Turaev-Viro invariants, colored Jones polynomials and volume

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1 Tuaev-Vio invaiants, coloed Jones polynomials and volume Renaud Detchey, Efstatia Kalfagianni and Tian Yang Abstact We obtain a fomula fo the Tuaev-Vio invaiants of a link complement in tems of values of the coloed Jones polynomial of the link. As an application we give the fist examples of 3-manifolds whee the lage asymptotics the Tuaev-Vio invaiants detemine the hypebolic volume. We veify the volume conjectue of Chen and the thid named autho [6] fo the Figue-eight knot and the Boomean ings. Ou calculations also exhibit new phenomena of asymptotic behavio of values of the coloed Jones polynomials that seem not to be pedicted by neithe the Kashaev-Muakami- Muakami volume conjectue and vaious of its genealizations no by Zagie s quantum modulaity conjectue. We then conjectue that the asymptotics of the Tuaev-Vio invaiants of any link complement detemine the simplicial volume of the link, and veify it fo all knots with zeo simplicial volume. Finally we obseve that ou simplicial volume conjectue is stable unde connect sum and split unions of links. 1 Intoduction In [33], Tuaev and Vio defined a family of 3-manifold invaiants as state sums on tiangulations of manifolds. The family is indexed by an intege, and fo each the invaiant depends on a choice of a -th oot of unity. In the last couple of decades these invaiants have been efined and genealized in many diections and shown to be closely elated to the Witten-Reshetikhin-Tuaev invaiants. (See [1, 15, 3, 0] and efeences theein.) Despite these effots, the elationship between the Tuaev-Vio invaiants and the geometic stuctues on 3-manifolds aising fom Thuston s geometization pictue is not undestood. Recently Chen and the thid named autho [6] conjectued that, evaluated at appopiate oots of unity, the asymptotic behavio of the Tuaev-Vio invaiants of a complete hypebolic 3-manifold, of finite volume, detemines the hypebolic volume of the manifold, and pesented compelling expeimental evidence to thei conjectue. In the pesent pape we focus mostly on the Tuaev-Vio invaiants of link complements in S 3. Ou main esult gives a fomula of the Tuaev-Vio invaiants of a link complement in tems of values of the coloed Jones polynomial of the link. Using the fomula we igoously veify the volume conjectue of [6] fo the Figue-eight knot and Boomean ings complement. These ae to the best of the authos knowledge the fist examples of this kind. Ou calculations exhibit new phenomena of asymptotic behavio of the coloed Jones polynomial that does not seem to be pedicted by the volume conjectues [14, 5, 7] o by Zagie s quantum modulaity conjectue [37]. 1.1 Relationship between knot invaiants To state ou esults we need to intoduce some notations. Fo a link L S 3, let T V (S 3 L, q) denote the -th Tuaev-Vio invaiant of the link complement evaluated at the oot of unity q such that q is E.K. is suppoted by NSF Gant DMS T.Y. is suppoted by NSF Gant DMS

2 pimitive of degee. Thoughout this pape, we will conside the case that q = A, whee A is eithe a pimitive 4-th oot fo any intege o a pimitive -th oot fo any odd intege. We use the notation i = (i 1,..., i n ) fo a multi-intege of n components (an n-tuple of integes) and use the notation 1 i m to descibe all such multi-integes with 1 i k m fo each k {1,..., n}. Given a link L with n components, let J L,i (t) denote the i-th coloed Jones polynomial of L whose k- th component is coloed by i k [19, 17]. If all the components of L ae coloed by the same intege i, then we simply denote J L,(i,...,i) (t) by J L,i (t). If L is a knot, then J L,i (t) is the usual i-th coloed Jones polynomial. The polynomials ae indexed so that J L,1 (t) = 1 and J L, (t) is the odinay ones polynomial, and ae nomalized so that J U,i (t) = [i] = Ai A i A A fo the unknot U, whee by convention t = A 4. Finally we define η = A A and η = A A. Befoe stating ou main esult, let us ecall once again the convention that q = A and t = A 4. Theoem 1.1. Let L be a link in S 3 with n components. (1) Fo an intege 3 and a pimitive 4-th oot of unity A, we have T V (S 3 L, q) = η J L,i (t). 1 i 1 () Fo an odd intege 3 and a pimitive -th oot of unity A, we have T V (S 3 L, q) = n 1 (η ) J L,i (t). 1 i 1 Extending an ealie esult of Robets [9], Beneditti and Petonio [1] showed that the invaiants T V (M, e πi ) of a 3-manifold M with bounday coincide up to a scala with the Witten-Reshetikhin- Tuaev invaiants of the double of M. In the poof of Theoem 1.1, we fist apply Beneditti-Petonio s agument to the case that is odd and A is a pimitive -th oot, extending this elation to the Tuaev- Vio invaiants and the SO(3) Reshetikhin-Tuaev invaiants [17, 3, 4]. See Theoem 3.1. Then the est of the poof follows fom the popeties of the Reshetikhin-Tuaev topological quantum field theoy as established by Blanchet, Habegge, Masbaum and Vogel [, 4]. Note that fo any pimitive -th oot of unity with 3, the quantities η and η ae eal and nonzeo. Since J L,1 (t) = 1, and with the notation as in Theoem 1.1, we have T V (S 3 L, q) η in case (1) and T V (S 3 L, q) n 1 (η ) in case (). In paticula, we have the following Coollay 1.. Fo any 3, any oot q = A and any link L in S 3, we have T V (S 3 L, q) > 0. We note that the values of the coloed Jones polynomials do not have such a positivity popety. In fact, all the values that ae involved in the Kashaev volume conjectue [14, 5] ae known to vanish fo the split links and the Whitehead chains [5, 35]. Anothe immediate consequence of Theoem 1.1 is that links with the same coloed Jones polynomials have the same Tuaev-Vio invaiants. In paticula, since the coloed Jones polynomial is invaiant unde Conway mutations and the genus mutations [3], we obtain the following. Coollay 1.3. Fo any 3, any oot q = A and any link L in S 3, the invaiants T V (S 3 L, q) emain unchanged unde Conway mutations and the genus mutations.

