Turaev-Viro invariants, colored Jones polynomials and volume
Renaud Detcherry Efstratia Kalfagianni Tian Yang






      
        
































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Turaev-Viro invariants, colored Jones polynomials and volume
    Renaud Detcherry, Efstratia Kalfagianni [E.K. is supported by NSF Grants DMS-1404754 and DMS-1708249]     and Tian Yang [T.Y. is supported by NSF Grant  DMS-1405066]
    
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We obtain a formula for the Turaev-Viro invariants of a link  complement in terms of values of the colored Jones polynomials of the link.  
As an application,  we  give  the first examples of 3-manifolds where the “large r" asymptotics   of the Turaev-Viro invariants determine the hyperbolic volume.
We verify  the volume conjecture of  Chen and the third named author <cit.>  for the figure-eight knot and the Borromean rings. Our calculations also exhibit new phenomena of asymptotic behavior of values of the colored Jones polynomials that seem  to be predicted neither by the Kashaev-Murakami-Murakami  volume conjecture and its generalizations nor by  Zagier's  quantum modularity conjecture. We conjecture that the asymptotics of the Turaev-Viro invariants of any link complement determine the simplicial volume of the link, and verify this conjecture for all knots with zero simplicial volume. Finally, we  observe that our simplicial volume conjecture is compatible with connected summations  and split unions of links.








§ INTRODUCTION
 In <cit.>, Turaev and Viro defined a family  of 3-manifold invariants as state sums on triangulations of  manifolds. 
The family is indexed by an integer r, and for each r the invariant depends on a choice of a  2r-th root of unity.
 In the last couple of decades these invariants have  been refined  and generalized in many directions and shown to be closely related to the Witten-Reshetikhin-Turaev invariants.
(See  <cit.> and references therein.) 
Despite these efforts, the relationship between the Turaev-Viro invariants and the geometric structures on 3-manifolds arising from Thurston's geometrization  picture is not understood.
Recently, Chen and the third named author <cit.> conjectured that, evaluated at appropriate roots of unity, the large-r asymptotics  of the Turaev-Viro invariants of a complete hyperbolic 3-manifold, with finite volume,
determine the hyperbolic volume of the manifold, and presented compelling experimental evidence to their conjecture.    

In the present paper, we focus mostly on the Turaev-Viro invariants of link complements in S^3.  Our main result  gives a formula of the Turaev-Viro invariants of a link  complement in terms of values of the colored Jones polynomials of the link. Using this formula we rigorously verify the volume conjecture of <cit.> for the figure-eight knot and Borromean rings complement. To the best of our knowledge these are first examples of this kind.
Our calculations exhibit  new phenomena of asymptotic behavior  of the colored Jones polynomials that does not seem to be predicted by the  volume conjectures <cit.> or by Zagier's quantum modularity conjecture <cit.>.






 §.§ Relationship between knot invariants
To state our results we need  to introduce some notation.
 For a link L⊂ S^3, let  TV_r(S^3∖ L,q) denote the r-th Turaev-Viro invariant of the link complement evaluated at a root of unity q such that q^2 is primitive of degree r.  Throughout this paper, we will consider the case that q=A^2, where A is either a primitive 4r-th root for any integer r or  a primitive 2r-th root for any odd integer r.

 
  
  We use the notation 𝐢=(i_1,…,i_n) for a multi-integer of n components (an n-tuple of integers) and use the notation 1⩽𝐢⩽ m
  to describe all such multi-integers with 1⩽ i_k ⩽ m for each k∈{1,…, n}.
  Given a link L with n components, 
  let  J_L,𝐢(t) denote the 𝐢-th colored Jones polynomial of L whose  k-th component is colored by i_k <cit.>. If all the components of L are colored by the same integer 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 colored Jones polynomial. The polynomials are
  indexed so that 
  J_L,1(t)=1 and  J_L, 2(t) is the ordinary Jones polynomial, and are normalized so that 

  
    J_U,i(t)=[i] =A^2i-A^-2i/A^2-A^-2

 for the unknot U, where by convention t=A^4.  Finally, we define

    η_r=A^2-A^-2/√(-2r)   and η_r'=A^2-A^-2/√(-r).


Before stating our main result, let us recall once again the convention that q=A^2 and t=A^4.



Let L be a link in S^3 with n components. 


  * For an integer r⩾ 3  and a primitive 4r-th root of unity A, we have

    TV_r(S^3∖ L,q)=η_r^21 ⩽𝐢⩽ r-1∑ 
    |  J_L, 𝐢 (t)   |^2.
 

  * For an odd integer  r⩾ 3 and a primitive 2r-th root of unity A, we have

    TV_r(S^3∖ L,q)=2^n-1(η_r')^21 ⩽𝐢⩽r-1/2∑
    |  J_L, 𝐢 (t)|^2.
 



Extending an earlier  result of Roberts <cit.>, Benedetti  and Petronio <cit.>  showed that the invariants TV_r(M,e^π i/r) of a 3-manifold M, with non-empty  boundary, coincide up to a scalar with the SU(2)  Witten-Reshetikhin-Turaev  invariants of the double of M.  The first step in our proof of 
 Theorem <ref> is to extend  this relation to the Turaev-Viro invariants and the SO(3) Reshetikhin-Turaev invariants <cit.>. See Theorem <ref>. 
For this we adapt the  argument of <cit.>  to the case that r is odd and A is a primitive 2r-th root of unity.
 Having this extension at hand,  the proof is completed by using the  properties of the
SO(3)  Reshetikhin-Turaev   Topological Qantum Field Theory (TQFT) developed by
Blanchet, Habegger, Masbaum and Vogel <cit.>.



Note that for any primitive r-th root of unity with r⩾ 3, the quantities η_r and η_r' are real and non-zero.
Since
J_L,1(t)=1, and with the notation as in Theorem <ref>, we have the following.

 For any r⩾ 3, any root q=A^2 and any link L in S^3, 
we have  
    TV_r(S^3∖ L,q) ⩾ H_r>0,

 where H_r=η_r^2 in case (1), and  H_r= 2^n-1(η_r')^2 in case (2).
 

Corollary <ref> implies that the invariants TV_r(q) do not vanish for any link in S^3.
In contrast to that,  the values of the colored Jones polynomials  involved in the Kashaev-Murakami-Murakami volume conjecture <cit.> are known to vanish for   split links and for a class of links called Whitehead chains <cit.>. 
 
Another immediate consequence of Theorem  <ref> is that links with the same colored Jones polynomials have the same Turaev-Viro invariants. In particular, since the colored Jones polynomials are invariant under Conway mutations and the genus 
2 mutations <cit.>, we obtain the following.

 For any r⩾ 3, any root q=A^2 and any link L in S^3, the invariants TV_r(S^3∖ L,q) remain unchanged under Conway mutations and the genus 2 mutations.







 §.§  Asymptotics of Turaev-Viro and colored Jones link invariants


We are interested in the large r asymptotics of the invariants TV_r(S^3∖ L,A^2) in the case that either 
A=e^π i/2r for integers r⩾ 3, or A=e^π i/r for odd integers r⩾ 3.
With these choices of A, we have in the former case that 

    η_r=2sin(π/r)/√(2r),
 and in the latter case that

    η_r'=2sin(2π/r)/√(r).



In <cit.>, Chen and the third named author presented experimental evidence and stated the following.

<cit.> For any 3-manifold M with a complete hyperbolic structure of finite volume, we have

    lim_r→∞2π/rlog (TV_r(M, e^2π i/r))=Vol(M),

where r runs over all odd integers. 


Conjecture <ref> impies that TV_r(M,e^2π i/r) grows exponentially in r.
This is particularly surprising since the corresponding growth of TV_r(M, e^π i/r) is expected, and in many cases known, to be polynomial by Witten's asymptotic expansion conjecture <cit.>. For closed 3-manifolds, this polynomial growth was established by  Garoufalidis <cit.>. Combining <cit.>
and the results of <cit.>, one has  that for every 3-manifold M with non-empty boundary, there exist  constants C>0 and N such that |TV_r(M, e^π i/r)|⩽ C r^N.
This together with Theorem <ref>(1) imply the following.

 For any link L in S^3, there exist constants C>0 and N such that for any integer r and  multi-integer 𝐢 with 1⩽𝐢⩽ r-1, the value of the 𝐢-th colored Jones polynomial at t=e^2π i/r satisfies 

    |J_L, 𝐢(e^2π i/r)|⩽ Cr^N.
  Hence, J_L, 𝐢(e^2π i/r) grows at most polynomially in r.
 

As a main application of Theorem <ref>, we provide the first rigorous evidence to Conjecture <ref>.
 
  Let L be either the figure-eight knot or the Borromean rings, and   let M be the complement of L in S^3.
Then

    lim_r→ +∞2π/rlog TV_r(M, e^2π i/r)=lim_m→ +∞4π/2m+1log|J_L, m(e^4π i/2m+1)|=Vol(M),

where r=2m+1 runs over all odd integers. 
 

