Turaev-Viro invariants, colored Jones polynomials and volume RenaudDetcherry,EfstratiaKalfagianni∗ andTianYang† 7 Abstract 1 0 We obtain a formula for the Turaev-Viro invariants of a link complement in terms of values of 2 the colored Jones polynomial of the link. As an application we give the first examples for which n thevolumeconjectureofChenandthethirdnamedauthor[6]isverified. Namely,weshowthatthe a asymptoticsoftheTuraev-ViroinvariantsoftheFigure-eightknotandtheBorromeanringscomple- J mentdeterminethecorrespondinghyperbolicvolumes.Ourcalculationsalsoexhibitnewphenomena 7 ofasymptoticbehaviorofvaluesofthecoloredJonespolynomialsthatseemnottobepredictedby 2 neithertheKashaev-Murakami-Murakamivolumeconjectureandvariousofitsgeneralizationsnor byZagier’squantummodularityconjecture. WeconjecturethattheasymptoticsoftheTuraev-Viro ] T invariants of any link complement determine the simplicial volume of the link, and verify it for all G knotswithzerosimplicialvolume.Finallyweobservethatoursimplicialvolumeconjectureisstable . underconnectsumandsplitunionsoflinks. h t a m 1 Introduction [ In [32], Turaev and Viro defined a family of 3-manifold invariants as state sums on triangulations of 2 v manifolds. The family is indexed by an integer r, and for each r the invariant depends on a choice of 8 a 2r-th root of unity. In the last couple of decades these invariants have been refined and generalized 1 in many directions and shown to be closely related to the Witten-Reshetikhin-Turaev invariants. (See 8 7 [1, 15, 31, 20] and references therein.) Despite these efforts, the relationship between the Turaev-Viro 0 invariantsandthegeometricstructureson3-manifoldsarisingfromThurston’sgeometrizationpictureis . 1 notunderstood. RecentlyChenandthethirdnamedauthor[6]conjecturedthat,evaluatedatappropriate 0 roots of unity, the asymptotic behavior of the Turaev-Viro invariants of a hyperbolic 3-manifold deter- 7 1 mines the hyperbolic volume of the manifold, and presented compelling experimental evidence to their : v conjecture. i In the presentpaper we focus mostly onthe Turaev-Viro invariants of link complements in S3.Our X mainresultgivesaformulaoftheTuraev-Viroinvariantsofalinkcomplementintermsofvaluesofthe r a coloredJonespolynomialofthelink. Usingtheformulawerigorouslyverifythevolumeconjectureof[6] fortheFigure-eightknotandBorromeanringscomplement. Thesearetothebestoftheauthorsknowl- edgethefirstexamplesofthiskind. Ourcalculationsexhibitnewphenomenaofasymptoticbehaviorof thecoloredJonespolynomialthatdoesnotseemtobepredictedbythevolumeconjectures[14,25,7]or byZagier’squantummodularityconjecture[36]. 