3 1. Asymptotics of Tuaev-Vio and coloed Jones link invaiants We ae paticulaly inteested in the lage asymptotics of the invaiants T V (S 3 L, A ) in the case that eithe A = e πi fo integes 3 o A = e πi fo odd integes 3. With these choices of A, we have in the fome case that η = sin( π ), and in the latte case that η = sin( π ). In [6], Chen and the thid named autho pesented compelling expeimental evidence to the following Conjectue 1.4. [6] Fo evey hypebolic 3-manifold M, we have whee uns ove all odd integes. π lim log(t V (M, e πi )) = V ol(m), Conjectue 1.4 impies that T V (M, e πi ) gows exponentially in. This is paticulaly supising since the coesponding gowth of T V (M, e πi ) is expected, and in many cases known, to be only polynomially by Witten s asymptotic expansion conjectue [36, 13]. Fo closed 3-manifolds, this polynomial gowth was established by Gaoufalidis [8]. Combining [8, Theoem.] and the esults of [1], one has that fo evey 3-manifold M with nonempty bounday, thee exist constants C > 0 and N such that T V (M, e πi ) C N. This togethe with Theoem 1.1 imply the following. Coollay 1.5. Fo any link L in S 3, thee exist constants C > 0 and N such that fo any intege and multi-intege i with 1 i 1, the value of the i-th coloed Jones polynomial at t = e πi satisfies Hence, J L,i (e πi ) gows at most polynomially in. J L,i (e πi ) C N. As a main application of Theoem 1.1, we povide the fist igoous evidences to Conjectue 1.4. Theoem 1.6. Let L be eithe the Figue-eight knot o the Boomean ings, and let M be the complement of L in S 3. Then π lim + log T V (M, e πi 4π ) = lim m + m + 1 log J L,m(e 4πi m+1 ) = V ol(m), whee = m + 1 uns ove all odd integes. m+ The asymptotic behavio of the values of J L,m (t) at t = e 1 was not pedicted peviously by eithe the oiginal volume conjectue [14, 5] o any of its genealizations [7, 4]. Theoem 1.6 seems to suggest that these values gow exponentially in m with gowth ate the hypebolic volume. This is somewhat supising because as noted in [9], and also in Coollay 1.5, that fo any positive intege l, J L,m (e πi m+l ) gows only polynomially in m. Question 1.7. Is it tue that fo any hypebolic link L, πi πi π lim m + m log J m+ L,m(e 1 ) = V ol(s 3 L)? 3

4 1.3 Knots with zeo simplicial volume Recall that the simplicial volume L of a link L is the sum of the volumes of the hypebolic pieces of the geometic decomposition of the link complement, divided by the volume of the egula ideal hypebolic tetahedon. In paticula, if the geometic decomposition has no hypebolic pieces, then L = 0. As a natual genealization of Conjectue 1.4, one can conjectue that fo evey link L the asymptotics of T V (S 3 L, e πi ) detemines L. See Conjectue 5.1. Using Theoem 1.1 and the positivity of the Tuaev-Vio invaiants (Coollay 1.), we have a poof of Conjectue 5.1 fo the knots with zeo simplicial volume. Theoem 1.8. Let K S 3 be a knot with simplicial volume zeo. Then π lim log T V (S 3 K, e πi ) = K = 0, whee uns ove all odd integes. We also obseve that, unlike the oiginal volume conjectue that is not tue fo split links [5, Remak 5.3], Conjectue 5.1 is closed unde the split unions of links, and unde some assumptions is also closed unde the connected sums. 1.4 Oganization The pape is oganized as follows. In Subsection.1, we eview the Reshetikhin-Tuaev invaiants [8] following the skein theoetical appoach by Blanchet, Habegge, Masbaum and Vogel [, 3, 4]. In Subsection., we ecall the definition of the Tuaev-Vio invaiants, and conside an SO(3)-vesion of them fo the pupose of extending Beneditti-Petonio s theoem [1] to othe oots of unity (Theoem 3.1). The elationship between the two vesions of the Tuaev-Vio invaiants is given in Theoem.9 whose poof is postponed to the Appendix. We pove Theoem 1.1 in Section 3, and pove Theoem 1.6 and Theoem 1.8 espectively in Sections 4 and Acknowledgement Pat of this wok was done duing the Advances in Quantum and Low-Dimensional Topology 016 at Univesity of Iowa and Knots in Hellas 016 at Intenational Olympic Academy, Geece. We would like to thank the oganizes fo suppot, hospitality and fo poviding excellent woking conditions. The authos ae also gateful to Fancis Bonahon, Chales Fohman, Stavos Gaoufalidis and Roland van de Veen fo discussions and suggestions. Peliminaies.1 Reshetikhin-Tuaev invaiants and TQFTs In this subsection we eview the definition and basic popeties of the Reshetikhin-Tuaev invaiants. Ou exposition follows the skein theoetical appoach of Blanchet, Habegge, Masbaum and Vogel [, 3, 4]. A famed link in an oiented 3-manifold M is a smooth embedding L of a disjoint union of finitely many thickened cicles S 1 [0, ɛ], fo some ɛ > 0, into M. Let Z[A, A 1 ] be the ing of Lauent polynomials in the indeteminate A. Then following [7, 31], the Kauffman backet skein module K A (M) of M is defined as the quotient of the fee Z[A, A 1 ]-module geneated by the isotopy classes of famed links in M by the following two elations: 4

5 (1) Kauffman Backet Skein Relation: = A + A 1. () Faming Relation: L = ( A A ) L. Thee is a canonical isomophism : K A (S 3 ) Z[A, A 1 ] between the Kauffman backet skein module of S 3 and Z[A, A 1 ] viewed a module ove itself. The Lauent polynomial L Z[A, A 1 ] detemined by the famed link L S 3 is called the Kauffman backet of L. The Kauffman backet skein module K A (T ) of the solid tous T = D S 1 is canonically isomophic to the module Z[A, A 1 ][z]. Hee we conside D as the unit disk in the complex plane, and call the famed link [0, ɛ] S 1 D S 1, fo some ɛ > 0, the coe of T. Then the isomophism above is given by sending i paallel copies of the coe of T to z i. A famed link L in S 3 of n components defines an Z[A, A 1 ]-multilinea map,..., L : K A (T ) n Z[A, A 1 ] on K A (T ), called the Kauffman multi-backet, as follows. Fo monomials z i k Z[A, A 1 ][z] = K A (T ), k = 1,..., n, let L(z i 1,..., z in ) be the famed link in S 3 obtained by cabling the k-th component of L by i k paallel copies of the coe. Then define z i 1,..., z in L. = L(z i 1,..., z in ), and extend Z[A, A 1 ]-multilinealy to the whole K A (T ). Fo the unknot U and any polynomial P (z) Z[A, A 1 ][z], we simply denote the backet P (z) U by P (z). The i-th Chebyshev polynomial e i Z[A, A 1 ][z] is defined by the ecuence elations e 0 = 1, e 1 = z, and ze j = e j+1 + e j 1, and satisfies e i = ( 1) i [i + 1]. The coloed Jones polynomials of an oiented knot K in S 3 ae defined using e i as follows. Let D be a diagam of K with withe numbe w(d), and fame D with the blackboad faming. Then the (i + 1)-st coloed Jones polynomial of K is J K,i+1 (t) = ( ( 1) i A i +i ) w(d) ei D. The coloed Jones polynomials fo an oiented link L in S 3 is defined similaly. Let D be a diagam of L with withe numbe w(d) and the blackboad faming. Fo a multi-intege i = (i 1,..., i n ), let i + 1 = (i 1 + 1,..., i n + 1). Then the (i + 1)-st coloed Jones polynomial of L is defined by whee s(i) = n (i k + i k). k=1 n J L,i+1 (t) = ( ka ( 1) k=1i s(i) ) w(d) ei1,..., e in D, We note that a change oientation on some o all components of L changes the withe numbe of D, and changes J L,i (t) only by a powe of A. Theefoe, fo an unoiented link L and a complex numbe A with A = 1, the modulus of the value of J L,i (t) at t = A 4 is well defined, and J L,i (t) = e i1 1,..., e in 1 D. (.1) 5

