Representation Varieties of Twisted Hopf Links

In this paper, we study the representation theory of the fundamental group of the complement of a Hopf link with n twists. A general framework is described to analyze the SLr(C)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\textrm{SL}\,}}_r({\mathbb {C}})$$\end{document}-representation varieties of these twisted Hopf links as byproduct of a combinatorial problem and equivariant Hodge theory. As application, close formulas of their E-polynomials are provided for ranks 2 and 3, both for the representation and character varieties.


Introduction
This work studies a special type of algebraic invariants of 3-dimensional links.To be precise, given a link L ⊂ S 3 and a complex affine algebraic group G, we can form the so-called G-representation variety of the link R(L, G) = Hom(π 1 (S 3 − L), G), which parametrizes representations of the fundamental group of the link complement into G.This set can be naturally equipped with an algebraic structure in such a way that R(L, G) becomes a complex affine variety.In particular, its cohomology is endowed with a mixed Hodge structure from which we can compute the E-polynomial e(R(L, G)) = k,p,q (−1) k h k,p,q c (R(L, G)) u p v q ∈ Z[u, v], where h k,p,q Since the fundamental group of the link complement does not vary under diffeotopy of the link, the E-polynomial e(R(L, G)) is an algebraic invariant of the link L up to link equivalence.This E-polynomial provides an invariant encoding the algebraic structure of the representation variety attached to L, and typically differs from other classical invariants of L such as its Jones polynomial [16] or its A-polynomial [4].In fact, the geometry of the representation variety has been exploited several times in the literature to prove striking results of 3-manifolds.For instance, in the foundational work of Culler and Shalen [3], the authors used some simple properties of the SL 2 (C)-representation variety to provide new proofs of Thurston's theorem stating that the space of hyperbolic structures on an acylindrical 3-manifold is compact, and of the Smith conjecture, which claims that any quotient with cyclic stabilizers of a closed oriented 3-manifold with nontrivial branch knot is not simply-connected [3,Section 5].
Representation varieties also play a central role in mathematical physics.In the very influential paper [28], Witten applied Chern-Simons theory to geometrically quantize SU(2)-representation varieties of knot complements, leading to a Topological Quantum Field Theory that computes the Jones polynomial of the knot.In some sense, our approach of looking at the E-polynomial of the representation variety can be understood as an alternative quantization of the representation varieties, more similar to Fourier-Mukai transforms in derived geometry [15], in the sense that it consists of a pull-push construction (with identity kernel), than to path integrals as arising in Chern-Simons theory [28] (see [11] for more information).
For these reasons, the computation of the E-polynomials e(R(L, G)) has been object of intense research in the recent years.The representation variety of torus knots for G = SL 2 (C) was studied in [25], and for G = SU(2) in [24]; whereas the G = SL 3 (C) case was accomplished in [26], and recently the case G = SL 4 (C) in [12] through a computer-aided proof.More exotic knots have also been studied, as the figure eight knot in [14].However, despite of these advances for representation varieties of knots, almost nothing is known in the case of links.The most studied case is the character variety of trivial links, i.e. representations of the free group, addressed in works such as [6,7,18] (focused on the topology) and [1,8,19] (computing the E-polynomials).Very recently, more complicated links were studied, such as the twisted Alexander polynomial for the Borromean link in [2].
The aim of this work is to give the first steps towards an extension of the techniques to links.In particular, we shall focus on the "twisted" Hopf link H n , obtained by twisting a classical Hopf link with 2 crossings to get 2n crossings, as depicted in Figure 1.In this sense, R(H n , G) should be understood as the variety counting "supercommuting" elements of G, generalizing the case n = 1 of the usual Hopf link that corresponds to commuting elements, as studied in [9,22].
