 − z2 + 4 lie at the “basement”. In Section 3, we investigate the “floors” starting with the Whitehead link and its cousins in the Bianchi group family. Then, we find that the affine E6 manifold—the two-covering of the affine E8 manifold (also the zero-surgery on the trefoil knot)—and affine D4 manifold—a three- covering—are other floors upon H and fH . The SL2(C) character variety of Ẽ6 is made of two K3 surfaces in addition to fH . In conclusion, we propose a few outlooks for future research. 2. Prolegomena 2.1. Algebraic Surfaces Given an ordinary projective surface S in the projective space P3 over a number field, if S is birationally equivalent to a rational surface, the software Magma [13] determines the map to such a rational surface and returns its type within five categories. The returned type of S can be the projective plane P2, a quadric surface for a degree two surface in P3, a rational ruled surface, a conic bundle, or a degree p Del Pezzo surface where 1 ≤ p ≤ 9. A further classification may be obtained for S in P3 if S has at most point singular- ities. Magma computes the type of the non-singular projective surfaces in its birational equivalence class from the classification of Kodaira and Enriques [5]. The first returned value is the Kodaira dimension of S, which is −∞, 0, 1, or 2. The second returned value further specifies the type within the Kodaira dimension −∞ or 0 cases (and is irrelevant in the other two cases). Birationally ruled surfaces have Kodaira dimension −∞. The second returned value is the irregularity q ≥ 0 of S. S is a rational surface that is birationally equivalent to a ruled surface over a smooth curve of genus q if and only if q is zero. Surfaces that are birationally equivalent to a torus, a bi-elliptic surface, a K3 surface, or an Enriques surface have Kodaira dimension 0. While an elliptic surface can have Kodaira dimension −∞, 0, or 1, all surfaces with Kodaira dimension 1 are elliptic surfaces (or quasi-elliptic in characteristics 2 or 3). Kodaira dimension two corresponds to algebraic surfaces of the general type. 2.2. The Hopf Link Let us anticipate the details of our approach of connecting knot/link theory, algebraic surfaces, and topological quantum computing. One takes the linking of two unknotted curves as in Figure 1 (left), and the obtained link is called the Hopf link H = L2a1, whose knot group is defined as the fundamental group of the knot complement in the three-sphere S3: π1(S3 \ L2a1) = 〈a, b|[a, b]〉 = Z2, (1) where [a, b] = abAB (with A = a−1, B = b−1) is the group theoretical commutator. There are interesting properties of the knot group π1 of the Hopf link that we would like to mention. First, the number of coverings of degree d of π1 (which is also the number of conjugacy classes of index d) is precisely the sum of divisor function σ(d) [14]. Second, there exists an invariance of π1 under a repetitive action of the Golden mean substitution (the Fibonacci map) ρ : a→ ab, b→ a or under the Silver mean substitution ρ : a→ aba, b→ a. The terms Golden and Silver refer to the Perron–Frobenius eigenvalue of the substitution matrix ([15], Examples 4.5 and 4.6). Such an observation links the Hopf link, the group π1 of the two-torus, and aperiodic substitutions. Using Sage software [4] developed by [3], the SL2(C) character variety is the polyno- mial: fH(x, y, z) = xyz− x2 − y2 − z2 + 4. (2) Symmetry 2022, 14, 915 3 of 10 As expected, the three-dimensional surface Σ : fH(x, y, z) = 0 is the trace of the com- mutator and is known to correspond to the reducible representations ([16], Theorem 3.4.1). A picture is given in Figure 1 (right). If we adopt the perspective of algebraic geometry by viewing Σ in the three-dimensional