6 If M is a closed oiented 3-manifold obtained by doing sugey along a famed link L in S 3, then the specialization of the Kauffman multi-backet at oots of unity yields invaiants of 3-manifolds. Fom now on, let A be eithe a pimitive 4-th oot of unity fo an intege 3 o a pimitive -th oot of unity fo an odd intege 3. To define the Reshetikhin-Tuaev invaiants, we need to ecall some special elements of K A (T ) = Z[A, A 1 ][z], called the Kiby coloing, defined by fo any intege, and ω = e i e i i=0 m 1 ω = e i e i i=0 fo any odd intege = m + 1. We also fo any intoduce κ = η ω U+, and fo any odd intoduce whee U + is the unknot with faming 1. κ = η ω U+, Definition.1. Let M be a closed oiented 3-manifold obtained fom S 3 by doing sugey along a famed link L with numbe of components n(l) and signatue σ(l). (1) Then the Reshetikhin-Tuaev invaiant of the M is defined by fo any intege 3, and by fo any odd intege 3. M = η 1+n(L) κ σ(l) ω,..., ω L M = (η ) 1+n(L) (κ ) σ(l) ω,..., ω L () Let also L be a famed link in M. Then the Reshetikhin-Tuaev invaiant of the pai (M, L ) is defined by M, L = η 1+n(L) κ σ(l) ω,..., ω, 1 L L fo any intege 3, and by fo any odd intege 3. M, L = (η ) 1+n(L) (κ ) σ(l) ω,..., ω, 1 L L Remak.. (1) We will call M the SO(3) Reshetikhin-Tueav invaiant of M. () Fo any element S in K A (M) epesented by a Z[A, A 1 ]-linea combinations of famed links in M, one can define M, S and M, S by Z[A, A 1 ]-linea extensions. (3) Since S 3 is obtained by doing sugey along the empty link, we have S 3 = η and S 3 = η. Moeove, fo any link L S 3 we have S 3, L = η L, and S 3, L = η L. 6

7 In [4], Blanchet, Habegge, Masbaum and Vogel also constucted the undeling topological quantum field theoies Z and Z of the Reshetikhin-Tuaev invaiants, which can be summaized as follows. Theoem.3. [4, Theoem 1.4] (1) Let Σ be a closed oiented suface, then fo any intege 3, thee exists a finite dimensional C-vecto space Z (Σ) satisfying Z (Σ 1 Σ ) = Z (Σ 1 ) Z (Σ ), and fo each odd intege 3, thee exists a finite dimensional C-vecto space Z (Σ) satisfying Z (Σ 1 Σ ) = Z (Σ 1 ) Z (Σ ). () If H is a handlebody with H = Σ, then Z (Σ) and Z (Σ) ae espectively quotients of the Kauffman backet skein module K A (H). (3) Evey compact oiented 3-manifold M with M = Σ and a famed link L in M defines fo any intege a vecto Z (M, L) in Z (Σ), and fo any odd intege a vecto Z (M, L) in Z (Σ). (4) Fo fo any intege, thee is a sesquilinea paiing, on Z (Σ) with the following popety: Given oiented 3-manifolds M 1 and M with bounday Σ = M 1 = M, and famed links L 1 M 1 and L M, we have M, L = Z (M 1, L 1 ), Z (M, L ), whee M = M 1 Σ ( M ) is the closed 3-manifold obtained by gluing M 1 and M along Σ and L = L 1 L. Similaly, fo any odd intege, thee is a sesquilinea paiing, on Z (Σ), such to any M and L as above, M, L = Z (M 1, L 1 ), Z (M, L ). Fo the pupose of this pape, we will only need to undestand the TQFT vecto spaces of the tous Z (T ) and Z (T ). These vecto spaces ae quotients of K A (T ) = Z[A, A 1 ][z], hence the Chebyshev polynomials {e i } define vectos in Z (T ) and Z (T ). We have the following Theoem.4. [4, Coollay 4.10, Remak 4.1] (1) Fo any intege 3, the vectos {e 0,..., e } fom a Hemitian basis of Z (T ). () Fo any odd intege = m + 1, the vectos {e 0,..., e m 1 } fom a Hemitian basis of Z (T ). (3) In Z (T ), we have fo any i with 0 i m 1 that e m+i = e m 1 i. (.) Theefoe, the vectos {e i } i=0,...,m 1 also fom a Hemitian basis of Z (T ). 7

8 . Tuaev-Vio invaiants In this subsection, we ecall the definition and basic popeties of the Tuaev-Vio invaiants [33, 15]. The appoach of [33] elies on quantum 6j-symbols while the definition of Kauffman and Lins [15] uses invaiants of spin netwoks. The two definitions wee shown to be equivalent in [6]. The fomalism of [15] tuns out to be moe convenient to wok with when using skein theoetic techniques to elate the Tuaev-Vio invaiants to the Reshetikhin-Tuaev invaiants. Fo an intege 3, let I = {0, 1,..., } be the set of non-negative integes less than o equal to. Let q be a -th oot of unity such that q is a pimitive -th oot. Fo example, q = A, whee A is eithe a pimitive 4-th oot o fo odd a pimitive -th oot, satisfies the condition. Fo i I, define i = ( 1) i [i + 1]. A tiple (i, j, k) of elements of I is called admissible if (1) i + j k, j + k i and k + i j, () i + j + k is an even, and (3) i + j + k ( ). Fo an admissible tiple (i, j, k), define i k j = ( 1) i+j+k [ i+j k ]![ j+k i ]![ k+i j ]![ i+j+k + 1]! [i]![j]![k]!. A 6-tuple (i, j, k, l, m, n) of elements of I is called admissible if the tiples (i, j, k), (j, l, n), (i, m, n) and (k, l, m) ae admissible. Fo an admissible 6-tuple (i, j, k, l, m, n), define m k l i n j 4 3 a=1 b=1 = [Q b T a ]! [i]![j]![k]![l]![m]![n]! min{q 1,Q,Q 3 } z=max{t 1,T,T 3,T 4 } ( 1) z [z + 1]! 4 a=1 [z T a]! 3 b=1 [Q b z]!, whee T 1 = i + j + k Remak.5. The symbols Q 1 = i + j + l + m i, T = i + m + n,, T 3 = j + l + n, Q = i + k + l + n i j k and m i n k j l, T 4 = k + l + m,, Q 3 = j + k + m + n., used above, ae examples of spin netwoks: tivalent ibbon gaphs with ends coloed by integes. The expessions on the ight hand sides of above equations give the Kauffman backet invaiant of the coesponding netwoks. See [15, Chapte 9]. In the language of [15], the second and thid spin netwoks above ae the tihedal and tetahedal netwoks, denoted by θ(i, j, k) and τ(i, j, k) theein, and the coesponding invaiants ae the tihedal and tetahedal coefficients, espectively. Definition.6. A coloing of a Euclidean tetahedon is an assignment of elements of I to the edges of, and is admissible if the tiple of elements of I assigned to the thee edges of each face of is admissible. See Figue 1 fo a geometic intepetation of tetahedal coefficients. Let T be a tiangulation of M. If M is with non-empty bounday, then we let T be an ideal tiangulation of M, i.e., a gluing of finitely many tuncated Euclidean tetaheda by affine homeomophisms between pais of faces. In this way, thee ae no vetices, and instead, the tiangles coming fom tuncations fom a tiangulation of the bounday of M. By edges of an ideal tiangulation, we only mean the ones coming fom the edges of the tetaheda, not the ones fom the tuncations. A coloing at level of 8

