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let @xmath1 be a hyperbolic knot and @xmath2 a conway sphere . namely @xmath3 intersects transversally @xmath4 in four points . we write @xmath5 to denote any of the three involutions in figure  [ fig : involutions ] , for @xmath6 . the knot @xmath7 obtained by cutting along @xmath3 and gluing again after composing with @xmath8 is called the mutant knot . we are interested in comparing invariants of @xmath4 and @xmath9 , thus we may assume that , if @xmath10 denotes the knot exterior , then the surface @xmath11 is essential in @xmath12 . ruberman @xcite showed that @xmath9 is also hyperbolic , and that @xmath13 has the same volume as @xmath14 . see @xcite for a recent account on invariants that distinguish or not @xmath4 from @xmath9 . let @xmath15 be a lift of the holonomy . if @xmath16 is a meridian , then @xmath17 , and there are precisely two lifts of the holonomy up to conjugation , one with @xmath18 and another with @xmath19 . by @xcite , @xmath20 is acyclic , namely the homology and cohomology of @xmath12 with coefficients twisted by @xmath20 vanish , hence the reidemeister torsion @xmath21 is well defined . moreover as the dimension of @xmath22 is even , there is no sign indeterminacy , thus @xmath21 is a well defined nonzero complex number , independent of the conjugacy class of @xmath20 . hence these torsions are two topological invariants of the hyperbolic knot . [ thm : main ] let @xmath4 , @xmath12 , @xmath8 and @xmath23 be as above . let @xmath15 and @xmath24 be lifts of the holonomy , with @xmath25 . then @xmath26    this is not true for any representation of @xmath27 . wada proved in @xcite that the twisted alexander polynomials could be used to distinguish mutant knots . n. dunfield , s. friedl and n. jackson @xcite computed the torsion for the representation @xmath20 twisted by the abelianization map ( namely , the corresponding twisted alexander polynomials ) and proved that it distinguishes mutant knots . however , the evaluation at @xmath28 of these polynomials gives numerical evidence of theorem  [ thm : main ] . in @xcite we prove that when we consider @xmath29 the @xmath30-dimensional irreducible representation , then the composition @xmath31 is acyclic , thus its torsion is well defined . we have computed that the torsion of @xmath32 distinguishes the conway and the kinoshita - terasaka mutants , see section  [ sec : concluding ] . the paper is organized as follows . in section  [ sec : mutation ] , we discuss the basic constructions for reidemeister torsion and representations of mutants , and we give a sufficient criterion in proposition  [ prop : criterion ] for invariance of the torsion under mutation . the sufficient criterion of proposition  [ prop : criterion ] is stated in terms of the action of @xmath8 on the cohomology of @xmath33 with twisted coefficients . this is applied for the proof when @xmath34 in section  [ sec : trace-2 ] . the proof when @xmath35 in section  [ sec : trace+2 ] is different , because the involved cohomology groups are different in each situation . in section  [ section : example ] we compute an example , the kinoshita - terasaka and conway mutants , and section  [ sec : concluding ] is devoted to further discussion . let @xmath36 and @xmath37 denote the components of @xmath38 , so that the pairs @xmath39 are tangles with two strings . the exterior of the knot is denoted by @xmath40 we also denote @xmath41 write a commutative diagram for the inclusions : @xmath42^{i_1 }   \ar[d]_{i_2 } & b_1 \ar[d ] \\ b_2 \ar[r ] & s^3 }   \qquad\textrm { and } \qquad \xymatrix {   s \ar   [ r]^{i_1 }   \ar[d]_{i_2 } & m_1 \ar[d ] \\ m_2 \ar[r ] & m } \ ] ] so that @xmath43 ) is an amalgamated product @xmath44 let @xmath45 , @xmath46 and @xmath47 denote the restriction of @xmath20 to @xmath48 , @xmath49 , and @xmath50 , respectively , so that @xmath51 using the notation @xmath52 there exists @xmath53 which is unique up to sign @xcite such that @xmath54 notice that @xmath53 corresponds to a rotation of order two in hyperbolic space , therefore , @xmath55 is conjugate to @xmath56    to construct the representation of @xmath57 , we also use the amalgamated product structure with the same inclusion @xmath58 , but with @xmath59 instead of @xmath60 . the representation @xmath24 is then defined by @xmath61 because @xmath62 .      