Volume Conjecture and Asymptotic Expansion of Q-Series Kazuhiro Hikami

Volume Conjecture and Asymptotic Expansion of q-Series Kazuhiro Hikami

CONTENTS We consider the “volume conjecture,” which states that an 1. Introduction asymptotic limit of Kashaev’s invariant (or, the colored Jones type 2. Kashaev’s Invariant and Volume Conjecture invariant) of knot K gives the hyperbolic volume of the com- 3. Torus Knots and q-Series plement of knot K. In the ﬁrst part, we analytically study an 4. Hyperbolic Knots asymptotic behavior of the invariant for the torus knot, and pro- Acknowledgments pose identities concerning an asymptotic expansion of q-series q N References which reduces to the invariant with being the -th root of unity. This is a generalization of an identity recently studied by Zagier. In the second part, we show that “volume conjecture” is numerically supported for hyperbolic knots and links (knots up to 6-crossing, Whitehead link, and Borromean rings).

1. INTRODUCTION In [Kashaev 95, Kashaev 97], Kashaev deﬁned an invariant KN for knot K using a quantum dilogarithm function at the N-th root of unity, and proposed the stimulating conjecture that for a hyperbolic knot K an asymptotic limit N →∞of the invariant KN gives a hyperbolic volume of a knot complement, 2 π 3 lim log KN = v3 · S \K , (2–8) N→∞ N

where v3 is the hyperbolic volume of the regular ideal tetrahedron, and · denotes the Gromov norm. It was later proved [Murakami and Murakami 01] that Kashaev’s invariant coincides with a specific value of the colored Jones polynomial. In several attempts since then, a geometrical aspect to relate Kashaev’s R-matrix with an ideal octahedron in the three-dimensional hyperbolic space has been clarified (see, e.g., [Thurston 99, Yokota 00, Hikami 01]). Furthermore, a relationship with the Chern–Simons invariant was pointed out [Murakami et al. 02]. In this paper, we are interested in an explicit form 2000 AMS Subject Classification: Primary 52M27, 11B65; Secondary 57M50, 11Mxx of Kashaev’s invariant for the knot K. In general, this invariant can be regarded as a reduction of certain q- Keywords: Jones polynomial, Rogers-Ramanujan identity, hyperbolic volume series. In [Zagier 01], Zagier derived a strange identity