9 Figue 1: The quantities T 1,..., T 4 coespond to faces and Q 1, Q, Q 3 coespond to quadilateals. the tiangulated 3-manifold (M, T ) is an assignment of elements of I to the edges of T, and is admissible if the 6-tuple assigned to the edges of each tetahedon of T is admissible. Let c be an admissible coloing of (M, T ) at level. Fo each edge e of T, let e c = c(e). Fo each face f of T with edges e 1, e and e 3, let f c = c 1 c c 3, whee c i = c(e i ). Fo each tetaheda in T with vetices v 1,..., v 4, denote by e ij the edge of connecting the vetices v i and v j, {i, j} {1,..., 4}, and let c = c 3 c 1 c 4 c 13 c 14 c 34, whee c ij = c(e ij ). Definition.7. Let A be the set of admissible coloings of (M, T ) at level, and let V, E F and T espectively be the sets of vetices, edges, faces and tetaheda in T. Then the -th Tuaev-Vio invaiant is defined by T V (M) = η V c A e E e c f E f c T c Fo an odd intege 3, one can also conside an SO(3)-vesion of the Tuaev-Vio invaiants T V (M) of M, which will elate T V (M) and the Reshetkin-Tuaev invaiants D(M) of the double of M (Theoems.9, 3.1). The invaiant T V (M) is defined as follows. Let I = {0,,..., 5, 3} be the set of non-negative even integes less than o equal to. An SO(3)-coloing of a Euclidean tetahedon is an assignment of elements of I to the edges of, and is admissible if the tiple assigned to the thee edges of each face of is admissible. Let T be a tiangulation of M. An SO(3)-coloing at level of the tiangulated 3-manifold (M, T ) is an assignment of elements of I to the edges of T, and is admissible if the 6-tuple assigned to the edges of each tetahedon of T is admissible. 9.

10 Definition.8. Let A be the set of SO(3)-admissible coloings of (M, T ) at level. Define T V (M) = (η ) V e E e c T c f E f. c c A The elationship between T V (M) and T V (M) is given by the following theoem. Theoem.9. Let M be a 3-manifold and let b 0 (M) and b (M) espectively be its zeoth and second Z -Betti numbe. (1) Fo any odd intege 3, T V (M) = T V 3 (M) T V (M). () (Tuaev-Vio [33]). If M = and A = e πi 3, then (3) If M is connected, M and A = e πi 3, then In paticula, T V 3 (M) is nonzeo. T V 3 (M) = b (M) b 0 (M). T V 3 (M) = b (M). We postpone the poof of Theoem.9 to Appendix A to avoid unnecessay distactions. 3 The coloed Jones sum fomula fo Tuaev-Vio invaiants In this Section, we fist establish a elationship between the SO(3) Tuaev-Vio invaiants of a 3-manifold with bounday to the SO(3) Reshetikhin-Tuaev invaiants of its double, following the agument of [1]. See Theoem 3.1. Then we pove Theoem 1.1 using the TQFT popeties of the Reshetikhin-Tuaev invaiants established in [, 4]. 3.1 Relationship between invaiants The elationship between Tuaev-Vio and Witten-Reshetikhin-Tuaev invaiants was studied by Tuaev- Walke [3] and Robets [9] fo closed 3-manifolds and by Beneditti and Petonio [1] fo 3-manifolds with bounday. Fo an oiented 3-manifold M with bounday, denote M with the opposite oientation by M, and let D(M) denote the double of M, i.e., D(M) = M M( M). We will need the following theoem of Benedetti and Petonio [1]. In fact [1] only teats the case of A = e πi, but, as we will explain below, the poof fo othe cases is simila. Theoem 3.1. Let M be a 3-manifold with bounday. Then fo any intege, and T V (M) = η χ(m) D(M) T V (M) = (η ) χ(m) D(M) fo any odd, whee χ(m) is the Eule chaacteistic of M. 10

11 We efe to [1] and [9] fo the SU() ( being any intege) case, and fo the eade s convenience include a sketch of the poof hee fo the SO(3) ( being odd) case. The main diffeence fo the SO(3) case comes fom to the following lemma due to Lickoish. Lemma 3. ([18, Lemma 6]). Let 3 be an odd intege and let A be a pimitive -th oot of unity. Then i = i if i = 0, 0 if i 0, I.e., the element of the i-th Tempeley-Lieb algeba obtained by cicling the i-th Jones-Wenzl idempotent f i by the Kiby coloing ω equals f i when i = 0 o, and equals 0 othewise. As a consequence, the usual fusion ule [19] should be modified to the following Lemma 3.3 (Fusion Rule). Let 3 be an odd intege. Then fo a tiple (i, j, k) of elements of I, i j k = (η ) i j k i j k i j k if (i, j, k) is -admissible, 0 if (i, j, k) is not -admissible. Hee the integes i, j and k being even is cucial, since it ules out the possibility that i+j+k =, which by Lemma 3. could be toublesome. This is the eason that we conside the invaiant T V (M) i j instead of T V (M). (The facto in the fomula above is also denoted by θ(i, j, k) in [1] and [9].) k Sketch of poof of Theoem 3.1. Following [1], we extend the chain-mail invaiant of Robets [9] to M with non-empty bounday using a handle decomposition without 3-handles. Fo such a handle decomposition, let d 0, d 1 and d espectively be the numbe of 0-, 1- and -handles. Let ɛ i be the attaching cuves of the -handles and let δ j be the meidians of the 1-handles. Thicken the cuves to bands paallel to the suface of the 1-skeleton H and push the ɛ-bands slightly into H. Embed H abitaily into S 3 and colo each of the image of the ɛ- and δ-bands by η ω to get an element in S M in K A (S 3 ). Then the chain-mail invaiant of M is defined by CM (M) = (η ) d 0 S M. Recall hee is the Kauffman backet. It is poved in [1, 9] that CM (M) is independent of the choice of the handle decomposition and the embedding, hence defines an invaiant of M. On the one hand, if the handle decomposition is obtained by the dual of an ideal tiangulation T of M, namely the -handles come fom a tubula neighbohood of the edges of T, the 1-handles come fom a tubula neighbohood of the faces of T and the 0-handles come fom the complement of the 1- and -handles. Since each face has thee edges, each δ-band encloses exactly thee ɛ-bands (see [9, Figue 11]). By elation (.), evey η ω on the ɛ-band can be witten as η ω = η 1 1 i=0 e i e i = η i=0 e i e i.

