Quantum Topol. 14 (2023), 407–428 © 2023 European Mathematical Society DOI 10.4171/QT/180 Published by EMS Press This work is licensed under a CC BY 4.0 license The adjoint Reidemeister torsion for the connected sum of knots Joan Porti and Seokbeom Yoon Abstract. Let K be the connected sum of knots K1 ; : : : ; Kn . It is known that the SL2 .C/- character variety of the knot exterior of K has a component of dimension  2 as the connected sum admits a so-called bending. We show that there is a natural way to define the adjoint Rei- demeister torsion for such a high-dimensional component and prove that it is locally constant on a subset of the character variety where the trace of a meridian is constant. We also prove that the adjoint Reidemeister torsion of K satisfies the vanishing identity if each Ki does so. 1. Introduction Let M be a compact oriented 3-manifold with tours boundary and X.M / be the char- acter variety of irreducible representations 1 .M / ! SL2 .C/. It happens very often that X.M / has a component of dimension 1. For instance, if the interior of M admits a hyperbolic structure of finite volume, then the distinguished component is 1-dimen- sional [15] and if M contains no closed essential surface, then every component is 1-dimensional [3]. Once we fix a simple closed curve  on the boundary torus @M , the adjoint Reidemeister torsion is defined as a meromorphic function on each 1-dimensional component of X.M / under some mild assumptions [6, 14]. It enjoys fruitful inter- action with quantum field theory and carries several conjectures consequently. See, for instance, [5, 7, 12]. Recently, it has been conjectured in [8] that the adjoint Reide- meister torsion satisfies a certain vanishing identity with respect to the trace function as follows. Conjecture 1.1. Suppose that the character variety X.M / consists of 1-dimensional components and that the interior of M admits a hyperbolic structure of finite volume. Then for generic c 2 C we have X 1 D0 (1) c  .M I / Œ2X .M / 2020 Mathematics Subject Classification. Primary 57K10; Secondary 57K31. Keywords. Connected sum, Reidemeister torsion, vanishing identity, character variety. J. Porti and S. Yoon 408 where Xc .M / is the pre-image of c 2 C under the trace function X.M / ! C of   @M and  .M I / is the adjoint Reidemeister torsion associated to  and a rep- resentation W 1 .M / ! SL2 .C/. As mentioned earlier, there are several 3-manifolds satisfying the conditions required in Conjecture 1.1. However, there are also several examples of 3-manifolds with torus boundary whose character varieties have high-dimensional components. The simplest one might be (the knot exterior of) the connected sum of knots. See [2, 4, 13] for other examples. Two immediate problems when we consider Conjec- ture 1.1 for such 3-manifolds are that (P1) the adjoint Reidemeister torsion is not defined for a component of dimension  2; (P2) the sum in equation (1) does not make sense as the level set Xc .M / is no longer finite. Related to these problems, we address the following question. Question 1.2. Is the adjoint Reidemeister torsion defined and locally constant on Xc .M /? If the answer of Question 1.2 is positive, then the sum in equation (1) makes sense for M in an obvious way: we understand the sum by taking one representative on