The asymptotic behavior of the values of J_L,m(t) at t=e^2π i/m+1/2 is not predicted  either by the original  volume conjecture <cit.> or by its generalizations  <cit.>. Theorem <ref> seems to suggest that these values  grow exponentially in m with growth rate the hyperbolic volume. This is somewhat surprising because as noted in  <cit.>, and also in Corollary <ref>,  that for any positive integer l, J_L,m(e^2π i/m+l) grows only polynomially in m. We ask the following.

 Is it true that for any hyperbolic link L in S^3, we have

    lim_m→ +∞2π/mlog|J_L,m(e^2π i/m+1/2)|=Vol(S^3∖ L)?








 §.§ Knots with zero simplicial volume
 Recall that the simplicial volume (or Gromov norm) ||L|| of a link L is the sum  of the volumes of the hyperbolic pieces in the JSJ-decomposition of the link complement, divided by the volume of the regular ideal hyperbolic tetrahedron. In particular,  if the geometric decomposition has no hyperbolic pieces, then ||L||=0 <cit.>.
 As a natural generalization of Conjecture <ref>, one can conjecture that for every link L the asymptotics of TV_r(S^3∖ L, e^2π i/r) determines ||L||. See Conjecture <ref>.
 
 


 Using Theorem <ref> and the positivity of the Turaev-Viro invariants (Corollary <ref>), we have a proof of  Conjecture <ref> for the knots with zero simplicial volume.  
  Let K⊂ S^3 be a knot with simplicial volume  zero. Then 


    lim_r→∞2π/rlog TV_r(S^3∖ K, e^2π i/r) =||K||=0,

where r runs over all odd integers.
 
 
 We also observe that, unlike the original volume conjecture that is not true for split links <cit.>, Conjecture <ref> is compatible with  split unions of links, and under some assumptions is also compatible  with  connected summations. 
 
 Since this article was first written there has been some further progress in the study of relations of  the Turaev-Viro invariants and geometric decompositions of 3-manifolds:  
 By work of Ohtsuki <cit.>
 Conjecture <ref> is true 
  for closed hyperbolic 3-manifolds obtained by integral surgeries along the figure-eight knot. In <cit.>, the authors of this paper verify Conjecture <ref> for infinite families of 
  cusped hyperbolic 3-manifolds. In <cit.>,  Detcherry and Kalfagianni establish a relation between Turaev-Viro invariants and simplicial volume of 3-manifolds with empty or toroidal boundary, and proved generalizations of Theorem <ref>. In <cit.>, Detcherry proves that Conjecture  <ref>  is stable under
certain link cabling operations.





 §.§ Organization
 The paper is organized as follows. 
In Subsection <ref>, we review the Reshetikhin-Turaev invariants <cit.>  following the skein theoretical approach by Blanchet, Habegger, Masbaum and Vogel <cit.>. In Subsection <ref>, we recall the definition of the Turaev-Viro invariants, and consider an SO(3)-version of them that facilitates our extension of the main theorem of <cit.> in the setting needed in this paper  (Theorem <ref>). The relationship between the two versions of the Turaev-Viro invariants is given in Theorem <ref> whose proof is included in  the Appendix. We prove Theorem <ref> in Section <ref>, and prove Theorem <ref> and Theorem <ref> respectively in Sections <ref> and <ref>.






 §.§ Acknowledgement
 Part of this work was done while the authors were attending the conferences  “Advances in Quantum and Low-Dimensional Topology 2016" at the University of Iowa, and “Knots in Hellas 2016" at  the International Olympic Academy in Greece. We would like to thank the organizers of these conferences for support, hospitality, and for providing excellent working conditions.

We are also grateful to Francis Bonahon, Charles Frohman, Stavros Garoufalidis and Roland van der Veen for discussions and suggestions. 






§ PRELIMINARIES
 





 §.§ Reshetikhin-Turaev invariants and TQFTs
 
In this subsection we review the definition and basic properties of the Reshetikhin-Turaev invariants.
Our exposition follows the skein theoretical approach of Blanchet, Habegger, Masbaum and Vogel <cit.>.


A framed link in an oriented 3-manifold M is a smooth embedding L 
 of a disjoint union of finitely many thickened circles S^1× [0,ϵ], for some ϵ >0, into M. Let [A,A^-1] be the ring of Laurent polynomials in the indeterminate A. Then following <cit.>,  the Kauffman bracket skein module K_A(M) of M is defined as the quotient of the free [A,A^-1]-module generated by the isotopy classes of framed links in M by the following two relations:


  * Kauffman Bracket Skein Relation:  < g r a p h i c s >
 = A 
    < g r a p h i c s >
  + A^-1 
    < g r a p h i c s >
. 


  * Framing Relation:  L ∪
    < g r a p h i c s >
=(-A^2-A^-2) L. 


There is a canonical isomorphism 

    ⟨ ⟩: K_A(S^3) →ℤ[A,A^-1]

between the Kauffman bracket skein module of S^3 and ℤ[A,A^-1] viewed a module over itself. The Laurent polynomial ⟨ L ⟩∈ℤ[A,A^-1] determined by a framed link L⊂ S^3 is called the Kauffman bracket of L.

The Kauffman bracket skein module K_A(T) of the solid torus T=D^2× S^1 is canonically isomorphic to the module [A,A^-1][z]. Here we consider D^2 as the unit disk in the complex plane, and call the framed link [0,ϵ]× S^1 ⊂ D^2× S^1, for some ϵ >0, the core of T. Then the isomorphism above is given by sending i parallel copies of the core of T to z^i. A framed link L in S^3 of n components defines an [A,A^-1]-multilinear map
 
    ⟨   , … ,   ⟩_L : K_A(T)^⊗ n→[A,A^-1],

 called the Kauffman multi-bracket, as follows. For monomials z^i_k∈[A,A^-1][z]≅ K_A(T), k = 1, …, n, let L(z^i_1, …, z^i_n) be the framed link in S^3 obtained by cabling the k-th component of L by i_k  parallel copies of the core. Then define 
    ⟨ z^i_1, …, z^i_n⟩_L ≐⟨ L(z^i_1, …, z^i_n) ⟩,
 and extend [A,A^-1]-multilinearly on the whole K_A(T).  For  the unknot U and  any polynomial P(z)∈[A,A^-1][z], we simply denote  the bracket ⟨ P(z) ⟩_U by ⟨ P(z) ⟩.
 
 The i-th Chebyshev polynomial e_i ∈[A,A^-1][z] is defined by the recurrence relations
 e_0=1, e_1=z, and  ze_j=e_j+1+e_j-1, and satisfies 
 
    ⟨ e_i ⟩ = (-1)^i[i+1].
 
The colored Jones polynomials of an oriented  knot K in S^3 are defined using e_i as follows. Let D be a diagram of K with writhe number w(D), equipped with the blackboard framing. Then the (i+1)-st colored Jones polynomial of K is 

    J_K,i+1(t)=((-1)^i A^i^2+2i)^w(D)⟨ e_i ⟩_D.


The colored Jones polynomials for an oriented link L in S^3 is defined similarly. Let D be a diagram of L with writhe number w(D) and equipped with the  blackboard framing. For a multi-integer 𝐢 = (i_1, …, i_n), let 𝐢+1 = (i_1+1,…, i_n+1). Then the (𝐢+1)-st colored Jones polynomial of L is defined by



    J_L, 𝐢+1(t) = ((-1)^k=1n∑i_k A^s(𝐢))^w(D)⟨ e_i_1, …, e_i_n⟩ _D,

where  s(𝐢)=k=1n∑(i_k^2+i_k).

We note that a change of orientation on some or all the components  of L changes the writhe number of D, and changes J_L,𝐢(t) only by a power of A. Therefore, for an unoriented link L and a complex number 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_i_1-1, …, e_i_n-1⟩ _D|.



 If M is a closed oriented 3-manifold obtained by doing surgery along a framed link L in S^3, then the specialization of the Kauffman multi-bracket at roots of unity yields invariants of 3-manifolds. From now on, let A be either a primitive 4r-th root of unity for an integer r⩾ 3 or a primitive 2r-th root of unity for an odd integer r⩾ 3. To define the Reshetikhin-Turaev invariants, we need to recall some special elements of K_A(T)≅[A,A^-1][z], called the Kirby coloring, defined by

    ω_r=i=0r-2∑⟨ e_i ⟩ e_i

for any integer r, and 

    ω_r'=i=0m-1∑⟨ e_2i⟩ e_2i

for any odd integer r=2m+1. We also for any r introduce

    κ_r=η_r ⟨ω_r ⟩_U_+,

and for any odd r introduce

    κ'_r=η_r' ⟨ω_r' ⟩_U_+,

where U_+ is the unknot with framing 1.

 Let M be a closed oriented 3-manifold obtained from S^3 by doing surgery along a framed link L with number of components n(L) and signature σ(L). 


  * The Reshetikhin-Turaev invariants  of M are  defined by 
    ⟨ M⟩_r=η_r^1+n(L) κ_r^-σ(L) ⟨ω_r, …, ω_r ⟩ _L
 for any integer r⩾ 3, and by

    ⟨ M⟩_r'=(η_r')^1+n(L) (κ_r')^-σ(L) ⟨ω_r', …, ω_r'⟩ _L
 for any odd integer r⩾ 3. 