to set notation , we recall the basic construction of cohomology with twisted coefficients . let @xmath63 be a compact @xmath64-complex and @xmath65 a representation . the cellular chains of its universal covering are denoted by @xmath66 , which is a chain complex of left @xmath67$]-modules of finite type . the cochains with twisted coefficients are then @xmath68 } ( c_*(\tilde x ; \mathbf z ) ,     \mathbf c^2_{\rho } ) , \ ] ] where @xmath69 is viewed as a left @xmath67$]-module by the action induced by @xmath20 . the corresponding cohomology groups are denoted by @xmath70 as they only depend on the underlying topological space @xmath71 of the @xmath64-complex @xmath63 . we shall mainly work with aspherical spaces , in this case the homology or cohomology of @xmath63 with twisted coefficients is naturally isomorphic to the group cohomology of @xmath72 . we shall also be interested in the de rham cohomology . assuming @xmath73 is a smooth manifold , let @xmath74 denote the flat bundle with monodromy @xmath20 . the space of @xmath75-forms valued on @xmath76 is @xmath77 . the de rham cohomology of @xmath78 is naturally isomorphic to @xmath79    many properties of cohomology without coefficients hold true when we have twisted coefficients : mayer - vietoris , long exact sequence of the pair , etc . poincar duality is discussed in section  [ subsection : perfectpairing ] . we will use mayer - vietoris for the pair @xmath80 to compute the torsion of @xmath12 and of @xmath23 . for this , we need to compute some cohomology groups . we start with the planar surface with four boundary components @xmath81 :    [ lem : cohoms ] @xmath82 for @xmath83 and @xmath84 . firstly @xmath85 is isomorphic to the subspace of @xmath22 of elements that are fixed by @xmath86 , hence it vanishes because @xmath45 is an irreducible representation . on the other hand , @xmath87 for @xmath88 , because @xmath33 has the homotopy type of a graph . finally @xmath89 [ lem : cohomm_i ] for @xmath90 , @xmath91 for @xmath92 and @xmath93 .    by mayer - vietoris , and using the fact that @xmath94 , then @xmath95 the lemma follows from lemma  [ lem : cohoms ] , because @xmath96 . mayer - vietoris for @xmath12 and @xmath23 give the isomorphisms : @xmath97    in order to relate @xmath98 and @xmath99 , we use the composition with @xmath55 , where @xmath100 denotes @xmath47 conjugated by @xmath55 . namely , recall that @xmath101 } ( c_*(\widetilde m_2;\mathbf z ) , \mathbf c^2_{\rho_2}).\ ] ] define : @xmath102 it is straightforward to check that this defines an isomorphism of complexes . thus @xmath103 is an isomorphism and we have a commutative diagram : @xmath104^{i_2^ * }   \ar[d]_{a _ * } & h^1(s;\rho_2 ) \ar[d]^ { a_*\circ\tau^ * } \\ h^1(m_2;\rho_2^a ) \ar[r]^{(i_2\circ\tau)^ * } & h^1(s;\rho_0 ) } \ ] ] write @xmath105 . since @xmath106 ( see equation  ( [ eqn : a ] ) ) and since @xmath107 , we have : @xmath108      let @xmath63 be a compact cw - complex equipped with a representation @xmath109 when @xmath110 , the reidemeister torsion can be defined , and it is an invariant of @xmath63 , up to subdivision , and the conjugacy class of @xmath20 . we will not recall the definition , that can be found in @xcite , for instance . there are two main issues for the torsion we are interested in . firstly , the torsion is only defined up to sign , but since we consider a two dimensional vector space , it is sign defined , hence a nonzero complex number . equivalently , any choice of homology orientation for turaev s refined torsion @xcite gives the same result . secondly , since we are working with three and two - dimensional manifolds , the pl - structure is not relevant . thus , for a two and three - dimensional manifold @xmath63 and an acyclic representation @xmath111 , its torsion is denoted by @xmath112 when @xmath20 is not acyclic , then we can also use the reidemeister torsion provided we specify a basis for @xmath113 .    