projective space as: ΣH(x, y, z, t) : xyz− t(x2 + y2 + z2) + 4t3 = 0, (3) then ΣH is a rational surface, more precisely a degree three Del Pezzo surface. It contains four simple singularities. In [9], the author proposed the representation π1 → SU(2)⊗ SU(2) as a model of two- qubit quantum computing in which each factor is associated with a single qubit located on each component of the Hopf link. Our project expands this idea by taking the representation π1 → SL(2,C) and the attached character variety ΣH as a model of topological quantum computing. Ideas in this direction are found in [17]. Figure 1. Left: the Hopf link. Right: a 3-dimensional picture of the SL2(C) character variety ΣH for the Hopf link complement. 2.3. Magic State Quantum Computing Since 2017, following the seminal paper [18], we have been developing a type of universal quantum computing based on magic states [19,20]. A magic state is a non- stabilizer pure state (a non-eigenstate of a Pauli group gate) that adds to stabilizer operations (Clifford group unitary operators, preparations, and measurements) in order to ensure the universality (the possibility of obtaining an arbitrary quantum gate). It has been recognized that some of the magic states are fiducial states for building a minimal informationally complete positive operator-valued measure (or MIC) of the corresponding Hilbert space dimension d, based on the action of the Pauli group Pd on the state. The lower-dimensional case is the qutrit MIC arising from the fiducial state fQT = (0, 1,±1). The next case is the two-qubit MIC arising from the fiducial state f2QB = (0, 1,−ω6, ω6 − 1) with ω6 = exp( 2iπ 6 ). For such magic/fiducial states, the geometry of triple products of projectors Πi = |ψi〉〈ψi| built with the d2 outcomes ψi is the Hesse configuration (for qutrits: in dimension d = 3) and the GQ(2, 2) configuration (for two qubits: in dimension d = 22) [19]. The latter configuration embeds the celebrated Mermin square configuration (a 3× 3 grid of observables) needed to prove the Kochen–Specker theorem. Our search for the magic states is performed with a two-generator infinite group G. A coset table over a subgroup H of index d is built by means of the Coxeter–Todd algorithm resulting in a permutation group. The latter may be seen as a d× d permutation matrix whose eigenstates are the candidates for a magic (and fiducial) state. We are dealing with low d values so that the choice is not large and many groups G do the job. Let us take G as the modular group Γ = PSL(2,Z) as in [20]. Then, the appropriate subgroups are in the family of congruence subgroups Γ0(N) of level N defined as the subgroups of upper Symmetry 2022, 14, 915 4 of 10 triangular matrices with entries taken modulo N. The index of Γ0(N) is the Dedekind function ψ(N). Thus, our relevant group for the qutrit magic state is the congruence subgroup Γ0(2), and for the two-qubit magic state, it is Γ0(3). Next, the two groups of interest Γ0(2) and Γ0(3) may be seen as fundamental groups of non-hyperbolic three-manifolds [11]. It is known that Γ is isomorphic to the funda- mental group π1(S3 \ 31) of the trefoil knot complement. The subgroups of interest of π1(S3 \ 31) ∼= Γ are attached to links L7n1 and L6a3 ([12], Table 4). Their presentation is as follows: π1(S3 \ L7n1) ∼= Γ0(2) = 〈 a, b|[a, b2] 〉 , π1(S3 \ L6a3) ∼= Γ0(3) = 〈 a, b|[a, b3] 〉 . Last but not least, in view of the presentation of the groups as commutators, the SL2(C) character variety of the two groups is that fH(x, y, z) of the Hopf link (apart for trivial factors y and y2 − 1), as shown in Table 1. One concludes that universal quantum computing based on the magic states fQT and f2QB is essentially topological quantum