Adjoint Reidemeister torsions of some 3-manifolds obtained by Dehn surgeries

We determine the adjoint Reidemeister torsion of a $3$-manifold obtained by some Dehn surgery along $K$, where $K$ is either the figure-eight knot or the $5_2$-knot. As in a vanishing conjecture, we consider a similar conjecture and show that the conjecture holds for the 3-manifold.


Introduction
Let g be the Lie algebra of a semisimple complex Lie group G, and M be a connected compact oriented manifold.Let R irr G (M ) be the (irreducible) character variety, that is, the set of conjugacy classes of irreducible representations π 1 (M ) → G. Given a homomorphism φ : π 1 (M ) → G, we can define the adjoint (Reidemeister) torsion τ φ (M ) under a mild assumption, which lies in C × and is determined by the conjugacy class of φ; see [15] or Section 2 for details.When dim M = 2, the torsion plays an interesting role as a volume form on the space R irr G (M ); see [11,18].In addition, if M is 3-dimensional and G = SL 2 (C), some attitudes of the torsions in R irr G (M ) are physically observed from the viewpoint of a 3D-3D correspondence, and some conjectures on the torsions are mathematically proposed in [1,4,5].
For instance, with reference to [5], the conjecture can be roughly described as follows.Suppose that dim M = 3 and M has a tori-boundary.For z ∈ C, introduce a finite subset "tr −1  γ (z)" of R irr G (M ) which is defined from a boundary condition, and discusse the sum of the n-th powers of the twice torsions, that is, φ∈tr −1 γ (z) (2τ φ (M )) n ∈ C for n ∈ Z with n ≥ −1.Then, the studies in [1,4,5] suggest that the sum lies in Z and, that if M is hyperbolic and n = −1, then the sum is zero.This conjecture is sometimes called the vanishing identity; see [12,14,19] and references therein for supporting evidence of this conjecture.
In this paper, we focus on the adjoint torsions in the case where dim M = 3 and M has no boundary.According to [1,4], it is seemingly reasonable to consider the following conjecture: Conjecture 1.1 [1,4].Take n ∈ Z with n ≥ −1.Suppose that M is a closed 3-manifold, and the set R irr G (M ) is finite.Then, the following sum lies in the ring of integers Z: Furthermore, if G = SL 2 (C), M is a hyperbolic 3-manifold, and n = −1, then the sum is zero.
In [2], when G = SL 2 (C), the adjoint torsion of certain Seifert 3-manifolds and torus bundles are explicitly computed; thus, we can easily check the conjecture for the non-hyperbolic 3-manifolds.In contrast, this paper provides supporting evidence on Conjecture 1.1 in hyperbolic cases.For p/q ∈ Q and a knot K in S3 , let S 3 p/q (K) be the closed 3-manifold obtained by (p/q)-Dehn surgery on K.
We similarly discuss whether Conjecture 1.1 is true for M = S 3 1/q (K) when the knot K is the 5 2 -knot; see Section 4.
The outline of the proof is as follows.While some computations of the adjoint torsions of 3manifolds with boundary are established (see, e.g., [3,14,19]), this paper employs a procedure of computing the adjoint torsions of closed 3-manifolds, which is established in [16], and we determine all the adjoint torsion (Theorems 3.3 and 3.4).As in the previous proof of the above supporting evidence, we apply Jacobi's residue theorem (see Lemma 3.7) to the sum (1) and demonstrate Theorem 1.2.Since it is complicated to check the condition for applying the residue theorem, we need some careful discussion (see Sections 3.2-3.3)2 .Finally, in Section 5, we also discuss the conjecture with n > 0, and see that some properties are needed to be addressed in future studies.Here, we show the 2 2n+1 -multiple of the conjecture with M = S 3 2m/1 (4 1 ); see Proposition 5.4.