choose @xmath114 a basis for @xmath115 as @xmath116-vector space . in particular @xmath117 is a basis for @xmath118 . by milnor s formula @xcite for the torsion of a long exact sequence applied to ( [ eqn : mv ] ) and ( [ eqn : mvtau ] ) : @xmath119 here @xmath120 denotes the disjoint union of basis . notice that milnor works with torsions up to sign in @xcite , but its formalism applies even with sign . since @xmath121 , we deduce @xmath122 namely , the determinant of the matrix whose entries are the coefficients of the basis @xmath123 with respect to @xmath124 . the following is a sufficient criterion for invariance of torsion with respect to mutation . [ prop : criterion ] if @xmath125 leaves invariant the image of @xmath126 , then @xmath127 . since @xmath128 by formula ( [ eqn : tau2=-id ] ) , @xmath129 diagonalizes with eigenvalues @xmath130 . hence , assuming that @xmath129 leaves invariant the image of @xmath131 , the matrix in equation  ( [ eqn : ratiotorsions ] ) is conjugate to @xmath132 hence it has determinant @xmath28 . we discuss first the case where @xmath133 . the proof has three parts . in subsection  [ subsection : perfectpairing ] we consider a perfect pairing on @xmath134 ( pulling back the cup product on @xmath135 by the isomorphism @xmath136 ) . we show that , for @xmath137 , @xmath138 , the images of @xmath139 are isotropic subspaces . then in subsection  [ subsec : quadric ] we analyze properties of isotropic planes of @xmath140 , that are viewed as lines in a ruled quadric in @xmath141 . the properties of this ruled quadric are used in a deformation argument in subsection  [ subsection : firstdfm argument ] to conclude the proof when @xmath133 .      for a closed oriented @xmath142-manifold @xmath143 and a representation @xmath144 , there is a nondegenrate pairing : @xmath145 which is the composition of the cup product and the map induced from the @xmath20-invariant pairing @xmath146 this pairing is bilinear , nondegenerate ( poincar duality ) and natural . in addition , when @xmath147 and @xmath137 it is symmetric ( because both the usual cup product and the determinant are antisymmetric ) . in particular , we have two symmetric pairings , for @xmath90 : @xmath148    the following lemma assumes that @xmath34 , as the whole section , though it only requires @xmath149 . [ lemma : spartial ] @xmath150 .    since @xmath34 , @xmath151 has no nontrivial fixed vectors in @xmath22 . thus @xmath152 and , by duality , @xmath150 . [ lemma : spartialmk ] for @xmath90 , the inclusion map induces an isomorphism @xmath153    this follows from the mayer - vietoris sequence applied to @xmath33 and @xmath154 , which is the union of the two annuli around the arcs of @xmath155 that have the homotopy type of a component of @xmath156 . hence by lemma  [ lemma : spartial ] , @xmath157 . pulling back the pairings ( [ eqn : pairingmk ] ) by the isomorphism ( [ eqn : isos ] ) , we obtain two symmetric perfect pairings @xmath158    [ lemma : cup1cup2 ] both pairings are the same : @xmath159 . we use de rham cohomology for the proof . we first claim that every cohomolgy class in @xmath160 is represented by a form compactly supported in the interior of @xmath33 , using that @xmath161 . namely , let @xmath162 be any closed form and @xmath163 a tubular neighborhood of @xmath156 . since @xmath161 , the restriction of @xmath164 to @xmath165 is exact : there exists a section @xmath166 satisfying @xmath167 where @xmath168 is the differential . consider a smooth function @xmath169 $ ] that extends smoothly to zero on @xmath170 and equals one in a smaller neighborhood of @xmath156 . then the form @xmath171 is cohomologous to @xmath164 and is supported on a compact subset of the interior of @xmath33 .    given two closed forms @xmath172 with compact support in the interior of @xmath33 , we may view them as smooth forms @xmath173 on @xmath174 by extending them trivially . as the isomorphism @xmath175 is induced by restriction , it maps the cohomology classes @xmath176 , [ \tilde \omega_2]\in   h^1(\partial m_k ; \rho_k)$ ] to @xmath177 , [ \omega_2]\in h^1(s;\rho_0)$ ] . thus the cup product @xmath178 of their cohomology classes is @xmath179\cup_k ' [ \omega_2]= [ \tilde \omega_1]\cup_k [ \tilde \omega_2]= \int_{\partial m_k } \det ( \tilde\omega_1\wedge\tilde \omega_2),\ ] ] where @xmath180 denotes the pairing associated to the determinant evaluated at the exterior product of the forms . by construction , the support of @xmath181 and @xmath182 is contained in @xmath33 , thus the previous integral can be evaluated on @xmath33 instead of @xmath174 , and in particular it does not depend on @xmath183 .    