each connected component of Xc .M /. The main purpose of the paper is to investigate Question 1.2 and Conjecture 1.1 for the connected sum of knots. To state our results, let K be the connected sum of knots K1 ; : : : ; Kn in S 3 with a meridian . We denote by M and Mj the knot exteriors of K and Kj , respectively. For technical reasons, we assume that for 1  j  n (C) the level set Xc .Mj / consists of finitely many -regular characters with the canonical longitude having trace other than ˙2 for generic c 2 C. For example, one may choose Kj as a two-bridge knot or a torus knot. It is known that the character variety X.M / has a component of dimension  2 as the connected sum admits a so-called bending. We refer to [10, 11, 13] for details on the bending construction. Our main theorems are as follows. Theorem 1.3. Let K be the connected sum of knots K1 ; : : : ; Kn in S 3 satisfying the above condition (C) and  be a meridian. Then there is a natural way to define the adjoint Reidemeister torsion on Xc .M / for generic c 2 C which is locally constant. Theorem 1.4. Let K be the connected sum of knots K1 ; : : : ; Kn in S 3 satisfying the above condition (C) and  be a meridian. Then the knot exterior M of K satisfies equation (1) if each Mj does so. The adjoint Reidemeister torsion for the connected sum of knots 409 It is proved in [18] that every hyperbolic two-bridge knot satisfies equation (1) for a meridian , hence we obtain the following corollary. Corollary 1.5. The knot exterior of the connected sum of hyperbolic two-bridge knots satisfies equation (1) for a meridian . Remark 1.6. Conjecture 1.1 was derived from the 3d–3d correspondence under the assumption that the interior of M admits a hyperbolic structure (see [8, Section 3] for details). It fails without the assumption, since torus knots are non-hyperbolic and do not satisfy equation (1). However, Theorem 1.4 and Corollary 1.5 suggest that one can relax the hyperbolicity condition, as the connected sum of knots is never hyperbolic. The paper is organized as follows. In Section 2, we briefly recall basic definitions on the sign-refined Reidemeister torsion. We define the adjoint Reidemeister torsion for the connected sum of knots in Sections 3.1 and 3.2, and prove Theorems 1.3 and 1.4 in Section 3.3. 2. Review on the sign-refined Reidemeister torsion 2.1. The Reidemeister torsion of a chain complex Let C be a chain complex of vector spaces over a field F @n @1 C D .0 ! Cn !    ! C1 ! C0 ! 0/ and H .C / be the homology of C . For a basis c of C and a basis h of H .C / the Reidemeister torsion is defined as follows. Here and throughout the paper, every basis and tuple is assumed to be ordered. For 0  i  n, we choose a lift hQ i of hi to Ci and a tuple bi of vectors in Ci such that @i bi is a basis of @i Ci . Then the tuple ci0 D .@i C1 biC1 ; hei ; bi / is another basis of Ci . Letting Ai be the basis transition matrix taking ci to ci0 , we have n 1/iC1 det A.i Y tor.C ; c ; h / D 2 F : i D0 Also, the sign-refined Reidemeister torsion is defined as n X Tor.C ; c ; h / D . 1/jC j tor.C ; c ; h / 2 F  ; jC j D ˛i .C /ˇi .C / i D0 Pi Pi where ˛i .C / D j D0 dim Cj and ˇi .C / D j D0 dim Hj .C /. J. Porti and S. Yoon 410 Suppose that we have a short exact sequence of chain complexes 0 ! C0 ! C ! C00 ! 