48

RECENT PROGRESSES ON THE VOLUME CONJECTURES FOR CERTAINQUANTUM INVARINATS

Tian Yang
Texas A&M University
1. VOLUME CONJECTURES

In [36], Witten interpreted values of the Jones polynomial using the Chern‐Simons gauge theory,and constructed a sequence of complex valued 3‐manifold invariants based on this idea. Thisapproach was mathematically rigorously formalized by Reshetikhin and Turaev [26, 27] though
the representation theory of quantum groups, where they generalized the Jones polynomial to asequence of polynomial invariants of a link, later called the colored Jones polynomials of that link.
They also defined a sequence of 3‐manifold invariants corresponding to Witten’s invariants. The
Reshetikhin‐Turaev construction of 3‐manifold invariants starts from a surgery description [21] of
the manifold, and evaluates the colored Jones polynomials of the surgery data at certain roots ofunity. In [33], Turaev and Viro developed a different approach from a triangulation of a 3‐manifold
constructing real valued invariants of the manifold. These Turaev‐Viro invariants tumed out to beequal to the square of the norm of the Reshetikhin‐Turaev invariants [28, 32, 35].
Using quantum dilogarithm functions, Kashaev [19, 20] defined for each integer n a complex
valued link invariant. He observed in a few examples, and conjectured in the general case, that theabsolute value of these invariants grow exponentially with n , and that the growth rate is given by
the hyperbolic volume of the complement of the link. In [18], Murakami and Murakami showedthat Kashaev’s invariants coincide with the values of the colored Jones polynomials at a certain
root of unity, and reformulated Kashaev’s conjecture as follows.

Volume Conjecture ([20, 18]). For a hyperbolic link L in S^{3} , let J_{n}(L, q) be its n‐th coloredJones
polynomial. Then\lim_{nar ow+\infty}\frac{2\pi}{n}\log|J_{n}(L, e^{\frac{2\pi\sqrt{-1} {n} )|= Vol(S^{3}\backslash L) ,where Vol(S^{3}\backslash L) is the hyperbolic volume of the complement of L.
This conjecture has now been proved for a certain number of cases: the figure‐eight knot [18], all
hyperbolic knots with at most seven crossings [22, 23, 24], the Borromean rings [14], the twisted
Whitehead links [37] and the Whitehead chains [34]. Various extensions of this conjecture havebeen proposed, and proved for certain cases.
In [6], Chen and the author investigated the asymptotic behavior of the Reshetikhin‐Turaev

and the Turaev‐Viro invariants evaluated at the root of unity q=e^{\frac{2\pi\sqrt{-1} {r} . Based on numericalcomputations, they made the followin g

Conjecture 1.1 ([6]). For a hyperbolic 3‐manifold M , let TV_{r}(M, q) be its Turaev-V\iota ro invariant
and let Vol(M) be its hyperbolic volume. Then for r running over all odd integers and for q=

49
RECENT PROGRESSES ON THE VOLUME CONJECTURES FOR CERTAIN QUANTUM INVARINATSe^{\frac{2\pi\sqrt{-1} {r} ,\lim_{rarrow+\infty}\frac{2\pi}{r}\log(TV_{r}(M, q))=Vol(M).

Conjecture 1.2 ([6]). Let M be a closed oriented hyperbolic 3‐manifold and let RT_{r}(M, q) be its

Reshetikhin‐Turaev invariants. Then for q=e^{\frac{2\pi\sqrt{-1} {r} with r odd and for a suitable choice of thearguments,r arrow+\infty 1\dot{ \imath} m\frac{4\pi\sqrt{-1} {r}\log(RT_{r}(M, q))=CS(M)+ Vol(M)\sqrt{-1} mod \pi^{2}\mathbb{Z},where CS(M) denotes the Chern‐Simons invariant ofthe hyperbolic metric of M multiplied by 2\pi^{2}.
2. RECENT PROGRESS

2.1. Ohtsuki’s result and method. The first family of examples for which Conjecture 1.1 and
1.2 hold was given by Ohtsuki [25], where he also obtained a full asymptotic expansion of theReshetikhin‐Turaev invariants.