by the previous lemma , we may omit the subindex and just write @xmath184    [ lem : isotropic ] the image of @xmath185 is an isotropic plane for the product ( [ eqn : ss ] ) . the fact that the image is a plane follows from lemma  [ lem : cohomm_i ] and its proof . then , by construction the lemma is equivalent to saying that the image of @xmath186 is an isotropic subspace . this is well known @xcite , but we sketch the argument for completeness .    using the long exact sequence of the pair @xmath187 , we have a commutative diagram with exact rows : @xmath188^{j_k^ * } \ar@{}[d]^*++{\!\!\!\!\times }   & h^1(\partial m_k;\rho_k ) \ar[r]^{\delta }   \ar@{}[d]^*++{\!\!\!\!\times } & h^2(m_k,\partial m_k;\rho_k )   \ar@{}[d]^*++{\!\!\!\!\times }   \\   h^2(m_k,\partial m_k;\rho_k ) \ar[d ] &    h^1(\partial m_k;\rho_k ) \ar[l]_{\delta } \ar[d ] & h^1(m_k;\rho_k ) \ar   [ l]_{j_k^ * }   \ar[d]\\ \mathbf c & \mathbf c & \mathbf c   } \ ] ] where the columns denote the pairings . this implies @xmath189 hence @xmath190 because @xmath191 , and we are done . let @xmath192 denote the projective space on @xmath84 . isotropic planes of @xmath160 are in bijection with projective lines in the quadric @xmath193 since @xmath194 is a nondegenrate paring , @xmath195 is the standard quadric , which is a ruled surface , with two rulings . we recall next its basic properties . [ prop : ruling ] there are two disjoint families of projective lines @xmath196 and @xmath197 in @xmath195 such that :    * every line in @xmath195 belongs to either @xmath196 or @xmath197 . * every point in @xmath195 belongs to precisely one line in @xmath196 and one in @xmath197 . * two lines in @xmath195 intersect if , and only if , one is in @xmath196 and the other one is in @xmath197 . * embedding those spaces of lines in the projective grassmannian , @xmath198 . we shall also use the action of @xmath199 , the isometry group of @xmath160 equipped with the quadratic form @xmath194 in ( [ eqn : ss ] ) . recall the isomorphism : @xmath200 after projectivizing , this induces an isomorphism : @xmath201    [ prop : product ] the action of @xmath202 is equivalent to the product action on @xmath203 ( by an equivalence that preserves the product ) . see @xcite for a proof . it follows from this proposition that @xmath204 acts trivially on @xmath197 , and @xmath205 acts trivially on @xmath196 . consider the involutions @xmath206 , @xmath207 and @xmath208 of the conway sphere @xmath3 , as in figure  [ fig : involutions ] . for @xmath6 let @xmath209 , satisfy @xmath210 , and define @xmath211 the corresponding actions on @xmath160 . the induced maps @xmath212 on @xmath141 lie in one factor @xmath204 or @xmath205 . in addition all @xmath213 , @xmath214 and @xmath215 lie in the same factor .    by looking at the action on @xmath84 , we have @xmath216 , by ( [ eqn : tau2=-id ] ) . this implies that @xmath212 projects to an involution of @xmath141 preserving @xmath194 , hence to an involution in @xmath202 . notice that if an involution of @xmath202 is nontrivial on each factor , it lifts to an element of @xmath217 whose square is @xmath218 , hence to an involution of @xmath219 . as @xmath220 , we deduce that each @xmath212 projects to an involution of one of the factors of @xmath202 , and is trivial on the other factor . this proves the first assertion of the lemma . for the last assertion , just use that @xmath221 .    