1.1 Relationshipbetweenknotinvariants To state our results we need to introduce some notations. For a link L ⊂ S3, let TV (S3(cid:114)L,q) denote r the r-th Turaev-Viro invariant of the link complement evaluated at the root of unity q such that q2 is ∗E.K.issupportedbyNSFGrantDMS-1404754 †T.Y.issupportedbyNSFGrantDMS-1405066 1 primitiveofdegreer.Throughoutthispaper,wewillconsiderthecasethatq = A2,whereAiseithera primitive4r-throotforanyintegerr oraprimitive2r-throotforanyoddintegerr. Weusethenotationi = (i ,...,i )foramulti-integerofncomponents(ann-tupleofintegers)and 1 n usethenotation1 (cid:54) i (cid:54) mtodescribeallsuchmulti-integerswith1 (cid:54) i (cid:54) mforeachk ∈ {1,...,n}. k Given a link L with n components, let J (t) denote the i-th colored Jones polynomial of L whose k- L,i th component is colored by i [19, 17]. If all the components of L are colored by the same integer k i, then we simply denote J (t) by J (t). If L is a knot, then J (t) is the usual i-th colored L,(i,...,i) L,i L,i Jones polynomial. The polynomials are indexed so that J (t) = 1 and J (t) is the ordinary ones L,1 L,2 polynomial,andarenormalizedsothat A2i−A−2i J (t) = [i] = U,i A2−A−2 fortheunknotU,wherebyconventiont = A4.Finallywedefine A2−A−2 A2−A−2 η = √ and η(cid:48) = √ . r −2r r −r Beforestatingourmainresult,letusrecallonceagaintheconventionthatq = A2 andt = A4. Theorem1.1. LetLbealinkinS3 withncomponents. (1) Foranintegerr (cid:62) 3andaprimitive4r-throotofunityA,wehave (cid:88) TV (S3(cid:114)L,q) = η2 |J (t)|2. r r L,i 1(cid:54)i(cid:54)r−1 (2) Foranoddintegerr (cid:62) 3andaprimitive2r-throotofunityA,wehave (cid:88) TV (S3(cid:114)L,q) = 2n−1(η(cid:48))2 |J (t)|2. r r L,i 1(cid:54)i(cid:54)r−1 2 Extending an earlier result of Roberts[28], Beneditti and Petronio[1] showed that the invariants πi TVr(M,er ) of a 3-manifold M with boundary coincide up to a scalar with the Witten-Reshetikhin- Turaev invariants of the double of M. In the proof of Theorem 1.1, we first apply Beneditti-Petronio’s argument to the case that r is odd and A is a primitive 2r-th root, extending this relation to the Turaev- ViroinvariantsandtheSO(3)Reshetikhin-Turaevinvariants[17,3,4]. SeeTheorem3.1. Thentherest of the proof follows from the properties of the Reshetikhin-Turaev topological quantum field theory as establishedbyBlanchet,Habegger,MasbaumandVogel[2,4]. Note that for any primitive r-th root of unity with r (cid:62) 3, the quantities η and η(cid:48) are real and r r non-zero. SinceJ (t) = 1,Theorem1.1immediatelyimpliesthefollowing L,1 Corollary1.2. Foranyr (cid:62) 3,anyrootq = A2 andanylinkLinS3,wehave TV (S3(cid:114)L,q) > 0. r We note that the values of the colored Jones polynomials do not have such a positivity property. In fact, all the values that are involved in the Kashaev volume conjecture[14, 25] are known to vanish for thesplitlinksandtheWhiteheadchains. See[25,34]. AnotherimmediateconsequenceofTheorem1.1isthatlinkswiththesamecoloredJonespolynomi- als have the same Turaev-Viro invariants. In particular, since the colored Jones polynomial is invariant underConwaymutationsandthegenus2mutations[23],weobtainthefollowing. Corollary 1.3. For any r (cid:62) 3, any root q = A2 and any link L in S3, the invariants TV (S3(cid:114)L,q) r remainunchangedunderConwaymutationsandthegenus2mutations. 