  * Let L' be a framed link in M. Then, the Reshetikhin-Turaev invariants of the pair (M,L') are defined by 
    ⟨ M,L' ⟩_r=η_r^1+n(L) κ_r^-σ(L) ⟨ω_r, …, ω_r, 1 ⟩ _L∪ L'
 for any integer r⩾ 3, and by

    ⟨ M, L' ⟩_r'=(η_r')^1+n(L) (κ_r')^-σ(L) ⟨ω_r', …, ω_r', 1 ⟩ _L∪ L'
 for any odd integer r⩾ 3.







  * The invariants  ⟨ M⟩_r and ⟨ M ⟩ _r' are called  the SU(2) and SO(3) Reshetikhin-Tureav invariants of M, respectively.

  * For any element S in K_A(M) represented by a [A,A^-1]-linear combinations of framed links in M, one can define ⟨ M, S ⟩_r and ⟨ M, S ⟩_r' by [A,A^-1]-linear extensions. 

  * Since S^3 is obtained by doing surgery along the empty link, we have ⟨ S^3 ⟩_r=η_r and ⟨ S^3 ⟩_r' =η_r'. Moreover, 
 for any link L ⊂ S^3 we have

    ⟨ S^3 ,L ⟩_r =η_r ⟨ L ⟩,    and   ⟨ S^3, L  ⟩_r'= η_r' ⟨ L ⟩.





In <cit.>, Blanchet, Habegger, Masbaum and Vogel gave a construction of  the topological quantum field theories underlying the SU(2) and SO(3) versions of the  Reshetikhin-Turaev invariants.
Below we will summarize the basic properties of the corresponding  topological quantum field functors denoted by
 Z_r and Z_r', respectively.  Note that for a  closed oriented 3-manifold M we will use -M to denote the manifolds with the orientation reversed.


 <cit.> 


  * For a closed oriented surface Σ  and  any integer r⩾ 3, there exists a finite dimensional ℂ-vector space Z_r(Σ) satisfying 
    Z_r(Σ_1 ∐Σ_2) ≅ Z_r(Σ_1)⊗ Z_r(Σ_2),

and, similarly,  for each odd integer r⩾ 3, 
there exists a finite dimensional ℂ-vector space Z_r'(Σ) satisfying 
    Z_r'(Σ_1 ∐Σ_2) ≅ Z_r'(Σ_1)⊗ Z_r'(Σ_2).



  * If H is a handlebody with ∂ H =Σ, then Z_r(Σ) and Z_r'(Σ) are  quotients of the Kauffman bracket skein module K_A(H).


  *  Every compact oriented 3-manifold M with ∂ M=Σ and a framed link L in M defines for any integer r a vector Z_r(M,L)  in Z_r(Σ),  and for any odd integer r a vector Z_r'(M,L)  in Z_r'(Σ). 


  * For  any integer r, there is a sesquilinear pairing ⟨ , ⟩ on Z_r(Σ) with the following property: Given  oriented 3-manifolds M_1 and M_2  with boundary  Σ=∂ M_1=∂ M_2, and framed links L_1⊂ M_1 and L_2⊂ M_2,   we have

 
    ⟨ M, L ⟩_r=⟨ Z_r(M_1, L_1),Z_r(M_2, L_2)⟩,


where M=M_1⋃_Σ (-M_2) is the closed 3-manifold obtained by gluing M_1 and -M_2  along  Σ and L= L_1∐ L_2. 
Similarly, for any odd integer r, there is a sesquilinear pairing ⟨ , ⟩ 
on  Z_r'(Σ),  such tor any  M and L as above,
  
    ⟨ M, L ⟩_r'=⟨ Z_r'(M_1, L_1),Z_r'(M_2, L_2)⟩.





For the purpose of this paper, we will only need to understand the TQFT vector spaces of the torus Z_r(T^2) and Z_r'(T^2). These vector spaces are quotients of K_A(T)≅[A,A^-1][z], hence the Chebyshev polynomials {e_i} define vectors in Z_r(T^2) and Z_r'(T^2).  We have the following


<cit.> 


  * For any integer r⩾ 3, the vectors {e_0, …, e_r-2} form a Hermitian basis of Z_r(T^2). 

  * For any odd integer r=2m+1, the vectors {e_0, …, e_m-1} form a Hermitian basis of Z_r'(T^2).

  * In Z_r'(T^2), we have for any i with  0⩽ i ⩽ m-1 that 
 
    e_m+i=e_m-1-i.
  Therefore, the vectors {e_2i}_i=0,… ,m-1 also form a Hermitian basis of Z_r'(T^2).
 










 §.§ Turaev-Viro invariants


In this subsection, we recall the definition and basic properties of the Turaev-Viro invariants <cit.>.
The approach of <cit.> relies on quantum 6j-symbols  while the definition of  Kauffman and Lins <cit.> uses invariants of spin networks.
The two definitions were shown to be equivalent in <cit.>.  The formalism of <cit.>  turns out to be more convenient to work with when using skein theoretic techniques to relate the Turaev-Viro invariants to the Reshetikhin-Turaev invariants.


For an integer r⩾ 3, let I_r={0,1,…, r-2} be the set of non-negative integers less than or equal to r-2.  Let q be a 2r-th root of unity such that q^2 is a primitive r-th root. For example, q=A^2, where A is either a primitive 4r-th root or for odd r a primitive 2r-th root, satisfies the condition.  For i∈ I_r, define 

    < g r a p h i c s >
    =(-1)^i[i+1].

A triple (i,j,k) of elements of I_r is called admissible if (1) i+j⩾ k, j+k⩾ i and k+i⩾ j, (2) i+j+k is an even,
and (3) i+j+k⩽ 2(r-2). For an admissible triple (i,j,k), define



    < g r a p h i c s >
     = (-1)^-i+j+k/2[i+j-k/2]![j+k-i/2]![k+i-j/2]![i+j+k/2+1]!/[i]![j]![k]!.



A 6-tuple (i,j,k,l,m,n) of elements of I_r is called admissible if the triples (i,j,k), (j,l,n), (i,m,n) and (k,l,m) are admissible. For an admissible 6-tuple (i,j,k,l,m,n),
define 

    < g r a p h i c s >
    =    ∏_a=1^4∏_b=1^3[Q_b-T_a]!/[i]![j]![k]![l]![m]![n]!∑_z=max{T_1, T_2, T_3, T_4}^min{ Q_1,Q_2,Q_3}(-1)^z[z+1]!/∏_a=1^4[z-T_a]!∏_b=1^3[Q_b-z]!,

where

 
    T_1=i+j+k/2,    T_2=i+m+n/2,  T_3=j+l+n/2,   T_4=k+l+m/2,
 

    Q_1=i+j+l+m/2,   Q_2=i+k+l+n/2,   Q_3=j+k+m+n/2.





  The symbols  < g r a p h i c s >
, 
    < g r a p h i c s > and < g r a p h i c s >, used above, 
are examples of spin networks: trivalent ribbon graphs with ends colored by integers.  The expressions on the right hand sides of above equations give the Kauffman bracket invariant of the
 corresponding networks. See <cit.>. In the language of <cit.>, the
second and third spin networks above are the  trihedral and tetrahedral networks, 
denoted by θ(i,j,k) and τ(i, j,k) therein, 
and the corresponding invariants are  the trihedral and  tetrahedral coefficients, respectively.





A coloring of a Euclidean tetrahedron Δ is an assignment of elements of I_r to the edges of Δ, and is admissible if the triple of elements of I_r assigned to the three edges of each face of Δ is admissible. See Figure <ref> for a geometric interpretation of tetrahedral coefficients. 



Let 𝒯 be a triangulation of M. If M has non-empty boundary, then we let 𝒯 be an ideal triangulation of M, i.e., a gluing of finitely many truncated Euclidean tetrahedra by affine homeomorphisms between pairs of faces. In this way, there are no vertices, and instead, the triangles coming from truncations form a triangulation of the boundary of M. By edges of an ideal triangulation, we only mean the ones coming from the edges of the tetrahedra, not the ones from the truncations.  A coloring at level r of the triangulated 3-manifold (M,𝒯) is an assignment of elements of I_r to the edges of 𝒯, and is admissible if the 6-tuple assigned to the edges of each tetrahedron of 𝒯 is admissible. Let c be an admissible coloring of (M,𝒯) at level r. For each edge e of 𝒯, let 

    |e|_c=
        < g r a p h i c s >
    .
 For each face f of 𝒯 with edges e_1, e_2 and e_3, let 


    |f|_c=
        < g r a p h i c s >
    ,

where c_i = c(e_i).


For each tetrahedra Δ in 𝒯 with vertices v_1,…,v_4, denote by e_ij the edge of Δ connecting the vertices v_i and v_j, {i,j}⊂{1,…,4}, and let

    |Δ|_c=
        < g r a p h i c s >
    ,

where c_ij=c(e_ij).