One of the main challenges we face in the study of the geometry of R(H n , G) is the analysis of the map p n : G → G, A → A n .In this paper, we propose to split this analysis into two different frameworks, that we call the combinatorial and the geometric.The combinatorial setting focuses on the study of the configuration space of possible eigenvalues, and how it can degenerate under the map p n .We will show in Section 4.1 that understanding these degenerations can be done systematically, and eventually it is performed by means of a thorough application of the inclusion-exclusion principle.
The geometric setting is discussed in Section 4.2, where we show how the E-polynomial of the representation variety can be obtained from the possible Jordan forms.To this aim, both the stabilizer of A n in SL r (C) under the conjugacy action (to parametrize the possible matrices B) and the stabilizer of A (through the conjugacy orbit of the Jordan form) play a role.Moreover, in the cases in which the Jordan form is not unique, but only unique up to permutation of eigenvalues, we show how the quotient by the corresponding symmetric group can be computed via equivariant Hodge theory, as developed in Section 2.2.
To show the feasibility of this approach, we apply it to the cases of rank 2 (Section 5) and rank 3 (Sections 7 and 8), obtaining the main result of this paper.
Theorem.The E-polynomials of the SL r (C)-representation variety of the twisted Hopf link H n with n twists for ranks r = 2, 3, are the following.
Additionally, in this paper we will go a step forward and also study the associated character varieties.The key point is that, if we want to obtain a genuine moduli space, we must identify isomorphic representations.This can be done by means of the GIT quotient of the representation variety R(L, G) under the adjoint action of G, giving rise to the so-called character variety It is well known [23] that every representation is equivalent, under the GIT quotient, to a semi-simple representation.The semi-simple representations are those that are direct sums of irreducible ones.Hence M(L, G) is stratified according to partitions of r, where G = SL r (C), corresponding to representations that are sums of irreducible representations of the ranks given in the partition.The E-polynomial of the reducible locus M red (L, G) is computed inductively from the irreducible representations of lower ranks.
In the case of the twisted Hopf link H n , to compute the E-polynomial e(M irr (H n , G)) we use the characterization that a representation is irreducible when A, B do not both leave invariant a proper subspace.This strategy is accomplished for rank 2 (Section 6) and rank 3 (Section 9), leading to the following result.
Theorem.The E-polynomials of the SL r (C)-character variety of the twisted Hopf link H n with n twists for ranks r = 2, 3, are the following.
It is worth mentioning that the strategies of computation described in this paper are not restricted to low rank, and work verbatim for arbitrary rank.However, the combinatorial analysis becomes exponentially more involved with increasing rank, so the higher rank cases are untreatable with a direct counting.An interesting future work would be to algorithmize the procedure of solving the combinatorial problem, so that the higher rank cases could be addressed via a computer aided-proof, as done in [12] for torus knots.
Finally, we would like to point out that this work is cornerstone to the understanding of representation varieties of general 3-manifolds.Recall that the Lickorish-Wallace theorem [21] states that any closed orientable connected 3-manifold can be obtained by applying Dehn surgery around a link L ⊂ S 3 .This highlights the importance of (i) studying representation varieties for general links, not only knots; and (ii) the key role that the maps p n (A) = A n play in this project, since they appear as part of the automorphism of the fundamental group of the torus around which surgery takes place.

