Turaev-Viro invariants, colored Jones polynomials and volume

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by  Renaud Detcherry

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