 Let A_r be the set of admissible colorings of (M,𝒯) at level r, and let V, E F and  T respectively be the sets of  (interior)  vertices, edges, faces and tetrahedra in 𝒯. Then the r-th Turaev-Viro invariant is defined by
 
    TV_r(M)= η_r^2|V|∑_c∈ A_r∏_e∈ E|e|_c∏_Δ∈ T|Δ|_c/∏_f∈ E|f|_c.




For an odd integer r⩾ 3, one can also consider an SO(3)-version of the Turaev-Viro invariants TV_r'(M)  of  M, which will relate  to the SU(2) invariants TV_r(M),  and to the Reshetikhin-Turaev invariants ⟨ D(M) ⟩'_r of the double of M (Theorems <ref>, <ref>). The invariant TV_r'(M) is defined as follows. Let I'_r={0,2,…, r-5, r-3} be the set of non-negative even integers less than or equal to r-2.  An SO(3)-coloring of a Euclidean tetrahedron Δ is an assignment of elements of I'_r to the edges of Δ, and is admissible if the triple assigned to the three edges of each face of Δ is admissible. Let 𝒯 be a triangulation of M. An SO(3)-coloring at level r of the triangulated 3-manifold (M,𝒯) is an assignment of elements of I'_r to the edges of 𝒯, and is admissible if the 6-tuple assigned to the edges of each tetrahedron of 𝒯 is admissible. 

 Let A'_r be the set of SO(3)-admissible colorings of (M,𝒯) at level r.  Define
 
    TV_r'(M)=(η_r')^2|V|∑_c∈ A'_r∏_e∈ E|e|_c∏_Δ∈ T|Δ|_c/∏_f∈ E|f|_c.



The relationship between TV_r(M) and TV'_r(M) is given by the following theorem.

 Let M be a 3-manifold and let b_0(M) and b_2(M) respectively be its zeroth and second ℤ_2-Betti number. 


  * For any odd integer r⩾ 3, 

    TV_r(M)=TV_3(M)· TV'_r(M).


  * (Turaev-Viro <cit.>). If ∂ M= ∅ and A=e^π i/3, then 
    TV_3(M)=2^b_2(M)-b_0(M).


  * If M is connected, ∂ M≠∅ and A=e^π i/3, then 
    TV_3(M)=2^b_2(M).


In particular, TV_3(M) is nonzero.


We postpone the proof of Theorem <ref> to Appendix <ref> to avoid unnecessary distractions.














§ THE COLORED JONES SUM FORMULA FOR TURAEV-VIRO INVARIANTS
 
In this Section, following the argument of <cit.>, we establish a relationship between the SO(3) Turaev-Viro invariants of a 3-manifold with boundary and the SO(3) Reshetikhin-Turaev invariants of its double. See Theorem <ref>. Then, we use  Theorem <ref> and results established  in <cit.>,  to prove Theorem <ref>.




 §.§ Relationship between invariants


The relationship between  Turaev-Viro and Witten-Reshetikhin-Turaev invariants 
was studied by Turaev-Walker <cit.> and Roberts <cit.> for closed 3-manifolds, and by Benedetti  and Petronio <cit.> for 3-manifolds with boundary.
For an oriented 3-manifold M with boundary, let -M  denote M with the orientation reversed, and  let D(M)
denote the double of M, i.e.,

    D(M)=M∂ M⋃(-M).

We will need the following theorem of Benedetti and Petronio <cit.>. In fact <cit.> only treats the case of A=e^π i/2r, but, as we will explain below, the proof for other cases is similar.


Let M be a 3-manifold with boundary.
Then,  

    TV_r(M)=η_r^-χ(M)⟨ D(M) ⟩_r

for any integer r, and

    TV_r'(M)=(η'_r)^-χ ( M)⟨ D(M) ⟩_r'

for any odd r, where χ(M) is the Euler characteristic of M.



We refer to <cit.> and <cit.> for the SU(2) (r being any integer) case, and for the reader's convenience include a sketch of the proof here for the SO(3) (r being odd) case. The main difference for the SO(3) case comes from to the following lemma due to Lickorish.


Let r⩾ 3 be an odd integer and let A be a primitive 2r-th root of unity.  Then 

    < g r a p h i c s >
    ={[ < g r a p h i c s >       if   i=0, r-2;                    ;                   0       if  i≠ 0, r-2 ].

I.e., the element of the i-th Temperley-Lieb algebra obtained by circling the i-th Jones-Wenzl idempotent f_i by the Kirby coloring ω_r' equals f_i when i=0 or r-2, and equals 0 otherwise. 

As a consequence, the usual fusion rule <cit.> should be modified to the following.

 Let r⩾ 3 be an odd integer. Then for a triple (i,j,k) of elements of I_r', 


    < g r a p h i c s >
    ={[ (η_r')^-2/
        < g r a p h i c s >
    
        < g r a p h i c s >                               if  (i,j,k)  is r-admissible,;                                                            ;                                                           0                           if  (i,j,k)  is not r-admissible. ].





Here the integers i,j and k being even is crucial, since it rules out the possibility that i+j+k=r-2, which by Lemma <ref> could create additional complications. This is the reason that we prefer to work with  the invariant TV_r'(M) instead of TV_r(M).  Note that the factor  < g r a p h i c s > in the formula above is also denoted by θ(i,j,k) in <cit.> and <cit.>.




Following <cit.>, we extend the “chain-mail" invariant of Roberts <cit.> to M with non-empty boundary using a handle decomposition without 3-handles. For such a handle decomposition, let d_0, d_1 and d_2 respectively be the number of 0-, 1- and 2-handles.  Let ϵ_i be the attaching curves of the 2-handles and let δ_j be the meridians of the 1-handles. Thicken the curves to bands parallel to the surface of the 1-skeleton H and push the ϵ-bands slightly into H.   Embed H arbitrarily into S^3 and color each of the image of the ϵ- and δ-bands by η_r'ω_r' to get an element in S_M in K_A(S^3). Then the chain-mail invariant of M is defined by

    CM_r(M)=(η_r')^d_0⟨ S_M ⟩,
  where, recall that, we use the notation ⟨ ⟩ for the Kauffman bracket.  It is proved in <cit.> that CM_r(M) is independent of the choice of the handle decomposition and the embedding, hence defines an invariant of M.

To prove the result we will compare the expressions of the invariant  CM_r(M)  obtained by considering two different handle decompositions of M.
On the one hand, suppose that  the handle decomposition is obtained by the dual of an ideal triangulation 𝒯 of M, namely the 2-handles come from a tubular neighborhood of the edges of 𝒯, the 1-handles come from a tubular neighborhood of the faces of 𝒯 and the 0-handles come from the complement of the 1- and 2-handles. Since each face has  three edges,  each δ-band encloses exactly three ϵ-bands (see <cit.>). By relation (<ref>), every η_r'ω_r' on the ϵ-band can be written as 

    η_r'ω_r'=η_r'∑_ i=0^r-1/2-1⟨ e_i⟩ e_i=η_r'∑_ i=0^r-1/2-1⟨ e_2i⟩ e_2i.

Next we apply Lemma <ref> to each δ-band. In this process the four δ-bands corresponding to each 
tetrahedron of 𝒯  give rise to a tetrahedral network
(see also <cit.>).  Then by Remark <ref> and equations preceding it, we may rewrite CM_r(M) in terms of trihedral and tetrahedral coefficients
to obtain 


    CM_r(M)=(η_r')^d_0-d_1+d_2∑_c∈ A'_r∏_e∈ E|e|_r^c∏_Δ∈ T|Δ|_r^c/∏_f∈ E|f|_r^c=(η_r')^χ(M)TV_r'(M).



On the other hand, suppose that the handle decomposition is standard, namely H is a standard handlebody in S^3 with exactly one 0-handle. Then we claim that the ϵ- and the δ-bands give a surgery diagram L of D(M). The way to see it is as follows. Consider the 4-manifold W_1 obtained by attaching 1-handles along the δ-bands (see Kirby <cit.>) and 2-handles along the ϵ-bands. Then W_1 is homeomorphic to M× I and ∂ W_1 =M×{0}∪∂ M× I ∪(-M)×{1}= D(M). Now if W_2 is the 4-manifold obtained by attaching 2-handles along all the ϵ- and the δ-bands, then ∂ W_2 is the 3-manifold represented by the framed link L. Then due to the fact that ∂ W_1=∂ W_2 and Definition <ref>, we have 
    CM_r(M)=η_r'⟨η_r'ω_r',…, η'ω' ⟩_L=(η_r')^1+n(L)⟨ω_r',…, ω_r' ⟩_L=⟨ D(M)⟩_r' (κ_r') ^σ(L).
 
We are left to show that σ(L)=0. It follows from the fact that the linking matrix of L has the form 

    LK(L)=[   0   A; A^T   0 ],

where the blocks come from grouping the ϵ- and the δ-bands together and A_ij=LK(ϵ_i, δ_j). Then, for any eigenvector v=(v_1,v_2) with eigenvalue λ, the vector v'=(-v_1,v_2) is an eigen-vector of eigenvalue -λ. 