hence , up to permuting @xmath197 and @xmath196 , we get :    [ coro : tauasigmab ]    * @xmath213 , @xmath214 and @xmath215 act trivially on @xmath197 . * there is no point in @xmath196 fixed by all @xmath213 , @xmath214 and @xmath215 . notice that , for assertion ( ii ) , we use that the subgroup of @xmath222 consisting of three involutions and the identity ( also called the 4-klein group ) has no global fixed point in @xmath223 . let @xmath224 denote the image . since @xmath225 , from proposition  [ prop : ruling ] ( iii ) we deduce :    [ cor : imi1i22 ] either both @xmath226 and @xmath227 belong to @xmath197 , or they both belong to @xmath196 . either @xmath228 belongs to @xmath197 and , by corollary  [ coro : tauasigmab ] , we may apply proposition  [ prop : criterion ] , or @xmath228 belongs to @xmath196 . to get rid of this last case we will use a deformation argument . the idea is to deform the hyperbolic structure on @xmath229 , so that it matches with another tangle which is invariant under the involutions @xmath230 . by corollary  [ coro : tauasigmab](ii ) , the tangle invariant by the involutions satisfies @xmath231 , hence @xmath232 for the deformed structure on @xmath229 , by corollary  [ cor : imi1i22 ] . then we shall use a continuity argument to have the same conclusion for the initial structure on @xmath229 . next subsection is devoted to this deformation argument . let @xmath233 be the pair of annuli , one around each arc of @xmath234 . the pair @xmath235 is a _ pared _ manifold . [ def : pared ] a _ pared _ manifold is a pair @xmath236 , where @xmath73 is a compact oriented 3-manifold , @xmath237 is a union of tori and annuli , such that    * no two components of @xmath238 are isotopic in @xmath239 , * every abelian noncyclic subgroup of @xmath240 is conjugate to a subgroup of a component or @xmath238 , and * there are no essential annuli @xmath241,s^1\times \partial [ 0,1])\to ( m , p)$ ] .    we say that a pared manifold @xmath236 is hyperbolic when the interior of @xmath73 admits a complete hyperbolic structure with cusps at @xmath238 . the rank of the cusp is one for an annulus , and two for a torus . [ lemma : tangle_hyp ] there exists a pared manifold @xmath242 , such that    1 .   @xmath243 is obtained from a @xmath138-tangle : namely @xmath244 is the exterior of two properly embedded arcs in a 3-ball , @xmath245 are the annuli around the arcs of the tangle , and @xmath246 . 2 .   for @xmath6 , @xmath247 extends to an involution of @xmath242 . 3 .   the pared manifolds @xmath243 and @xmath248 are both hyperbolic . we take @xmath244 to be the exterior of a simple 2-tangle that is symmetric with respect to @xmath206 , @xmath207 and @xmath208 . here simple means that @xmath244 is irreducible , @xmath249-irreducible , atoroidal and anannular . in @xcite , wu gives a criterion for deciding when a rational tangle is simple and provides examples of simple tangles with the required symmetries . in particular , the pared manifold @xmath243 admits a hyperbolic structure with totally geodesic boundary in @xmath250 ( and rank one cusps in @xmath245 ) . since @xmath243 is simple and @xmath235 hyperbolic , standard arguments in 3-dimensional topology prove that @xmath251 is irreducible , acylindrical , atoroidal and not seifert fibered ( @xmath33 should be horizontal in a seifert fibration ) , hence hyperbolic .    the variety or representations of @xmath50 to @xmath0 is denoted by @xmath252 and it is an algebraic subset of affine space @xmath253 . [ lemma : almostequal-2 ] if @xmath34 , then @xmath228 belongs to @xmath197 . we connect @xmath254 to @xmath255 , a lift of the holonomy representation of @xmath229 that matches with the tangle @xmath244 of lemma  [ lemma : tangle_hyp ] , which is a symmetric tangle . namely we want to find a path or representations @xmath256&\to & r(m_2 ) \\        t & \mapsto & \varphi_t \end{array}\ ] ] that satisfies :    * @xmath257 . * @xmath258 $ ] , @xmath259 is the lift of the holonomy of a hyperbolic structure on @xmath229 , with rank one cusps at the arcs @xmath234 , and satisfying @xmath260 * @xmath258 $ ] , @xmath261 . * @xmath262 is the lift of the holonomy of a hyperbolic structure on @xmath229 that matches with @xmath244 in lemma  [ lemma : tangle_hyp ] . assuming we have this path of representations , then since @xmath244 is @xmath206 and @xmath207-invariant , the image of @xmath263 is a subspace @xmath212-invariant . hence the image of @xmath264 must be contained in @xmath197 , by corollary  [ coro : tauasigmab ] ( ii ) . this implies that for this