2 1.2 AsymptoticsofTuraev-ViroandcoloredJoneslinkinvariants We are particularly interested in the large r asymptotics of the invariants TV (S3(cid:114)L,A2) in the case r thateitherA = e2πri forintegersr (cid:62) 3orA = eπri foroddintegersr (cid:62) 3.WiththesechoicesofA,we haveintheformercasethat 2sin(π) η = √ r , r 2r andinthelattercasethat 2sin(2π) η(cid:48) = √ r . r r In[6],Chenandthethirdnamedauthorpresentedcompellingexperimentalevidencetothefollowing Conjecture1.4. [6]Foreveryhyperbolic3-manifoldM,wehave 2π 2πi lim log(TVr(M,e r )) = Vol(M), r→∞ r wherer runsoveralloddintegers. 2πi Conjecture 1.4 impies that TVr(M,e r ) grows exponentially in r. This is particularly surprising πi sincethecorrespondinggrowthofTVr(M,er )isexpected,andinmanycasesknown,tobeonlypoly- nomiallybyWitten’sasymptoticexpansionconjecture[35,13]. Forclosed3-manifolds,thispolynomial growthwasestablishedbyGaroufalidis[8]. Combining[8,Theorem2.2]andtheresultsof[1],onehas that for every 3-manifold M with nonempty boundary, there exist constants C > 0 and N such that |TVr(M,eπri)| (cid:54) CrN.ThistogetherwithTheorem1.1implythefollowing. Corollary1.5. ForanylinkLinS3,thereexistconstantsC > 0andN suchthatforanyintegerr and multi-integeriwith1 (cid:54) i (cid:54) r−1,thevalueofthei-thcoloredJonespolynomialatt = e2rπi satisfies |JL,i(e2rπi)| (cid:54) CrN. 2πi Hence,JL,i(e r )growsatmostpolynomiallyinr. AsamainapplicationofTheorem1.1,weprovidethefirstrigorousevidencestoConjecture1.4. Theorem 1.6. Let L be either the Figure-eight knot or the Borromean rings, and let M be the comple- mentofLinS3.Then 2π 2πi 4π 4πi lim logTVr(M,e r ) = lim log|JL,m(e2m+1)| = Vol(M), r→+∞ r m→+∞ 2m+1 wherer = 2m+1runsoveralloddintegers. 2πi The asymptotic behaviour of the values of JL,m(t) at t = em+12 was not predicted previously by either the original volume conjecture[14, 25] or any of its generalizations [7, 24]. Theorem 1.6 seems to suggest that these values grow exponentially in m with growth rate the hyperbolic volume. This is somewhat surprising because as noted in [9], and also in Corollary 1.5, that for any positive integer l, 2πi JL,m(em+l)growsonlypolynomiallyinm. Question1.7. IsittruethatforanyhyperboliclinkL, 2π 2πi lim log|JL,m(em+12)| = Vol(S3(cid:114)L)? m→+∞ m 3 1.3 Knotswithzerosimplicialvolume Recallthatthesimplicialvolume||L||ofalinkListhesumofthevolumesofthehyperbolicpiecesofthe geometricdecompositionofthelinkcomplement,dividedbythevolumeoftheregularidealhyperbolic tetrahedron. In particular, if the geometric decomposition has no hyperbolic pieces, then ||L|| = 0. As a natural generalization of Conjecture 1.4, one can conjecture that for every link L the asymptotics of TVr(S3(cid:114)L,e2rπi)determines||L||.SeeConjecture5.1. UsingTheorem1.1andthepositivityoftheTuraev-Viroinvariants(Corollary1.2),wehaveaproof ofConjecture5.1fortheknotswithzerosimplicialvolume. Theorem1.8. LetK ⊂ S3 beaknotwithsimplicialvolumezero. Then 2π lim logTVr(S3(cid:114)K,e2rπi) = ||K|| = 0, r→∞ r wherer runsoveralloddintegers. Wealsoobservethat,unliketheoriginalvolumeconjecturethatisnottrueforsplitlinks[25,Remark 5.3],Conjecture5.1isclosedunderthesplitunionsoflinks,andundersomeassumptionsisalsoclosed undertheconnectedsums. 