 Theorems <ref> and <ref> together with the main result of <cit.>  imply that if
Conjecture <ref> holds for M with totally geodesic or toroidal boundary, then it holds for D(M).









 §.§ Proof of Theorem <ref>

We are now ready to prove Theorem <ref>.  For the convenience of the reader we restate the theorem.

Theorem <ref>
Let L be a link in S^3 with n components. 


  * For an integer r⩾ 3  and a primitive 4r-th root of unity A, we have

    TV_r(S^3∖ L,q)=η_r^21 ⩽𝐢⩽ r-1∑ 
    |  J_L, 𝐢 (t)   |^2.
 

  * For an odd integer  r=2m+1⩾ 3 and a primitive 2r-th root of unity A, we have

    TV_r(S^3∖ L,q)=2^n-1(η_r')^21 ⩽𝐢⩽ m∑
    |  J_L, 𝐢 (t)|^2.
 

Here, in both cases we have t=q^2=A^4.










We first consider the case that  r=2m+1 is odd. For a framed link L in S^3 with n components, we let M=S^3∖ L.
Since, by Theorem <ref>, we have
TV_r(M)=2^n-1 TV_r'(M), from now on we will work with TV_r'(M).

Since the Euler characteristic of M is zero,
by  Theorem <ref>, we obtain

    TV_r'(M)=⟨ D(M) ⟩_r'=⟨ Z_r'(M),Z_r'(M) ⟩,

where Z_r(M) is a vector in Z_r(T^2)^⊗ n.
Let {e_i}_i=0,…, m-1 be the basis of Z_r'(T^2) described in Theorem <ref> (2). 
Then the vector space Z_r(T^2)^⊗ n has a Hermitian basis given by {e_𝐢=e_i_1⊗ e_i_2… e_i_n} for all 𝐢=(i_1,i_2,…,i_n) with 0⩽𝐢⩽ m-1.
 
We write ⟨ e_𝐢⟩_L for the multi-bracket ⟨ e_i_1,e_i_2,…,e_i_n⟩_L. Then, by relation (<ref>), to establish the desired formula in terms of the colored Jones polynomials, it is suffices to show that
 
    TV_r'(M)=(η_r')^20 ⩽𝐢⩽ m-1∑|⟨ e_𝐢⟩_L|^2.
 
By writing 
    Z_r'(M)=0 ⩽𝐢⩽ m-1∑λ_𝐢e_𝐢

and  using equation (<ref>), we have that 

    TV_r'(M)=0 ⩽𝐢⩽ m-1∑|λ_𝐢|^2.


The computation of the coefficients  λ_𝐢 of Z_r(M)   relies on the TQFT properties of the invariants <cit.>. (Also compare with the argument in <cit.>). Since  {e_𝐢} is a Hermitian basis of Z_r(T^2)^⊗ n, we have
 
    λ_𝐢=⟨ Z_r'(M),e_𝐢⟩.

 A tubular neighborhood N_L of L  is a disjoint union of solid tori  k=1n∐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 using  the i_k-th Chebyshev polynomial e_i_k. 
Then in Z_r'(T)^⊗ n, we have

    e_𝐢= Z_r'(N_L, L(e_𝐢)).

Now by Theorem <ref> (4),  since S^3=M ∪ (-N_L), we have
 
    ⟨ Z_r'(M),e_𝐢⟩= ⟨ Z_r'(M), Z_r'(N_L, L(e_𝐢))  ⟩=  ⟨   M∪(-N_L ),  L(e_𝐢))  ⟩_r'=⟨ S^3 , L(e_𝐢) ⟩_r'.

Finally, by Remark <ref> (2),  we have

    ⟨ S^3 , L(e_𝐢) ⟩_r'=η_r' ⟨ e_𝐢⟩_L.

Therefore, we have 

    λ_𝐢=η_r' ⟨ e_𝐢⟩_L,

which finishes the proof in the case of r=2m+1.



 
The argument of the remaining case is very similar.
By Theorem <ref>, we obtain

    TV_r(M)=⟨ D(M) ⟩_r=⟨ Z_r(M),Z_r(M) ⟩.

Working with the Hermitian basis
 {e_i}_i=0,… ,r-2  of Z_2r(T^2) given in Theorem <ref> (1),  we have
 
    TV_r(M)=0 ⩽𝐢⩽ r-2∑|λ_𝐢|^2,

 where
  λ_𝐢=⟨ Z_r(M),e_𝐢⟩   and  e_𝐢= Z_r(N_L, L(e_𝐢)).
Now by Theorem <ref> (4)
 and Remark <ref>,
one sees

    λ_𝐢=η_r⟨ e_𝐢⟩_L,

which finishes the proof.















§ APPLICATIONS TO CONJECTURE <REF>
 

In this section we use Theorem <ref> to determine the asymptotic behavior of the Turaev-Viro invariants for some  hyperbolic 
 knot and link complements. In particular, we verify  Conjecture <ref>
 for the complement of the figure-eight knot and the  Borromean rings. To the best of our knowledge these are the first calculations of this kind. 




 §.§ The figure-eight complement
 




The following theorem verifies Conjecture <ref> for the figure-eight knot.


Let K be the figure-eight knot and   let M be the complement of K in S^3.
Then

    lim_r→ +∞2π/rlog TV_r(M, e^2π i/r)=lim_m→ +∞4π/2m+1log|J_K, m(e^4π i/2m+1)|=Vol(M),

where r=2m+1 runs over all odd integers.  



 By Theorem <ref>, and for odd r=2m+1, we have that

    TV_r(S^3∖ K,e^2π i/r)=(η_r')^2∑_i=1^m|J_i(K,t)|^2,
 where t=q^2=e^4π i/r.
Notice that (η_r')^2 grows only polynomially in r. 

By Habiro and Le's formula <cit.>, we have 

    J_K,i(t)=1+∑_j=1^i-1∏_k=1^j(t^i-k/2-t^-i-k/2)(t^i+k/2-t^-i+k/2),

where t=A^4=e^4π i/r.

For each i define the function g_i(j) by

    g_i(j)=   ∏_k=1^j|(t^i-k/2-t^-i-k/2)(t^i+k/2-t^-i+k/2)|
    
    =   ∏_k=1^j4|sin2π(i-k)/r||sin2π(i+k)/r|.

Then 


    |J_K,i(t)|⩽ 1+∑_j=1^i-1g_i(j).

Now let i be such that i/r→ a∈[0,1/2] as r →∞.  For each i, let j_i∈{1, …, i-1} such that g_i(j_i) achieves the maximum. We have that j_i/r converges to some j_a∈ (0,1/2)  which varies continuously in a when a is close to 1/2. Then

    lim_r →∞1/rlog|J_K,i|⩽lim_r →∞1/rlog(1+∑_j=1^i-1g_i(j))= lim_r →∞1/rlog(g_i(j_i)),

where the last term equals 

    lim_r →∞1/r(∑_k=1^j_ilog|2sin2π(i-k)/r|+∑_k=1^j_ilog|2sin2π(i+k)/r|)
    
    =   1/2π∫_0^j_aπlog(2|sin(2π a-t)|)dt+ 1/2π∫_0^j_aπlog(2|sin(2π a+t)|)dt
    
    =    -1/2π(Λ(2π (j_a- a))+Λ(2π a)) -1/2π(Λ(2π (j_a+ a))-Λ(2π a))
    
    =    -1/2π(Λ(2π (j_a- a))+Λ(2π (j_a+ a))).

Here Λ denotes the  Lobachevsky function.
Since Λ(x) is an odd function and achieves the maximum at π/6, the last term above is less than or equal to 
    Λ(π/6)/π=3Λ(π/3)/2π=Vol(S^3∖ K)/4π.

We also notice that for i=m, i/r = m/2m+1→1/2, j_1/2=5/12 and all the inequalities above become equalities. Therefore, the term |J_K,m(t)|^2 grows the fastest, and 

    lim_r→ +∞2π/rlog TV_r(S^3∖ K,A^2)=lim_r→ +∞2π/rlog|J_K,m(t)|^2=Vol(S^3∖ K).









 §.§ The Borromean rings complement
 

In this subsection we prove the following theorem that verifies Conjecture <ref> for the 3-component   Borromean rings.
 Let L be the 3-component  Borromean rings, and   let M be the complement of L in S^3.
Then

    lim_r→ +∞2π/rlog TV_r(M, e^2π i/r)=lim_m→ +∞4π/2m+1log|J_L, m(e^4π i/2m+1)|=Vol(M),

where r=2m+1 runs over all odd integers.  Here,  J_L, m denotes the colored Jones polynomial where all the components of L  are colored by m.  


The proof  relies on the following formula for the colored Jones polynomials of the Borromean rings given by Habiro <cit.>. Let L be the Borromean rings and k, l and n be non-negative integers. Then

    J_L,(k,l,n)(t)=j=0min(k,l,n)-1∑ (-1)^j [k+j]![l+j]![n+j]!/[k-j-1]![l-j-1]![n-j-1]!( [j]!/[2j+1]!)^2.

