QUANTUM INVARIANTS FOR LINKS AND 3-MANIFOLDS KAZUHIRO HIKAMI

1. INTRODUCTION Studies on quantum invariants for knots and links have been developed since seminal work of Jones [25]. The Jones polynomial for link L, which has n knot components Ki embedded in M, is reinterpreted as correlation functions of the Wilson loop [61] (see also Ref. 1)

 Jk1 ,...,kn (L) = Wk1 (K1 ) · · · Wkn (Kn ) (1.1) where the expectation is for the SU(2) Chern–Simons exp (2 π i k CS( A)) with k ∈ Z and   Z 1 2 CS( A) = Tr A ∧ d A + A ∧ A ∧ A 8 π2 3

functional

(1.2)

M

The Wilson loop operator Wk j (K) with color k j is defined by   I Wk j (K) = Tr P exp  A K

where the path ordering P is performed along K. Correspondingly the quantum invariant for 3-manifolds is defined as the partition function, and is called the Witten invariant; Z Z k (M) =

exp (2 π i k CS( A)) D A

(1.3)

The Witten invariant was later given the mathematically rigorous definition by Reshetikhin and Turaev [53, 54], and it has the following normalization in the case of SU(2); τk+2 (M)
Z k (M) = (1.4) τk+2 S 2 × S 1 Date: This Note is based on the author’s two talks at Rikkyo University on December 23 (2003) and October 18 (2004). 1

englishKAZUHIRO HIKAMI

2

where 

2

τN S × S We note that

1



=

r

N 1 2 sin (π/N )

  τN S3 = 1

Asymptotic behavior of the Witten–Reshetikhin–Turaev quantum invariant in k → ∞ has been studied since Witten’s original work [9, 10, 35–37, 56–59, 61], and it is expected that 1 3 Xp Z k (M) ∼ e− 4 πi Tα (M) e−2πiIα /4 e2πi(k+2) CS(A) (1.5) 2 α

where α denotes a flat connection, and Tα and Iα are respectively the Reidemeister–Ray–Singer torsion and the spectral flow defined modulo 8. From the viewpoint of the knot invariant, the asymptotic expansion of the quantum invariants receives renewed interests since “Volume Conjecture” was proposed. According to Refs. 27, 47, 48, we have 2π log hKi N = Vol(S 3 \ K) + i CS(K) (1.6) lim N→∞ N where hKi N is the Kashaev invariant [27, 29] for knot K, and Vol denotes the hyperbolic volume. Later proved [47] is that the Kashaev invariant hKi N coincides with a special value of the N -colored Jones polynomial at q = e2π i/N . It is noted that we still have another intriguing “AJ conjecture” [12] concerning the geometric structure of the colored Jones polynomial (see Ref. 39 for recent results). In this Note, we shall present a method to calculate an exact asymptotic expansion of the SU(2) quantum invariants for certain classes of links and 3-manifolds. Computations of explicit forms of quantum invariants and the exact asymptotic expansions thereof are generally the difficult task. We adopt a method of Ref. 38 where realized was that the WRT invariant for the Poincaré homology sphere is regarded as the Eichler type integral of certain modular form with half-integral weight. Therein it was further demonstrated that the nearly modular property of the Eichler integral gives the exact asymptotics of the quantum invariant. We thus see a close connection between topological invariants and modular forms. Content of this Note is as follows. • classical theory on the modular forms and the Eichler integrals • modular forms with half-integral weight • colored Jones polynomial for torus link T2,2P

englishQUANTUM INVARIANTS FOR LINKS AND 3-MANIFOLDS

3

• WRT invariant for the Brieskorn homology sphere 2. MODULAR FORM

AND

EICHLER INTEGRAL

We review classical results on the Eichler integral (see, e.g., Ref. 34). Hereafter we use q = e2πiτ

where τ is in the upper half plane, τ ∈ H. The Eichler integral is originally defined for the modular form with integral weight; let the function F(τ ) be defined by F(τ ) =

∞ X

an q n

(2.1)

n=1

and be the modular form with weight k ∈ Z≥2 , i.e.,

F(γ (τ )) = (c τ + d)k · F(τ )

where we mean



a b γ = c d



(2.2)

∈ S L(2; Z)

e ) is set to be k − 1 times integration of F(τ ) w.r.t. τ ; Then the Eichler integral F(τ   d k−1 e 1 · F(τ ) = F(τ ) (2.3) 2 π i dτ We thus have a q-series ∞ X an n e F(τ ) = q (2.4) n k−1 n=1

We should note that the Eichler integral (2.3) can also be written in an integral form as Z (2 π i)k−1 i∞ e F(τ ) = − F(z) (τ − z)k−2 dz (2.5) (k − 2)! τ Once we know this expression, we can prove that the Eichler integral has a nearly modular property; e (τ )) − F(τ e ) = G γ (τ ) (c τ + d)k−2 F(γ

(2.6)

where G γ (τ ) is defined by

(2 π i)k−1 G γ (τ ) = (k − 2)!

Z

i∞

γ −1 (i∞)

F(z) (τ − z)k−2 dz

(2.7)

Note that the function G γ (τ ) is a polynomial of τ with order up to k − 2. It is then called the period polynomial whose coefficients are called the periods.

englishKAZUHIRO HIKAMI

4

3. MODULAR FORM

WITH

HALF-INTEGRAL WEIGHT

We fix P ∈ Z≥2 . We introduce the q-series by 1X n2 (a) 9 (a) (τ ) = n ψ2P (n) q /4P P 2

(3.1)

n∈Z

(a)

for 1 ≤ a < P. Here ψ2P (n) is an odd periodic function with modulus 2 P ( ±1 for n ≡ ±a mod 2 P (a) ψ2P (n) = 0 otherwise (a)

and we have a mean value zero. In the following we sometimes use ψ2P (n) for a ∈ Z satisfying (a) (2P+a) (−a) ψ2P (n) = ψ2P (n) = −ψ2P (n) (a)

Proposition 1. A set of the q-series 92P (τ ) is a modular form with weight 3/2:  3/2 X P−1 i (a) (b) 9 P (τ ) = M(P)ba 9 P (−1/τ ) (3.2) τ b=1

a2

(a) 2P πi 9 9 (a) P (τ ) P (τ + 1) = e

(3.3)

where M(P) is a (P − 1) × (P − 1) matrix whose element is given by r   ab 2 b M(P)a = sin π P P Proof. It is easy to prove eq. (3.3). By use of the Poisson summation formula, X XZ ∞ f (n) = e−2πitn f (t) dt n∈Z

n∈Z −∞

we get eq. (3.2).