computing over the Hopf link. The underlying algebraic geometry for these models is the surface drawn in Figure 1. Table 1. Character varieties of fundamental groups whose reducible representations are that of the Hopf link. Column 1 identifies the group, as well as the corresponding link and 3- or 4-manifold. Column 2 is the name of the link or the relation it has to magic state quantum computing based on qutrits (QT) or two qubits (2QB). Column 3 is for the relation(s) of the two-generator fundamental groups. When the link is not the Hopf link, Column 4 is for the canonical component(s) of the representations and its (their) type as a surface in the 3-dimensional projective space. Link L Name Rel(s) Link Group π1(L) Character Variety f (x, y, z) L2a1 Hopf [a, b] = abAB fH = xyz− x2 − y2 − z2 + 4 - - - deg 3 Del Pezzo Γ0(2), L7n1 QT related [a, b2] y fH Γ0(3), L6a3 2QB related [a, b3] (y2 − 1) fH Γ−1(12), L5a1 Whitehead ab3a2bAB3A2B xy2z− y3 − x2y− xz+ 2y ooct0100001 WL - conic bundle, K3 type Γ−1(12), L13n5885 sister WL a2bAb2A2BaB2 x2y2 − xyz− x2 + 1 ooct0100000 - - deg 4 Del Pezzo, K3 type Γ−3(24), L6a2 Bergé ab3a2b2AB3A2B2 xy3z− x2y2 − y4 − xyz+ 3y2 − 1 otet0400001 - - conic bundle, general type Γ−7(6), L6a1 abABa2BAb3ABabA2baB3 undetermined Ẽ6 IV∗ a3b3, ab2aBA2B xy3 − y2z− x2 − 2xy+ z+ 2, - - - y4 − x2z+ xy− 4y2 + z+ 2 - - - K3 type 3. Character Varieties for Fundamental Groups of Three-Manifolds and the Related Algebraic Surfaces 3.1. The SL2(C) Character Varieties of Knot Groups Whose Reducible Component Is that of the Hopf Link We refer to some torsion-free subgroups of rank one of Bianchi groups. A Bianchi group Γk = PSL(2,Ok) < PSL(2,C) acts as a subset of orientation-preserving isometries of the three-dimensional hyperbolic space H3 with Ok the ring of integers of the imaginary quadratic field I = Q( √ −k). A torsion-free subgroup Γk(l) is the fundamental group π1 of a three-manifold defined by a knot or a link such as the figure-of-eight knot (with Γ−3(12)), the Whitehead link (with Γ−1(12)), or the Borromean rings (with Γ−1(24)). See [21,22] for more cases. Symmetry 2022, 14, 915 5 of 10 Looking at [22] (Table 1), We see that both the Whitehead link L5a1 and its sister L13n5885 (which have the same volume) are of the type Γ−1(12). They correspond to the three-manifolds ooct0100001 and ooct0100000, respectively. We are also interested in the links L6a2 = 622 (the Bergé link) and L6a1 = 6 2 3 [2], which are of the type Γ−3(24) and Γ−7(6), respectively. The latter link is related to the three-manifold otet0400001. The four links have fundamental groups of rank one as follows: π1(S3 \ L5a1) = 〈 a, b|ab3a2bAB3A2B 〉 , π1(S3 \ L13n5885) = 〈 a, b|a2bAb2A2BaB2 〉 , π1(S3 \ L6a2) = 〈 a, b|a2bAb2A2BaB2 〉 , π1(S3 \ L6a1) = 〈 a, b|ab3a2b2AB3A2B2 〉 . Remarkably, as for the Hopf link, we find that the cardinality structure of conjugacy classes of subgroups (card seq) of the fundamental group π1(S3 \ L) for the four links L is invariant under the repetitive action of the Golden mean and the Silver mean substitution. This points out an unexpected relationship of rank one Bianchi groups to aperiodicity. The card seq of such four groups is ηd(π1(S3 \ L5a1)) = [1, 3, 6, 17, 22, 79, 94, 412, 616, 1659, 2938, 10641, · · · ], ηd(π1(S3 \ L13n5885)) = [1, 3, 5, 12, 19, 60, 44, 153, 221, 517, 632, 2223, · · · ], ηd(π1(S3 \ L6a2)) = [1, 3, 4, 9, 24, 59, 71, 156, 262, 1208 · · · ], ηd(π1(S3 \ L6a1)) = [1, 3, 7, 23, 28, 134, 184, 694, 1353, 3466 · · · ]. Unlike the case of π1(L) for the Hopf link L =L2a1, with links of the Bianchi family, the Golden and Silver mean maps do not preserve the original group. Only the card seq of π1(L) is invariant for the above four links. We find that