c A K Peters, Ltd. 1058-6458/2003 $ 0.50 per page Experimental Mathematics 12:3, page 319 320 Experimental Mathematics, Vol. 12 (2003), No. 3 (3,2) ∞ for a q-series, F (q)= n=0(q)n, which was origi- 2. KASHAEV’S INVARIANT nally introduced in [Stoimenow 98] as an upper bound AND VOLUME CONJECTURE of the number of linearly independent Vassiliev invari- The quantum invariant of knot K can be constructed once F (3,2) −t half- ants. He showed that (e ) is related to the we have the enhanced Yang–Baxter operator [Turaev 88], differential η of the Dedekind -function. From our view- (R, µ, α, β), satisfying point, F (3,2)(q) with q → e2πi/N is nothing but Kashaev’s (3,2) 2πi/N invariant for the trefoil, 31N = F (e ). This (R ⊗ 1)(1 ⊗ R)(R ⊗ 1) = (1 ⊗ R)(R ⊗ 1)(1 ⊗ R), (2–1) motivates us to study an asymptotic expansion of the q- µ ⊗ µ R R µ ⊗ µ , series which, when q is the N-th root of unity, reduces = (2–2) to Kashaev’s invariant for the torus knot. We introduce ±1 ±1 Tr2 R (1 ⊗ µ) = α β. (2–3) the q-series F (2m+1,2)(q) as a generalization of Zagier’s q-series and prove an identity, The operators R±1 and µ±1 are usually depicted as follows; F (2m+1,2)(e2πi/N ) 2 2 3 πi − πi (2m−1) j j √ N 2 e 4 N 4(2m+1) i i 2 m +1 m−1 ij −1 ij 2 R R j 2 j +1 −Nπi (2j+1) k = k = × − m − j π 4(2m+1) ( 1) ( )sin m e × × j=0 2 +1 k k ∞ 2 (2m+1,2) n − πi (2m−1) Tn π N 4(2m+1) , +e n m N n=0 ! 4(2 +1) i (3–15) / k k −1 k and then propose a conjecture, µ = µ = k / F (2m+1,2)(e−t) ∞ 2 (2m+1,2) n (2m−1) t Tn t 8(2m+1) . =e n 3 m (3–24) K ξ n=0 ! 2 (2 +1) When the knot is given as a closure of a braid with n strands, the invariant τ1(K) is computed as for the (2m+1,2) (2m+1,2) K Here F (q), respectively T -number Tn ,are (1,1)-tangle of knot as defined in Equations (3–22), respectively (3–10). τ K α−w(ξ) β−n b ξ ⊗ µ⊗(n−1) . In Section 2, we review the volume conjecture and 1( )= Tr2,...,n R( )(1 ) explain how to construct Kashaev’s invariant from the (2–4) b ξ enhanced Yang–Baxter operator. In Section 3, we study Here we have associated the homomorphism R( )byre- σ±1 ξ R±1 w ξ analytically in detail Kashaev’s invariant for the torus placing the braid group i in with ,and ( ) knot. We consider an asymptotic expansion of the invari- denotes a writhe (a sum of exponents). ant following Kashaev-Tirkkonen [Kashaev and Tirkko- Kashaev’s invariant is originally defined by use of nen 00], and derive an asymptotic formula for q-series the quantum dilogarithm function with a deformation 2πi/N N with q → e . In Section 4, we study numerically an parameter being the -th root of unity [Fadeev and asymptotic behavior of invariants for hyperbolic knots Kashaev 94], ω π /N . and links. We use PARI/GP [PARI 00], and show that =exp(2 i ) (2–5) there is a universal logarithmic correction to invariants. The invariant is then defined as follows: We use the q- We then propose a conjecture as an extension of Equa- product, tion (2–8), n i (ω)n = (1 − ω ), 3 N 3 0 log KN ∼ v3 · S \K · + #(K) · log N + O(N ), i=1 2 π 2 (4–1) n ∗ −i where #(K) is the number of prime factors of a knot (ω)n = (1 − ω ). considered as a connected-sum of prime knots. i=1 Hikami: Volume Conjecture and Asymptotic Expansion of q-Series 321

Theorem 2.1. [Kashaev 95, Kashaev 97] (See also Conjecture 2.3. [Kashaev 97, Murakami and Murakami [Murakami and Murakami 01].) Kashaev’s invariant 01] The asymptotic behavior of Kashaev’s invariant gives KN for the knot K is deﬁned by Equation (2–4) with the hyperbolic volume of the knot complement of knot K; R µ the following and matrices; π 2 K v · S3 \K, 1−(k−j+1)(−i) lim log N = 3 (2–8) ij Nω ij N→∞ N Rk = ∗ ∗ · θ , (ω)[−k−1] (ω) (ω)[i−j] (ω) k [j−] [k−i] where v3 is the hyperbolic volume of the regular ideal (2–6a) tetrahedron, and · denotes the Gromov norm. −1+(−i−1)(k−j) −1 ij Nω ij (R )k = ∗ ∗ · θ , A mathematically rigorous proof of this conjecture has (ω) (ω)[j−] (ω) (ω)[k−i] k [−k−1] [i−j] not been established yet (only thea case of the ﬁgure- (2–6b) eight knot was proved (see, e.g., [Murakami 00]). How- k 1 ever, several geometrical studies have been done; the µ = −δk,+1 ω 2 . (2–6c) relationship between Kashaev’s R-matrix and the ideal Here [x] ∈{0, 1,...,N − 1} modulo N,and hyperbolic octahedron has been established [Thurston  99, Yokota 00, Hikami 01], and it was found that the i ≤ k<≤ j, saddle point equation of the invariant coincides with the  ij j ≤ i ≤ k<, hyperbolicity consistency condition. θ =1, if and only if k  ≤ j ≤ i ≤ k (with