 (a)

The modular form 9 P (τ ) is related to the affine su(2) b P−2 character (see, e.g., Refs. 7, 26) 9 (a) (τ ) P (3.4) cha (τ ) = P 3 (η(τ )) where η(τ ) is the Dedekind η-function, η(τ ) = q

1/24

∞ Y

k=1

(1 − q k )

englishQUANTUM INVARIANTS FOR LINKS AND 3-MANIFOLDS

5

It should be noted that, due to the Jacobi triple product identity, we have 92(1) (τ ) = (η(τ ))3 and that we have (1)

93 (τ ) =

(η(2 τ ))5

(2)

93 (τ ) = 2

(η(4 τ ))2

(η(τ ) η(4 τ ))2 η(2 τ )

(a)

We have interests in the Eichler integral of 9 P (τ ). As our modular form has a half-integral weight, so the classical definition (2.3) cannot be applied. Although, we can define the Eichler integral naïvely based on eq. (2.4) as was pointed out in Ref. 38; ∞ X n2 (a) (a) e (τ ) = 9 ψ2P (n) q /4P (3.5) P n=0

e (a) (τ ) is defined for τ ∈ H, and we have a limiting value as The q-series 9 P follows. Proposition 2. Limiting value of the Eichler integral at τ → M/N ∈ Q is given by e (a) ( M/N ) = − 9 P

2P N X

M (a) ψ2P (k) eπi N

k2 2P

k=0

B1



k 2PN



(3.6)

Here we assume M and N are relatively prime integers, and N > 0. We use Bk (x) as the k-th Bernoulli polynomial whose generating function is ∞

X tk t ext = B (x) k et − 1 k!

(3.7)

k=0

and we have B0 (x) = 1 B1 (x) = x −

1 2

B2 (x) = x 2 − x +

1 6

To prove this proposition, we use the following formulae for asymptotic expansion (see, e.g., Refs. 38, 62);

englishKAZUHIRO HIKAMI

6

Proposition 3. Let C f (n) be a periodic function with mean value 0 and modulus f . Then we have asymptotic expansions as t & 0; ∞ X

C f (n) e

−nt

n=1

∞ X

C f (n) e

−n 2 t

n=1

'

∞ X

L(−k, C f )

'

∞ X

L(−2 k, C f )

k=0

(−t)k k!

k=0

(3.8)

(−t)k k!

(3.9)

Here L(k, C f ) is the Dirichlet L-function associated with C f (n), and by analytic continuation it is given for −k < 0 by f fk X L(−k, C f ) = − C f (n) Bk+1 (n/ f ) k+1 n=1

where Bn (x) denotes the n-th Bernoulli polynomial (3.7).

Proof. We first prove eq. (3.8). By the periodicity of C f (n), we have ∞ X n=1

C f (n) e−nt =

f X

C f (m)

m=1

=−

f X

e−mt 1 − e− f t

C f (m)

m=1

∞ X

Bk+1 (m/ f )

k=0

(− f t)k (k + 1)!

where in the second equality we have used eq. (3.7) and mean value zero condition for C f (n). For a proof of eq. (3.9), we recall Z √ z2 hw2 πhe = dz e− h +2wz C

englishQUANTUM INVARIANTS FOR LINKS AND 3-MANIFOLDS

7

where a path C passing through the origin is chosen for the convergence of the integral. We then have ∞ X n=1

C f (n) e

−n 2 t

Z f X 1 e2mz z2 =√ C f (m) dz e /t 1 − e2 f z −π t m=1 C

1 =√ −π t =−

∞ X

f X

C f (m)

m=1

f X

∞ X k=0

−(2 f )k−1 Bk (m/ f ) k!

Z

dz z k−1 e

z 2/t

C

C f (m) B2l+1 (m/ f )

l=0 m=1

f 2l (−t)l 2 l + 1 l!

Here in the first equality we have used a periodicity of the function C f (n), and have applied eq. (3.7) in the second equality. In the last equality, we have performed integration using that the integrand should be even. These identities can be checked by use of the Mellin transformation. Proof of Prop. 2. Substituting τ = e (a) 9 P



M N

+

y 2π



i with y > 0 for eq. (3.5), we have

 X ∞ n2 n2 M M y (a) + i = ψ2P (n) e 2P N πi− 4P y N 2π n=0

We see that the function (a)

n2 M N πi

(a)

C 2P N (n) = ψ2P (n) e 2P

is a periodic function with modulus 2 P N . So in a limit y & 0, we can apply the formula (3.9), and then we obtain eq. (3.6).  We note that for N ∈ Z we have e (a) (1/N ) 9 P

=−

2P XN

2

k (a) ψ2P (k) e 2P N πi

B1

k=0

 a  a 2 πiN e (a) (N ) = 1 − 9 e 2P P P



k 2PN



(3.10)

(3.11)

These two limits of the Eichler integral are related to each other by the nearly modular property as follows;

englishKAZUHIRO HIKAMI

8

e (a) ( M/N ) of the modular Proposition 4. Limiting values of the Eichler integral 9 P (a) form 9 P (τ ) with weight 3/2 has a nearly modular property. Especially an asymp(a) totic expansion of 9 P (1/N ) in N → ∞ is given by r P−1  ∞ (a)  X L(−2 k, ψ2P ) N X πi k (a) b e (b) e 9 P (1/N ) + (3.12) M(P)a 9 P (−N ) ' i k! 2PN b=1

(a) L(k, ψ2P )

where in eq. (3).

k=0

(a)

denotes the Dirichlet L-function associated with ψ2P (n) defined

Proof. We introduce another Eichler integral Z ∞ (a) 9 P (τ ) 1 (a) b (z) = √ 9 dτ (3.13) √ P τ −z 2 P i z¯ which is an analogue of eq. (2.5). To avoid singularity, it is defined for z in the lower half plane, z ∈ H− , and ¯· denotes a complex conjugate. We see that the modular property of 9 (a) P (τ ), especially the modular S-transformation (3.2), leads

where

P−1 1 X (a) b (b) (−1/z ) = r (a) (z; 0) b M(P)ab 9 9 P (z) + √ 9P P i z b=1

(3.14)

Z

∞ 9 (a) (τ ) =√ √P dτ τ −z 2Pi α with α ∈ Q. Substituting definition (3.1) of modular form for eq. (3.13) and performing integration, we find that (a) r9 P (z; α)

1

e (a) (1/N ) = 9 b (a) (1/N ) 9 P P

where l.h.s. are the limiting value from the upper plane H while r.h.s. are from the lower half plane H− . Taking asymptotic expansions of r.h.s. of eq. (3.14), we obtain eq. (3.12).  The right hand side of eq. (3.12) plays a role of the period polynomial G 0 −1 (τ ) in eq. (2.6), but now it is not a polynomial any more and may be γ=

1 0

called the period function [38, 62].

We note that a generating function of coefficients of the period functions is given by ∞ (a) sh((P − a) z) X L(−2 k, ψ2P ) 2k = z (3.15) sh(P z) (2 k)! k=0

englishQUANTUM INVARIANTS FOR LINKS AND 3-MANIFOLDS

4. COLORED JONES POLYNOMIAL

FOR

9

TORUS LINK T2,2P

The N -colored Jones polynomial for torus knot Ts,t with coprime integers s and t was studied in Refs. 41, 44, 55 (see also Ref. 16). Following these methods, we compute the colored Jones polynomial for torus link T2,2P in this section (we assume P > 0).