Theorem 2.1 ([25]). Let p be an integer and let M_{p} be the 3‐manifold obtained from the figure‐8
knot by a p Dehn‐filling. Then for |p|>4 . the Reshetikhin‐Turaev invariant RT_{r}(M_{p}, e^{\frac{2\pi i}{r} ) is
expanded as rarrow\infty in the following form,

RT_{r}(M_{p}, e^{\frac{2\pi }{r} )=e^{\frac{r}{4\pi x}(CS(M_{p})+iVol(M_{p}) }\epsilon(p, r)\omega(M_{p})r^{\frac{3}{2} (1+\sum_{k=1}^{d}\kap a_{k}(M_{p}) (\frac{4\pi }{r})^{k}+O(\frac{1}{r^{d+1} ),where \epsilon(p, r) is a concrete root of unity depending only on p and r , and \omega(M_{p}) and \kappa_{k}(M_{p}) areconstants determined by M_{p}.
In particular, Theorem 2.1 implies that Conjecture 1.1 and 1.2 hold for M_{p} . Based on this, hefurther conjectured that RT_{r}(M, e^{\frac{2\pi\dot{i} {r} ) of any closed hyperbolic 3‐manifold M can be expanded in
the same form as above, and the sub‐leading coefficient \omega(M) is closely related to the Reidemeister
torsion of M . He also expected that \kappa_{k}(M) define new invariants of M.
The method Ohtsuki used to get the result can be roughly summarized in the following threesteps. In the first step he wrote the quantum invariants as the sum of values f(n) of a complex holo‐
morphic function f at the (multi)‐integers. The function f comes from the quantum dilogarithm
function which serves an integral representation of quantum factorials. In the second step, by using
Poisson Summation Formula, he wrote the sum of f(n) into the sum of the Fourier coefficients\hat{f}(m) of f, each of which is an integral of complex holomorphic function. By carefully verifying
a technical condition he posed, he can prove that among all these Fourier coefficients \hat{f}(m) , twoof them dominate the asymptotics. Finally, by rigorously using the Saddle Point Method, he man‐aged to analyze the growth rate of the two dominating Fourier coefficients, and hence obtained thegrowth rate of the invariants.

2.2. Another approach. By studying the Reshetikhin‐Turaev and the Turaev‐Viro TQFTs [3, 4,33], Detcherry, Kalfagianni and the author [11] gave a formula relating the Turaev‐Viro invariantsof the complement of a link L in S^{3} to the values of the colored Jones polynomials of L , and Bel‐letti, Detcherry, Kalfagianni and the author [2] gave a formula relating the Turaev‐Viro invariants
the complement of a link L in an oriented closed 3‐manifold M to the relative Reshetikhin‐Turaev
invariants of the pair (M, L) . Using these relations, they verified Conjecture 1.1 for the figur‐8
knot, the Borromean rings complement [11] and the fundamental shadow link complements [2].

50RECENT PROGRESSES ON THE VOLUME CONJECTURES FOR CERTAIN QUANTUM INVARINATS

For a multi‐integer n=(n_{1}, \ldots, n_{l}) of l components, we use the notation 1\leq n\leq m todescribe all such multi‐integers with 1\leq n_{k}\leq m for each k\in\{1, l\} . Given a link L inS^{3} with l components, let J_{n}(L, t) denote the n‐th colored Jones polynomial of L whose k‐thcomponent is colored by n_{k} . If all the components of L are colored by the same integer n , then we
simply denote J_{(n,\ldots,n)}(L, t) by J_{n}(L, t) . If L is a knot, then J_{n}(L, t) is the usual i‐th colored Jones
polynomial. The polynomials are indexed so that J_{1}(L, t)=1 and J_{2}(L, t) is the ordinary Jones
polynomial, and are normalized so thatfor the unknot

J_{n}(U, t)=[n]= \frac{q^{n}-q^{-n} {q-q^{-1}

U, where by convention t=q^{2}.