0 (2) with bases c ; c0 ; and c00 of C ; C0 , and C00 , respectively. It is proved in [17, Lemma 3.4.2] that if c ; c0 ; and c00 are compatible with respect to sequence (2), i.e., c D .c0 ; c00 /, then Tor.C ; c ; h / D . 1/vCu Tor.C0 ; c0 ; h0 / Tor.C00 ; c00 ; h00 / tor.H / (3) where h ; h0 ; and h00 are bases of H .C /; H .C0 /, and H .C00 /, respectively. Here X vD ˛i 1 .C0 /˛i .C00 /; (4) i X uD ..ˇi .C / C 1/.ˇi .C0 / C ˇi .C00 // C ˇi 0 00 1 .C /ˇi .C //; (5) i and tor.H / is the Reidemeister torsion of the long exact sequence induced from (2) with respect to h ; h0 , and h00 . We refer to [16, 17] for details. 2.2. The adjoint Reidemeister torsion of a CW-complex Let g be the Lie algebra of SL2 .C/ and fix a basis of g as       0 1 1 0 0 0 e1 D ; e2 D ; e3 D : 0 0 0 1 1 0 Note that the Killing form h; i on g is given by   0 a0   b a b ; D 8bb 0 C 4ac 0 C 4ca0 : c b c0 b0 Let X be a finite CW-complex and W 1 .X/ ! SL2 .C/ be a representation. We con- sider a cochain complex z Z/; g/ C  .X I g / D HomZŒ1X  .C .XI where Xz is the universal cover of X . Here g is viewed as a ZŒ1 .X /-module through the adjoint action Ad W 1 .X/ ! Aut.g/ associated to . We denote the cohomology of C  .X I g / by H  .XI g / and call it the twisted cohomology. Note that H 0 .X I g / coincides with the set of invariant vectors in g under the 1 .X /-action. Let c1 ; : : : ; cn be all the cells of X and fix their order by cX D .c1 ; : : : ; cn /. We assume that each cell ci has a preferred orientation and a lift cQi to X. z We define an The adjoint Reidemeister torsion for the connected sum of knots 411 element ci.k/ 2 C  .XI g / for 1  i  n and 1  k  3 by assigning cQi to ek and every cell of Xz that is not a lift of ci to 0. Then the tuple cX D .c1.1/ ; c1.2/ ; c1.3/ ; : : : ; cn.1/ ; cn.2/ ; cn.3/ / is a basis of C  .XI g /. We refer to it as the geometric basis. Let C .XI R/ be the ordinary chain complex of X with the real coefficient. Note that the tuple cX is a basis of C .XI R/. For an orientation oX of the R-vector space H .XI R/ we define ".oX / D sgn.Tor.C .X I R/; cX ; hX // 2 ¹˙1º where hX is any basis of H .X I R/ positively oriented with respect to oX and sgn.x/ is the sign of x 2 R . Definition 2.1. For a basis hX of H  .XI g / and an orientation oX of H .XI R/ the adjoint Reidemeister torsion is defined as  .XI ; hX ; oX / D ".oX /  Tor.C  .XI g /; cX ; hX / 2 C  : The above definition does not depend on the order, orientations, and lifts of ci ’s. Moreover, it does not depend on the choice of a basis of g if the Euler characteristic of X is zero. Note that every notion in this section associated to  is invariant under conjugating  up to an appropriate isomorphism. In particular, the adjoint Reidemeister torsion is invariant under the conjugation. Example 2.2. Let † be a 2-torus with a usual CW-structure: one 0-cell p, two 1-cells  and , and one 2-cell † as in Figure 1 (left). We choose their lifts (to the universal cover of †) as in Figure 1 (right) and fix an order of the cells by c† D .p; ; ; †/. Let o† be the orientation of H .†I R/ induced from c† so that ".o† / D 1. † Q z † p pQ Q Figure 1. The cells of a 2-torus and their lifts. J. Porti and S. Yoon 412 Let W 1 .†/ ! SL2 .C/ be a representation with tr ./ ¤ ˙2. Up to conjugation we have     m 0 l 0 ./ D ; ./ D 0 m 1 0 l 1 for some m ¤ ˙1 and l 2 C  . With respect to the geometric basis, the boundary maps ı 0 W C 0 .