Theorem 2.2. [Murakami and Murakami 01] Kashaev’s where CS denotes the Chern–Simons invariant, invariant KN coincides with the colored Jones polyno- 2 mial at the N-th root of unity, whose R-matrix is given CS(M)=2π cs(M), by 1 2 csM(A)= Tr A ∧ dA + A ∧ A ∧ A . min(N−1−i,j) 8 π2 3 ij i+j+n M Rk = δ,i+n δk,j−n (−1) n=0 ∗ (ω)i+n (ω)j ij+ 1 (i+j−n) × ω 2 , ∗ ∗ 3. TORUS KNOTS AND q-SERIES (ω)i (ω)j−n (ω)n (2–7a) 3.1 Invariant of the Torus Knot m, p min(N−1−j,i) We consider the ( )-torus knot, where we suppose that R−1 ij δ δ − i+j+n m and p are coprime integers. The knot is expressed in k = ,i−n k,j+n ( 1) n=0 terms of generators of Artin’s braid group as ∗ (ω)i (ω)j+n −ij− 1 (i+j−n) × ω 2 , p ∗ ξ = σ1 σ2 ···σm−1 . (ω)i−n (ω)j (ω)n (2–7b) Hereafter, we denote it as Trs(m, p). For (m, p)=(3, 2) k k+ 1 and (5, 2) case, they are called the trefoil knot and the µ = −δk, ω 2 . (2–7c) Solomon’s Seal knot, respectively (see Figure 1). Using results from quantum groups, the explicit form In this article, we focus on the following stimulating of the colored Jones polynomial of the torus knot is ob- conjecture. tainable. 322 Experimental Mathematics, Vol. 12 (2003), No. 3

We rewrite the r.h.s. using the Gauss integral formula 2 √ 2 z πh hw z − wz , e = d exp h +2 C where a path C is to be chosen by the convergence condition. We apply the Gaussian integral formula to Equa- tion (3–1) and get FIGURE 1. Trefoil Knot and Solomon’s Seal Knot. h ( m + p ) Nh JK(h; N) 2e4 p m sh 2 JO(h; N) [Morton 95, Rosso and Jones 93] The (N−1)/2 Proposition 3.1. 2 hmp r+ m+p colored Jones polynomial JK(h; N) for K =Trs(m, p) is = e 2mp given by =±1 r=−(N−1)/2 (N−1)/2 2 Nh J h N 1 − z +z(2r+ 1 + ) K( ; ) = √ dz e hmp p m 2sh πhmp JO h N 2 ( ; ) =±1 r=−(N−1)/2 C (N−1)/2 z 2 Nz hmpr2+hr(m+p)+ 1 h 2 − z + z sh( )sh(m ) = e 2 , (3–1) = √ dz e hmp p . πhmp sh(z) =±1 r=−(N−1)/2 C where a parameter q issettobeq =exp(h). Unknot is Summing the integrand with one replacing z →−z,we denoted by O, and we have have z z 2 Nz sh(Nh/2) 2 − z sh( )sh(m )sh(p ) JO(h; N)= . = √ dz e hmp . sh(h/2) πhmp sh z C

Nz Nz − −Nz / By use of the relationship between the colored Jones Decomposing sh( )into(e e ) 2 and using an z →−z polynomial and Kashaev’s invariant (Theorem 2.2), we invariance under , we see that z z can give an asymptotic expansion of the invariant of the 2 2 − z +Nz sh( m )sh(p ) torus knot. = √ dz e hmp πhmp sh z C [Kashaev and Tirkkonen 00] For the 2 Proposition 3.2. mp mp(Nz− z ) 2sh(mz)sh(pz) K m, p m p = dz e h . torus knot =Trs( ) with and being coprime, πh sh(mpz) Kashaev’s invariant is represented by the following inte- C gral: To obtain Kashaev’s invariant Trs(m, p)N defined in Equation (3–3), we differentiate the above integral with m, p Trs( ) N respect to h, and we obtain Equation (3–2). 3/2 mpN πi(N+ 1 )− πi ( m + p )− πi = e N 2N p m 4 2 Proposition 3.3. An asymptotic expansion of Kashaev’s invariant for K =Trs(m, p) is given by mpNπ(z+ i z2) 2 sh(mπz)sh(pπz) × dz e 2 z . (3–2) sh(mpπz) C Trs(m, p)N 3/2 mpN πiN+ πi 1− 1 ( p + m ) − πi e N 2 m p 4 Proof: We follow [Kashaev and Tirkkonen 00]. As 2 Kashaev’s invariant is defined for the (1,1)-tangle of × Res(m, p) knots due to Theorem 2.2, we have for K =Trs(m, p) (m+1)(p+1) πiN(1+ 1 mp)+ πi 1− 1 ( p + m ) +(−1) e 2 N ( 2 m p ) that ∞ T (m,p) π n × n . πi(N+ 1 ) JK(h; N) (3–4) m, p N . n mpN Trs( ) N =e lim (3–3) n=0 ! 2 i h→2πi/N JO(h; N) Hikami: Volume Conjecture and Asymptotic Expansion of q-Series 323