1.4 Organization The paper is organized as follows. In Subsection 2.1, we review the Reshetikhin-Turaev invariants[27] following the skein theoretical approach by Blanchet, Habegger, Masbaum and Vogel[2, 3, 4]. In Sub- section2.2,werecallthedefinitionoftheTuraev-Viroinvariants,andconsideranSO(3)-versionofthem forthepurposeofextendingBeneditti-Petronio’stheorem[1]tootherrootsofunity(Theorem3.1). The relationshipbetweenthetwoversionsoftheTuraev-ViroinvariantsisgiveninTheorem2.8whoseproof ispostponedtotheAppendix. WeproveTheorem1.1inSection3,andproveTheorem1.6andTheorem 1.8respectivelyinSections4and5. 1.5 Acknowledgement Part of this work was done during the Advances in Quantum and Low-Dimensional Topology 2016 at UniversityofIowaandKnotsinHellas2016atInternationalOlympicAcademy,Greece. Wewouldlike tothanktheorganizersforsupport,hospitalityandforprovidingexcellentworkingconditions. TheauthorsarealsogratefultoFrancisBonahon,CharlesFrohman,StavrosGaroufalidisandRoland vanderVeenfordiscussionsandsuggestions. 2 Preliminaries 2.1 Reshetikhin-TuraevinvariantsandTQFTs InthissubsectionwereviewthedefinitionandbasicpropertiesoftheReshetikhin-Turaevinvariants. Our expositionfollowstheskeintheoreticalapproachofBlanchet,Habegger,MasbaumandVogel[2,3,4]. A framed link in an oriented 3-manifold M is a smooth embedding L of a disjoint union of finitely manythickenedcirclesS1 ×[0,(cid:15)],forsome(cid:15) > 0,intoM.LetZ[A,A−1]betheringofLaurentpoly- nomialsintheindeterminateA.Thenfollowing[26,30],theKauffmanbracketskeinmoduleK (M)of A M is defined as the quotient of the free Z[A,A−1]-module generated by the isotopy classes of framed linksinM bythefollowingtworelations: 4 (1) KauffmanBracketSkeinRelation: = A + A−1 . (2) FramingRelation: L∪ = (−A2−A−2)L. Thereisacanonicalisomorphism (cid:104)(cid:105) : K (S3) → Z[A,A−1] A between the Kauffman bracket skein module of S3 and Z[A,A−1] viewed a module over itself. The Laurent polynomial (cid:104)L(cid:105) ∈ Z[A,A−1] determined by the framed link L ⊂ S3 is called the Kauffman bracketofL. TheKauffmanbracketskeinmoduleK (T)ofthesolidtorusT = D2×S1 iscanonicallyisomor- A phic to the module Z[A,A−1][z]. Here we consider D2 as the unit disk in the complex plane, and call the framed link [0,(cid:15)]×S1 ⊂ D2 ×S1, for some (cid:15) > 0, the core of T. Then the isomorphism above is givenbysendingiparallelcopiesofthecoreofT tozi.AframedlinkLinS3 ofncomponentsdefines anZ[A,A−1]-multilinearmap (cid:104) ,..., (cid:105) : K (T)⊗n → Z[A,A−1] L A on KA(T), called the Kauffman multi-bracket, as follows. For monomials zik ∈ Z[A,A−1][z] ∼= KA(T),k = 1,...,n,letL(zi1,...,zin)betheframedlinkinS3 