12 Next we apply Lemma 3.3 to each δ-band. In this pocess the fou δ-bands coesponding to each tetahedon of T give ise to a tetahedal netwok (see also [9, Figue 1]). Then by Remak.5 and equations peceding it, we may ewite CM (M) in tems of tihedal and tetahedal coefficients to obtain CM (M) = (η ) d 0 d 1 +d c A T c f E f c e E e c = (η ) χ(m) T V (M). On the othe hand, if the handle decomposition is standad, namely H is a standad handlebody in S 3 with exactly one 0-handle, then the ɛ- and the δ-bands gives a sugey diagam L of D(M). The way to see it is as follows. Conside the 4-manifold W 1 obtained by attaching 1-handles along the δ-bands (see Kiby [16]) and -handles along the ɛ-bands. Then W 1 is homeomophic to M I and W 1 = M {0} M I ( M) {1} = D(M). Now if W is the 4-manifold obtained by attaching -handles both the ɛ- and the δ-bands, i.e. L, then W is the 3-manifold epesented by the famed link L. Then due to the fact that W 1 = W and Definition.1, we have CM (M) = η η ω,..., η ω L = (η ) 1+n(L) ω,..., ω L = D(M) (κ ) σ(l). We ae left to show that σ(l) = 0. It follows fom that the linking matix of L has the fom [ ] 0 A LK(L) = A T, 0 whee the blocks come fom gouping the ɛ- and the δ-bands togethe and A ij = LK(ɛ i, δ j ). Then, fo any eigen-veco v = (v 1, v ) with eigenvalue λ, the vecto v = ( v 1, v ) is an eigen-vecto of eigenvalue λ. Remak 3.4. Theoems 3.1 and.9 togethe with the main esult of [9] imply that if Conjectue 1.4 holds fo M with totally geodesic bounday, then it holds fo D(M). 3. Poof of Theoem 1.1 We ae now eady to pove Theoem 1.1. Fo the convenience of the eade we estate the theoem. Theoem 1.1. Let L be a link in S 3 with n components. (1) Fo an intege 3 and a pimitive 4-th oot of unity A, we have T V (S 3 L, q) = η J L,i (t). 1 i 1 () Fo an odd intege = m and a pimitive -th oot of unity A, we have T V (S 3 L, q) = n 1 (η ) J L,i (t). Hee, in both cases we have t = q = A 4. 1 i m Poof. We fist conside the case that = m + 1 is odd. Fo a famed link L in S 3 with n components, we let M = S 3 L. Since by Theoem.9, T V (M) = n 1 T V (M), we will wok fom now on with T V (M). 1

13 Since the Eule chaacteistic of M is zeo, by Theoem 3.1, we obtain T V (M) = D(M) = Z (M), Z (M), (3.1) whee Z (M) is a vecto in Z (T ) n. Let {e i } i=0,...,m 1 be the basis of Z (T ) descibed in Theoem.4 (). Then the vecto space Z (T ) n has a Hemitian basis given by {e i = e i1 e i... e in } fo all i = (i 1, i,..., i n ) with 0 i m 1. We wite e i L fo the multi-backet e i1, e i,..., e in L. Then by elation (.1), to establish the desied fomula in tems of the coloed Jones polynomials, it is suffices to show the following By witing and equation (3.1), we have that T V (M) = (η ) Z (M) = T V (M) = 0 i m 1 0 i m 1 0 i m 1 e i L. λ i e i λ i. The computation of the coefficients λ i of Z (M) elies on the TQFT popeties of the invaiants [4]. (Also compae with the agument in [5, Section 4.]). Since {e i } is a Hemitian basis of Z (T ) n, we have λ i = Z (M), e i. n A tubula neighbohood N L of L is a disjoint union of solid toi T k. We let L(e i ) be the element of K A (N L ) obtained by cabling the component of the L in T k the i k -th Chebyshev polynomial e ik. Then in Z (T ) n, we have e i = Z (N L, L(e i )). Now by Theoem.3 (4), since S 3 = M ( N L ), we have Z (M), e i = Z (M), Z (N L, L(e i )) = M ( N L ), L(e i )) = S 3, L(e i ). Finally, by Remak. (), we have Theefoe, we have which finishes the poof in the case of = m + 1. S 3, L(e i ) = η e i L. λ i = η e i L, The agument of the emaining case is vey simila. By Theoem 3.1, we obtain k=1 T V (M) = D(M) = Z (M), Z (M). Woking with the Hemitian basis {e i } i=0,..., of Z (T ) given in Theoem.4 (1), we have T V (M) = λ i, 0 i whee λ i = Z (M), e i and e i = Z (N L, L(e i )). Now by Theoem.3 (4) and Remak., one sees which finishes the poof. λ i = η e i L, 13

14 4 Applications to Conjectue 1.4 In this section we use Theoem 1.1 to detemine the asymptotic behavio of the Tuaev-Vio invaiants fo some hypebolic knot and link complements. In paticula, we veify Conjectue 1.4 fo the complement of the Figue-eight knot and the Boomean ings. To the best of ou knowledge these ae the fist calculations of this kind. 4.1 The Figue-eight complement The following theoem veifies Conjectue 1.4 fo the Figue-eight knot. Theoem 4.1. Let K be the Figue-eight knot and let M be the complement of K in S 3. Then π lim + log T V (M, e πi 4π ) = lim m + m + 1 log J K,m(e 4πi m+1 ) = V ol(m), whee = m + 1 uns ove all odd integes. Poof. By Theoem 1.1, we have fo odd = m + 1 that m T V (S 3 K, e πi ) = (η ) J i (K, t), whee t = q = e 4πi. Notice that (η ) gows only polynomially in. By Habio and Le s fomula [11], we have i 1 J K,i (t) = 1 + j j=1 k=1 whee t = A 4 = e 4πi. Fo each i define the function g i (j) by Then g i (j) = = j k=1 (t i k j 4 sin k=1 i=1 ) (t i k t i k )(t i+k t i+k, t i k π(i k) ) )(t i+k t i+k sin i 1 J K,i (t) 1 + g i (j). j=1 π(i + k) Now let i be such that i a [0, 1 ] as. Fo each i, let j i {1,..., i 1} such that g i (j i ) achieves the maximum. We have that j i conveges to some j a (0, 1/) which vaies continuously in a when a is close to 1. Then 1 lim log J 1 ( K,i lim log i 1 j=1. ) 1 g i (j) = lim log ( g i (j i ) ),