Here we have set where χ2mp(n) is a periodic function modulo 2 mp:

n mod 2 mp χ2mp(n) m, p Res( ) mp− m − p 1 mp−1 nπ nπ mp− m + p -1 2i n+1 2 (3–9) = (−1) n sh i sh i mp+ m − p -1 (mp)3 p m n=1 mp+ m + p 1 2 Nπi(n− n ) × e 2mp , (3–5) others 0

and the T -series is given by We apply the Mellin transformation to Equa- 1 ∞ −nw tions (3–6) and (3–8), χ2mp(n)e ∞ (m,p) 2 n=0 sh(mw)sh(pw) Tn ∞ T (m,p) = (−1)n w2n+1. (3–6) n (−1)n w2n+1. The left-hand side is inte- sh(mpw) (2 n + 1)! n=0 (2n+1)! n=0 grated as ∞ ∞ 1 s−1 −nw χ2mp(n) w e dw Proof: We use an integral representation (3–2) of the 2 n=0 0 C C invariant. When we shift the path to +i,weget s ∞ Γ( ) χ n 1 = 2mp( ) ns 3/2 2 n=0 mpN πi(N+ 1 )− πi ( m + p )− πi Trs(m, p)N = e N 2N p m 4 Γ(s) 2 = L(s, χ2mp), 2 × Res(m, p) while the right-hand side is ∞N−1 (m,p) i 2 mπz pπz Tn mpNπ(z+ z ) 2 sh( )sh( ) − n w2n+1 + dz e 2 z . n ( 1) sh(mpπz) 0 n=0 (2 + 1)! C +i + O(w2N+1) ws−1 dw m, p Here, the ﬁrst term, Res( ), comes from residues of N−1 (m,p) n z π n , ,...,mp− Tn n 1 the integral at = mp ifor =12 1, and − R s , = n ( 1) n s + 2N+1( ) it is computed as Equation (3–5). In the second term, n=0 (2 + 1)! 2 + +1 we introduce z = w + i, and using a fact that the even with RN (s) holomorphic in (s) > −N. Comparing the functions only survive in the integrand, we get residues at s = −2 n − 1, we ﬁnd that the T -numbers (m,p) Tn can be given in terms of the associated L-series as Trs(m, p)N (m,p) 1 n+1 3/2 T − L − n − ,χ2mp mpN πi(N+ 1 )− πi ( m + p )− πi n = ( 1) ( 2 1 ) (3–10) = e N 2N p m 4 2 2 mp 2n+1 1 − n (2 ) mp+m+p 1 mpNπi = ( 1) × m, p − 2 2 2 n +2 Res( )+2i( 1) e 2mp a 1 impNπw2 sh(mπw)sh(pπw) × χ a B , × dw e 2 w . (3–7) 2mp( ) 2n+2 mp sh(mpπw) a=1 2 C where Bn(x) is the Bernoulli polynomial. It is noted that Substituting the expansion (3–6) into an integrand, we the T -number with (m, p)=(3, 2) is called the Glaisher recover Equation (3–4). T -number [Sloane 02]. A table of T -numbers is given in Table 1. Remark 3.4. The T -numbers can be written in terms We now give an explicit form of the invariant for the of the L-series. The left-hand side of Equation (3–6) is (2 m +1, 2)-torus knot (for m ≥ 1) by use of Kashaev’s expanded as R-matrix in Theorem 2.1.