 Recall that in this formula [n]=t^n/2-t^-n/2/t^1/2-t^-1/2 and [n]!=[n][n-1]… [1]. 
 From now on we specialize at t=e^4π i/r where r=2m+1.  
 We have

    [n]=2sin(2nπ/r)/2sin(2π/r)={ n }/{ 1 },

where we write
{ j } =2sin(2jπ/r).
 We can rewrite  formula (<ref>) as 

    J_L,(k,l,n)(e^4iπ/r)=j=0min(k,l,n)-1∑ (-1)^j 1/{ 1 }{ k+j }!{ l+j }!{ n+j}!/{ k-j-1}!{ l-j-1}!{ n-j-1}!( { j} !/{ 2j+1} !)^2.



Next we establish three lemmas needed for the proof of Theorem <ref>.


For any integer j with 0<j<r, we have

    log(| j !|)=-r/2πΛ(2jπ/r)+O(log (r)),

where O(log(r)) is uniform: there is a constant C independent of j and  r, such that O(log r)≤ C log r.


This result is an adaptation of the result in <cit.> for r odd. 
By the Euler-Mac Laurin summation formula, for any twice differentiable function f on [a,b] where a and b are integer, we have

    k=ab∑ f(k)=∫_a^b f(t)dt +1/2f(a)+1/2f(b)+R(a,b,f),

where 
    |R(a,b,f)|⩽3/24∫_a^b |f”(t)|dt.
 Applying this to 

    log(| j !|)=k=1j∑log(2|sin(2kπ/r)|),

we get

    [                                                     log(| j !|)                                                               =            ∫_1^jlog(2|sin(2tπ/r)+1/2(f(1)+f(j))+R(2π/r,2jπ/r,f);                                                                                                                               = r/2π∫_2π/r^2jπ/rlog(2|sin(2tπ/r)+1/2(f(1)+f(j))+R(2π/r,2jπ/r,f);                                                               =         r/2π(-Λ(2jπ/r)+Λ(2π/r))+1/2(f(1)+f(j))+R(2π/r,2jπ/r,f), ]

where f(t)=log(2|sin(2tπ/r)|).

Since we have |rΛ(2π/r)|⩽ C' log(r) and |f(1)+f(j)|⩽ C”log(r) for constants C' and C” independent of j, and since

    R(1,j,f)=∫_1^j |f”(t)|dt=∫_1^j 4π^2/r^21/sin(2π t/r)^2=2π/r( (2jπ/r)-(2π/r))=O(1),

we get 
    log(| j !|)=-r/2πΛ(2jπ/r)+O(log (r))
 as claimed.


Lemma <ref> allows us to get an estimation of terms that appear in Habiro's sum for the multi-bracket of Borromean rings. We find that

    log| 1/{ 1 }{ k+j }!{ l+j }!{ n+j}!/{ k-i-1}!{ l-i-1}!{ n-i-1}!( { i} !/{ 2i+1} !)^2 | 
     =-r/2π(f(α,θ)+f(β,θ)+f(γ,θ))+O(log(r)),

where α=2kπ/r, β=2lπ/r, γ=2nπ/r and  θ=2jπ/r, and 

    f(α,θ)=Λ(α+θ)-Λ(α-θ)+2/3Λ(θ)-2/3Λ(2θ).


The minimum of the function f(α,θ) is -8/3Λ(π/4)=-v_8/3. This minimum is attained for α=0 modulo π and θ=3π/4 modulo π.


The critical points of f are given by the conditions

    Λ'(α+θ)-Λ'(α-θ)=0 
    
    Λ'(α+θ)+Λ'(α-θ)+2/3Λ'(θ)-4/3Λ'(2 θ)=0.

As Λ'(x)=2log|sin(x)|, the first condition is equivalent to α+θ=±α-θπ. Thus, either θ=0 π/2 in which case f(α,θ)=0, or α=0  or π/2π.


In the second case, as the Lobachevsky function has the symmetries Λ(-θ)=-Λ(θ) and Λ(θ +π/2)=1/2Λ(2θ)-Λ(θ), we get 

    f(0,θ)=8/3Λ(θ)-2/3Λ(2θ),
 
and 

    f(π/2,θ)=1/3Λ(2θ)-4/3Λ(θ).

 We get critical points when 2Λ'(θ)=Λ(2θ) which is equivalent to (2 sin (θ))^2=2|sin (2θ)|. This happens only for θ=π/4 or 3π/4π and the minimum value is -8/3Λ(π/4), which is obtained only for α=0 π and θ=3π/4π only. 

 If r=2m+1, we have that 

    log(|J_L,(m,m,m)(e^4iπ/r)|)=r/2πv_8 +O(log(r)).



Again, the argument is very similar to the argument of the usual volume conjecture for the Borromean ring in Theorem A.1 of  <cit.>.
We remark that quantum integer n admit the symmetry that 
    m+1+i =- m-i
 for any integer i.
 
  
 Now, for k=l=n=m, Habiro's formula for the colored Jones polynomials turns into

    J_L,(m,m,m)(t)   =j=0m-1∑ (-1)^j m ^3/ 1 ( k=1j∏ m+k  m-k )^3 ( { j} !/{ 2j+1} !)^2
       =j=0m-1∑ m ^3  m+j+1 / 1  m+1 ( k=1j∏ m+k )^6 ( { j} !/{ 2j+1} !)^2.

Note that as n =sin (2 π n/2m+1)<0 for n ∈ m+1,m+2, …,2m ,  the factor m+j+1 will always be negative for 0 ⩽ m-1. Thus all terms in the sum have the same sign. Moreover, there is only a polynomial in r number of terms in the sum as m=r-1/2.  Therefore, log(|J_L,(m,m,m)|) is up to O(log (r)) equal to the logarithm of the biggest term. But the term j=⌊3r/8⌋ corresponds to α=2(m-1)π/r=0+O(1/r) π and θ =2 jπ/r=3π/4+O(1/r) π, so 

    log| m ^3/ 1 ( k=1m-1∏ m+k  m-k )^3 ( { m-1} !/{ 2m-1} !)^2|=r/2πv_8+O(log(r)),

and

    2π/rlog |J_L,(m,m,m)|=v_8+O(log(r)/r).




 By Theorem <ref>, we have 

    TV_r'(S^3∖ L,e^2π i/r)=(η_r')^21 ⩽ k,l,n⩽ m∑|J_L,(k,l,n))(e^4iπ/r)|^2.

This is a sum of m^3=(r-1/2)^3 terms, the  logarithm of all of which are less than r/2π(2v_8)+O(log(r)) by Lemma <ref>.  Also, the term |J_L,(m,m,m)(e^4iπ/r)|^2 has logarithm r/2π(2v_8)+O(log(r)).
 Thus we have 

    r→∞lim2π/rlog (TV_r'(S^3∖ L),e^2π i/r)=2v_8= (S^3∖ L).



Finally we note that Theorem <ref> stated in the introduction follows by Theorems <ref> and <ref>.





§ TURAEV-VIRO INVARIANTS AND SIMPLICIAL VOLUME
 

Given a link L in S^3, there is a unique up to isotopy collection T of essential embedded tori in M=S^3∖ L so that each component of M cut along T is either hyperbolic or a Seifert fibered space.
This is the toroidal or JSJ-decomposition of M <cit.>.
Recall that the simplicial volume (or Gromov norm) of L, denoted by ||L||,  is the sum  of the volumes of the hyperbolic pieces of the decomposition, divided by v_3; the volume of the regular ideal tetrahedron in the hyperbolic space.
In particular,  if the toroidal decomposition has  no hyperbolic pieces, then we have ||L||=0.  It is known <cit.>  that the simplicial volume is additive under  split unions and connected summations of links.
That is, we have

    ||L_1 ⊔ L_2||=  ||L_1 # L_2||= ||L_1||+ ||L_2||.

We note that the connected sum for multi-component  links is not uniquely defined, it depends on the components of links being connected. 

   For every link L⊂ S^3, we have


    lim_r→∞2π/rlog (TV_r(S^3∖ L, e^2π i/r)) = v_3 ||L||,

where r runs over all odd integers.







Theorem <ref>  suggests that the Turaev-Viro invariants are a  better object to study for the volume conjecture for links.
As remarked in <cit.>,  all the Kashaev invariants of a split link are zero. As a result, the original simplicial volume conjecture <cit.>
is not true for split links. On the other hand, Corollary <ref> 
implies that  TV_r'(S^3∖ L,q)≠ 0 for any r⩾ 3 and any primitive root of unity q=A^2. 



Define the double of a knot complement to be the double of the complement of a tubular neighborhood of the knot. Then Theorem <ref> and the main result of <cit.> implies that if Conjecture <ref> holds for a link, then it holds for the double of its complement. In particular, by Theorem <ref>,  we have 

 Conjecture <ref> is true for the double of the figure-eight and the Borromean rings complement.


Since colored Jones polynomials are multiplicative under split union of links, Theorem <ref> also implies that TV_r'(S^3∖ L,q) is up to a factor multiplicative under split union.