15 whee the last tem equals lim = 1 π = 1 1 log sin ( j i k=1 jaπ 0 π = 1 π π(i k) j i + log sin k=1 ( log sin ( πa t ) ) dt + 1 π ( Λ ( π(j a a) ) + Λ ( πa )) 1 π ( Λ ( π(j a a) ) + Λ ( π(j a + a) )). jaπ 0 π(i + k) ) ( log sin ( πa + t ) ) dt (Λ ( π(j a + a) ) Λ ( πa )) Since Λ(x) is an odd function and achieves the maximum at π 6, the last tem above is less than o equal to Λ( π 6 ) = 3Λ( π 3 ) π π We also notice that fo i = m, i = m m+1 1, j 1 Theefoe, the tem J K,m (t) gows the fastest, and = V ol(s3 K). 4π = 5 1 and all the inequalities above become equalities. π lim + log T V (S 3 K, A π ) = lim + log J K,m(t) = V ol(s 3 K). 4. The Boomean ings complement In this subsection we pove the following theoem that veifies Conjectue 1.4 fo the 3-component Boomean ings. Theoem 4.. Let L be the 3-component the Boomean ings, and let M be the complement of L in S 3. Then π lim + log T V (M, e πi 4π ) = lim m + m + 1 log J L,m(e 4πi m+1 ) = V ol(m), whee = m + 1 uns ove all odd integes. Hee, J L,m denotes the coloed Jones polynomial whee all the components of L ae coloed by m. The poof elies on the following fomula fo the coloed Jones polynomials of the Boomean ings given by Habio [11, 1]. Let L be the Boomean ings and k, l and n be non-negative integes. Then J L,(k,l,n) (t) = min(k,l,n) 1 j=0 ( 1) j [k + j]![l + j]![n + j]! [k j 1]![l j 1]![n j 1]! ( ) [j]!. (4.1) [j + 1]! Recall that in this fomula [n] = tn/ t n/. Fom now on we specialize at t = e 4πi t 1/ t 1/ whee = m + 1. We have that nπ sin( [n] = ) sin( π ) = {n} {1}, whee we wite {j} = sin( jπ ). Then we can ewite fomula (4.1) as J L,(k,l,n) (e 4iπ min(k,l,n) 1 ) = ( 1) j 1 {k + j}!{l + j}!{n + j}! {1} {k j 1}!{l j 1}!{n j 1}! j=0 Next we establish thee lemmas needed fo the poof of Theoem ( ) {j}!. {j + 1}!

16 Lemma 4.3. Fo odd, let ev denote the evaluation of a Lauent polynomial at A = e πi. Then fo any intege j with 0 < j <, we have whee O(log()) is unifom. log( ev ({ j }!) ) = π Λ(jπ ) + O(log()), Poof. This esult is an adaptation of the esult in [9] fo odd. We ecall the agument fo the sake of completeness. By the Eule-Mac Lauin summation fomula, fo any twice diffeentiable function f on [a, b] whee a and b ae intege, we have whee Applying this to we get b f(k) = k=a ( ev ({ j }!) ) = j 1 b a f(t)dt + 1 f(a) + 1 f(b) + R(a, b, f), R(a, b, f) 3 4 log( ev ({ j }!) ) = = π = π whee f(t) = log( sin( tπ ) ). j k=1 b a f (t) dt. ( log sin( kπ ) ), tπ log( sin( ) + 1 (f(1) + f(j)) + R( π, jπ, f) jπ π ( Λ( jπ log( sin( tπ ) + 1 (f(1) + f(j)) + R( π, jπ, f) ) + Λ( π )) + 1 (f(1) + f(j)) + R( π, jπ As Λ( π ) C log() and f(1) + f(j) C log() fo constants C and C independent of j, and as j j R(1, j, f) = f 4π 1 (t) dt = 1 1 sin( πt = π ( cot( jπ ) ) ) cot(π ) = O(1), we get as claimed. log( ev ({ j }!) ) = π Λ(jπ ) + O(log()) Lemma 4.3 allows us to get an estimation of tems that appea in Habio s sum fo the multi-backet of Boomean ings. We find that ( ) 1 {k + j}!{l + j}!{n + j}! {i}! log {1} {k i 1}!{l i 1}!{n i 1}! {i + 1}! = (f(α, θ) + f(β, θ) + f(γ, θ)) + O(log()), π whee α = kπ, β = lπ, γ = nπ and θ = jπ, and f(α, θ) = Λ(α + θ) Λ(α θ) + 3 Λ(θ) 3 Λ(θ)., f), 16

17 Lemma 4.4. The minimum of the function f(α, θ) is 8 3 Λ( π 4 ) = v 8 3. This minimum is attained fo α = 0 modulo π and θ = 3π 4 modulo π. Poof. The citical points of f ae given by the conditions { Λ (α + θ) Λ (α θ) = 0 Λ (α + θ) + Λ (α θ) + 3 Λ (θ) 4 3 Λ (θ) = 0. As Λ (x) = log sin(x), the fist condition is equivalent to α + θ = ±α θ mod π. Thus, eithe θ = 0 mod π in which case f(α, θ) = 0, o α = 0 o π mod π. In the second case, as the Lobachevski function has the symmeties Λ( θ) = Λ(θ) and Λ(θ+ π ) = 1 Λ(θ) Λ(θ), we get f(0, θ) = 8 3 Λ(θ) 3 Λ(θ), and f( π, θ) = 1 3 Λ(θ) 4 3 Λ(θ). We get citical points when Λ (θ) = Λ(θ) which is equivalent to ( sin(θ)) = sin(θ). This happens only fo θ = π 4 o 3π 4 mod π and the minimum value is 8 3 Λ( π 4 ), which is obtained only fo α = 0 mod π and θ = 3π 4 mod π only. Lemma 4.5. If = m + 1, we have that log( J L,(m,m,m) (e 4iπ ) ) = π v 8 + O(log()). Poof. Again, the agument is vey simila to the agument of the usual volume conjectue fo the Boomean ing in Theoem A.1 of [9]. We emak that quantum intege { n } admit the symmety that { m i } = { m i } fo any intege i. Now, fo k = l = n = m, Habio s fomula fo the coloed Jones polynomial tuns into ( m 1 j ) 3 ( J L,(m,m,m) (t) = ( 1) j { m }3 {j}! { m + k } { m k } { 1 } {j + 1}! j=0 j=0 k=1 m 1 { m } 3 { m + j + 1 } = { 1 } { m + 1 } ) ( j ) 6 ( ) {j}! { m + k }. {j + 1}! Note that as { n } = sin( πn m+1 ) < 0 fo n { m + 1, m +,..., m }, the facto { m + j + 1 } will always be negative fo 0 m 1. Thus all tems in the sum have the same sign. Moeove, thee is only a polynomial in numbe of tems in the sum as m = 1. Theefoe, log( J L,(m,m,m) ) is up to O(log()) equal to the logaithm of the biggest tem. But the tem j = 3 8 coesponds to α = (m 1)π = 0 + O( 1 jπ ) mod π and θ = = 3π 4 + O( 1 ) mod π, so ( log { m } 3 m 1 ) 3 ( {m { m + k } { m k } { 1 } {m 1}!) = π v 8 + O(log()), and k=1 k=1 π log J L,(m,m,m) = v 8 + O( log() ). 17