Theorem 2.2 ([11]). Let L be a link in S^{3} with l components.
(1) For an integer r\geq 3 , we haveTV_{r}(S^{3}\backslash L, e^{\frac{\pi }{r} )=\frac{2s\dot{ \imath} n^{2} (\frac{\pi\dot{i} {r}) {r}\sum_{1\leq n\leq r-1}|J_{n}(L, t)|^{2}(2) For an odd integer r\geq 3 , a primitive 2r ‐th root of unity A and q=A^{2} , we haveTV_{r}(S^{3}\backslashL,e^{\frac{2\pi\dot{\iota} {r} )=\frac{2^{l+1} s\dot{ \imath} n^{2}(\frac{2\pi }{r )}{r \sum_{1\leqn\leq\frac{r-1}{2} |J_{n} (L,t)|^{2}

As a consequence, they got the following

Theorem 2.3 ([11]). Let L be either the figure‐eight knot or the Borromean rings, and let M be
the complement of L in S^{3} . Thenr ar ow+\infty 1\dot{ \imath} m\frac{2\pi}{r}\log TV_{r}(M, e^{\frac{2\pi i}{r} })=\lim_{mar ow+\infty}\frac{4\pi}{2m+1}\log|J_{m}(L, e^{\frac{4\pi\dot{i} {2m+ 1}})|=Vol(M),where r=2m+1 runs over all odd integers.

Remark 2.4. The asymptotic behavior of the values of J_{m}(L, t) at t=e^{\frac{2\pi i}{m+z1}} is not predicted either
by the originaı volume conjecture [20, 18] or by its generalizations [15, 17]. Theorem 2.3 seems
to suggest that these values grow exponentially in m with growth rate the hyperbolic volume. Thisis somewhat surprising because as noted in [14] that for any positive integer k, J_{m}(L, e^{\frac{2\pi\dot{i} {m+k}}) grows
only polynomially in m . They ask the following question.
Question 2.5. ls it true thatfor any hyperbolic link L in S^{3} , we havem ar ow+\infty 1\dot{ \imath} m\frac{2\pi}{m}\log|J_{m}(L, e^{\frac{2\pi\dot{i} }{m+z1} )|=Vol(S^{3}\backslash L)?
Given a link L in closed oriented 3‐manifold M , let RT_{r}(M, L, n) denote the relative Reshetikhin‐
Turaev invariants [3] of the pair (M, L) with the k‐th component of L colored by n_{k}.

Theorem 2.6 ([2]). Let L be a link in a closed oriented 3‐manifold M with l components.

TV_{r}(M\backslash L)=\sum_{0\leq n\leq r-2}|RT_{r}(M, L, n)|^{2}
where the sum is over multi‐integers with l components.

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RECENT PROGRESSES ON THE VOLUME CONJECTURES FOR CERTAIN QUANTUM INVARINATS

As a consequence, they got the following
Theorem 2.7 ([2]). Conjecture 7.1 holds for the complements of all fundamental shadow links.

Remark 2.8. As proved by Costantino and Thurston [8], the fundamental shadow link complements
form a universal class in the sense that any orientable 3‐mamifold with empty or toroidal boundary
is obtained from a complement of a fundamental shadow by Dehn filıing. Therefore, if Ohtsuki’s
method mentioned in Section 2.1 could be made to work in this situation, one would be able tosolve Conjectures 1.1 and 1.2 for all closed and cusped hyperbolic 3‐manifolds.
A key observation in their proof of Theorem 2.7 besides Theorem 2.6 is below a sharp upperbound on the growth of the quantum 6j ‐symbol.
Proposition 2.9 ([2]).