Figure 1: Hopf link T2,2 and torus link T2,4

We use the Jones–Wenzl idempotent in the Temperley–Lieb algebra, and use of following formulaePSfrag (see, e.g., Refs. 31, 40); PSfrag make replacements replacements a

b

a = (−1)(a+b−c)/2 Aa+b−c+

PSfrag replacements

a 2 +b2 −c2 2

(4.1)

PSfrag replacements

c

b

c a

a b

=

X

c: (a, b, c) is admissible

1c θ (a, b, c)

a c

b

(4.2) b

where each label denotes a color, and we mean that 1n = (−1)n

A2(n+1) − A−2(n+1) A2 − A−2

  a ≤ b + c (a, b, c) is admissible ⇔ a + b + c is even, and b ≤ c + a  c ≤ a + b

englishKAZUHIRO HIKAMI PSfrag replacements

10

We also need a θ -net

θ (a, b, c) =

a

b

(4.3)

c

which is given as θ (a, b, c) = with a = y+z

1x+y+z ! 1x−1 ! 1 y−1 ! 1z−1 ! 1 y+z−1 ! 1z+x−1 ! 1x+y−1 ! b=z+x

c=x+y

and we mean 1n ! = 1n 1n−1 · · · 11 By these formulae, we can compute the N -colored Jones polynomial for torus link T2,2P as follows; Proposition 5. The N -colored Jones polynomial J N (h; K) for the torus link K = T2,2P is given by 2 sh ( N h/2)

X N−1 X 1 J N (h; K) 2 − 12 P(N 2 −1)h =e ε e Ph j +(P+ε) h j + 2 hε J N (h; O)

(4.4)

ε=±1 j =0

where a parameter q is set to be

q = A 4 = eh and O denotes unknot whose invariant is given by sh(N h/2) J N (h; O) = sh(h/2) Proof. We first apply eq. (4.2) in the torus link T2,2P (see Fig. 2), and untangle crossings recursively using eq. (4.1). We see that the θ -net θ (a, b, c) vanishes at the end. We have   X c 2 1 2 2P J N (h; K) = 1c (−1) N−1− 2 A−2(N−1)+c−(N−1) + 2 c c:(N − 1, N − 1, c) is admissible

2 N−1  A−2P(N −1) X 4P j ( j +1)  2(2 j +1) −2(2 j +1) = 2 A A − A A − A−2

j =0

This proves eq. (4.4).



englishQUANTUM INVARIANTS FOR LINKS AND 3-MANIFOLDS

11

H⇒ PSfrag replacements c n

n

n

n

Figure 2: We apply eq. (4.2) to the torus link T2,4 . We set n = N − 1.

Motivated by the volume conjecture, we have an interest in a specific value of the N -colored Jones polynomial, and we define the Kashaev invariant by hKi N = J N (2 π i/N ; K)

(4.5)

Here we have normalized the invariant such that hOi N = 1

This knot invariant hKi N as a function of N was originally investigated by Kashaev [27, 28] by use of the quantum dilogarithm function [3–5]. Coincidence with a specific value of the colored Jones polynomial was later proved in Ref. 47. Proposition 6. The Kashaev invariant for the torus link T2,2P is written as hT2,2P i N = P N e−

(P−1)2 2P N πi

e (P−1) (1/N ) 9 P

(4.6)

Proof. By L’Hopital theorem we get from eq. (4.4)

T2,2P



N

2P N 2 X j2 1 (P−1) − (P−1) πi 2P N =− e j 2 ψ2P ( j ) e 2P N πi 4PN j =0

(a) Using the anti-periodicity of the periodic function ψ2P (n), we get  2P N 2P N X X j2 j2 j P (P−1) 2 (P−1) πi 2 2P N j ψ2P ( j ) e = −4 P N − ψ2P ( j ) e 2P N πi 2 2N j =0

j =0

= (2 P N )

2

2P N X j =0

B1



j 2PN



(P−1)

ψ2P

j2

( j ) e 2P N πi

englishKAZUHIRO HIKAMI

12

Recalling eq. (3.10), we complete proof.



The coincidence with a limiting value of the Eichler integral enables us to have an exact asymptotic expansion of the N -colored Jones polynomial from eq. (3.12). Corollary 7 ( [15]). The asymptotic expansion of the Kashaev invariant hT2,2P i N in N → ∞ is hT2,2P i N '

r

P−1 a  a2 2 − (P−1)2 πi 3/2 X (−1)a (P − a) sin e 2P N π e− 2P πiN N iP P a=1  ∞ (P−1)  2 X L(−2 k, ψ2P ) πi k − (P−1) πi 2P N +PNe (4.7) k! 2PN k=0

This asymptotic expansion proves that the invariant hT2,2P i is dominated by P − 1 exponential terms, and that we have the Chern–Simons invariant for torus link T2,2P CS( A) = −

a2 4P

mod 1 for a = 1, 2, . . . , P − 1

As will be clarified in the next section, a tail part, i.e., the second term in eq. (4.7), corresponds to the Ohtsuki invariant. Eq. (3.15) indicates that a generating function of these invariants is written as ∞

(P−1) ez/2 − e−z/2 X L(−2 k, ψ2P ) 2k = z 1T2,2P (ez ) (2 k)!

(4.8)

k=0

where 1T2,2P ( A) is the Alexander polynomial for torus link T2,2P 1T2,2P ( A) =

5. WRT INVARIANT

FOR

A P − A−P A1/2 + A−1/2

(4.9)

BRIESKORN HOMOLOGY SPHERE

The Brieskorn homology sphere 6( #» p ) = 6( p1 , p2 , p3 ), where pi are pairwise coprime positive integers, is the intersection of the singular complex surface p

p

p3

z1 1 + z2 2 + z3

=0

englishQUANTUM INVARIANTS FOR LINKS AND 3-MANIFOLDS

13

in C3 with the unit five-sphere, |z 1 |2 + |z 2 |2 + |z 3 |2 = 1 [42]. This manifold has a rational surgery description as p1/q1

0

p3/q3

p2/q2

which means a link in Fig. 3. The fundamental group has the presentation E
D pk #» −qk π1 6( p ) = x 1 , x 2 , x 3 , h h center, x k = h , x 1 x 2 x 3 = 1

(5.1)

where qk is coprime integer to pk satisfying

3 X qk =1 P pk k=1

with

P = p 1 p2 p3 0

PSfrag replacements p1 /q1

p3 /q3

p2 /q2 Figure 3: Rational surgery description of the Brieskorn homology sphere 6( p1 , p2 , p3 )

When the 3-manifold M is constructed by the rational surgeries p j /q j on the j -th component of n-component link L, it was shown [24,54] that the SU(2) WRT invariant τ N (M) is given by τ N (M) =e

π i N −2 4 N

P

( p j ,q j ) n )−3 sign(L) j =1 8(U



N−1 X

k1 ,...,kn =1

Jk1,...,kn (L)

n Y

j =1

Here the surgery p j /q j is described by an S L(2; Z) matrix   pj rj ( p j ,q j ) U = qj sj

ρ(U ( p j ,q j ) )k j ,1 (5.2)

englishKAZUHIRO HIKAMI

14

and 8(U ) is the Rademacher 8-function defined by (see, e.g., Ref. 51)  p+s     − 12 s( p, q) for q 6 = 0 p r 8 = r q q s   for q = 0 s