18 Poof of Theoem 4.. Theoem 1.1, we have T V (S 3 L, e πi ) = (η ) 1 k,l,n m J L,(k,l,n) )(e 4iπ ). This is a sum of m 3 = ( ) 1 3 tems, the logaithm of all of which ae less than Lemma 4.4. Also, the tem J L,(m,m,m) (e 4iπ π (v 8) + O(log()) by ) has logaithm π (v 8) + O(log()). Thus we have π lim log(t V (S 3 L), e πi ) = v8 = Vol(S 3 L). Finally we note that Theoem 1.6 stated in the intoduction follows by Theoems 4.1 and Tuaev-Vio Invaiants and simplicial volume Given a link L in S 3 one can conside the tooidal decomposition of its complement. Recall that the simplicial volume (o Gomov nom) of L, denoted by L, is the sum of the volumes of the hypebolic pieces of the decomposition, divided by v 3 the volume of the egula ideal tetahedon in the hypebolic space. In paticula, if the tooidal decomposition has no hypebolic pieces, then we have L = 0. Soma [30] has shown that the simplicial volume is additive unde split union and connected sums of links. That is L 1 L = L 1 #L = L 1 + L. We note that the connected sum fo links is not uniquely defined, it depends on the components of links being connected. Conjectue 5.1. Fo evey link L S 3, we have π lim log(t V (S 3 L, e πi )) = v3 L, whee uns ove all odd integes. Theoem 1.1 suggests that the Tuaev-Vio invaiants ae a moe natual object to study fo the volume conjectue fo links. As emaked in [5] that all the Kashaev invaiants of a split link ae zeo. As a esult, the oiginal volume conjectue [5] is not tue fo split links. On the othe hand, Coollay 1. implies that T V (S 3 L, q) 0 fo any 3 and any pimitive oot of unity q = A. Define the double of a knot complement to be the double of the complement of a tubula neighbohood of the knot. Then Theoem 3.1 and the main esult of [9] implies that if Conjectue 5.1 holds fo a link, then it holds fo the double of its complement. In paticula, as a consequence of Theoem 1.6, we have Coollay 5.. Conjectue 5.1 is tue fo the double of the Figue-eight and the Boomean ings complement. Since coloed Jones polynomials ae multiplicative unde split union of links, Theoem 1.1 also implies that T V (S 3 L, q) is up to a facto multiplicative unde split union. Coollay 5.3. Fo any odd intege 3 and q = A fo a pimitive -th oot of unit A, T V (S 3 (L 1 L ), q) = (η ) 1 T V (S 3 L 1, q) T V (S 3 L, q). 18

19 The additivity of simplicial volume implies that Conjectue 5.1 is tue fo L 1 and L, then it is tue fo the split union L 1 L. We next discuss the behavio of the Tuaev-Vio invaiants unde taking connected sums of links. With ou nomalization of the coloed Jones polynomials, we have fo a connected sum of two like L 1 #L that J L1 #L,i(t) = [i]j L1,i 1 (t) J L,i (t), whee i 1 and i ae espectively the estiction of i to L 1 and L, and i is the component of i coesponding to the component of L 1 #L coming fom the connected sum. This implies the following Coollay 5.4. Fo any odd intege 3, q = A and t = A 4 fo a pimitive -th oot of unit A, T V (S 3 L 1 #L, q) = (η ) [i] J L1,i 1 (t) J L,i (t), 1 i m whee i 1 and i ae espectively the estiction of i to L 1 and L, and i is the component of i coesponding to the component of L 1 #L coming fom the connected sum. In the est of this section, we focus on the value q = e iπ fo odd = m + 1. Notice in this case that the quantum integes [i] fo 1 i m ae non-zeo and thei log ae of ode O(log ). Coollay 5.4 implies that π lim sup log T V (S 3 L 1 #L, q) π lim sup log T V (S 3 L 1, q) + lim sup π log T V (S 3 L, q). Moeove if we assume a positive answe to Question 1.7 fo L 1 and L, then the tem J L1 #L,m(t) of the sum fo L 1 #L satisfies π lim log J L 1 #L,m(t) = Vol(S 3 L 1 #L ). It follows that if the answe to Question 1.7 is positive, and Conjectue 5.1 is tue fo links L 1 and L, then Conjectue 5.1 is tue fo thei connected sum. In paticula, Theoem 1.6 implies the following Coollay 5.5. Conjectue 5.1 is tue fo any link obtained by connected sum of the Figue-eight and the Boomean ings. We finish the section with the poof of Theoem 1.8, veifying Conjectue 5.1 fo knots of simplicial volume zeo. Theoem 1.8. Let K S 3 be knot with simplicial volume zeo. Then, we have π lim log(t V (S 3 K, e πi )) = K = 0, whee uns ove all odd integes. Poof. By pat () of Theoem 1.1 we have T V (S 3 K, e iπ ) = (η ) 19 1 i m J L,i (e 4iπ ). (5.1)

20 Since J K,1 (t) = 1, we have T V (S 3 K) η > 0 fo any knot K. Thus fo >> 0 the sum of the values of the coloed Jones polynomials in (5.1) is lage o equal to 1. On the othe hand, we have η 0 and log( η ) 0 as. Theefoe, log T V (S 3 K) lim inf 0. Now we only need to pove fo simplicial volume zeo knots that log T V (S 3 K) lim sup 0. By Theoem 1.1, () again, it suffices to pove that the L 1 -nom J K,i (t) of the coloed Jones polynomials of any knot K of simplicial volume zeo is bounded by a polynomial in i. By Godon [10], the set of knots of simplicial volume zeo is geneated by the tous knots, and is closed unde taking connected sums and cablings. Theefoe, it suffices to pove that the set of knots whose coloed Jones polynomials have L 1 -nom gowing at most polynomially contains the tous knots, and is closed unde taking connected sums and cablings. Fom Moton s fomula [1], fo the tous knot T p,q, we have J Tp,q,i(t) = t pq(1 i ) i 1 k = i 1 t 4pqk 4(p+q)k+ t 4pqk 4(p q)k t t. Each faction in the summation can be simplified to a geometic sum of powes of t, and hence has L 1 -nom less than qi + 1. Fom this we have J Tp,q,i(t) = O(i ). Fo a connected sum of knots, we ecall that the L 1 -nom of a Lauent polynomial is Fo a Lauent polynomial R(t) = c f t f, we let f Z a d t d = d. d Z d Z a deg(r(t)) = max({d /c d 0}) min({d /c d 0}). Then fo two Lauent polynomials P (t) = a d t d and Q(t) = e t d Z e Zb e, we have ( ( ) P Q = d t d Za d) b d t d a d b e t f d Z f Z d+e=f deg(p Q) a d b e d+e=f deg(p Q) P Q. Since the L 1 -nom of [i] gows polynomially in i, if the L 1 -noms of J K1,i(t) and J K,i(t) gow polynomially, then so does that of J K1 #K,i(t) = [i]j K1,i(t) J K,i(t). Finally, fo the (p, q)-cabling K p,q of a knot K, the cabling fomula [, 34] says J Kp,q,i(t) = t pq(i 1)/4 i 1 k= i 1 0 t pk(qk+1) J K,qn+1 (t),