 For any odd integer r⩾ 3 and q=A^2 for a primitive 2r-th root of unity A, 

    TV_r'(S^3∖ (L_1 ⊔ L_2),q)=  (η_r')^-1  TV_r'(S^3∖ L_1,q)·  TV_r'(S^3∖ L_2,q).





The additivity of simplicial volume implies that if Conjecture <ref> is true for L_1 and L_2,
then it is true for the split union L_1⊔ L_2. 



Next we discuss the behavior of the Turaev-Viro invariants under taking connected sums of links.   With our normalization of the colored Jones polynomials,  we have  that 
    J_L_1 # L_2,𝐢(t)= [i] J_L_1 ,𝐢_1 (t)· J_L_2,𝐢_2(t),
 where 𝐢_1 and 𝐢_2 are respectively  the restriction of 𝐢 to L_1 and L_2, and i is the component of 𝐢 corresponding to the component of L_1#L_2 coming from the connected summation. This implies the following.

  Let A be a primitive 2r-th root of unity.  For any odd integer r⩾ 3, q=A^2 and t=A^4, we have

    TV_r'(S^3∖ L_1#L_2,q)=(η_r')^2∑_1⩽𝐢⩽ m[i]^2|J_L_1,𝐢_1(t)|^2|J_L_2,𝐢_2(t)|^2,

where 𝐢_1 and 𝐢_2 are respectively  the restriction of 𝐢 to L_1 and L_2, and i is the component of 𝐢 corresponding to the component of L_1#L_2 coming from the connected summation.



In the rest of this section, we focus on the value q=e^2 i π/r for odd r=2m +1.
Notice that in this case,  the quantum integers [i] for 1⩽ i ⩽ m are non-zero and their logarithms  are of order O(log r).
Corollary <ref> implies that 



    r →∞lim sup2π/rlog TV_r'   (S^3∖ L_1# L_2,q)
       ⩽r →∞lim sup2π/rlog TV_r'(S^3∖ L_1 ,q)+r →∞lim sup2π/rlog TV_r'(S^3∖ L_2, q).


Moreover if we  assume a positive answer to Question  <ref> for L_1 and L_2, then the  term |J_L_1#L_2,m(t)|^2 of the sum for L_1#L_2 satisfies 

    r →∞lim2π/rlog |J_L_1#L_2,m(t)|^2= (S^3∖ L_1#L_2).

It  follows that if the answer to Question <ref> is positive, and Conjecture <ref> is true for links  L_1 and L_2, then Conjecture <ref> is true for their connected sum. In particular, Theorem <ref> implies the following.

 
 Conjecture <ref> is true for any 
 link obtained by connected sum of the figure-eight and the Borromean rings.
 
 
 
We finish the section with the proof of Theorem <ref>, verifying Conjecture <ref> for knots of simplicial volume zero. 



Theorem <ref> Let K⊂ S^3 be knot with simplicial volume  zero. Then, we have
 


    lim_r→∞2π/rlog (TV_r(S^3∖ K, e^2π i/r))  =||K||=0,

where r runs over all odd integers.
 

 By part (2) of Theorem <ref>  we have


    TV_r(S^3∖ K, e^2iπ/r)=(η_r')^21 ⩽ i⩽ m∑
    |  J_L, i (e^4iπ/r)|^2.

Since 
J_K,1(t)=1, we have
 TV_r(S^3∖ K)⩾η_r'^2>0 for any knot K.
 Thus for r> > 0 the sum of the values of the colored Jones polynomials in (<ref>)  is larger or equal to 1.
 On the other hand, we have η_r'≠ 0  and 
log (|η_r'|^2)/r→ 0 as r →∞.
Therefore, 

    r →∞lim inflog|TV_r(S^3 ∖ K)|/r⩾ 0.


Now we only need to prove that for simplicial volume zero knots, we have  
    r →∞lim suplog|TV_r(S^3 ∖ K)|/r⩽ 0.

By Theorem <ref>, part (2) again, it suffices to prove that the L^1-norm ||J_K,i(t)|| of the  colored Jones polynomials of any knot K of simplicial volume zero is bounded by a polynomial in i. By Gordon <cit.>, the set of  knots of simplicial volume zero is generated by torus knots, and is closed under taking connected sums and cablings. Therefore, it suffices to prove that the set of knots whose colored Jones polynomials have  L^1-norm growing at most polynomially contains the torus knots, and is closed under taking connected sums and cablings.




From Morton's formula <cit.>, for the torus knot T_p,q, we have

    J_T_p,q,i(t)=t^pq(1-i^2)|k|=-i-1/2i-1/2∑t^4pqk^2-4(p+q)k+2-t^4pqk^2-4(p-q)k-2/t^2-t^-2.

Each fraction in the summation can be simplified to a geometric sum of powers of t^2, and hence has L^1-norm less than 2qi+1. From this we have ||J_T_p,q,i(t)||=O(i^2).

For a connected sum of knots, we recall that the L^1-norm of a Laurent polynomial is

    ||d ∈∑ a_d t^d||=d ∈∑|a_d|.

For a Laurent polynomial R(t)=f ∈∑ c_f t^f, we let

    deg(R(t))=max ({ d  / c_d≠ 0})-min ( { d  / c_d≠ 0}).

Then for two Laurent polynomials P(t)=d ∈∑ a_d t^d and Q(t)=e ∈∑ b_e t^e, we have

    ||PQ||=||(d ∈∑a_d t^d)(d ∈∑b_d t^d)||   ⩽    ||f ∈∑ (d+e=f∑a_d b_e)t^f||
    
       ⩽   deg(PQ) d+e=f∑|a_d b_e|
    
       ⩽   deg(PQ)||P|| ||Q||.

Since the  L^1-norm  of [i] grows polynomially in i, if the   L^1-norms of J_K_1,i(t) and J_K_2,i(t) grow polynomially, then so does that of  J_K_1#K_2,i (t)=[i] J_K_1 ,i(t)· J_ K_2 ,i(t).





Finally, for the (p,q)-cabling K_p,q of a knot K, the cabling formula <cit.> says

    J_K_p,q,i(t)=t^pq(i^2-1)/4k=-i-1/2i-1/2∑t^-pk(qk+1)J_K,2qn+1(t),

where k runs over integers if i is odd and over half-integers if i is even.
It  implies that if ||J_K,i(t)||=O(i^d), then ||J_K_p,q,i(t)||=O(i^d+1).


By Theorem  <ref> and the argument in the beginning of the proof of Theorem <ref> applied to links we obtain the following.

  For every link L⊂ S^3, we have

    r →∞lim inflog|TV_r(S^3 ∖ L)|/r⩾ 0,

where r runs over all odd integers.

 As said earlier, there is no lower bound for the growth rate of the Kashaev invariants that holds for all links; and no such bound is known for knots as well.
 






§ THE RELATIONSHIP BETWEEN TV_R(M) AND TV_R'(M)



The goal of this appendix is to prove Theorem <ref>. To this end, it will be convenient  to modify the definition
of the Turaev-Viro invariants given in Subsection 2 and use the formalism of quantum 6j-symbols as in <cit.>.

For i∈ I_r, we  let 
    |i|=(-1)^i[i+1],

and for each admissible triple (i,j,k), we let

    |i, j, k| =(-1)^-i+j+k/2[i+j-k/2]![j+k-i/2]![k+i-j/2]!/[i+j+k/2+1]!.

Also for each admissible 6-tuple (i,j,k,l,m,n), we let 

    |[ i j k; l m n;  ]|=∑_z=max{T_1, T_2, T_3, T_4}^min{ Q_1,Q_2,Q_3}(-1)^z[z+1]!/[z-T_1]![z-T_2]![z-T_3]![z-T_4]![Q_1-z]![Q_2-z]![Q_3-z]!.











Consider a triangulation 𝒯 of M, and let c be an admissible coloring of (M,𝒯) at level r. For each edge e of 𝒯, 
 we let 

    |e|_c=|c(e)|,
 
and for each face f with edges e_1, e_2 and e_3, we let 

    |f|_c=|c(e_1), c(e_2), c(e_3)|.

Also for each tetrahedra Δ  with edges e_ij, {i,j}⊂{1,…, 4}, we let

    |Δ|_c=|
          [ c(e_12) c(e_13) c(e_23); c(e_34) c(e_24) c(e_14);         ]|.


Now recall the invariants TV_r(M) and  TV_r'(M) given in Definitions <ref> and <ref>, respectively.
Then we have the following.

 


  * For any integer r⩾ 3,
 
    TV_r(M)= η_r^2|V|∑_c∈ A_r∏_e∈ E|e|_c∏_f∈ E|f|_c∏_Δ∈ T|Δ|_c.


  * For any odd integer r⩾ 3,
 
    TV_r'(M)= (η_r')^2|V|∑_c∈ A_r'∏_e∈ E|e|_c∏_f∈ E|f|_c∏_Δ∈ T|Δ|_c.




 The proof is  a straightforward calculation. 