Review; the adjoint Reidemeister torsion
After reviewing algebraic torsions in Section 2.1, we briefly recall the definition of the adjoint Reidemeister torsion in Section 2.2.We note that our definition of the adjoint torsion is of sign-refined type.Section 2.3 explains cellular complexes of M .Throughout this paper, we assume that any basis of a vector space is ordered.

Algebraic torsion of a cochain complex
Let C * be a bounded cochain complex consisting of finite dimensional vector spaces over a commutative field F, that is, Let H i = H i (C * ) be the i-th cohomology group.Choose a basis c i of C i and a basis h i of H i .The Reidemeister torsion Tor(C * , c * , h * ) is defined as follows.Let h i ⊂ C i be a representative cocycle of h i in C i .Let b i be a tuple of vectors in C i such that δ i (b i ) is a basis of B i+1 = Im δ i .Then the union of the sequences of the vectors where α i (C * ) := i j=0 dim C j and β i (C * ) := i j=0 dim H j .Let c * be (c 0 , . . ., c m ) for C * and h * be (h 0 , . . ., h m ) for H * .Then, the torsion is defined to be the alternating product of the form It is known that the torsion Tor(C * , c * , h * ) does not depend on the choices of h i and b i , but depends only on c * and h * .We refer to [7,15] for the details.Note that, if C * is acyclic (i.e., H * (C * ) = 0), then the torsion Tor(C * , c * , h * ) is usually denoted by Tor(C * , c * ).
Remark 2.1.In [7] and [15], the torsion was defined from a chain complex; however, for convenience of computation, we define the torsion from a cochain complex in this paper.

Adjoint Reidemeister torsion of a 3-manifold
Let M be a connected oriented closed 3-manifold, and let G be a semisimple Lie group with Lie algebra g.Let φ : π 1 (M ) → G be a representation, that is, a group homomorphism.Suppose that G injects SL n (C) for some n ∈ N.
First, we introduce the cochain complex.Choose a finite cellular decomposition of M and consider the universal covering space M .We can canonically obtain a cellular structure of M as a lift of the decomposition of M , and define the cellular complex (C * ( M ; Z), ∂ * ).We regard the covering transformation of M as a left action of π 1 (M ) on M , and naturally regard C * ( M ; Z) as a left Z[π 1 (M )]-module.Since g is a left Z[π 1 (M )]-module via the composite of φ and the adjoint action G → Aut(g), we have the cochain complex of the form where δ i is defined by Next, we define an ordered basis of Here, c i,j is a lift of c i,j to M .Since g is semisimple, the Killing form B is non-degenerate, and we can fix an ordered basis B = (e 1 , e 2 , . . ., e dim g ) of g that is orthogonal with Here, δ j,ℓ is the Kronecker delta.Then the tuple provides an ordered basis of C i φ (M ; g) as desired.We next consider the cellular cochain complex C * (M ; R) with the real coefficient.Let c i j : Then, the adjoint Reidemeister torsion of M associated with φ is defined to be As is known [3,11], the definition of τ φ (M ) does not depend on the choices of the orthogonal basis B, finite cellular decompositions of M , c i , and h i R , but depends only on M and the conjugacy class of φ.Finally, we give a sufficient condition for the acyclicity, which might be known: Lemma 2.2.As in Conjecture 1.1, assume that R irr G (M ) is of finite order.Then, for any irreducible representation φ : π 1 (M ) → G, the associated cohomology H * φ (M ; g) is acyclic.
Proof.Since it is classically known [17] that the first cohomology H 1 φ (M ; g) is identified with the cotangent space of the variety R irr G (M ), it vanishes by assumption; by Poincaré duality, the second one does.Meanwhile, by definition, the zeroth cohomology H 0 φ (M ; g) equals the invariant part {a ∈ g | a • φ(g) = a for any g ∈ π 1 (M )}, which is zero by the irreducibility.Hence, the third one also vanishes by Poincaré duality again.