(1) For any sequence of r ‐admissible 6‐tuples

have\{(n_{1}^{(r)}, \ldots, n_{6}^{(r)})\} , we

1\dot{\imath}m\sup_{ra ow+\infty}\frac{2\pi}{r\log|\begin{ar y}{l } n_{1}^{(r)} n_{2}^{(r)} n_{3}^{(r)} n_{4}^{(r)} n_{5}^{(r)} n_{6}^{(r)} \end{ar y}|\leqv_{8},where v_{8} is the volume of the regular ideal hyperbolic octahedron.
(2) If n_{i}^{(r)}= \frac{r\pm 1}{r} for each i\in\{1 ,6 \} , we haverarow+\infty1\dot{\imath}m\frac{2\pi}{r\log|\begin{ar y}{l } n_{1}^{(r)} n_{2}^{(r)} n_{3}^{(r)} n_{4}^{(r)} n_{5}^{(r)} n_{6}^{(r)} \end{ar y}|=v_{8}.

Part (2) of Proposition 2.9 is actually a special case of a result originally due to Costantino [7]\pi iat the root q=e\overline{r} and recaptured independently by Chen and Murakami [5] and Detcherry andthe author [12] at the root q=e^{\frac{2\pi i}{r} for odd r.

Theorem 2.10 ([7, 5, 12]). Let \{(n_{1}^{(r)}, n_{6}^{(r)})\}_{r} be a sequence of r ‐admissible 6‐tuples with\alpha_{i}=\lim_{rar ow\infty}\frac{2\pi n_{i}^{(r)} {r},and let \theta_{i}=|\pi-\alpha_{i}| . lf (\theta_{1}\ldots, \theta_{6}) are the dihedral angles of a hyperideal hyperbolic tetrahedral\triangle , thenr ar ow+\infty 1\dot{ \imath} m\frac{2\pi}{r}\log |\begin{ar y}{l n_{1}^(r)} n_{2}^(r)} n_{3}^(r)} n_{4}^(r)} n_{5}^(r)} n_{6}^(r)} \end{ar y}| q=e^{\frac{2\pi i}{r} =Vol(\triangle).

It worth mentioning that in [5], Chen and Murakami also computed the second leading term of
the asymptotic expansion of quantum 6j ‐symbols, which is closely related to the Gram matrix of\triangle.

Remark 2.11. Since quantum 6j ‐symbols are the main building blocks of the Turaev‐Viro in‐variants and hyperbolic tetrahedral are building blocks of the hyperbolic structure, Theorem 2.10
essentially says that Conjecture 1.1 is ture at least at the level of building blocks.
3. RELATIONSHIP To THE AMU CONJECTURE

According to Nielsen‐Thurston’s classification of the elements of the mapping class group of
surfaces, every irreducible orientation preserving self‐homeomorphism of a surface of finite type iseither periodic (of finite order) or pseudo‐Anosov (preserving two transverse measure laminations).
Here a self‐homeomorphism being irreducible means that it does not restrict of a proper subsurface.
In [1], Andersen, Masbaum and Ueno made the following

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RECENT PROGRESSES ON THE VOLUME CONJECTURES FOR CERTAIN QUANTUM INVARINATS

Conjecture 3.1 ([1]). Let \Sigma be a orientable surface offinite type, let \phi be a pseudo‐Anosov self‐
homeomorphism of \Sigma , and let \{\rho_{r}\}_{r} be the sequence of the Turaev-V\iota ro representations of the
mapping class group of \Sigma . Then for r sufficiently large, \rho_{r}([\phi]) is a linear transformation of infinite
order.