the three former links have a character variety with two components. The reducible component corresponds to the character variety of the Hopf link complement and, as described in the Introduction, is associated with a degree three Del Pezzo surface. The irreducible (or canonical component) is characterized below; see Table 1 for a summary. Using Sage software [4] developed by [3], the SL2(C) character variety for the three links L5a1, L13n5885, and L6a2 factorizes as the product of two polynomials: fH(x, y, z) f (x, y, z), where fH(x, y, z) is the character variety for the Hopf link complement as obtained in Section 2.2. The polynomial f (x, y, z) consists of the irreducible SL2(C) representations of π1(L). For the Whitehead link, one obtains f (x, y, z) = xy2z− y3 − x2y− xz + 2y. It is important to mention that the character variety for the group π1(L) depends on the selected Wirtinger representation. In [3] (Section 4.2), the relation for the fundamental group of the Whitehead link complement was taken to be abaB[A, B]ABAb[a, b] instead of the one obtained from SnapPy [23] so that the canonical component of the character variety contains an extra third-order term. Passing to the description of the surface f (x, y, z) in the three-dimensional projective space as the homogeneous polynomial Σ(x, y, z, t) = 0, the main algebraic properties of Σ remain the same whatever the choice of the Wirtinger representation of π1(L). For the Whitehead link, we find that the surface Σ is birationally equivalent to a conic bundle with a Kodaira dimension zero. More precisely, Σ belongs to the family of K3 surfaces. Figure 2 provides the canonical component of the character variety for the links L13n5885 and L6a2, whose algebraic description is a degree four Del Pezzo surface of the K3 family and a conic bundle of the general type, respectively. Unfortunately, we could not determine the character variety attached to the link L6a1. Symmetry 2022, 14, 915 6 of 10 Figure 2. The canonical component of character varieties for (a) the Whitehead link L5a1, (b) the Whitehead link sister L13n5885, and (c) the Bergé link L6a2. 3.2. The SL2(C) Character Variety of Singular Fiber IV∗ = Ẽ6 In [24], we found connections between Kodaira singular fibers and magic state quan- tum computing. The starting point of this viewpoint is the affine Coxeter–Dynkin diagram Ẽ8, which corresponds to the fiber I I∗ in Kodaira’s classification of minimal elliptic surfaces ([25], p. 320); see Figure 3. Alternatively, one can see Ẽ8 as the 0-surgery on the trefoil knot 31. The fundamental group of affine E8 manifold has the card seq: ηd(Ẽ8) = [1, 1, 2, 2, 1, 5, 3, 2, 4, 1, 1, 12, 3, 3, 4, . . .] (4) where the bold characters mean that at least one of the subgroups of the corresponding index leads to a MIC. The Seifert fibered toroidal manifold is the boundary of the manifold associated with Ẽ8 [26], denoted Σ′ in [11] (Table 5). Figure 3. A few singular fibers in Kodaira’s classification of minimal elliptic surfaces. (a) Fiber I∗0 (alias D̃4), (b) fiber IV∗ (alias Ẽ6), and (c) fiber I I∗ (alias Ẽ8). Symmetry 2022, 14, 915 7 of 10 For this sequence, the coverings are fundamental groups of ([24], p. 20): [Ẽ8, Ẽ6, {D̃4, Ẽ8}, {Ẽ6, Ẽ8}, Ẽ8, {BR0, D̃4, Ẽ6}, {Ẽ8}, {Ẽ6}, {D̃4, Ẽ8}, Ẽ6, · · · ] The subgroups/coverings are fundamental groups for Ẽ8 Ẽ6, D̃4, or BR0, where BR0 is the manifold found from zero-surgery on all circles of Borromean rings. One sees that the singular fiber IV∗ = Ẽ6 appears as the degree two covering of I I∗ = Ẽ8. The fundamental group is π1(S4 \ Ẽ6) = 〈 a, b|a3b3, ab2aBA2B 〉 , (5) where S4 is the four-sphere. We already found an invariance of the card seq of π1(L) under the Golden mean substitution (the