(5.3)

where s(b, a) denotes the Dedekind sum (see, e.g., Ref. 52, and also Ref. 32) s(b, a) = sign(a) with ((x)) =

|a|−1 X  k=1

  x − bxc −

k   k b  · a a

1 2

(5.4)

if x 6 ∈ Z

0

if x ∈ Z

and bxc is the greatest integer not exceeding x. It is known that the Dedekind sum is rewritten as     a−1 k 1 X kb cot s(b, a) = π cot π 4a a a k=1

An n × n matrix L is a linking matrix

L j,k = lk( j, k) + p j /q j · δ j,k and sign(L) is a signature of L, i.e., the difference between the number of positive and negative eigenvalues of L. The polynomial Jk1 ,...,kn (L) is the colored Jone polynomial for link L with color k j for the j -th component link, and ρ(U ( p,q) ) is a representation ρ of P S L(2; Z); ρ(U ( p,q) )a,b sign(q) − π i 8(U ( p,q) ) 2Nπ iq sb2 = −i √ e 4 e 2 N |q|

X

e

πi 2N q

γ mod 2Nq γ =a mod 2N

pγ 2



e

πi Nq γ b

−e

− Nπqi γ b



(5.5)

for 1 ≤ a, b ≤ N − 1 [24]. It is remarked that the representation ρ corresponds to that we have constructed in eqs. (3.2)—(3.3); r   2 abπ sin ρ(S)a,b = N N a2

πi

ρ(T )a,b = e 2N πi− 4 δa,b

englishQUANTUM INVARIANTS FOR LINKS AND 3-MANIFOLDS

with



0 −1 S= 1 0 satisfying





1 1 T = 0 1

15



(5.6)

S 2 = (S T )3 = 1 In the case of the Brieskorn homology sphere, we need the colored Jones polynomial for a link in Fig. 3
Q3 1 j =1 sin k0 k j π/N · Jk0,k1 ,k2 ,k3 (L) = sin(π/N ) sin2 (k0 π/N ) where k0 is a color of an unknotted component whose linking number with other components is 1, and k j (for j = 1, 2, 3) denotes a color of a component of link L which is to be p j /q j -surgered. We can compute the SU(2) WRT invariant following a recipe of Ref. 54, and the final expression is as follows. Proposition 8 ( [37]). The WRT invariant for M = 6( #» p ) is given by   p) 1 2π i φ( #» 2π i e N ( 4 − 2 ) e /N − 1 τ N (M) Q3 2 P N−1 −2 eπ i/4 X − 1 n 2 πi j =1 sin n π/N e 2P N =√ sin (n π/N ) 2 P N n=0 N -n

Here the function φ( #» p ) is defined by
1 φ( #» p ) = 3 − + 12 s( p2 p3 , p1 ) + s( p1 p3 , p2 ) + s( p1 p2 , p3 ) P where s(b, a) is the Dedekind sum (5.4).

pj




(5.7)

(5.8)

Proof. We omit proof. We only stress that the Gauss reciprocity formula plays a crucial role; s X X N πi πi πi 2 2 e N Mn +2πikn = e 4 sign(N M) e− M N(n+k) (5.9) M n

mod N

where N , M ∈ Z with N k ∈ Z and N M being even.

n

mod M



Note that the Casson invariant λC (M) for M = 6( #» p ) is written in terms of the #» function φ( p ) as [11] ! 1 1 1 (5.10) − 24 λC (M) = φ( #» p) + P 1 − 2 − 2 − 2 p1 p2 p3

englishKAZUHIRO HIKAMI

16

Our purpose is to relate the WRT invariant τ N (M) with the Eichler integral of the modular form with half-integral weight. We define the odd periodic function #» ` χ2P (n) with modulus 2P by  3  X `j   #» −ε1 ε2 ε3 for n = P + P εj ` pj χ2P (n) = (5.11) j =1   0 others

#» Here ε j = ±1, and a triple ` ∈ Z3 satisfies 1 ≤ ` j ≤ p j − 1. The number of the independent functions is 1 D( #» p ) = ( p1 − 1)( p2 − 1)( p3 − 1) (5.12) 4 #»

#»

` ` As the periodic function χ2P (n) is odd, the function χ2P (n) can be written as a (a) sum of ψ2P (n), P X #» (P+P j ε j ` j/ p j ) 1 ` ε1 ε2 ε3 ψ2P (n) (5.13) χ2P (n) = − 2 ε1 ,ε2 ,ε3 =±1

We give some examples. • #» p = (2, 3, 5); we have D(2, 3, 5) = 2, and we can choose `E = (1, 1, 1), (1, 1, 2) which give (1,1,1)

χ60

(1)

(11)

(19)

(29)

(n) = −ψ60 (n) − ψ60 (n) − ψ60 (n) − ψ60 (n)

(1,1,2) (7) (13) (17) (23) χ60 (n) = −ψ60 (n) − ψ60 (n) − ψ60 (n) − ψ60 (n)

(5.14)

See that the subscript 60 is twice the Coxeter number of E 8 , and that a set of superscripts coincides with the exponents of E 8 . #» • #» p = (2, 3, 7); we have D(2, 3, 7) = 3, and choose ` = (1, 1, 1), (1, 1, 2), and (1, 1, 3). We see that (1,1,1)

(n) = ψ84 (n) − ψ84 (n) − ψ84 (n) + ψ84 (n)

(1,1,2)

(n) = −ψ84 (n) − ψ84 (n) − ψ84 (n) − ψ84 (n)

χ84 χ84

(1)

(5)

(13)

(19)

(29)

(41)

(23)

(37)

(1,1,3) (11) (17) (25) (31) χ84 (n) = −ψ84 (n) − ψ84 (n) − ψ84 (n) − ψ84 (n)

We define the (vector) modular forms #» #» 1X n 2/4P ` ` n χ 8 #» (τ ) = 2P (n) q p 2

(5.15)

n∈Z

(a)

which can also be written as a sum of the modular form 9 P (τ ) due to eq. (5.13).

englishQUANTUM INVARIANTS FOR LINKS AND 3-MANIFOLDS

17

#»

` (τ ) is a modular form with weight 3/2; namely Proposition 9. The function 8 #» p  3/2 X #» #» #» i 0 0 ` S `#» 8 #»p (τ ) = 8 `#»p (−1/τ ) (5.16) ` τ #» `0

#»

#»

#»

` ` ` 8 #» p (τ ) p (τ + 1) = T 8 #»

(5.17)

 3  2 X P ` j/ p j 1+ π i T = exp  2

(5.19)

#» where the sum of `0 runs over D( #» p ) distinct triples. A D( #» p ) × D( #» p ) matrix S and a diagonal matrix T are respectively given by ! r 3 0 P3 ` j +`0j P ` j `0k Y #»0 ` ` j 32 1+P+P +P j j =1 p j j 6 =k p j pk S `#» (−1) = sin P π (5.18) 2 ` P p j j =1 #» `



j =1

Proof. We easily obtain eq. (5.17). For a proof of eq. (5.16) we compute as follows; P X #» (P+P j ε j ` j/ p j ) 1 ` (τ ) ε1 ε2 ε3 9 P 8 #» p (τ ) = − 2 ε1 ,ε2 ,ε3 =±1     3/2 P−1 r X X X `j i 2 −1 1 = ε1 ε2 ε3 sin 1 + ε j  b π  9 (b) P (− /τ ) τ 2 P pj  3/2 i = τ  3/2 i = τ

ε j =±1

j

b=1

  P−1 r 3 Y `j −1 X 2 (b) 1+b 3 (−1) 2 b π 9 P (−1/τ ) sin 2 P pj b=1 j =1 r `0 P X X 2 1+P+P j ε j p j j (−1) (−2) P 0

×

` j ε j =±1

Y j

`j sin P pj

1+

X k

`0k εk pk

!