Representation varieties and character varieties
Let Γ be a finitely generated group, and let G be a complex reductive Lie group.A representation of Γ in G is a homomorphism ρ : Γ → G. Consider a presentation Γ = ⟨γ 1 , . . ., γ k |{r λ } λ∈Λ ⟩, where Λ is the (possibly infinite) indexing set of relations of Γ.Then ρ is completely determined by the k-tuple (A 1 , . . ., A k ) = (ρ(γ 1 ), . . ., ρ(γ k )) subject to the relations r λ (A 1 , . . ., A k ) = Id, for all λ ∈ Λ.The representation variety is Therefore R(Γ, G) is an affine algebraic set.Even though Λ may be an infinite set, R(Γ, G) is defined by finitely many equations, as a consequence of the noetherianity of the coordinate ring of G k .
We say that two representations ρ and ρ ′ are equivalent if there exists g ∈ G such that ρ ′ (γ) = g −1 ρ(γ)g, for every γ ∈ Γ.The moduli space of representations, also known as the character variety, is the GIT quotient Recall that by definition of the GIT quotient for an affine variety, if we write R(Γ, G) = Spec A, then M (Γ, G) = Spec A G , where A G is the finitely generated k-algebra of invariant elements of A under the induced action of G.
A representation ρ is reducible if there exists some proper linear subspace W ⊂ V such that for all γ ∈ Γ we have ρ(γ)(W ) ⊂ W ; otherwise ρ is irreducible.If ρ is reducible, then there is a flag of subspaces 0 = W 0 ⊊ W 1 ⊊ . . .⊊ W r = V such that ρ leaves W i invariant, and it induces an irreducible representation ρ i in the quotient V i = W i /W i−1 , i = 1, . . ., r.Then ρ and ρ = ρ i define the same point in the quotient M(Γ, G).We say that ρ is a semi-simple representation, and that ρ and ρ are S-equivalent.The space M(Γ, G) parametrizes semi-simple representations [23,Thm. 1.28] up to conjugation.
The name 'character variety' for M(Γ, G) is justified by the following fact.Suppose now that G = SL r (C).Given a representation ρ : Γ → G, we define its character as the map χ ρ : Γ → C, χ ρ (g) = tr ρ(g).Note that two equivalent representations ρ and ρ ′ have the same character.There is a character map χ : R(Γ, G) → C Γ , ρ → χ ρ , whose image It turns out that this map is an isomorphism for G = SL r (C) cf.[23,Chapter 1].This is the same as to say that A G is generated by the traces χ ρ , ρ ∈ R(Γ, G).In other words, in this case M(Γ, G) is made of characters, justifying its name.However, for other reductive groups the map (2) may not be an isomorphism, as for For a general discussion on this issue, see [20].
2.1.Hodge structures and E-polynomials.A pure Hodge structure of weight k consists of a finite dimensional complex vector space H with a real structure, and a decomposition H = k=p+q H p,q such that H q,p = H p,q , the bar meaning complex conjugation on H.A Hodge structure of weight k gives rise to the so-called Hodge filtration, which is a descending filtration F p = s≥p H s,k−s .We define Gr p F (H) := F p /F p+1 = H p,k−p .
A mixed Hodge structure consists of a finite dimensional complex vector space H with a real structure, an ascending (weight) filtration ) and a descending (Hodge) filtration F such that F induces a pure Hodge structure of weight k on each Gr W k (H) = W k /W k−1 .We define H p,q := Gr p F Gr W p+q (H) and write h p,q for the Hodge number h p,q := dim H p,q .