Fibonacci map) or under the Silver mean substitution when L is the Hopf link and when π1(L) is in the Bianchi family of two-generator groups. We now observe that this invariance is preserved when L is the trefoil knot 31, its surgery Ẽ8 = 31(0, 1) and Ẽ6. Aperiodicity is a feature of all the fundamental groups we have encountered so far. Using Sage software [4], the SL2(C) character variety of group π1(S4 \ Ẽ6) factorizes as the polynomial product: fH(x, y, z)(x− y)(xy− z+ 1)(x2 + xy+ y2 − 3) f1(x, y, z) f2(x, y, z), where fH(x, y, z) is the SL2(C) character variety for the fundamental group of the Hopf link complement, f1(x, y, z) = xy3 − y2z− x2 − 2xy + z + 2 and f2(x, y, z) = y4 − x2z + xy− 4y2 + z+ 2. A plot of the latter surfaces is in Figure 4. Passing to the description of the surfaces f1(x, y, z) and f2(x, y, z) in the three-dimensional projective space as Σ1(x, y, z, t) and Σ2(x, y, z, t), we find that Σ1 is birationally equivalent to a conic bundle and Σ2 to the projective plane P2. Both surfaces show a Kodaira dimension 0 characteristic of K3 surfaces. The magic states from ηd(Ẽ8) at index 3 and 4 are fQT and f2QB, as in Section 2.3 for the manifolds L7n1 and L6a3, but their algebraic geometry is not the Hopf link. The associated SL2(C) character varieties are found to contain quadric surfaces y− z2 + 2 (as for ηd(Ẽ8) itself) and x2 + xy+ y2 − 3, respectively. Thus, the existence of a magic state is not sufficient for the issue of topological quantum computing. The concept of the SL2(C) character variety of the fundamental group adds topological and algebraic features not present in the fundamental group of the manifold. In this respect, the affine E6 manifold (the singular fiber IV∗) is a potential candidate. Figure 4. The surfaces f1(x, y, z) in (a) and f2(x, y, z) in (b) belonging to the character variety of singular fiber IV∗ = Ẽ6. Both surfaces are birationally equivalent to K3 surfaces. 3.3. The SL2(C) Character Variety of Singular Fiber I∗0 = D̃4 Singular fibers occur inside a (minimal) elliptic fibration. Let us pass to the generic elliptic fiber I0, a torus, and to the singular fiber I∗0 = D̃4 shown in Figure 3a. A neighbor- hood of the singular fiber inside a K3 surface leads to a plumbing diagram that is precisely I∗0 ([25], Figure 3.15, p. 133). Symmetry 2022, 14, 915 8 of 10 As shown in the previous section, the link D̃4 corresponds to the cyclic covering of degree three of π1(Ẽ8). The fundamental group is π1(S4 \ D̃4) = 〈 a, b, c|a2c2, b2c2, aBCaBc 〉 . (6) For this the three-generator group, the SL2(C) character variety is made of seven variable polynomials. Making use of the software available in [4], it has the form f (k, x, y, z, u, v, w) = ( fH(x, y, z) + wxk− 2k2) (uk2 + vx− 2u)(vk2 + ux− 2v)(wk2 + xk− 2W)(k3 + wx− 2k) (u2 − k2)(uv− wk)(v2 − k2)(uw− vk)(vw− uk)(w2 − k2)(uy− 2w) (vy− 2k)(wy− 2u)(uz− 2k)(vz− 2w)(wz− 2v)(yk− 2v)(zk− 2u), where fH(x, y, z) is the Hopf link polynomial of Section 2.2. Thus, a section at constant w and k of the character variety for the link D̃4 is simply a deformation of the character variety for the Hopf link, apart from trivial linear or quadratic polynomials. As for the other links L encountered so far, there exists an invariance of the card seq of π1(L) when L = D̃4. Let us apply the map a → b, b → abc, c → a on the three generators of L; the substitution map T = 0 1 1 1 1 0 0 1 0 is primitive since T3 >> 0, and the Perron–Frobenius eigenvalue is the real root of the polynomial λ3 − 2λ + 1 = 0, that is λPF ∼ 1.83928, the Tribonacci constant [27]; see also [15,28] (Section 4) for the mathematical details. This reveals the aperiodicity of the fundamental group. 