Presentations of the cellular complexes of M
From now on, we assume that G = SL 2 (C) and M is one of S 3 p/1 (4 1 ) and S 3 1/q (4 1 ) for some integers p and q ̸ = 0 as in Theorem 1.2.According to [8], group presentations of π 1 (M ) are given as follows: Here, [x, y] is xyx −1 y −1 .Let g be the number of generators of the group presentation above.Replace m by x 3 , m ′ by x 4 , and let r i denote the i-th relator in (2).Under the identifications , g), the cochain complex (C * φ (M ; g), δ * ) is isomorphic to the dual of the following chain complex: We now describe the differentials δ * in detail.Let F and P be the free groups ⟨x 1 , . . ., x g | ⟩ and ⟨ρ 1 , . . ., ρ g | ⟩, respectively.We define the homomorphism ψ : P * F → F by setting ψ(ρ j ) = r j and ψ(x i ) = x i .Let µ denote the natural surjection from F to π 1 (M ).According to [8, §3.1], we can describe δ * by the words of the presentations (2) as follows: let W ∈ P * F be ).Then, each δ * can be written as the matrices where ∂ * ∂ * is Fox derivative, see [15, §16] for the definition.Although each entry of the matrices is described in Z[π 1 (M )], we regard the entry as an automorphism of g via the adjoint action.
3 Proof of Theorem 1.2 The purpose of this section is to show the proof of Theorem 1.2.First, Section 3.1 determines the torsion with respect to every irreducible representation.Next, Section 3.2 establishes two key lemmas, and Section 3.3 completes the proof.Throughout this section, E 2 means the (2 × 2)-identity matrix, and we let G be SL 2 (C).

Preliminary
as in Figure 1, and define the Laurent polynomial ) for some integer p.If p ̸ = 0, then there is a bijection when the eigenvalues of φ(m) ) is a conjugacy class with a representation φ defined by If p = 0, then there is a bijection Proposition 3.2.Let M = S 3 1/q (4 1 ) for some integer q ̸ = 0.Then, there is a bijection Here, Φ M is defined by (5) as in Proposition 3.1.Note that, since ), ( 6) can be excluded from the definition in this case.Proof of Proposition 3.1.Let p ̸ = 0.For an irreducible representation φ : π 1 (M ) → SL 2 (C), take x, y, z, w ∈ C so that φ(x 1 ) = x y z w and xw −yz = 1.We first claim that φ(m) is diagonalizable.
Thus, x = w = 1 and y = 0; therefore, φ(x 1 ) and φ(x 2 ) are upper triangular matrices, which leads to a contradiction to the irreducibility.
By the above claim, we may suppose φ(m) = a 0 0 a −1 for some a ∈ D\{0}.Since we consider φ up to conjugacy, we may suppose y = 1.Thus, z = xw − 1.Since φ(r 1 ) = φ(r 2 ) = φ(r 3 ) = E 2 , with the help of a computer program of Mathematica, we have and Here, we fix a branch of the 1/2-th power on C × \ R, and define the signs η ∈ {±1} by setting When a = ε √ −1 for some ε ∈ {±1}, we have by the condition φ(r 1 ) = φ(r 2 ) = φ(r 3 ) = E 2 .In summary, the map Φ M is well-defined and injective.Finally, we can easily show the surjectivity of Φ M by following the reverse process of the above calculation.
Proof of Proposition 3.2.It can be proved in the same manner as Proposition 3.1.In this case, instead of (9), we have Theorem 3.3.Let M = S 3 p/1 (4 1 ) for some integer p ̸ = 0.For a ∈ Q −1 M (0) ∩ D as in Proposition 3.1, we denote the representative SL 2 (C)-representation of Φ −1 M (a) by φ a .Then, the adjoint Reidemeister torsion of M with respect to φ a is computed as Theorem 3.4.Let M = S 3 1/q (4 1 ) for some integer q ̸ = 0.For a ∈ Q −1 M (0) ∩ D as in Proposition 3.2, we denote the representative SL 2 (C)-representation of Φ −1 M (a) by φ a .Then, the adjoint Reidemeister torsion of M with respect to φ a is computed as Proof of Theorems 3.3 and 3.4.Under the identification of g ∼ = C 3 , we can concretely describe each δ i as the matrices according to (4) and the description of Φ M in the proofs of Propositions 3.1 and 3.2.Applying the τ -chain method in [15, §2.1] to the chain complex C * φ (M ; g), with the help of a computer program of Mathematica, we can directly obtain the resulting τ φa (M ).