Let Z be any quasi-projective algebraic variety (possibly non-smooth or non-compact).The cohomology groups H k (Z) and the cohomology groups with compact support H k c (Z) are endowed with mixed Hodge structures [5].We define the Hodge numbers of Z by The key property of Hodge-Deligne polynomials that permits their calculation is that they are additive for stratifications of Z.If Z is a complex algebraic variety and Z = n i=1 Z i , where all Z i are locally closed in Z, then e(Z) = n i=1 e(Z i ).Also e(X × Y ) = e(X)e(Y ) or, more generally, e(X) = e(F )e(B) for any fiber bundle F → X → B in the Zariski topology [10,Proposition 4.6].Moreover, by [22,Remark 2.5] if G → X → B is a principal fiber bundle with G a connected algebraic group, then e(X) = e(G)e(B).
When h k,p,q c = 0 for p ̸ = q, the polynomial e(Z) depends only on the product uv.This will happen in all the cases that we shall investigate here.In this situation, it is conventional to use the variable q = uv.If this happens, we say that the variety is of balanced type.Some cases that we shall need are: • e(P r ) = q r + . . .+ q 2 + q + 1.
2.2.Equivariant E-polynomial.We enhance the definition of E-polynomial to the case where there is an action of a finite group (see [19,Section 2]).
Definition 2.1.Let X be a complex quasi-projective variety on which a finite group F acts.Then F also acts on the cohomology H * c (X) respecting the mixed Hodge structure.So [H * c (X)] ∈ R(F ), the representation ring of F .The equivariant E-polynomial is defined as e F (X) = p,q,k Note that the map dim : R(F ) → Z recovers the usual E-polynomial as dim(e F (X)) = e(X).Moreover, let T be the trivial representation.Let ℓ be the number of irreducible representations of F , which coincides with the number of conjugacy classes of F .Let T = T 1 , T 2 , . . ., T ℓ be the irreducible representations in R(F ).Write e F (X) = ℓ j=1 a j T j .Then e(X/F ) = a 1 , the coefficient of T in e F (X).
We need specifically the case of the symmetric group S r .For instance, for an action of S 2 , there are two irreducible representations T, N , where T is the trivial representation, and N is the non-trivial representation.Then e S 2 (X) = aT We shall use later also the case of the symmetric group F = S 3 .Denote by α = (1, 2, 3) the 3-cycle and τ = (1, 2) a transposition.There are three irreducible representations T, S, D, where T is the trivial one, S is the sign representation, and D is the standard rerpresentation.The sign representation is one-dimensional S = R, where α ) is easily checked to be given by Let X be a variety with an S 3 -action.Then e S 3 (X) = aT + bS + cD.Then e(X) = a + b + 2c and e(X/S 3 ) = a.For the transposition τ = (1, 2) ∈ S 3 , we have An interesting case that we will also apply in Section 8 is the following.Proposition 2.2.Let G be a complex algebraic group equipped with an action of a finite group ρ : also is so.Hence, any inner automorphism is connected to the identity through a path, meaning that any inner automorphism is homotopic to the identity.Then, for all τ ∈ F , the map τ • : G → G is null-homotopic, so it induces a trivial action in cohomology.□