π

!

(P+P

9P

P

j

εj

`0j

pj

)

(−1/τ )

In the first equality, we have used eq. (5.13), and have applied eq. (3.2) in the second equality. We then replace a sum of b by that of `0j and ε j in the last equality. This can be done due to conditions that p j are coprime integers and
that we have a term sin (` j/ p j ) b π . Finally we obtain eq. (5.16).  We give examples of the S and T-matrices below.

englishKAZUHIRO HIKAMI

18

• #» p = (2, 3, 5);

2 S= √ 5



sin(π/5) sin(2 π/5) sin(2 π/5) − sin(π/5)   (1/60)πi e T= e(49/60)πi

• #» p = (2, 3, 7);





 π/7) 2 π/7) 3 π/7) sin( sin( sin( 2 sin(π/7)  S = − √ sin(2 π/7) − sin(3 π/7) 7 sin(3 π/7) sin(π/7) − sin(2 π/7)   (1/84)πi e  T= e(25/84)πi 47 e−( /84)πi

e (a) (τ ), we can set the Recalling the definition (3.5) of the Eichler integral 9 P #» ` e #» (τ ) as [38, 62] Eichler integral 8 p ∞ X

#»

` e #» 8 p (τ ) =

#»

` (n) q χ2P

n 2/4P

(5.20)

n=0

#»

` e #» Limiting values of the Eichler integrals 8 p (α) in α ∈ Q are as given in Prop. 2, and especially we have for N ∈ Z #»

` e #» 8 p (1/N ) = − #»

2PX N−1

#»

` χ2P (n) e(

n=0 #» ` e #» 8 p (N ) = −

n 2/2P N )πi

B1

N 1 #»  #» ` #» C p(`) T 2P



n  2PN

where T ` is defined in eq. (5.19), and we have set 2P

X #» #» ` C #»p ( ` ) = n χ2P (n)

(5.21)

n=1

Then we have the following nearly modularity as an analogue of eq. (2.6); Proposition 10 ( [38]). For N ∈ Z>0 , we have r #»   ∞ ` X #» L(−2 k, χ2P ) N X `#»0 `#»0 πi k ` e #» (−N ) + e #» (1/N ) ' − S #» 8 8 p i #» ` p k! 2PN `0

k=0

#» where the sum of `0 runs over D( #» p ) distinct triples.

(5.22)

englishQUANTUM INVARIANTS FOR LINKS AND 3-MANIFOLDS

19

As in the case of the torus link, we can prove that the WRT SU(2) invariant #» ` (1/N ). e #» for the Brieskorn homology sphere is related to the Eichler integrals 8 p #» Theorem 11 ( [17, 38]). The WRT invariant for M = 6( p ) is written in terms of limiting values of the Eichler integral;

• for the Poincaré homology sphere, #» p = (2, 3, 5);   1 πi 2π i 2π i e(1,1,1) (1/N ) e /N e /N − 1 τ N (M) = 1 + e− /60N · 8 #» p 2 P • for #» p s.t. 3j =1 1/ p j < 1;   2π i φ( #» 1 (1,1,1) 2π i/N ( 4p ) − 12 ) e #» N e e − 1 τ N (M) = 8 (1/N ) 2 p

(5.23)

(5.24)

Proof. We first prove eq. (5.24). We recall the formula for the Gauss sum; G(N ) = which gives G(N ) =

2N−1 X n=0

2N−1 X

n2

e− 2N πi =

1

e− 2N (k−n)

√

2 N e−

π i/4

(5.25)

2 πi

k=0

=e

1 2 − 2N n πi

2N−1 X

1

e− 2N k

2 πi+ 1 knπi N

(5.26)

k=0

These follow from the reciprocity formula (5.9). We further use a generating function of the periodic function
Q3 P/ p j ∞ − P/ p j X z − z j =1 (1,1,1) = χ2P (n) z n (5.27) z P − z −P n=0

We then have

e(1,1,1) (1/N ) = lim 8 p

t&0

∞ X

(1,1,1)

χ2P

n2

(n) e 2P N πi e−nt

n=0

  ∞ 2PX N−1 X 1 − 2P1 N k 2 πi+n PkN πi−t (1,1,1) = lim χ2P (n) e t&0 G(P N ) n=0 k=0  k  Q3 − Nkp πi N p j πi j −e 2PX N−1 j =1 e 1 − 2P1 N k 2 πi = e k k G(P N ) e N πi − e− N πi k=0 N-k

englishKAZUHIRO HIKAMI

20

In the first equality we have used eq. (3.9), and applied eq. (5.26) in the second equality. In the last equality we have used eq. (5.27) and the fact that a sum for N |k vanishes. Proof for eq. (5.23) can be done in the same manner when we notice that the generating function of the periodic function becomes ∞

X (1,1,1) 1 (z 6 − z −6 ) (z 10 − z −10 ) (z 15 − z −15 ) χ60 (n) z n = + z + z 30 − z −30 z

(5.28)

n=0

in place of eq. (5.27).



This result proves that the SU(2) WRT invariant for the Poincaré homology sphere 6(2, 3, 5) is written in terms of the Eichler integral of the modular form which is related to the Lie algebra E 8 pointed out in eq. (5.14). This result may imply the fact that the Poincaré homology sphere can also be constructed from link whose linking matrix has −2


−2


−2


−2

−2







−2


−2





−2 which is nothing but the Dynkin diagram for E 8 . Combining with the nearly modularity (5.22) of the Eichler integral, we have the following asymptotic expansion of the invariant; Corollary 12. For the Poincaré homology sphere M = 6(2, 3, 5), we have an exact asymptotic expansion in N → ∞ of the WRT invariant as e

121π i/60N



e

2π i/N



− 1 τ N (M) '

r

+e

 π   2 π  49 1 − 60 πiN − 60 πiN sin e + sin e 5 5  ∞ (1,1,1)  ) 1 X L(−2 k, χ60 πi k (5.29) + 2 k! 60 N

N 2 √ i 5

π i/60N



k=0

englishQUANTUM INVARIANTS FOR LINKS AND 3-MANIFOLDS

P In other cases M = 6( #» p ) s.t. j 1/ p j < 1, we have e

p) 1 2π i φ( #» N ( 4 −2)



e

2π i/N



− 1 τ N (M) '

r

N 1 X #» #» − 1 πiP N ` C #»p ( ` ) e 2 S1,1,1 i 4 P #» `

1 + 2

∞ X k=0

(1,1,1) L(−2 k, χ2P ) k!