21 whee k uns ove integes if i is odd and ove half-integes if i is even. It implies that if J K,i (t) = O(i d ), then J Kp,q,i(t) = O(i d+1 ). By Theoem 1.1 and the agument in the beginning of the poof of Theoem 1.8 applied to links we obtain the following. Coollay 5.6. Fo evey link L S 3, we have whee uns ove all odd integes. log T V (S 3 L) lim inf 0, As said ealie, thee is no lowe bound fo the gowth ate of the Kashaev invaiants that holds fo all links; and no such bound is known fo knots as well. A The elationship between T V (M) and T V (M) The goal of this appendix is to pove Theoem.9. To this end, it will be convenient to modify the definition of the Tuaev-Vio invaiants given in Subsection and use the fomalism of quantum 6jsymbols as in [33]. Fo i I, we let i = ( 1) i [i + 1], fo each admissible tiple (i, j, k), we let i, j, k = ( 1) i+j+k [ i+j k ]![ j+k i ]![ k+i j ]! [ i+j+k, + 1]! and fo each admissible 6-tupe (i, j, k, l, m, n), we let i j k l m n = min{q1,q,q3} z=max{t 1,T,T 3,T 4 } ( 1) z [z + 1]! [z T 1 ]![z T ]![z T 3 ]![z T 4 ]![Q 1 z]![q z]![q 3 z]!. Conside a tiangulation T of M. Let c be an admissible coloing of (M, T ) at level. Fo each edge e of T, we let e c = c(e), fo each face f with edges e 1, e and e 3, we let f c = c(e 1 ), c(e ), c(e 3 ), and fo each tetaheda with edges e ij, {i, j} {1,..., 4}, we let c = c(e 1) c(e 13 ) c(e 3 ) c(e 34 ) c(e 4 ) c(e 14 ). Now ecall the invaiants T V (M) and T V (M) given in Definitions.7 and.8, espectively. Then we have the following. Poposition A.1. (a) Fo any intege 3, T V (M) = η V c A e E e c f c c. f E T 1

22 (b) Fo any odd intege 3, T V (M) = (η ) V Poof. The poof is a staightfowad calculation. c A e E e c f c c. Next we establish fou lemmas on which the poof of Theoem.9 will ely. We will use the notations i, i, j, k and i j k l m n espectively mean the values of i, i, j, k and i j k l m n at a pimitive -th oot of unity A. f E T Lemma A = 1 3 = 1, 0, 0, 0 3 = 1, 1, 0 3 = 1 and = = = 1. 3 Poof. A diect calculation. The following lemma can be consideed as a Tuaev-Vio setting analogue of Theoem.4 (3). Lemma A.3. Fo i I, let i = i. (a) If i I, then i I. Moeove, i = i. (b) If the tiple (i, j, k) is admissible, then so is (i, j, k). Moeove, i, j, k = i, j, k. (c) If the 6-tuple (i, j, k, l, m, n) is admissible, then so ae (i, j, k, l, m, n ) and (i, j, k, l, m, n). Moeove, i j k l m n = i j k and l m n i j k l m n = i j k l m n. Poof. (a) (b) ae staightfowad fom definition. To see the fist identity of (c), let T i and Q j be the coesponding sums fo (i, j, k, l, m, n ), namely T 1 = i + j + k = T 1, T = j + l + n and Q = i + k + l + n, etc. Fo the tems in the summation, let us leave T 1 alone fo now, and conside the othe T i s and Q j s. The key obsevation is that, if without loss of geneality Q 3 Q Q 1 T 4 T 3 T, then one can easily check (1) Q 3 Q 1 = T 4 T, Q Q 1 = T 4 T 3, Q 1 T 4 = Q 1 T 4, T 4 T 3 = Q Q 1 and T 4 T = Q 3 Q 1, which implies () Q 3 Q Q 1 T 4 T 3 T.

23 Fo z in between max{t 1,..., T 4 } and min{q 1, Q, Q 3 }, let P (z) = ( 1) z [z + 1]! [z T 1 ]![z T ]![z T 3 ]![z T 4 ]![Q 1 z]![q z]![q 3 z]!, and similaly fo z in between max{t 1,..., T 4 } and min{q 1, Q, Q 3 } let P (z) = ( 1) z [z + 1]! [z T 1 ]![z T ]![z T 3 ]![z T 4 ]![Q 1 z]![q z]![q 3 z]!. Then fo any a {0, 1,..., Q 1 T 4 = Q 1 T 4 } one veifies by (1) above that P (T 4 + a) = P (Q 1 a). (A.1) Thee ae the following thee cases to conside. Case 1. T 1 T 4 and T 1 T 4. In this case T max = T 4, Q min = Q 1, T max = T 4 and Q min = Q 1. By (A.1), we have Q 1 z=t 4 P (z) = Q 1 T 4 a=0 P (T 4 + a) = Q 1 T 4 a=0 P (Q 1 a) = Q 1 z=t 4 P (z). Case. T 1 > T 4 but T 1 < T 4, o T 1 < T 4 but T 1 > T 4. But symmety, it suffices to conside the fome case. In this case T max = T 1, Q min = Q 1, T max = T 4 and Q min = Q 1, and Q 1 ( ) = i + j l m As a consequence Q 1 >. By (A.1), we have = T 1 T 4. Q 1 z=t 1 P (z) = Q 1 T 4 a=t 1 T 4 P (T 4 + a) = Q 1 T 4 a=q 1 ( ) P (Q 1 a) = The last equality is because that P (z) = 0 fo z >. z=t 4 P (z) = Q 1 z=t 4 P (z). Case 3. T 1 > T 4 and T 1 > T 4. In this case T max = T 1, Q min = Q 1, T max = T 1 and Q min = Q 1. We have Q 1 ( ) = i + j l m = T 1 T 4 > 0, hence Q 1 >. Also, we have Q 1 T 1 = l + m k = T 4. As a consequence, Q 1 ( ) = T 1 T 4 = T 1 T 4 > 0, and hence Q 1 >. By (A.1), we have Q 1 z=t 1 P (z) = z=t 1 P (z) = T 4 a=t 1 T 4 P (T 4 + a) = Q 1 T 1 a=q 1 ( ) P (Q 1 a) = The fist and the last equality ae because that P (z) = P (z) = 0 fo z >. The second identity of (c) is a consequence of the fist. z=t 1 P (z) = Q 1 z=t 1 P (z). 3

24 As an immediate consequence of the two lemmas above, we have Lemma A.4. (a) Fo all i I, i = 0 3 i and i = 1 3 i. (b) If the tiple (i, j, k) is admissible, then i, j, k = 0, 0, 0 3 i, j, k and i, j, k = 1, 1, 0 3 i, j, k. (c) Fo all admissible 6-tuple (i, j, k, l, m, n), i j k l m n = i j k 3 l m n, i j k l m n = i j k 3 l m n, i j k l m n = i j k 3 l m n. Now we ae eady to pove Theoem.9. Poof of Theoem.9. Fo (a), we obseve that thee is a bijection φ : I 3 I I defined by φ(0, i) = i and φ(1, i) = i. This induces a bijection Then by Poposition A.1, we have T V 3 (M) T V (M) ( = η V e c f c 3 c A 3 e E = (η 3 η ) V = η V = T V (M), f F (c,c ) A 3 A e E φ(c,c ) A e E φ : A 3 A A. T c )( η V f F ) e c f c c c A e E f F e c e c f c f c c c f F T e φ(c,c ) f φ(c,c ) φ(c,c ) whee the thid equality comes fom η = η 3 η and Lemma A.4. This finishes the poof of pat (a) of the statement of the theoem. Pat (b) is given in [33, 9.3.A]. To deduce (c), by Lemma A. we have that T V 3 (M) = c A 3 1 = A 3. Note that c A 3 if and only if c(e 1 ) + c(e ) + c(e 3 ) is even fo the edges e 1, e, e 3 of a face. Now conside the handle decomposition of M dual to the ideal tiangulation. Then thee is a one-to-one coespondence between 3-coloings and maps T c : { handles} Z, T 4

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