obtainedbycablingthek-thcompo- nentofLbyi parallelcopiesofthecore. Thendefine k . (cid:104)zi1,...,zin(cid:105) = (cid:104)L(zi1,...,zin)(cid:105), L andextendZ[A,A−1]-multilinearlytothewholeK (T).FortheunknotU andanypolynomialP(z) ∈ A Z[A,A−1][z],wesimplydenotethebracket(cid:104)P(z)(cid:105) by(cid:104)P(z)(cid:105). U The i-th Chebyshev polynomial e ∈ Z[A,A−1][z] is defined by the recurrence relations e = 1, i 0 e = z,andze = e +e ,andsatisfies 1 j j+1 j−1 (cid:104)e (cid:105) = (−1)i[i+1]. i ThecoloredJonespolynomialsofanorientedknotK inS3 aredefinedusinge asfollows. LetD bea i diagramofK withwrithenumberw(D),andframeDwiththeblackboardframing. Thenthe(i+1)-st coloredJonespolynomialofK is J (t) = (cid:0)(−1)iAi2+2i(cid:1)w(D)(cid:104)e (cid:105) . K,i+1 i D The colored Jones polynomials for an oriented link L in S3 is defined similarly. Let D be a diagram of L with writhe number w(D) and the blackboard framing. For a multi-integer i = (i ,...,i ), let 1 n i+1 = (i +1,...,i +1).Thenthe(i+1)-stcoloredJonespolynomialofLisdefinedby 1 n n JL,i+1(t) = (cid:0)(−1)k(cid:80)=1ikAs(i)(cid:1)w(D)(cid:104)ei1,...,ein(cid:105)D, n wheres(i) = (cid:80)(i2 +i ). k k k=1 WenotethatachangeorientationonsomeorallcomponentsofLchangesthewrithenumberofD, andchangesJ (t)onlybyapowerofA.Therefore,foranunorientedlinkLandacomplexnumberA L,i with|A| = 1,themodulusofthevalueofJ (t)att = A4 iswelldefined,and L,i |J (t)| = |(cid:104)e ,...,e (cid:105) |. (2.1) L,i i1−1 in−1 D 5 If M is a closed oriented 3-manifold obtained by doing surgery along a framed link L in S3, then thespecializationoftheKauffmanmulti-bracketatrootsofunityyieldsinvariantsof3-manifolds. From now on, let A be either a primitive 4r-th root of unity for an integer r (cid:62) 3 or a primitive 2r-th root of unity for an odd integer r (cid:62) 3. To define the Reshetikhin-Turaev invariants, we need to recall some specialelementsofK (T) ∼= Z[A,A−1][z],calledtheKirbycoloring,definedby A r−2 (cid:88) ω = (cid:104)e (cid:105)e r i i i=0 foranyintegerr,and m−1 (cid:88) ω(cid:48) = (cid:104)e (cid:105)e r 2i 2i i=0 foranyoddintegerr = 2m+1.Wealsoforanyr introduce κ = η (cid:104)ω (cid:105) , r r r U+ andforanyoddr introduce κ(cid:48) = η(cid:48)(cid:104)ω(cid:48)(cid:105) , r r r U+ whereU istheunknotwithframing1. + Definition 2.1. Let M be a closed oriented 3-manifold obtained from S3 by doing surgery along a framedlinkLwithnumberofcomponentsn(L)andsignatureσ(L). (1) ThentheReshetikhin-TuraevinvariantoftheM isdefinedby (cid:104)M(cid:105) = η1+n(L) κ−σ(L) (cid:104)ω ,...,ω (cid:105) r r r r r L foranyintegerr (cid:62) 3,andby (cid:104)M(cid:105)(cid:48) = (η(cid:48))1+n(L) (κ(cid:48))−σ(L) (cid:104)ω(cid:48),...,ω(cid:48)(cid:105) r r r r r L foranyoddintegerr (cid:62) 3. (2) Let also L(cid:48) be a framed link in M. Then the Reshetikhin-Turaev invariant of the pair (M,L(cid:48)) is definedby (cid:104)M,L(cid:48)(cid:105) = η1+n(L) κ−σ(L) (cid:104)ω ,...,ω ,1(cid:105) r r r r r L∪L(cid:48) foranyintegerr (cid:62) 3,andby (cid:104)M,L(cid:48)(cid:105)(cid:48) = (η(cid:48))1+n(L) (κ(cid:48))−σ(L) (cid:104)ω(cid:48),...,ω(cid:48),1(cid:105) r r r r r L∪L(cid:48) foranyoddintegerr (cid:62) 3. Remark2.2. (1) Wewillcall(cid:104)M(cid:105)(cid:48) theSO(3)Reshetikhin-TureavinvariantofM. r (2) For any element S in K (M) represented by a Z[A,A−1]-linear combinations of framed links in A M,onecandefine(cid:104)M,S(cid:105) and(cid:104)M,S(cid:105)(cid:48) byZ[A,A−1]-linearextensions. r r (3) Since S3 is obtained by doing surgery along the empty link, we have (cid:104)S3(cid:105) = η and (cid:104)S3(cid:105)(cid:48) = η(cid:48). r r r r Moreover,foranylinkL ⊂ S3 wehave (cid:104)S3,L(cid:105) = η (cid:104)L(cid:105), and (cid:104)S3,L(cid:105)(cid:48) = η(cid:48) (cid:104)L(cid:105). r r r r 6 In[4],Blanchet,Habegger,MasbaumandVogelalsoconstructedtheunderlingtopologicalquantum fieldtheoriesZ andZ oftheReshetikhin-Turaevinvariants,whichcanbesummarizedasfollows. r r(cid:48) Theorem2.3. [4,Theorem1.4] (1) Let Σ be a closed oriented surface, then for any integer r (cid:62) 3, there exists a finite dimensional C-vectorspaceZ (Σ)satisfying r (cid:97) ∼ Z (Σ Σ ) = Z (Σ )⊗Z (Σ ), r 1 2 r 1 r 2 andforeachoddintegerr (cid:62) 3,thereexistsafinitedimensionalC-vectorspaceZ(cid:48)(Σ)satisfying r Z(cid:48)(Σ (cid:97)Σ ) ∼= Z(cid:48)(Σ )⊗Z(cid:48)(Σ ). r 1 2 r 1 r 2 (2) IfH isahandlebodywith∂H = Σ,thenZ (Σ)andZ(cid:48)(Σ)arerespectivelyquotientsoftheKauff- r r manbracketskeinmoduleK (H). A (3) Every compact oriented 3-manifold M with ∂M = Σ and a framed link L in M defines for any integerr avectorZ (M,L)inZ (Σ),andforanyoddintegerr avectorZ(cid:48)(M,L)inZ(cid:48)(Σ). r r r r (4) Forforanyintegerr,thereisasesquilinearpairing(cid:104), (cid:105)onZ (Σ)withthefollowingproperty: Given r oriented3-manifoldsM andM withboundaryΣ = ∂M = ∂M ,andframedlinksL ⊂ M and 1 2 1 2 1 1 L ⊂ M ,wehave 2 2 (cid:104)M,L(cid:105) = (cid:104)Z (M ,L ),Z (M ,L )(cid:105), r r 1 1 r 2 2 (cid:83) where M = M (−M ) is the closed 3-manifold obtained by gluing M and M along Σ and 1 Σ 2 1 2 L = L (cid:96)L . Similarly,foranyoddintegerr,thereisasesquilinearpairing(cid:104), (cid:105)onZ(cid:48)(Σ),such 1 2 r toranyM andLasabove, (cid:104)M,L(cid:105)(cid:48) = (cid:104)Z(cid:48)(M ,L ),Z(cid:48)(M ,L )(cid:105). r r 1 1 r 2 2 For the purpose of this paper, we will only need to understand the TQFT vector spaces of the torus Z (T2)andZ(cid:48)(T2).ThesevectorspacesarequotientsofK (T) ∼= Z[A,A−1][z],hencetheChebyshev r r A polynomials{e }definevectorsinZ (T2)andZ(cid:48)(T2).Wehavethefollowing i r r Theorem2.4. [4,Corollary4.10,Remark4.12] (1) Foranyintegerr (cid:62) 3,thevectors{e ,...,e }formaHermitianbasisofZ (T2). 0 r−2 r (2) Foranyoddintegerr = 2m+1,thevectors{e ,...,e }formaHermitianbasisofZ(cid:48)(T2). 0 m−1 r (3) InZ(cid:48)(T2),wehaveforanyiwith0 (cid:54) i (cid:54) m−1that r e = e . (2.2) m+i m−1−i Therefore,thevectors{e } alsoformaHermitianbasisofZ(cid:48)(T2). 