21



πi 2PN

1+

k

P

`j j pj

2

(5.30)

#» where the sum of ` runs over D( #» p ) triples.

We note that the generating functions of the L-function which have appeared in tail part of the asymptotic expansions are given by • #» p = (2, 3, 5);

• #» p s.t.

P

∞

(1,1,1) ) 2n ch(9 z) ch(5 z) X L(−2 n, χ60 = z −2 ch(15 z) (2 n)! n=0

j

< 1; Q3 j =1 sh(P z/ p j )

1/ p j

4

sh(P z)

∞ (1,1,1) X L(−2 n, χ2P ) 2n = z (2 n)! n=0

See that these generating functions have already appeared in summand of the WRT invariant (5.7), and that it may play a role of inverse of the Alexander polynomial when we recall results (4.8) in the previous section. Eqs. (5.29) – (5.30) indicate that the asymptotic expansion of the WRT invariant τ N (M) in N → ∞ is dominated by exponential terms (see the first term in r.h.s. of those equations). The number of these non-vanishing exponential terms #» #» is that of triples ` satisfying C #»p ( ` ) 6 = 0, and by construction it is bounded by D( #» p ). #» #» Proposition 13 ( [17]). We have C #»p ( ` ) 6 = 0 iff triple ` ∈ Z3 is inside the tetrahedron depicted in Fig. 4 P ` Proof. As we have 0 < 3j =1 p jj < 3 by definition, it is rather straightforward to prove this statement once we take an adequate classification. As a result we have #» #» C #»p ( ` ) = 0 if ` is inside the tetrahedron, and #» C #»p ( ` ) = 4 P for other cases.



englishKAZUHIRO HIKAMI

22

`3 

(0, 0, p3 )





( p1 , 0, 0)

(0, p2 , 0) `2



`1 Figure 4: The number of the integral lattice points in the tetrahedron coincides with that of triples E 6= 0. `E such that C pE (`)

The number of lattice points inside the tetrahedron in Fig. 4 was essentially computed by Mordell [43], and it is written in terms of the Dedekind sum (5.4). #» #» ` e #» Theorem 14. The number γ ( #» p ) of triples ` s.t. 8 p (N ) 6 = 0 for N ∈ Z, i.e., #» C #»p ( ` ) 6 = 0, is ! 3 X P 1 1 1 1 1 #» γ( p) = s( P/ p j , p j ) + 1− 2 − 2 − 2 − + (5.31) 12 12 P 4 p1 p2 p3 j =1

We show examples. • the Poincaré homology sphere, #» p = (2, 3, 5); We have D(2, 3, 5) = 2, and the vector modular forms are spanned by #» #» ` = (1, 1, 1) and (1, 1, 2). We see that both triples fulfill C 2,3,5 ( ` ) 6 = 0, and we have γ (2, 3, 5) = 2 as seen from eq. (5.29). • #» p = (2, 3, 7); We have D(2, 3, 7) = 3, and 3-dimensional vector space is spanned by #» ` = (1, 1, 1), (1, 1, 2), and (1, 1, 3). Though, we see that 1/2 + 1/3 + 1/7 = #» 41/42 < 1, and that ` = (1, 1, 1) is outside the tetrahedron in Fig. 4. We indeed have γ (2, 3, 7) = 2. We see that this number of the lattice points inside the tetrahedron is proportional to the Casson invariant (5.10) for 6( #» p ).

englishQUANTUM INVARIANTS FOR LINKS AND 3-MANIFOLDS

23

Corollary 15. The Casson invariant λC (M) for M = 6( #» p ) is proportional to the number of the lattice points given in eq. (5.31); 1 p) λC (M) = − γ ( #» 2

(5.32)

As the Casson invariant is naïvely the signed sum of the number of inequiv#» alent irreducible SU(2) representation α of π1 (M), the triple ` which orig#» ` (τ ) should be related inally labels the half-integral weight modular form 8 #» p #» to the irreducible representation of the fundamental group. In fact, for ` #» s.t. C #»p ( `) 6 = 0 we have SU(2) representation of generators in eq. (5.1), e(1−`k/ pk )πi up to conjugation. Correspondingly the α(x k ) ∼ e−(1−`k/ pk )πi Chern–Simons invariant is interpreted as follows; #» #» Proposition 16. For ` s.t. C #»p ( ` ) 6 = 0, the Chern–Simons invariant is related to the T-matrix (5.19); #»

T ` = e2πi CS(A)

(5.33)

We can expect that the S-matrix (5.18) may also be related to topological invariants of M. We recall that the Ray–Singer torsion and the spectral flow of 6( #» p ) are respectively computed as [6, 8, 33]

#»
2 2 e( ` ) Iα = −3 − P 

p

p j −1 3 X 2 X cot + pj j =1

k=1

3 8 Y Tα = √ sin P j =1

k P π/ p

j

2




cot

k π/ p j

P ` j/ p




j

2





#» sin2 k e( ` ) π/ p j

(5.34)



(5.35) mod 8

where 3

X #» e( ` ) = P (1 − ` j/ p j ) j =1

#» #» Theorem 17 ( [17]). For ` s.t. C #»p ( ` ) 6 = 0, the S-matrix (5.18) gives both the torsion and the spectral flow of 6( #» p ) as follows; p √ ` ,` ,` 1 2 3 2 S1,1,1 (5.36) = Tα e−2πiIα /4

englishKAZUHIRO HIKAMI

24

Proof. The torsion part is trivial from absolute value of explicit form of the Smatrix. The phase factor is checked as follows;   p j −1 #»  3 #» 2
Y      X (e( ` )) πi kπ πi kPπ k e( ` ) π 2π iI  cot exp  e− α/4 = e 2 3+2 P cot sin2 2 pj p p p j j j j =1 k=1 ! 3  q e( #»  e( #» #» Y `)  `)  j πie( ` ) sign sin π sin π =e pj pj j =1

Here in the second equality we have used pPj ·q j = 1 (mod p j ), and an identity [24, 57],      nπ rnπ − i sign sin sin eπin p p          p−1 2 k n π  kqπ 2X kπ 2r n πi  cot + cot sin2 −2 + = exp − 2 p p p p p k=1

where we suppose n ∈ Z and q r = 1 (mod p). We further see that ! #» ! 3 3 P Y Y ` e( ` ) j sin P 2 π sin π = (−1)1+ j
j =1

and 3 Y

j =1

#» !! q j e( ` ) π =1 sign sin pj

which follows from   #» ! q1 e( ` ) `1 P 1+q1 ( p2 p3 +`2 p3 + p2 `3 ) sin π = (−1) sin π · q1 p1 p1 p1   `1 π 1+q1 ( p2 p3 +`2 p3 + p2 `3 )−`1 ( p2 q3 +q2 p3 ) = (−1) sin p1 Collecting these results, we obtain e−2πiIα /4 = (−1)

1+P+P

P3

`j j =1 p j

+

P

j
which proves a phase part of eq. (5.36).

3 Y

j =1

sign sin

P `j π p j2

!! 