hyperbolic structure @xmath265 by corollary  [ cor : imi1i22 ] . now , since there exists the path @xmath259 , the ruled quadric of @xmath266 is also deformed continuously ( notice that as @xmath267 is irreducible and @xmath259 of a meridian has trace @xmath268 , by ( iii ) , lemmas  [ lem : cohoms ] , [ lemma : spartial ] , and [ lem : isotropic ] apply to @xmath266 ) . hence along the deformation , the image of @xmath131 is contained in @xmath197 , as @xmath269 . hence @xmath270 as claimed . let us justify the existence of the path @xmath271 between @xmath47 and @xmath272 . if both @xmath273 and @xmath274 are geometrically finite , then they can be connected along the space of geometrically finite structures of the pared manifold , because by ahlfors - bers theorem this space is isomorphic to the teichmller space of @xmath33 , cf . in addition , this is an open subset of the variety of representations , and since the dimension of de cohomology is upper semi - continuous ( it can only jump in a zariski closed subset ) , ( iii ) can be achieved by avoiding a proper zariski closed subset ( hence of real codimension @xmath275 ) . if any of @xmath273 and @xmath274 is not geometrically finite , then it lies in the closure of geometrically finite structures ( cf . @xcite though this is a particular case of the density theorem ) , thus there is still a path in the space of representations satisfying ( ii ) and ( iii ) .    by lemma  [ lemma : almostequal-2 ] , corollary  [ cor : imi1i22 ] and proposition  [ prop : criterion ] , @xmath276 we shall prove that there is also equality of signs :    [ proposition : equal-2 ] if @xmath34 , then @xmath277    to remove the sign ambiguity , we use again the deformation @xmath259 of the proof of lemma  [ lemma : almostequal-2 ] . since @xmath258 $ ] , @xmath259 satisfies the sufficiency criterion of proposition  [ prop : criterion ] , the eigenvalues of @xmath212 restricted to the image of @xmath131 belong to @xmath278 , and they do not change as we deform @xmath279 , hence the determinant of @xmath212 restricted to the image of @xmath131 is @xmath280 , because this holds for @xmath281 ( as @xmath244 is @xmath230-invariant ) . when @xmath282 , then lemma  [ lemma : spartial ] does not apply , hence we can not use the argument of section  [ sec : trace-2 ] . recall that @xmath283 denotes the variety of representations of @xmath27 in @xmath0 . we will consider representations @xmath284 to which the arguments of section  [ sec : trace-2 ] apply and such that @xmath285 converges to @xmath20 in @xmath286 , as @xmath287 . let @xmath288 be a lift of the holonomy with @xmath282 . by thurston s hyperbolic dehn filling and for @xmath289 large enough , the orbifold with underlying space @xmath290 , branching locus @xmath4 and ramification index @xmath142 is hyperbolic . it induces a representation of @xmath27 in @xmath291 that lifts to @xmath284 . the lift satisfies @xmath292 , and there is precisely one lift for every choice of sign . we chose the lift satisfying @xmath293 . [ prop : orbifold ] for @xmath289 large enough , there exist @xmath284 which is a lift of the holonomy of the orbifold with underlying space @xmath290 , branching locus @xmath4 and ramification index @xmath142 , so that @xmath294 . these orbifolds can also be considered for the mutant knot , and there exist the corresponding mutant representations @xmath295 namely , the lifts of the holonomies of the orbifold structures on @xmath9 are the `` mutant representations '' of @xmath285 . moreover , @xmath296 .    [ lem : orbif ] for @xmath289 large enough , @xmath297 and @xmath298    we use semi - continuity of cohomology to say that @xmath297 . more precisely , the dimension of cohomology is an upper semi - continuous function on @xmath286 , cf . @xcite . hence , as @xmath20 and @xmath299 are acyclic , then all representations in a zariski open subset containing @xmath20 and @xmath299 are acyclic , and so are @xmath285 and @xmath300 , as claimed .    