2i i=0,...,m−1 r 7 2.2 Turaev-Viroinvariants Inthissubsection,werecallthedefinitionandbasicpropertiesoftheTuraev-Viroinvariants[32,15]. For anintegerr (cid:62) 3,letI = {0,1,...,r−2}bethesetofnon-negativeintegerslessthanorequaltor−2. r Let q be a 2r-th root of unity such that q2 is a primitive r-th root. For example, q = A2, where A is eitheraprimitive4r-throotorforoddr aprimitive2r-throot,satisfiesthecondition. Fori ∈ I ,define r i = (−1)i[i+1]. A triple (i,j,k) of elements of I is called admissible if (1) i+j (cid:62) k, j +k (cid:62) i and k +i (cid:62) j, (2) r i+j +k isaneven,and(3)i+j +k (cid:54) 2(r−2).Foranadmissibletriple(i,j,k),define i [i+j−k]![j+k−i]![k+i−j]![i+j+k +1]! j = (−1)−i+j2+k 2 2 2 2 . [i]![j]![k]! k A 6-tuple (i,j,k,l,m,n) of elements of I is called admissible if the triples (i,j,k), (j,l,n), r (i,m,n)and(k,l,m)areadmissible. Foranadmissible6-tuple(i,j,k,l,m,n),define mk inj = (cid:81)[i4a]=![k3b]=![1l[]Q![mb−]![nT]a!]! min{Q(cid:88)1,Q2,Q3} (cid:81)4 [z(−−T1)z]![z(cid:81)+3 1][!Q −z]!, l z=max{T1,T2,T3,T4} a=1 a b=1 b where i+j +k i+m+n j +l+n k+l+m T = , T = , T = , T = , 1 2 3 4 2 2 2 2 i+j +l+m i+k+l+n j +k+m+n Q = , Q = , Q = . 1 2 3 2 2 2 Remark 2.5. We choose the notations i , i j and mk inj because the quantities are respectively k l theKauffmanbracketofthecorrespondingspinnetworks. See[15,Chapter9]. AcoloringofaEuclideantetrahedron∆isanassignmentofelementsofI totheedgesof∆,andis r admissibleifthetripleofelementsofI assignedtothethreeedgesofeachfaceof∆isadmissible. See r Figure1: InthissettingeachofT ,...,T correspondstoafaceandeachofQ ,Q ,Q correspondsto 1 4 1 2 3 aquadrilateralinthetetrahedron. Figure1 Let T be a triangulation of M. If M is with non-empty boundary, then we let T be an ideal trian- gulationofM,i.e.,agluingoffinitelymanytruncatedEuclideantetrahedrabyaffinehomeomorphisms 8 betweenpairsoffaces. Inthisway,therearenovertices,andinstead,thetrianglescomingfromtrunca- tions form a triangulation of the boundary of M. By edges of an ideal triangulation, we only mean the onescomingfromtheedgesofthetetrahedra,nottheonesfromthetruncations. Acoloringatlevelrof thetriangulated3-manifold(M,T)isanassignmentofelementsofI totheedgesofT,andisadmis- r sibleifthe6-tupleassigned totheedges ofeachtetrahedron ofT isadmissible. Letcbean admissible coloringof(M,T)atlevelr.ForeachedgeeofT,let c(e) |e| = . c Foreachfacef ofT withedgese ,e ande ,let 1 2 3 c 1 c |f|c = 2 , c 3 wherec = c(e ). i i For each tetrahedra ∆ in T with vertices v ,...,v , denote by e the edge of ∆ connecting the 1 4 ij verticesv andv ,{i,j} ⊂ {1,...,4},andlet i j c c 23 12c |∆|c = c c24 , 13 14 c 34 wherec = c(e ). ij ij Definition 2.6. Let A be the set of admissible colorings of (M,T) at level r, and let V, E F and T r respectivelybethesetsofvertices,edges,facesandtetrahedrainT.Thenther-thTuraev-Viroinvariant isdefinedby (cid:81) (cid:81) TV (M) = η2|V| (cid:88) e∈E|e|c ∆∈T |∆|c. r r (cid:81) |f| c∈Ar f∈E c For an odd integer r (cid:62) 3, one can also consider an SO(3)-version of the Turaev-Viro invariants TV(cid:48)(M)ofM,whichwillrelateTV (M)andtheReshetkin-Turaevinvariants(cid:104)D(M)(cid:105) ofthedouble r r r ofM (Theorems2.8,3.1). TheinvariantTV(cid:48)(M)isdefinedasfollows. LetI(cid:48) = {0,2,...,r−5,r−3} r r be the set of non-negative even integers less than or equal to r−2. An SO(3)-coloring of a Euclidean tetrahedron∆isanassignmentofelementsofI(cid:48) totheedgesof∆,andisadmissibleifthetripleassigned r tothethreeedgesofeachfaceof∆isadmissible. LetT beatriangulationofM.AnSO(3)-coloringat levelr ofthetriangulated3-manifold(M,T)isanassignmentofelementsofI(cid:48) totheedgesofT,and r isadmissibleifthe6-tupleassignedtotheedgesofeachtetrahedronofT isadmissible. Definition2.7. LetA(cid:48) bethesetofSO(3)-admissiblecoloringsof(M,T)atlevelr.Define r (cid:81) (cid:81) TV(cid:48)(M) = (η(cid:48))2|V| (cid:88) e∈E|e|c ∆∈T |∆|c. r r (cid:81) |f| c∈A(cid:48) f∈E c r TherelationshipbetweenTV (M)andTV(cid:48)(M)isgivenbythefollowingtheorem. r r 9 Theorem 2.8. Let M be a 3-manifold and let b (M) and b (M) respectively be its zeroth and second 0 2 Z -Bettinumber. 2 (1) Foranyoddintegerr (cid:62) 3, TV (M) = TV (M)·TV(cid:48)(M). r 3 r πi (2) (Turaev-Viro[32]). If∂M = ∅andA = e3 ,then TV (M) = 2b2(M)−b0(M). 3 πi (3) IfM isconnected,∂M (cid:54)= ∅andA = e3 ,then TV (M) = 2b2(M). 3 Inparticular,TV (M)isnonzero. 3 WepostponetheproofofTheorem2.8toAppendixAtoavoidunnecessarydistractions. 3 The colored Jones sum formula for Turaev-Viro invariants InthisSection,wefirstestablisharelationshipbetweentheSO(3)Turaev-Viroinvariantsofa3-manifold withboundarytotheSO(3)Reshetikhin-Turaevinvariantsofitsdouble,followingtheargumentof[1]. See Theorem 3.1. Then we prove Theorem 1.1 using the TQFT properties of the Reshetikhin-Turaev invariantsestablishedin[2,4]. 3.1 Relationshipbetweeninvariants TherelationshipbetweenTuraev-ViroandWitten-Reshetikhin-TuraevinvariantswasstudiedbyTuraev- Walker[31] and Roberts[28] for closed 3-manifolds and by Beneditti and Petronio[1] for 3-manifolds withboundary. Foranoriented3-manifoldM withboundary,denoteM withtheoppositeorientationby −M,andletD(M)denotethedoubleofM,i.e., (cid:91) D(M) = M (−M). ∂M We will need the following theorem of Benedetti and Petronio[1]. In fact [1] only treats the case of πi A = e2r,but,aswewillexplainbelow,theproofforothercasesissimilar. Theorem3.1. LetM bea3-manifoldwithboundary. Then TV (M) = η−χ(M)(cid:104)D(M)(cid:105) r r r foranyintegerr,and TV(cid:48)(M) = (η(cid:48))−χ(M)(cid:104)D(M)(cid:105)(cid:48) r r r foranyoddr,whereχ(M)istheEulercharacteristicofM. We refer to [1] and [28] for the SU(2) (r being any integer) case, and for the reader’s convenience includeasketchoftheproofherefortheSO(3)(r beingodd)case. ThemaindifferencefortheSO(3) casecomesfromtothefollowinglemmaduetoLickorish. 10