Twisted Hopf links
In this paper, we shall focus on the twisted Hopf link, which is the link formed by two circles knotted as the Hopf link but with n twists, as depicted in Figure 2. We will denote this knot by H n .Notice that the link H 1 is the usual Hopf link.
Proposition 3.1.The fundamental group of the (complement of the) twisted Hopf link with n ≥ 1 twists is Proof.Let us compute the Wirtinger presentation of the fundamental group of H n , which is a presentation of the fundamental group of the complement of a knot that can be obtained algorithmically from the crossings of a planar representation of the knot [27, Chapter III.D].Let us orient the two strands of H n as shown in Figure 2. From these orientation, we observe that π 1 (S 3 − H n ) is generated by 2n elements, namely x 1 , y 1 , . . ., x n , y n , corresponding to the 2n arcs of overpassing strands.For the relations, the knot has n double crossings of the form of Figure 3.  From each of these crossings, we obtain two relations y k x k = x k+1 y k and x k+1 y k = y k+1 x k+1 for k = 1, 2, . . ., n, writing x n+1 = x 1 and y n+1 = y 1 .Therefore, we get that From this relations, we can solve for y k and x k , for k ≥ 2, from x 1 and y 1 .Thus, the group can be also written as Making the change a = x 1 y 1 and b = y 1 we get the desired presentation.□ Remark 3.2.For n = 1, the group π 1 (S 3 − H 1 ) = Z × Z coincides with the fundamental group of the 2-dimensional torus, which is generated by two commuting elements.In some sense, π 1 (S 3 − H n ) generalizes this result by considering 'supercommutation' relations instead, of the form a n b = ba n .
Using the description (1) we directly get from Proposition 3.1 that the G-representation variety of the twisted Hopf link with n twists is To emphasize the role of the twisted Hopf link in the representation variety, throughout this paper we shall denote R(H n , G) = R(Γ n , G).
We can see ∆ r as a (coarse) configuration space, which is naturally stratified by equalities α i = α j .Here, we only need to consider a simple case of the Fulton-MacPherson stratification.Given an equivalence relation σ on {1, . . ., r} (equivalently, a partition of the set {1, . . ., r}), let us denote by ∆ r σ ⊂ ∆ r the collection of (α 1 , . . ., α r ) such that α i = α j if and only if i ∼ σ j.Observe that if σ = {υ 1 , . . ., υ s }, then there is a natural action of the group S σ := S t 1 × . . .× S tr on ∆ r σ by permutation of blocks, where t i is the number of subsets υ j of size |υ j | = i.
For n ≥ 1, there is a natural map p n : ∆ r → ∆ r given by p n (α 1 , . . ., α r ) = (α n 1 , . . ., α n r ).We say that σ ′ refines σ if the partition of σ ′ is obtained from that of σ by extra subdivisions.We indicate this as σ A key observation is that we can compute e(∆ r σ ′ →σ ) recursively.For instance, for notational simplicity, let us suppose that σ = {υ 1 , . . ., υ s }, In that case, we get that if (λ 1 , . . ., λ s+t ) ∈ ∆ r σ ′ →σ , since λ n s = λ n s+1 = . . .= λ n s+t , we must have λ s+k = λ s ε k for all k = 1, . . ., t and some pairwise different roots of unit ε k ∈ µ * n .Here, we denote Now, observe that if we remove any of the two later inequality conditions, we get a space of the form ∆ r σ′ →σ for some coarser σ′ and σ.Therefore, we can simply compute the E-polynomial of the space and then remove the strata corresponding to the different equalities using the inclusionexclusion principle.
Finally, observe that the action of S σ ′ does not restrict to an action on ∆ r σ ′ →σ .Instead, we find that there is a maximal subgroup S σ ′ →σ < S σ ′ acting on ∆ r σ ′ →σ namely, those permutations that preserve the refinement.4.2.The geometric setting.In this section, we show how the combinatorial set-up previously developed can be used to control important strata that appear in the representation varieties of Hopf links.To control the Jordan structure of the matrices, we need the following definition.Definition 4.2.A Jordan type of rank r ≥ 1 is a tuple ξ = (σ, κ) where σ = {υ 1 , . . ., υ s } is a partition of {1, . . ., r} and κ = {τ 1 , . . ., τ s } is a collection where τ i is a partition of υ i into linearly ordered sets.