†I g / ! C 1 .†I g / and ı 1 W C 1 .†I g / ! C 2 .†I g / are given by   0 Ad ./ I3 ı D ; Ad ./ I3 ı 1 D .Ad ./ I3 I3 Ad .//: Here Ik is the identity matrix of size k. It follows that dim H i .†I g / D 1 for i D 0; 2, dim H i .†I g / D 2 for i D 1, and dim H i .†I g / D 0 otherwise. Note that H 0 .†I g / is spanned by e2 2 g, as it is invariant under the actions of  and . Let P D 81 e2 2 H 0 .†I g / and define maps 0 W C 0 .†I g / ! C; ˛ 7! h˛.p/; Q P i; 1 1 W C .†I g / ! C ;2 ˛ 7! .h˛./; Q P i/; Q P i; h˛./; 2 W C 2 .†I g / ! C; z P i: ˛ 7! h˛.†/; One easily checks that i induces an isomorphism H i .†I g / ! C for i D 1; 3 and ' 2 induces an isomorphism H 2 .†I g / ! C 2 . For simplicity, we use the same notation i for these isomorphisms. We choose a basis hi† of H i .†I g / by the pre-image of the standard basis of C (C 2 if i D 2) under i . Explicitly, we have h0† D p .2/ , h1† D ..2/ ; .2/ /, and h2† D †.2/ . Choosing a tuple b i of vectors in C i .†I g / as b 0 D .p .1/ ; p .3/ /, b 1 D ..1/ ; .3/ /, and b 2 D ;, we obtain  .†I ; h† ; o† / D 1  .m2 1/.m 2 1/  . .m2 1/.m 2 1// 1 D 1: Note that a different choice of P 2 H 0 .†I g / changes the basis h† but still we have  .†I ; h† ; o† / D 1. 2.3. The adjoint Reidemeister torsion of a knot exterior Let M be the knot exterior of a knot K  S 3 with any given triangulation. It is well known that dim Hi .M I R/ D 1 for i D 0; 1 and dim Hi .M I R/ D 0 otherwise. We choose the orientation oM of H .M I R/ induced from a basis hM D .pt; / of H .M I R/ where pt is a point in M and  is a meridian of K oriented arbitrarily. Let W 1 .M / ! SL2 .C/ be a representation of the knot group. For the sake of simplicity, we assume that m ¤ ˙1 and K .m2 / ¤ 0; The adjoint Reidemeister torsion for the connected sum of knots 413 where m is an eigenvalue of ./ and K is the Alexander polynomial of K. It follows that if  is reducible, then it should be abelian (see e.g. [1]). Therefore,  is either irreducible (Section 2.3.1) or abelian (Section 2.3.2). 2.3.1. Irreducible representations. Suppose that  is irreducible. In this case we further assume that  is -regular [14, Definition 3.21], i.e., dim H 1 .M I g / D 1 and the inclusion  ,! M induces an injective map H 1 .M I g / ! H 1 .I g /. We choose an element P 2 H 0 .†I g /, where † D @M , and define maps 1 W C 1 .M I g / ! C; ˛ 7! h˛./; Q P i; 2 W C 2 .M I g / ! C; z P i; ˛ 7! h˛.†/; where Q and † z are lifts of  and † (to the universal cover of M ) respectively satisfy- z ing Q  †. Here the boundary torus † is oriented as in Stokes’ theorem. It is proved in [14] that the -regularity implies that i induces an isomorphism H i .M I g / ! C for i D 1; 2. We define  .M I / D .M I ; hM ; oM / (6) i where hM is a basis of H i .M I g / given by the pre-image of the standard basis of C under i . Note that a different choice of P 2 H 0 .†I g / changes the basis hM but not the value of  .M I /. 