∞ mw pw m , sh( )sh( ) 1 χ n −nw, Lemma 3.5. Kashaev’s invariant for the (2 +1 2)-torus mpw = 2mp( )e (3–8) sh( ) 2 n=0 knot is given explicitly as follows: 324 Experimental Mathematics, Vol. 12 (2003), No. 3

n 01 2 3 4 5 (3,2) Tn 1 23 1681 257543 67637281 27138236663 (5,2) Tn 1 71 14641 6242711 4555133281 5076970085351 (7,2) Tn 1 143 58081 48571823 69471000001 151763444497103 (9,2) Tn 1 239 160801 222359759 525750911041 1898604115708079 (11,2) Tn 1 359 361201 746248439 2635820840161 14219082731542919 (13,2) Tn 1 503 707281 2041111463 10069440665761 75868751534107223 (15,2) Tn 1 671 1256641 4828434911 31713479172481 318124890738776351 (17,2) Tn 1 863 2076481 10248374303 86458934113921 1113984641517368543 (19,2) Tn 1 1079 3243601 19997487719 210737173733281 3391720107333707159 (21,2) Tn 1 1319 4844401 36486145079 469706038871521 9234991712596896839

TABLE 1. T -numbers.

• Trefoil 31 (m =1): Proof: This is a tedious but straightforward computation. The following identities are useful [Yokota 00, Mu- N−1 rakami and Murakami 01]: , ω , Trs(3 2) N = ( )a (3–11a) ∗ a=0 (ω)[i−1] (ω)[−i] = N, (3–12) ω−(m−+1)k ∗ • m ω [m−k] ω Solomon’s Seal Knot 51 ( =2): k∈[,m] ( ) ( )[k−] − [m−] ω([m−]+1)([m−]−2m)/2, N−1 =( 1) (3–13) , ω−ab ω , Trs(5 2) N = ( )a+b (3–11b) ω−k(i−j) a,b=0 ∗ = δi,j . (3–14) 0≤a+b≤N−1 ω [i−k] ω k∈[i,j] ( ) ( )[k−j]

• (2 m +1, 2)-torus knot (m>2): Recalling a result of Proposition 3.3, we obtain an asymptotic expansion for the above set of the ω-series. Trs(2 m +1, 2)N 2m−2 a = N (−1) j=1 j Corollary 3.6. We have an asymptotic expansion for the

1≤a2m−2≤···≤a1≤N−1 ω-series with limit N →∞: 1 2m−2 a (a −1) ω 2 j=1 j j • Trefoil (m =1): × 2m−3 (ω)a −a N−1 j=1 j j+1 3 π i π i N π i ω N 2 − − ( )a exp 2m−2 c 4 12 12 N = (−1) j=3 j a=0 ∞ (3,2) n 0≤c2m−2≤···≤c2≤N−c1−1≤N−2 πi Tn π − 12N +e . 1 2m−2 ω −c c + c (c +1) ( )c1+c2 n! 12 i N × ω 1 2 2 j=3 j j n=0 2m−3 ω j=2 ( )cj −cj+1 (3–15a)

N−1 ω • Solomon’s Seal Knot (m =2): ( )a1+a2+···+a2m−2 = 2m−3 ω a −ab a1,a2,...,a2m−2=0 j=2 ( ) j ω (ω)a+b 0≤a +a +···+a ≤N−1 1 2 2m−2 0≤a+b≤N−1 2m−2 jaj × − j=3 2 3 π i− 9iπ − Nπi − 9Nπi ( 1) √ N 2 4 20N a 20 − b 20 e 2 e e −a a + 2m−2 j −1−a a + 1 2m−2(a +a +···+a )2 × ω 1 2 j=3 ( 2 1) j 2 j=3 j j+1 2m−2 . 5 ∞ (5,2) n − 9iπ Tn π (3–11c) 20N , +e n N (3–15b) n=0 ! 20 i Hikami: Volume Conjecture and Asymptotic Expansion of q-Series 325