Next we establish four lemmas on which the  proof  of Theorem <ref>  will rely. We will use the notations |i|_r, |i,j,k|_r and |[ i j k; l m n;  ]|_r respectively to mean the values of |i|, |i,j,k| and |[ i j k; l m n;  ]| at a primitive 2r-th root of unity A.

|0|_3= |1|_3=1, |0, 0, 0|_3=|1, 1, 0|_3=1 and

    |[ 0 0 0; 0 0 0;  ]|_3
    = |[ 0 0 0; 1 1 1;  ]|_3=|[ 1 1 0; 1 1 0;  ]|_3=1.



 A direct calculation.


The following lemma can be considered as a Turaev-Viro setting  analogue  of Theorem <ref> (3).
For i∈ I_r, let i'=r-2-i.
  
 
  * If i∈ I_r, then i'∈ I_r. Moreover, |i'|_r=|i|_r.
 
  * If the triple (i,j,k) is admissible, then so is the triple (i', j', k). Moreover, 
    |i',j',k|_r=|i,j,k|_r.


  * If the 6-tuple (i,j,k,l,m,n) is admissible, then so are the 6-tuples (i,j,k, l',m',n') and (i',j', k, l', m', n). Moreover, 
    |[  i  j  k; l' m' n';  ]|_r
    =|[ i j k; l m n;  ]|_r
       and  |[ i' j'  k; l' m'  n;  ]|_r
    =|[ i j k; l m n;  ]|_r.




  Parts (a) (b) follow easily from the definitions. 

To see the first identity of (c), let T_i' and Q_j' be the  sums for (i,j,k,l',m',n'), involved in the expression of the corresponding 6j-symbol. 
Namely, let
 
    T'_1=i+j+k/2=T_1,  T'_2=j+l'+n'/2   and   Q'_2=i+k+l'+n'/2,    etc.
  For the terms in the summations defining the two 6j-symbols, let us leave T_1 alone for now, and consider the other T_i's and Q_j's. Without loss of generality  we assume that, Q_3⩾ Q_2⩾ Q_1 ⩾ T_4⩾ T_3⩾ T_2. One can easily check that
 


  * Q_3-Q_1=T'_4-T'_2, Q_2-Q_1=T'_4-T'_3, Q_1-T_4=Q'_1-T'_4, T_4-T_3=Q'_2-Q'_1 and T_4-T_2=Q'_3-Q'_1, which implies 

  * Q'_3⩾ Q'_2 ⩾ Q'_1⩾ T'_4⩾ T'_3⩾ T'_2.
 

For z in between max{T_1,… , T_4} and min{Q_1, Q_2, Q_3}, let 


    P(z)=(-1)^z[z+1]!/[z-T_1]![z-T_2]![z-T_3]![z-T_4]![Q_1-z]![Q_2-z]![Q_3-z]!,
 
and similarly for z in between max{T'_1,… , T'_4} and min{Q'_1, Q'_2, Q'_3} let 


    P'(z)=(-1)^z[z+1]!/[z-T'_1]![z-T'_2]![z-T'_3]![z-T'_4]![Q'_1-z]![Q'_2-z]![Q'_3-z]!.


Then for any a∈{0, 1, …, Q_1-T_4=Q'_1-T'_4} one verifies by (1) above that 

    P(T_4+a)=P'(Q'_1-a).



There are the following three cases to consider.  

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 (<ref>), we have 

    ∑_z= T_4^Q_1P(z)=∑_a=0^Q_1-T_4P(T_4+a)=∑_a=0^Q'_1-T_4P'(Q'_1-a)=∑_z= T'_4^Q'_1P'(z).

  

Case 2. T_1 > T_4 but  T'_1 < T'_4, or T_1 < T_4 but  T'_1 > T'_4. 

By symmetry, it suffices to consider the former case. In this case T_max=T_1, Q_min=Q_1, T'_max=T'_4 and Q'_min=Q'_1, and 
    Q'_1 - (r-2) = i + j - l -m/2 = T_1 -T_4.
 As a consequence Q'_1>r-2 . By (<ref>), we have 



    ∑_z= T_1^Q_1P(z)=∑_a=T_1-T_4^Q_1-T_4P(T_4+a)=∑_a=Q'_1-(r-2)^Q'_1-T'_4P'(Q'_1-a)=∑_z= T'_4^r-2P'(z)=∑_z= T'_4^Q'_1P'(z).

The last equality is because we have P'(z)=0, for z> r-2.

 
Case 3. T_1 > T_4 and  T'_1 >T'_4. 

In this case we have  T_max=T_1, Q_min=Q_1, T'_max=T'_1 and Q'_min=Q'_1. 
We have

 
    Q'_1-(r-2)= i+j-l-m/2 = T_1-T_4>0,

  hence Q_1>r-2.  Also, we have 
    Q'_1-T'_1 = l'+m'-k /2=r-2-T_4.
 As a consequence, Q'_1-(r-2)=T'_1-T_4=T_1-T_4>0, and hence Q'_1>r-2. By (<ref>), we have 


    ∑_z= T_1^Q_1P(z)=∑_z= T_1^r-2P(z)=∑_a=T_1-T_4^r-2-T_4P(T_4+a)=∑_a=Q'_1-(r-2)^Q'_1-T'_1P'(Q'_1-a)=∑_z= T'_1^r-2P'(z)=∑_z= T'_1^Q'_1P'(z).

The first and the last equality are because P(z)=P'(z)=0, for z>r-2. 

The second identity of (c) is a consequence of the first.



As an immediate consequence of the two lemmas above, we have
 


  * For all i∈ I_r, |i|_r=|0|_3|i|_r and |i'|_r=|1|_3|i|_r.

  * If the triple (i,j,k) is admissible, then 
    |i,j,k|_r=|0,0,0|_3|i,j,k|_r   and  |i',j',k|_r=|1,1,0|_3|i,j,k|_r.


  * For every admissible 6-tuple (i,j,k,l,m,n)  we have the following.

    |[ i j k; l m n;  ]|_r   = |[ 0 0 0; 0 0 0;  ]|_3
    |[ i j k; l m n;  ]|_r,
    
    |[  i  j  k; l' m' n';  ]|_r   =|[ 0 0 0; 1 1 1;  ]|_3|[ i j k; l m n;  ]|_r,
    
    |[ i' j'  k; l' m'  n;  ]|_r   =|[ 1 1 0; 1 1 0;  ]|_3|[ i j k; l m n;  ]|_r.


 



Now we are ready to prove Theorem <ref>.



 For (a), we observe that there is a bijection ϕ: I_3× I'_r → I_r defined by 
ϕ (0, i)=i and ϕ (1, i)=i'. This induces a bijection 
    ϕ: A_3 ×  A'_r →  A_r.

Then, by Proposition <ref>, we have 

    TV_3(M)   · TV'_r(M)
       = (η_3^2|V|∑_c∈ A_3∏_e∈ E|e|_c∏_f∈ F|f|_c∏_Δ∈ T|Δ|_c)(η_r'^2|V|∑_c'∈ A'_r∏_e∈ E|e|_c'∏_f∈ F|f|_c'∏_Δ∈ T|Δ|_c')
       =(η_3η'_r)^2|V|∑_(c,c')∈ A_3× A'_r∏_e∈ E |e|_c|e|_c'∏_f∈ F |f|_c|f|_c'∏_Δ∈ T |Δ|_c|Δ|_c'
       = η_r^2|V|∑_ϕ(c,c')∈ A_r∏_e∈ E |e|_ϕ(c,c')∏_f∈ F |f|_ϕ(c,c')∏_Δ∈ T |Δ|_ϕ(c,c')
       =TV_r(M),

where the third equality comes from the fact that  η_r = η_3 ·η_r' and Lemma <ref>. This finishes the proof of part (a) of the statement of the theorem.
Part  (b) is given in <cit.>.

To deduce  (c), note that  by Lemma <ref> we have  that

    TV_3(M)=∑_c∈ A_31=|A_3|.
 
Also note that c∈ A_3 if and only if c(e_1)+c(e_2)+c(e_3) is even for the edges e_1, e_2, e_3 of a face. Now consider the handle decomposition of M dual to the ideal triangulation. Then there is a one-to-one correspondence between 3-colorings and maps

    c:{ 2-handles}→ℤ_2,

and c∈ A_3 if and only if c is a 2-cycle; that is if and only if c∈ Z_2(M, ℤ_2).
Hence  we get |A_3|=(Z_2(M, ℤ_2)). Since there are no 3-handles, H_2(M,ℤ_2)≅ Z_2(M, ℤ_2). Therefore, 
    TV_3(M)=|A_3|=  (H_2(M, ℤ_2))=2^b_2(M).





hamsplain


Effstratia Kalfagianni

Department of Mathematics, Michigan State University

East Lansing, MI 48824

(kalfagia@math.msu.edu)


  
 Renaud Detcherry

Department of Mathematics, Michigan State University

East Lansing, MI 48824

(detcherry@math.msu.edu)

 
  
  
Tian Yang

Department of Mathematics, Texas A &M University

College Station, TX 77843

( tianyan@math.tamu.edu)