2.3.2. Abelian representations. Suppose that  is abelian. This case might not be that interesting, as it essentially reduces to the Alexander polynomial. However, we present it explicitly as a setup for Section 3. Lemma 2.3. We have dim H i .M I g / D 1 for i D 0; 1 and dim H i .M I g / D 0 otherwise. Proof. We choose any Wirtinger presentation of the knot group 1 .M / D hg1 ; : : : ; gn j r1 ; : : : ; rn 1 i: Recall that the corresponding 2-dimensional cell complex X consists of one 0-cell p, n 1-cells g1 ; : : : ; gn , and n 1 2-cells r1 ; : : : ; rn 1 . It is known that X is simple homotopy equivalent to M and thus we may use X instead of M . We choose a lift of the base point p arbitrarily and the lifts of other cells accordingly. Then with respect to the geometric basis, the boundary maps ı 0 W C 0 .XI g / ! C 1 .XI g / and ı 1 W C 1 .XI g / ! C 2 .XI g / J. Porti and S. Yoon 414 are given as 0 1 ˆ.g1 1/ 0 ı D@ B :: C : A ˆ.gn 1/ and  0 1 ˆ.@r1 =@g1 / ˆ.@r1 =@gn / 1 ı D@ B :: :: :: C : : : A ˆ.@rn 1 =@g1 /  ˆ.@rn 1 =@gn /; where ˆ is the Z-linear extension of Ad  and @rj =@gi denotes the Fox free differen- tial. Similarly, as in Example 2.2, we have   m 0 .g1 / D    D .gn / D ; m ¤ ˙1 0 m 1 up to conjugation. It is clear that Im ı 0 ' C 2 , Ker ı 0 ' C and dim H 0 .XI g / D 1. On the other hand, ı 1 is surjective since K .1/ ¤ 0 and K .m˙2 / ¤ 0. It follows that dim H 2 .XI g / D 0 and dim H 1 .XI g / D 1, since the Euler characteristic of X is zero. Explicitly, the twisted cohomology of X is generated by ˛ 2 C 0 .X I g / such that ˛.p/ Q D e2 ; (7a) 1 ˛ 2 C .XI g / such that ˛.e gi / D e2 for all 1  i  n: (7b) Once again, we choose an element P 2 H 0 .†I g / D H 0 .M I g / and define 0 W C 0 .M I g / ! C; ˛ 7! h˛.p/; Q P i; 1 1 W C .M I g / ! C; ˛ 7! h˛./; Q P i; where pQ and Q are lifts of p and  (to the universal cover of M ) respectively satisfying Q It is clear from equation (7) that i induces an isomorphism H i .M I g / ! C pQ  . for i D 0; 1. We define  .M I / D .M I ; hM ; oM / i where hM is a basis of H i .M I g / given by the pre-image of the standard basis of C i under . In fact, one can compute that K .m2 /K .m 2 /  .M I / D .m m 1 /2 up to sign, but we would not use this fact in this paper. The adjoint Reidemeister torsion for the connected sum of knots 415 3. The connected sum of knots Let K be the connected sum of knots K1 ; : : : ; Kn in S 3 . We denote by M and Mj the knot exteriors of K and Kj , respectively. It is known that the JSJ decomposition of M consists of a composing space and M1 ; : : : ; Mn . 3.1. A composing space Let D1 ; : : : ; Dn be mutually disjoint discs in the interior of a disc D 2 and let W D D 2 n int.D1 t    t Dn / be a planar surface. Here int.X/ denotes the interior of X . Let Y D W  S 1 , called a composing space, having n C 1 boundary tori †j D @Dj  S 1 .1  j  n/ and † D @D 2  S 1 . Letting  D ¹ptº  S 1 and j D @Dj  ¹ptº, we have 1 .Y / D h; 1 ; : : : ; n j Œ; 1  D    D Œ; n  D 1i: One can check that H0 .Y I R/ ' R is generated by a point p 2 Y , H1 .Y I R/ ' RnC1 is generated by ; 1 ; : : : ; n , and H2 .Y I R/ ' Rn is generated by †1 ; : : : ; †n . We choose the orientation oY of H .Y I R/ induced from a basis hY D .p; ; 1 ; : : : ; n ; †1 ; : : : ; †n / of H .Y I R/. Here we orient , j , and †j as in Example 2.2 and Stokes’ theorem. Let W 1 .Y / ! SL2 .C/ be a representation with tr ./ ¤ ˙2 and tr .j / ¤ ˙2 for some 1  j  n. Since  commutes with all j ’s, we have up to conjugation     m 0 lj 0 ./ D ; .j / D (8) 0 m 1 0 lj 1 for some m ¤ ˙1 and lj 2 C  . Note that there is no relation among m, l1 ; : : : ; ln . Proposition 3.1. We have 8 ˆ ˆ 1 if i D 0; ˆ