where Equations (3–15b) and (3–15c) indicate a decomposition √ into several terms labelled by ﬂat connections, and we π 5 1 1 a =sin = 1 − √ , have 5 2 2 5 √ m , CS Trs(2 +1 2) 2 π 5 1 1 b =sin = 1+√ . (2 j +1)2 5 2 2 5 = − π2 j =0, 1,...,m− 1 . (3–19) 2(2m +1) • (2 m +1, 2)-torus knot (m>2): This decomposition may be explained as follows. The N−1 fundamental group of S3 \ Trs(m, p) has a presentation ω 2m−2 ( )a1+a2+···+a2m−2 ja (−1) j=3 j 2m−3 ω a ,a ,...,a =0 j=2 ( )aj 3 m p 1 2 2m−2 π1 S \ Trs(m, p) = x, y | x = y . (3–20) 0≤a1+a2+···+a2m−2≤N−1 −a a + 2m−2 j −1−a a + 1 2m−2(a +a +···+a )2 1 2 j=3 ( 2 1) j 2 j=3 j j+1 2m−2 × ω As was discussed in [Klassen 91], there are (m − 1) (p − m−1 3 (2m−1)2 1)/2 disjoint irreducible representations, ρ : π1(S \ 2 3 πi − πi j √ N 2 e 4 N 4(2m+1) (−1) (m − j) m Trs(m, p)) → SU(2), up to conjugacy. This corresponds 2 +1 j=0 to a decomposition in Equation (3–5). Especially, in 2 2 j +1 −Nπi (2j+1) m , m × sin π e 4(2m+1) the case of Trs(2 +1 2), we have representations 2 m +1 in which the eigenvalues of ρ(y), respectively ρ(x), are ∞ 2 j+1 2 (2m+1,2) n − πi (2m−1) Tn π given by exp(±π i/2), respecively exp(± 2 m+1 π i) with N 4(2m+1) . +e n m N j =0, 1,...,m− 1. The Chern–Simons invariant may be n=0 ! 4(2 +1) i (3–15c) computed by considering a path of representation along a line of [Kirk and Klassen 93].

Remark 3.7. Equation (3–15a) was conjectured in [Za- For our later discussion, we comment on asymp- gier 01] (according to this reference, it was due to Kont- totics of the invariant which simply follows from Equa- sevich), and it was discussed that a power exponent 3/2 tion (3–4). of N 3/2 which appeared on the right-hand side is related K m, p with a weight of the “nearly modular function.” Namely, Corollary 3.9. For the torus knot =Trs( ), we have N →∞ we define in the limit that 2 (2m+1) (2m−1) πiα (2m+1,2) 2πiα 3 Φ (α)=e4(2m+1) F (e ), (3–16) log Trs(m, p)N ∼ log N. (3–21) 2 where F (2m+1,2)(q) will be defined in Equation (3–22). Then, from Equation (3–15), we have the modular trans- 3.2 q-Series formation property, We define the q-series based on Kashaev’s invariant of ∞ (3,2) 1 3 Tn π n the (2 m +1, 2)-torus knot which was given in Equa- Φ(3)( )+ −i N 2 Φ(3)(−N)= . N n N tion (3–11). n=0 ! 12 i (3–17) • Trefoil (m = 1):

∞ A generalization of this property will be discussed be- (3,2) F (q)= (q)n, (3–22a) low. n=0

Remark 3.8. The torus knot is not hyperbolic, and we • m have S3 \ Torus = 0. In view of complexiﬁcation of the Solomon’s Seal Knot ( = 2): volume conjecture (Conjecture 2.4), Equation (3–15a) ∞ shows F (5,2) q q−ab q , π2 ( )= ( )a+b (3–22b) CS(Trefoil) = − . (3–18) a,b=0 6 326 Experimental Mathematics, Vol. 12 (2003), No. 3