4. Conclusions We discovered connections between the SL2(C) character varieties for the fundamental groups of some links, the theory of algebraic surfaces, and topological quantum computing. Our study was based on the Hopf link and links showing the Hopf link character variety as a component. In particular, we were concerned with links in the Bianchi family (such as the Whitehead link) and links for singular fibers in an elliptic fibration. The former define three-dimensional manifolds, while the latter correspond to four-dimensional manifolds. Our approach may connect to some theories of topological quantum field theory [29] and quantum gravity [30,31]. In particular, in loop quantum gravity [31], the quantum states of the gravitational field are described by a map between SU(2) and SL2(C) representations, the topological quantum computing aspects of which can be exploited. One starting point for future investigations may start from [32], where several mathe- matical connections of character varieties to other branches of mathematics were proposed. In particular, Cayley’s nodal cubic surface described as Equation (1) is in the family of smooth symmetric Fricke cubic surfaces [33]. The latter are isomorphic to a two-parameter family of character varieties for the exceptional group G2(C). The group G2(C) arises as the group of automorphisms of the complex octonions; its (unique) semisimple conjugacy class is six-dimensional and relates to the E8 root lattice thanks to the Fano plane represen- tation of octonions. Three elements of G2(C), obtained from three lines passing through a single point in the Fano plane, generate a finite simple subgroup of G2(C) isomorphic to G2(2)′ ∼= U3(3) of order 6048. Next, the group U3(3) stabilizes the split Cayley hexagon GH(2, 2)—a [633] configuration —and its dual ([34], Table 8). The 63 points of the hexagon may be encoded with three-qubit Pauli observables [35]; the hexagon embeds 12, 096 = 2× 6048 Mermin pentagrams (proofs of the Kochen–Specker theorem), which correspond to the number of automorphisms of G2(2) [36], and finally, the dual of the hexagon is obtained from the triple products of projectors defining the Hoggar SIC-POVM (symmetric informationally complete positive operator-valued measure) ( [19,37], Section 2.6 and Figure 3). Symmetry 2022, 14, 915 9 of 10 Author Contributions: Conceptualization, M.P., F.F., and K.I.; methodology, M.P., D.C., and R.A.; software, M.P.; validation, R.A., F.F., D.C., and M.M.A.; formal analysis, M.P. and M.M.A.; investiga- tion, M.P., D.C., F.F., and M.M.A.; writing—original draft preparation, M.P.; writing—review and editing, M.P.; visualization, F.F. and R.A.; supervision, M.P. and K.I.; project administration, K.I.; funding acquisition, K.I. All authors have read and agreed to the published version of the manuscript. Funding: Funding was obtained from Quantum Gravity Research in Los Angeles, CA, USA. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not Applicable. Data Availability Statement: Computational data are available from the authors. Conflicts of Interest: The authors declare no conflict of interest. References 1. Culler, M.; Shalen P.B. Varieties of group representations and splitting of 3-manifolds. Ann. Math. 1983, 117, 109–146. [CrossRef] 2. Harada, S. Canonical components of character varieties of arithmetic two-bridge link complements. arXiv 2012, arXiv:1112.3441. 3. Ashley, C.; Burelle J.P.; Lawton, S. Rank 1 character varieties of finitely presented groups. Geom. Dedicata 2018, 192, 1–19. [CrossRef] 4. Python Code to Compute Character Varieties. Available online: http://math.gmu.edu/~slawton3/Main.sagews (accessed on 1 May 2021). 5. Enriques–Kodaira Classification. 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