Remark 3.5. (i) While this paper deals with the adjoint torsion via adjoint action, the classical
Reidemeister torsion of M = S 3 p/q (4 1 ) with respect to the SL 2 (C)-representation was computed in [6].
(ii) When M = S 3 p/1 (4 1 ), the torsion τ φ (M ), up to sign, was computed in [10].The advantage of Theorem 3.3 is that the sign of the torsion is recovered; thus, we can compute the sum of τ φ (M ) n 's, as is seen later.

Two key lemmas
As preliminaries of the proof of Theorem 1.2, we prepare two lemmas: for some m ∈ Z.Then, Q M (x) with M = S 3 p/1 (4 1 ) is divisible by κ p (x), and the quotient Q M (x)/κ p (x) has no repeated roots.On the other hand, Q M (x) with M = S 3  1/q (4 1 ) is divisible by (1 + x) 2 , and the quotient Q M (x)/(1 + x) 2 also has no repeated roots.
Proof.The required statement with |p| ≤ 4 and |q| ≤ 4 can be directly shown, we may assume |p| ≥ 5 and |q| ≥ 5. We first focus on the case M = S 3 p/1 (4 1 ).By a computation of we can easily verify the multiplicity of Q M (x).To elaborate, if p = 2m + 1, then are all nonzero, which implies that 1, ± √ −1 are not roots of Q M (x).Furthermore, Applying ( 16) to (15) to kill the term a p , we equivalently have Since a 2 ̸ = ±1, the last quartic term equation can be solved as for some ε, η ∈ {±1}.Let F be the field extension Q(a) of degree 8. Let us regard (15) as a quadratic equation in F of a p .Since the discriminant is not zero and |p| > 4, a p does not lie in F .This is a contradiction.In summary, Q M (x)/κ p (x) has no repeated roots as required.
On the other hand, if M = S 3 1/q (4 1 ), we can easily show that Q M (x) is divisible not by (1 + x) 3 but by (1 + x) 2 .Similarly, suppose Q M (x) has a repeated root a ∈ C with a ̸ = ±1.Then, 2(1 + a −1 ) = (2q + 1)a −2q + (−2q + 1)a 2q − (4q + 1)a −4q − (−4q + 1)a 4q . ( the substitution of ( 17) and ( 18) into (19) gives the equation where we replace a 2q by b.If ω 2q = ±1 and ω ∈ C, we can easily check Q M (ω) ̸ = 0 by definition.Thus, a q is a solution of the quartic equation in (20) and does not lie in Q, for any q ∈ Z.Let F/Q be the field extension by the quartic equation.By definition, F does not contain a and 2+a+a Then, the following sum is zero: Proof.If ζ = 0, the statement is Jacobi's residue theorem exactly (see, e.g., [14,Section 6]).Thus, we may suppose ζ = 2.Note that the derivative of (1+x Hence, the left hand side of (21) is computed as a∈k −1 (0) g(a)/k ′ (a), which is equal to zero by the residue theorem.
The proof of the cases of p ≤ −5 and q ≤ −2 can be shown in the same manner; so we here do not carry out the detailed proof.
Finally, in the remaining cases of |p| ≤ 4 for M = S 3 p/1 (4 1 ), we can obtain the following by a direct calculation: For example, we now discuss the detail in the case p = 4 for M = S 4 Surgeries on the 5 2 -knot We discuss Conjecture 1.1 in the case of M = S 3 1/q (K), when K is the 5 2 -knot and |q| ≥ 3. Since the outline of the discussion in this section is almost the same as that in Section 3, we now roughly describe the discussion.