since @xmath285 and @xmath300 are acyclic , @xmath301 . in addition , up to conjugation , @xmath302 hence @xmath22 has no @xmath303-invariant proper subspaces and lemma  [ lemma : spartial ] applies . thus the pairing  ( [ eqn : ss ] ) , lemma  [ lem : isotropic ] and all the results of section  [ subsection : perfectpairing ] hold true for @xmath285 . to conclude , for the deformation argument , we use that @xmath285 is the lift of the holonomy of an orbifold . instead of working with pared structures on @xmath235 , we work with orbifold structures with underlying space the ball @xmath37 , branching locus @xmath234 and branching index @xmath142 . the results on the space of hyperbolic structures ( geometrically finite or infinite ) apply , and we may use the deformation argument of lemma  [ lemma : almostequal-2 ] .    by proposition  [ prop : orbifold ] and lemma  [ lem : orbif ] , by taking the limit when @xmath287 we get :    [ cor : equal-2 ] if @xmath35 , then @xmath277    the proofs for @xmath282 and @xmath34 are quite different , because @xmath304 is non zero when the trace is + 2 , and vanishes when it is -2 . the generic case is @xmath305 . the proof of section  [ sec : trace-2 ] applies to the following situation . [ prop : generic ] let @xmath15 be a representation satisfying :    1 . @xmath306 ; 2 .   @xmath149 @xmath20 restricted to @xmath48 is irreducible ; 4 .   the representation @xmath20 is in the same irreducible component of @xmath286 as some representation such that the image of @xmath131 is contained in @xmath307 .    then @xmath308 . [ cor : generic ] for a generic representation @xmath20 of the irreducible component of @xmath286 that contains a lift of the holonomy , @xmath309 . the holonomy representation of a hyperbolic knot has two lifts to @xmath0 , each one with a different sign for the image of the meridian . do they belong to the same irreducible component of the variety of representations ? this happens to be true for instance if the component of the variety of representations contains a dihedral representation , as this is a ramification point or the map from the variety of representations in @xmath0 to those in @xmath222 . let @xmath310 and @xmath3 be the kinoshita - terasaka knot and the conway knot respectively . it is well known that they are mutant hyperbolic knots . using the snap program @xcite , based of j. weeks snappea @xcite , we have obtained all the necessary information to compute their torsion . the fundamental groups of these knots have the following presentations : @xmath311 as usual , capital letters denote inverse .    the image of the holonomy representation is contained in @xmath312 where @xmath313 is the number field generated by a root @xmath164 of the following polynomial : @xmath314 the torsions then are elements of @xmath313 . in order to express elements in @xmath313 , we use the @xmath315-basis @xmath316 . tables  [ table : tra2 ] and [ table : tra-2 ] give the coefficients of the torsions of @xmath310 and @xmath3 with respect to this @xmath317-basis . on each table , the first column gives the element of the basis . we let @xmath142 denote the dimension of the irreducible representation of @xmath0 used to compute the torsion , and the tables show the values for @xmath147 ( i.e. the standard representation ) , but also @xmath318 and @xmath319 . in order to compare them , the coefficients of the torsion for kinoshita - terasaka ( @xmath310 ) and conway ( @xmath3 ) knots are tabulated side by side . we give a table for each lift of the holonomy , one when the trace of the meridian is @xmath138 ( table  [ table : tra2 ] ) and another when it is @xmath268 ( table  [ table : tra-2 ] ) . cc|r|r|r|r|r|r| & & & & + & & & & & & + & @xmath320 & @xmath320 & @xmath321 & @xmath321 & @xmath322 & @xmath323 + & @xmath324 & @xmath324 & @xmath325 & @xmath325 & @xmath326 & @xmath327 + & @xmath328 & @xmath328 & @xmath329 & @xmath329 & @xmath330 & @xmath331 + & @xmath332 & @xmath332 & @xmath333 & @xmath333 & @xmath334 & @xmath335 + & @xmath336 & @xmath336 & @xmath337 & @xmath337 & @xmath338 & @xmath339 + & @xmath340 & @xmath340 & @xmath341 & @xmath341 & @xmath342 & @xmath343 + & @xmath344 & @xmath344 & @xmath345 & @xmath345 & @xmath346 & @xmath347 + & @xmath348 & @xmath348 & @xmath349 & @xmath349 & @xmath350 & @xmath351 + & @xmath352 & @xmath352 & @xmath353 & @xmath353 & @xmath354 & @xmath355 + & @xmath356 & @xmath356 & @xmath357 & @xmath357 & @xmath358 & @xmath359 + & @xmath360 & @xmath360 & @xmath361 & @xmath361 & @xmath362 & @xmath363 +    cc|r|r|r|r|r|r| & & & & + & & & & & & + & @xmath364 & @xmath364 & @xmath365 & @xmath366 & @xmath367 & @xmath368 + & @xmath369 & @xmath369 & @xmath370 & @xmath371 & @xmath372 & @xmath373 + & @xmath374 & @xmath374 & @xmath375 & @xmath376 & @xmath377 & @xmath378 + & @xmath379 & @xmath379 & @xmath380 & @xmath381 & @xmath382 & @xmath383 + & @xmath384 & @xmath384 & @xmath385 & @xmath386 & @xmath387 & @xmath388 + & @xmath389 & @xmath389 & @xmath390 & @xmath391 & @xmath392 & @xmath393 + & @xmath394 & @xmath394 & @xmath395 & @xmath396 & @xmath397 & @xmath398 + & @xmath399 & @xmath399 & @xmath400 & @xmath401 & @xmath402 & @xmath403 + & @xmath404 & @xmath404 & @xmath405 & @xmath406 & @xmath407 & @xmath408 + & @xmath409 & @xmath409 & @xmath410 & @xmath411 & @xmath412 & @xmath413 + & @xmath414 & @xmath414 & @xmath415 & @xmath416 & @xmath417 & @xmath418 +    of course , for @xmath147 and for any lift of the holonomy , the torsion of @xmath310 and the torsion of @xmath3 is the same . notice that for the 4-dimensional representation , they are also the same for one lift but different for the other , and that they differ for both lifts when we use the 6-dimensional representation .    as said in the introduction , when @xmath147 , these had been computed by dunfield , friedl and jackson in @xcite . they computed numerically a twisted alexander invariant ( which are not mutation invariant ) for all knots up to 15 crossings , and the torsions computed here are just the evaluations at @xmath28 . the mutation considered in this paper is called @xmath419-mutation , because the involved surface is planar and has 4 boundary components . by tubing along invariant arcs of the knot , this is a particular case of the so called @xmath420-mutation , namely , the mutation along a closed surface of genus 2 and using the hyperelliptic involution .    in @xcite ruberman proved that a @xmath420-mutant of a hyperbolic manifold is again hyperbolic . the behavior of invariants under @xmath420-mutation has been investigated by many authors , see @xcite for instance . unfortunately , our arguments do not apply , as in section  [ sec : trace-2 ] we require two involutions , and in genus two mutation we can use only the hyperelliptic involution . so we arise :      the three dimensional representation of @xmath0 is conjugate to the adjoint representation in the automorphism group of the lie algebra @xmath422 . the representation @xmath423 is not acyclic , but a natural choice of basis for homology has been given in @xcite , hence its torsion is well defined . moreover , we have :      the proof is straightforward , as @xmath425 is the cotangent space to the variety of characters of @xmath33 , and the action of the hyperelliptic involution is trivial on the variety of characters of @xmath33 . we have seen that if we compose the lift of the holonomy with the 6-dimensional representation of @xmath0 ( or the 4-dimensional one when the trace of the meridian is @xmath268 ) , the torsion is not invariant under @xmath420-mutation , as it is not invariant under @xmath419-mutation , see the example of the previous section .      to conclude , we notice that our arguments do not apply if we tensorize @xmath15 with the abelianization map @xmath426 . this torsion gives the twisted polynomial in @xmath427 $ ] studied in @xcite , where it is proved not to be mutation invariant . if we could apply the arguments of this paper , then we would be in the generic situation of proposition  [ prop : generic ] and section  [ sec : trace-2 ] , because @xmath428 . notice that two of the involutions in figure  [ fig : involutions ] reverse the orientation of the meridian , and only one preserves them . thus we can use only a single involution , and at least two involutions are required in our argument from section  [ sec : trace-2 ] , more precisely in corollary  [ coro : tauasigmab](ii ) .
given a hyperbolic knot , we prove that the reidemeister torsion of any lift of the holonomy to @xmath0 is invariant under mutation along a conway sphere .