A type ξ ′ = (σ ′ , κ ′ ) is said to refine ξ = (σ, κ), and we shall denote it by ξ ′ → ξ, if σ ′ is a refinement of σ and for any υ i ∈ σ that decomposes as The rationale behind a Jordan type is that it codifies the block structure of a Jordan matrix.On the one hand, the partition σ identifies the multiplicities of the eigenvalues: each of the sets υ i of the partition corresponds to a collection of equal eigenvalues (each number identifies the column of the eigenvalue).On the other hand, κ determines the inner structure of the Jordan blocks for each eigenvalue: each set of the partition τ i corresponds to a block of the Jordan matrix associated to an eigenvalue.The total order within this set is needed to identify the eigenvector column (the last element) and the off-diagonal elements of the Jordan form: if a is a succesor of b, then there is a 1 at the (a, b)-entry of the matrix.3), (4)}, {(5, 6)}} corresponds to Jordan matrices of the form , (4)}, {(2, 6)}} corresponds to Jordan matrices of the form There is a natural action of the symmetric group S r on the set of types by 'permutation of columns': relabel each element of {1, . . ., r} according to the permutation both in σ and τ i .We will say that two types are equivalent if they lie in the same S r -orbit.Notice that, by definition, in principle, for a type ξ = (σ, κ), the partition σ of {1, . . ., r} may not be into segments (i.e.equal eigenvalues may be sparse in the matrix).However, by 'putting together' the Jordan blocks, there always exists an equivalent type for which σ and each τ i are made of segments, and the total order in τ i agrees with the natural order in {1, . . ., r}.Given a type ξ of rank r, let us denote by A ξ the collection of Jordan matrices (with 1's as off-diagonal elements according to the total orders given by κ) of SL r (C) whose block structure is ξ.If we allow any non-zero off-diagonal element in the entries determined by κ, we will refer to these matrices as generalized Jordan matrices.The space of generalized Jordan matrices will be denoted A g ξ .We also consider Ãξ = SL r (C)•A ξ , where the action is by conjugation.If we pick a collection ξ 1 , . . ., ξ N of non-equivalent representatives of all the types of rank r, we get a decomposition Given a type ξ = (σ, κ), we have a group S ξ < S σ .This is the group of permutations of σ = (υ 1 , . . ., υ s ) that permute blocks υ i whose decompositions τ i are equivalent (of the same number of sets and of the same sizes).The group S ξ acts on A ξ as follows: let φ ∈ S ξ and A ∈ A ξ .So φ is a permutation of the blocks υ i of σ, that is, of the eigenvalues λ i .For such eigenvalues, the Jordan forms match exactly since the corresponding decompositions τ i are equivalent, therefore the matrix A ′ with the eigenvalues λ φ(i) lies in A ξ as well.Moreover, as this is given by a change of the basis by permutation of the vectors, there is a matrix P φ such that A ′ = P −1 φ AP φ .This P φ is well defined in PGL r (C)/ Stab(A) (quotient by action on the left).The action on PGL r (C)/ Stab(A) is by product of P φ on the right.Now, we consider the map and we set Moreover, if ξ ′ refines ξ, then there is a group S ξ ′ →ξ of permutations of ξ ′ (that is, permutations φ of σ ′ that respect the blocks of ξ ′ ) that induce permutations of ξ (it is enough that they induce a permutation on σ under the refinement σ ′ → σ).
Proof.A direct computation shows that, since taking powers preserves the Jordan block structure, if A ∈ A g ξ ′ , then A n can only lie in strata of the form A g ξ when ξ ′ refines ξ.
does not refine ξ.On the other hand, given A ∈ A ξ ′ →ξ with ξ ′ refining ξ, we observe that the off-diagonal elements of A are fixed to coincide with the Jordan structure.In this manner, the only freedom we have is to choose the eigenvalues of A, and this is precisely ∆ r σ ′ →σ .□ Let us denote by Observe that we have a natural stratification where each of the indices ξ ′ and ξ runs over a collection of non-equivalent representatives of all the types of rank r.We denote by Stab(ξ ′ ) the stabilizer in PGL r (C) of any matrix of A ξ ′ , and Stab(ξ) the stabilizer in SL r (C) of any matrix of A ξ .Note that the action of an element φ ∈ S ξ ′ →ξ is by permutation of columns in either PGL r (C), SL r (C), PGL r (C)/ Stab(ξ ′ ) and Stab(ξ).
Proposition 4.7.For any types ξ = (σ, κ) and ξ ′ = (σ ′ , κ ′ ), we have that The matrix P is determined up to Stab(A 0 ).On the other hand, B commutes with A n = P −1 A n 0 P , hence B 0 = P BP −1 lies in Stab(A n 0 ) = Stab(ξ).The pair (A, B) is determined by (A 0 , P, B 0 ).This is not unique, the matrix A 0 can be changed by an equivalent matrix A ′ 0 via an element φ ∈ S ξ ′ .The associated triple is In order for B ′ 0 to lie in the stabilizer of a matrix of A ξ , it is needed that whenever two eigenvalues λ i , λ j satisfy λ n i = λ n j , the permutation φ moves then to eigenvalues λ ′ i , λ ′ j such that (λ ′ i ) n = (λ ′ j ) n .This means that φ respects the partition σ, that is, it lies in S ξ ′ →ξ .□