Combining these results and recalling the definition (1.4) of the partition function Z k (M), we have the following results;

englishQUANTUM INVARIANTS FOR LINKS AND 3-MANIFOLDS

Corollary 18. The WRT invariant for M = 6( #» p ) in N → ∞ behaves as   X √ ` ,` ,` − P 1+P ` j/ p j 2 πiN 1 − 3 πi j 1 2 3 2 4 2 S1,1,1 e Z N−2 (M) ∼ e 2 #» #»

25

(5.37)

` s.t.C #» p ( ` )6 =0

which coincides with eq. (1.5). This supports observations in Refs. 9, 61. We see that exponentially divergent terms give invariants such as the Chern– Simons invariant, torsion, and the spectral flow. Tail part, which may be regarded as period of the modular form with half-integral weight, gives the Ohtsuki invariant [45, 46, 49, 50] when we replace 2πi/N by log q. Proposition 19. Let the function τ (M) for M = 6( #» p ) be defined by the formal ∞

series as follows.

• for the Poincaré sphere, M = 6(2, 3, 5), q

121 120

(q − 1) τ∞ (M) = q

• for M = 6( #» p ) s.t. q

φ( #» p) 1 4 −2

1/120

P3

j =1

∞

(1,1,1)

1 X L(−2 k, χ60 + 2 k!

)

k=0

1/ p j



log q 120

k

(5.38)

< 1, ∞

(1,1,1)

1 X L(−2 k, χ2P (q − 1) τ∞ (M) = 2 k!

)

k=0



log q 4P

k

Then the Ohtsuki invariant λn (M) is given by ∞ X τ∞ (M) = λn (M) · (q − 1)n

(5.39)

(5.40)

n=0

Explicitly we have λn (M) = Here we have defined

   3n ( #» p)

   3n (2, 3, 5) + (−1)n+1

for #» p s.t.

3 X j =1

1/ p j

<1

(5.41)

for #» p = (2, 3, 5)

  n+1 X 1 2 − φ( #» p) m 1 (m) #» 3n ( p ) = Sn+1 2 (n + 1)! 4 m=1 k m    X 1 m (1,1,1) L(−2 k, χ2P ) (5.42) × #» k P (2 − φ( p )) k=0

englishKAZUHIRO HIKAMI

26 (m)

where Sn

is the Stirling number of the first kind satisfying ∞ X (log q)m (q − 1)n = Sn(m) m! n! n=m

As was proved in Ref. 46, we have λ1 (M) = 6 λC (M) which originates from eq. (5.10). See also Refs. 37, 60 for studies on higher terms. 6. CONCLUDING REMARKS We have studied an exact asymptotic expansion of the colored Jones polynomial for torus link T2,2P , and the SU(2) WRT invariant for the Brieskorn homology sphere 6( p1 , p2 , p3 ). Based on the method proposed by Lawrence and Zagier [38], we have shown that these quantum invariants can be rewritten in terms of the Eichler integrals of modular forms with weight 3/2. Though we do not have a modular property for the partition function unlike Ref. 2 as a consequence, we can evaluate the asymptotic behavior based on the nearly modular property of the Eicher integrals. Summarized below are our results indicating the correspondence between the topological invariants for the Brieskorn homology sphere M = 6( #» p ) and the modular form; Topological Invariant

Modular Form

SU(2) WRT invariant Eichler integral in τ → 1/N for N ∈ Z #» #» ` s.t. C #»p ( ` ) 6 = 0 irreducible SU(2) rep. of π1 (M) Casson invariant # of non-vanishing Eichler integrals in τ → N Chern–Simons invariant T-matrix torsion, spectral flow S-matrix period Ohtsuki invariant Dominating terms of the SU(2) WRT invariant τ N (M) for the Seifert sphere M = 6( p1 , . . . , p M ) in N → ∞ can also be rewritten in terms of (differentials of) the Eichler integral of the modular form with half-integral weight. In this case, the weight is 1/2 for a case of even M, while it is 3/2 for odd M [19, 20]. We also find the correspondence between the quantum invariants and the modular forms in the case of torus knot Ts,t [22] (see also Refs. 13, 14, 18, 23, 30), and the rational

englishQUANTUM INVARIANTS FOR LINKS AND 3-MANIFOLDS

27

homology sphere [21]. Studies on exact asymptotic expansion of the quantum invariants for other knots/links and manifolds such as hyperbolic manifolds would be absorbing.

REFERENCES [1] M. F. Atiyah, The Geometry and Physics of Knots, Cambridge Univ. Press, Cambridge, 1990. [2] A. Cappelli, C. Itzykson, and J. B. Zuber, The A-D-E classification of minimal and A 1(1) conformal invariant theories, Commun. Math. Phys. 113, 1–26 (1987). [3] L. D. Faddeev, Discrete Heisenberg–Weyl group and modular group, Lett. Math. Phys. 34, 249–254 (1995). [4] ———, Modular double of quantum group, in G. Dito and D. Sternheimer, eds., Conference Mosh Flato 1999 — Vol. I. Quantization, Deformations, and Symmetries, Math. Phys. Stud. 21, pp. 149–156, Kluwer, Dordrecht, 2000. [5] L. D. Faddeev and R. M. Kashaev, Quantum dilogarithm, Mod. Phys. Lett. A 9, 427–434 (1994). [6] R. Fintushel and R. Stern, Instanton homology of Seifert fibered homology three spheres, Proc. Lond. Math. Soc. 61, 109–137 (1990). [7] P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory, Springer, New York, 1997. [8] D. S. Freed, Reidemeister torsion, spectral sequences, and Brieskorn spheres, J. Reine Angew. Math. 429, 75–89 (1992). [9] D. S. Freed and R. E. Gompf, Computer calculation of Witten’s 3-manifold invariant, Commun. Math. Phys. 141, 79–117 (1991). [10] ———, Computer tests of Witten’s Chern–Simons theory against the theory of threemanifolds, Phys. Rev. Lett. 66, 1255–1258 (1991). [11] S. Fukuhara, Y. Matsumoto, and K. Sakamoto, Casson’s invariant of Seifert homology 3spheres, Math. Ann. 287, 275–285 (1990). [12] S. Garoufalidis, On the characteristic and deformation varieties of a knot, Geom. Topol. Monographs 7, 291–309 (2004). [13] K. Hikami, Volume conjecture and asymptotic expansion of q-series, Exp. Math. 12, 319–337 (2003). [14] ———, q-series and L-functions related to half-derivatives of the Andrews–Gordon identity, Ramanujan J. (2004), to appear. [15] ———, Quantum invariant for torus link and modular forms, Commun. Math. Phys. 246, 403–426 (2004). [16] ———, Difference equation of the colored Jones polynomial for the torus knot, Int. J. Math. 15, 959–965 (2004). [17] ———, On the quantum invariant for the Brieskorn homology spheres, Int. J. Math. (2005), to appear (math-ph/0405028). [18] ———, Quantum invariant for torus knot and modular form, in T. Ibukiyama, K. Saito, E. Bannai, and M. Miyamoto, eds., The 3rd Spring Conference on “Modular Forms and Related Topics”, pp. 26–35, Ryusi-do, 2004. [19] ———, Quantum invariant, modular form, and lattice points, IMRN 2005, 121–154 (2005), math-ph/0409016. [20] ———, in preparation. [21] ———, in preparation.