Combined with the fact that the image of a finite order element under any group representa‐tion is of finite order, the AMU Conjecture essentially claims that the sequence of Turaev‐Viro
representations of the mapping class groups respects the Nielsen‐Thurston classification. The sim‐
ilar conjecture can be made for the Reshetikhin‐Turaev representations, which are a sequence of
projective representations of mapping class group of surfaces. The AMU conjecture is known tobe true for punctured spheres [1, 13] and the once‐punctured torus [29]. Recently, Marché andSantharoubane [16] related the Turaev‐Viro representations to representations of the fundamentalgroup of surfaces, and provide an efficient algorithm of determining whether an element of thefundamental group can be represented by a simple closed curve on the surface, assuming that the
AMU Conjecture is true.
Observed by Santharoubane [30] (see also Detcherry and Kalfagianni [10]), the AMU Conjec‐
ture is a consequence of the following a weaker version of Conjecture 1.1.
Conjecture 3.2. Let M be a hyperbolic 3‐manifold with finite volume, and let TV_{r}(M, q) be its
r‐th Turaev‐Vro invariant at the root of unity q . Then for r running over all the odd integers,\lim\dot{ \imath} nf\frac{1}{r}\ln TV_{r}(M, e^{\frac{2\pi }{r} )rar ow+ \infty>0.
The relationship between the two conjectures mentioned above is given by the underlying TQFTs.
Roughly speaking, the mapping cylinder MC_{\phi} of \phi can be considered as a cobordism from \Sigma toitself. Hence for each r , the Turaev‐Viro TQFT assigns MC_{\phi} a linear map, which is exactly \rho_{r}([\phi])by the construction of the Turaev‐Viro representation. By the TQFT axioms, the trace of \rho_{r}([\phi])equals to the Turaev‐Viro invariant TV_{r}(M_{\phi}) of the mappin g torus M_{\phi} of \phi . Since \phi is pseudo‐
Anosov, Thurston’s result shows that M_{\phi} is hyperbolic. Then Conjecture 3.2 implies that TV_{r}(M)grows at least exponentially at particular roots of umity. On the other hand, if \rho_{r}([\phi]) was of finite
order, the each of its eigenvalues should be a root of unity. As a consequence, the trace of \rho_{r}([\phi])
is at most the dimension of the TQFT vector space of \Sigma , which by the Verlinde formula is only apolynomial in r . That is a contradiction.

In a recent work [9], Detcherry and Kalfagianni showed that the behavior of the Turaev‐Viroinvariant is “similar to” that of the hyperbolic volume, in the sense that it does not increase underDehn‐fillings.
Theorem 3.3 ([9]). Let M and M' be compact, oriented 3‐manifolds with empty or toroidal bound‐ary. Then we have the following.
(1) lf T is an embedded torus in M and M' is the manifold obtained by cutting M along T,
thenTV_{r}(M)\leq(\frac{r-1}{2})TV_{r}(M').

(2) Suppose that \partial M'\neq 0 . If M is obtained from M' by Dehn‐filling some components of\partial M' , thenTV_{r}(M)\leq TV_{r}(M').

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RECENT PROGRESSES ON THE VOLUME CONJECTURES FOR CERTAIN QUANTUM INVARINATS

Therefore, if one could prove Conjecture 3.2 for a 3‐manifold M , then Conjecture 3.2 is auto‐
matically true for all the 3‐manifolds obtained from M by removing a link inside it. By the work of
Ohtsuki [25] and Belletti, Detcherry, Kalfagianni and the author [2] mentioned in Section 2, Con‐jecture 3.2 hold for all the 3‐manifolds obtained from the examples mentioned above by remove alink inside them, and the AMU Conjecture holds for the fibered ones obtained from those examples
by doing the same operation.
Corollary 3.4 ([10, 2]). Let M_{p} be the 3‐manifold obtained by doing p Dehn‐filling along thefigure‐8 knot, and let M be a fibered 3‐manifold obtainedfrom M_{p} orfrom a fundamental shadow
link complement by removing a link. Then Conjecture [1] holds for the holonomy of any fiber
structure of M.

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