Representation variety of the twisted Hopf link of rank 2
In this section, we shall compute the E-polynomial of the SL 2 (C)-representation variety of the twisted Hopf link H n , that is, the space R(H n , SL 2 (C)).For that purpose, we analyze the combinatorial and geometric settings as outlined in Section 4.

Combinatorial setting.
In rank 2 we have only 2 possible partitions, up to equivalence.These are The first one corresponds to SL 2 (C) matrices with equal eigenvalues, and the second one with distinct eigenvalues.Hence, we have σ 2 has a non-trivial action of S 2 , given by (λ 1 , λ 2 ) → (λ 2 , λ 1 ).For the degenerations, we have e(∆ 2 Now, let us analyze the quotients by S 2 .Recall that the action of S 2 on C * given by λ → λ −1 has quotient C * /S 2 = C, given by the invariant function s = λ + λ −1 .We have that Therefore, using (3) we get the equivariant E-polynomials:

Geometric setting.
From the two partitions σ 1 and σ 2 , we can create three types (up to equivalence), namely Recall that ξ 1 is the type of the diagonalizable matrices with equal eigenvalues (namely, ±Id), ξ 2 is the type of the two Jordan type matrices , and ξ 3 is the type of diagonalizable matrices with different eigenvalues.

Rank 2 character variety of the twisted Hopf link
In this section, we compute the E-polynomial e(M(H n , G)), for G = SL 2 (C).First, we deal with reducible representations (A, B).These are S-equivalent to representations of the form where (λ, µ) ∈ (C * ) 2 .These are defined modulo the action (λ, µ) Now we move to the irreducible representations, which form a space (1) For a type ξ and G = SL r (C), the space R irr (H n , G) ξ→ξ = ∅.
(3) For a type ξ not corresponding to a multiple of the identity, R irr (H n , G) ξ ′ →ξ = ∅.
Proof.(1) Let (A, B) ∈ R(H n , G) ξ→ξ , then A and A n are of the same type.Therefore, the eigenvectors of A and A n are the same.If A and A n are a multiple of the identity, then any vector subspace W ⊊ C r fixed by B is also fixed by A. Otherwise, take an eigenvalue λ of A such that the eigenspace and hence (A, B) is a reducible representation.
(2) Let (A, B) ∈ R(H n , G) ξ ′ →ξ , and suppose that λ is an eigenvalue of A such that the Jordan λ-block is not diagonal.Then E λ (A) is a proper subspace of the λ-block, and hence Again [A n , B] = Id implies that B(W ) = W , and hence (A, B) is a reducible representation.

Rank 3 representation variety of the twisted Hopf link. Combinatorial setting
In this section and the next one, we shall compute the E-polynomial of the SL 3 (C)representation variety of the twisted Hopf link H n .For that purpose, we will analyze the combinatorial and geometric settings as described in Section 4. We start in this section by analyzing the combinatorial setting.

Let us focus first on the natural extension of this action to {λ
).These are n − 1 different punctured lines and the action depends on the value of ε.If ε = −1 (which can only happen if n is even) the action is λ 1 → −λ 1 whose quotient is C * ; for ε ̸ = ±1, the action interchanges the pair of lines C * × {ε} to C * × {ε −1 } so the quotient is just one of them.In this way, where ⌊x⌋ is the floor function (the greatest integer less than or equal to x).Now we have to remove, the action is clearly free, and if ε = −1 then the action is free as well.Hence this accounts form 3n(n−1)/2 points.Therefore, putting all together we get Thus, we finally find that ( 16) • To study the remaining configuration, ∆ 3 σ 3 →σ 3 , observe that regarding decomposition (13) the action of S 3 on ∆ 3 σ 3 leaves invariant ∆ 3 σ 3 →σ 3 , ∆ 3 σ 3 →σ 1 , and . For this later action, we have Hence, using the previous computations we get Similarly, for the action of τ we have that ∆ 3 σ 3 →σ 3 , ∆ 3 σ 3 →σ 2 and ∆ 3 σ 3 →σ 1 are invariant, whereas it permutes ∆ 3 and therefore In this way, using (4), we get 8. Rank 3 representation variety of the twisted Hopf link.Geometric setting From the three partitions σ 1 , σ 2 , σ 3 in (12), we can create six types (up to equivalence), namely The type ξ j is given by the matrix A j , j = 1, 2, 3, 4, 5, 6, where Only in the case ξ 6 we have an action of the group S 3 given by the permutation of the eigenvalues λ i .
Only in the case of ξ 6 there is an action of S 3 .Let us give the equivariant Epolynomials.The quotient D/S 3 is parametrized by (s = and hence e(D/⟨τ ⟩) = q 2 − q.All together (4) yields (19) e S 3 (D) = q 2 T + S − qD.
Remark 9.1.There is an error in the calculation of the E-polynomial of M red 2,1 (H n , G) in the published version of this manuscript.In the case ε = −1, the equivariant Epolynomial of B/C * should be e S 2 (B/C * ) = (q − 1)T , and not e S 2 (B/C * ) = qT − N as computed in previous versions.The current version of the manuscript corrects this mistake and recomputes the final E-polynomial.

Figure 1 .
Figure 1.The twisted Hopf link of n twists.

Figure 2 .
Figure 2. The twisted Hopf link of n twists with oriented strands.

Figure 3 .
Figure 3.A crossing of the H n .

4 . 1 .
The SL r (C)-representation variety of the twisted Hopf link 4.The combinatorial setting.Given r ≥ 1, let us consider the space of possible eigenvalues of a matrix of SL r (C),