28

englishKAZUHIRO HIKAMI

[22] K. Hikami and A. N. Kirillov, Torus knot and minimal model, Phys. Lett. B 575, 343–348 (2003). [23] ———, Hypergeometric generating function of L-function, Slater’s identities, and quantum knot invariant, math-ph/0406042 (2004). [24] L. C. Jeffrey, Chern–Simons–Witten invariants of lens spaces and torus bundles, and the semiclassical approximation, Commun. Math. Phys. 147, 563–604 (1992). [25] V. F. R. Jones, A polynomial invariant for knots via von Neumann algebras, Bull. Amer. Math. Soc. 12, 103–111 (1985). [26] V. G. Kac, Infinite Dimensional Lie Algebras, Cambridge Univ. Press, Cambridge, 1990, 3rd ed. [27] R. M. Kashaev, A link invariant from quantum dilogarithm, Mod. Phys. Lett. A 10, 1409– 1418 (1995). [28] ———, The hyperbolic volume of knots from quantum dilogarithm, Lett. Math. Phys. 39, 269–275 (1997). [29] ———, Quantum hyperbolic invariants of knots, in A. Bobenko and R. Seiler, eds., Discrete Integrable Geometry and Physics, pp. 343–360, Oxford Univ. Press, Oxford, 1999. [30] R. M. Kashaev and O. Tirkkonen, Proof of the volume conjecture for torus knots, J. Math. Sci. 115, 2033–2036 (2003). [31] L. H. Kauffman and S. L. Lins, Temperley–Lieb Recoupling Theory and Invariants of 3Manifolds, Ann. Math. Stud. 134, Princeton Univ. Press, Princeton, 1994. [32] R. Kirby and P. Melvin, Dedekind sums, µ-invariants and the signature cocycle, Math. Ann. 299, 231–267 (1994). [33] P. A. Kirk and E. P. Klassen, Representation space of Seifert fibered homology spheres, Topology 30, 77–95 (1991). [34] S. Lang, Introduction to Modular Forms, Grund. math. Wiss. 222, Springer, Berlin, 1976. [35] R. Lawrence, Asymptotic expansions of Witten–Reshetikhin–Turaev invariants for some simple 3-manifolds, J. Math. Phys. 36, 6106–6129 (1995). [36] ———, Witten–Reshetikhin–Turaev invariants of 3-manifolds as holomorphic functions, in J. E. Andersen, J. Dupont, H. Pedersen, and A. Swann, eds., Geometry and Physics, Lect. Notes Pure Appl. Math. 184, pp. 363–377, Dekker, New York, 1996. [37] R. Lawrence and L. Rozansky, Witten–Reshetikhin–Turaev invariants of Seifert manifolds, Commun. Math. Phys. 205, 287–314 (1999). [38] R. Lawrence and D. Zagier, Modular forms and quantum invariants of 3-manifolds, Asian J. Math. 3, 93–107 (1999). [39] T. T. Q. Le, The colored Jones polynomial and the A-polynomial of two-bridge knots, math.GT/0407521 (2004). [40] R. Lickorish, An Introduction to Knot Theory, Graduate Texts in Mathematics 175, Springer, New York, 1997. [41] P. M. Melvin and H. R. Morton, The coloured Jones function, Commun. Math. Phys. 169, 501–520 (1995). [42] J. Milnor, On the 3-dimensional Brieskorn manifolds M( p, q, r ), in L. P. Neuwirth, ed., Knots, Groups, and 3-Manifolds, pp. 175–225, Princeton Univ. Press, 1975, papers Dedicated to the Memory of R. H. Fox. [43] L. J. Mordell, Lattice points in a tetrahedron and generalized Dedekind sums, J. Indian Math. Soc. (NS) 15, 41–46 (1951). [44] H. R. Morton, The coloured Jones function and Alexander polynomial for torus knots, Proc. Cambridge Philos. Soc. 117, 129–135 (1995). [45] H. Murakami, Quantum SU(2)-invariants dominate Casson’s SU(2)-invariant, Math. Proc. Camb. Phil. Soc. 115, 253–281 (1993).

englishQUANTUM INVARIANTS FOR LINKS AND 3-MANIFOLDS

29

[46] ———, Quantum SO(3) invariants dominate the SU(2) invariant of Casson and Walker, Math. Proc. Cambridge Philos. Soc. 117, 237–249 (1995). [47] H. Murakami and J. Murakami, The colored Jones polynomials and the simplicial volume of a knot, Acta Math. 186, 85–104 (2001). [48] H. Murakami, J. Murakami, M. Okamoto, T. Takata, and Y. Yokota, Kashaev’s conjecture and the Chern–Simons invariants of knots and links, Exp. Math. 11, 427–435 (2002). [49] T. Ohtsuki, A polynomial invariant of integral homology 3-spheres, Math. Proc. Camb. Phil. Soc. 117, 83–112 (1995). [50] ———, A polynomial invariant of rational homology 3 spheres, Invent. Math. 123, 241–257 (1996). [51] H. Rademacher, Topics in Analytic Number Theory, Grund. Math. Wiss. 169, Springer, New York, 1973. [52] H. Rademacher and E. Grosswald, Dedekind Sums, Carus Mathematical Monographs 16, Mathematical Association of America, Washington DC, 1972. [53] N. Yu. Reshetikhin and V. G. Turaev, Ribbon graphs and their invariants derived from quantum groups, Commun. Math. Phys. 127, 1–26 (1990). [54] ———, Invariants of 3-manifolds via link polynomials and quantum groups, Invent. Math. 103, 547–597 (1991). [55] M. Rosso and V. Jones, On the invariants of torus knots derived from quantum groups, J. Knot Theory Ramifications 2, 97–112 (1993). [56] L. Rozansky, A large k asymptotics of Witten’s invariant of Seifert manifolds, in D. N. Yetter, ed., Proceedings of the Conference on Quantum Topology, pp. 307–354, World Scientific, Singapore, 1994. [57] ———, A large k asymptotics of Witten’s invariant of Seifert manifolds, Commun. Math. Phys. 171, 279–322 (1995). [58] ———, A contribution of the trivial connection to Jones polynomial and Witten’s invariant of 3d manifolds I, Commun. Math. Phys. 175, 275–296 (1996). [59] ———, A contribution of the trivial connection to Jones polynomial and Witten’s invariant of 3d manifolds II, Commun. Math. Phys. 175, 297–318 (1996). [60] C. Sato, Casson–Walker invariant of Seifert fibered rational homology spheres as quantum SO(3)-invariant, J. Knot Theory Ramifications 6, 79–93 (1997). [61] E. Witten, Quantum field theory and Jones’ polynomial, Commun. Math. Phys. 121, 351–399 (1989). [62] D. Zagier, Vassiliev invariants and a strange identity related to the Dedekind eta-function, Topology 40, 945–960 (2001). Department of Physics, Graduate School of Science, University of Tokyo, Hongo 7–3–1, Bunkyo, Tokyo 113–0033, Japan. URL: http://gogh.phys.s.u-tokyo.ac.jp/~hikami/ E-mail address: [email protected]