• (2 m +1, 2)-Torus Knot (m>2): Proof: We outline a proof following [Zagier 01] (see also (2m+1,2) [Andrews et al. 01] for a generalization of this identity). F (q) We define a function S(x)by 2m−2 c −c c + 1 2m−2 c (c +1) − j=3 j q 1 2 2 j=3 j j = ( 1) ∞ 0≤c <∞ n 1 S(x)= (x)n+1 x 0≤c2m−2≤···≤c2<∞ n=0 (q)c +c × 1 2 ∞ 2m−3 q xq − x xq − xq xn. j=2 ( )cj −cj+1 =( )∞ +(1 ) ( )n ( )∞ ∞ n=0 q 2m−2 ( )a1+a2+···+a2m−2 ja j=3 j (3–26) = 2m−3 (−1) (q)a a1,...,a2m−2=0 j=2 j xq The subtraction of ( )∞ in the summation is to avoid −a a + 2m−2( j −1−a )a + 1 2m−2(a +a +···+a )2 × q 1 2 j=3 2 1 j 2 j=3 j j+1 2m−2 . divergence in the limit x → 1, and the second equality is (3–22c) proved using the Euler identity, ∞ xn 1 In this section, we use the following notation: = . n q x n=0 ( )n ( )∞ x x q − xqi−1 . ( )n =( ; )n = (1 ) q i=1 We can check that it solves the -difference equation, 2 2 3 (2m+1,2) S x − qx − q x S qx . Note that generally the q-series functions F (q) ( )=1 ( ) (3–27) q → ω ≡ do not converge in any open set, but in the limit On the other hand, we can easily see that a function π /N exp(2 i ) the functions reduce to the invariant of the ∞ 1 (n−1) 1 (n2−1) torus knot: S(x)= χ12(n) x 2 q 24 , (3–28) F (2m+1,2) ω m , . n=1 ( )= Trs(2 +1 2) N (3–23) also solves the same q-difference equation (3–27). Here Collecting these observations, we propose the follow- χ12(n) is the Dirichlet character which follows from ing conjecture on the asymptotic expansion of the q- Equation (3–9) with (m, p)=(3, 2): series. We have numerically checked the validity of this n conjecture for several n and m. mod 12 1 5 7 11 others . χ12(n) 1 −1 −11 0 Conjecture 3.10. We have the asymptotic expansions of It is remarked that S(x = 1) coincides with the Dedekind (2m+1,2) the q-series F (q) defined in Equation (3–22) as η-function, q −t ( =e ) ∞ 1 (n2−1) q χ n q 24 , F (2m+1,2) −t ( )∞ = 12( ) (3–29) (e ) n=1 ∞ 2 (2m+1,2) n (2m−1) t Tn t 8(2m+1) , where the equality follows from the Jacobi triple product =e n 3 m (3–24) n=0 ! 2 (2 +1) identity. Thus, from Equations (3–26) and (3–28), we find that where the T -number is defined by Equation (3–10) (or ∞ Equation (3–6)). n (xq)∞ +(1− x) (xq)n − (xq)∞ x n=0 This conjecture is proved in [Zagier 01] for the case ∞ m 1 (n−1) 1 (n2−1) = 1 as follows. = χ12(n) x 2 q 24 . (3–30) n=0 m Theorem 3.11. [Zagier 01] Conjecture 3.10 for =1is By differentiating with respect to x and setting x → 1, correct. we get ∞ −t −2t −nt ∞ n ∞ (1 − e )(1− e ) ···(1 − e ) 1 q (q)∞ · − − (q)n − (q)∞ n=0 − qn 2 n=1 1 n=0 ∞ (3,2) n t/24 Tn t ∞ =e . (3–25) 1 1 (n2−1) n = nχ12(n) q 24 . (3–31) n=0 ! 24 2 n=0 Hikami: Volume Conjecture and Asymptotic Expansion of q-Series 327

Thus, in the limit t → 0, we obtain It is noted that the right-hand side is now a “half- ∞ differential” of the infinite q-product defined by −t/24 (3,2) −t − 1 n2t −2e F (e ) nχ12(n)e 24 , (3–32) ∞ n=0 1 (n2−(2m−1)2) χ8m+4(n) q 8(2m+1) because (q)∞ induces an infinite order of t when we set n=1 q −t 2m 2m+1 2m+1 =e . Applying the Mellin transformation to an equal- =(q,q ,q ; q )∞ (3–37) ∞ − 1 n2t ∞ n nχ n 24 ∼ γ t ity n=0 12( )e n=0 n ,weget ∞ k (m+ 1 )k2+(m− 1 )k (−1)n = (−1) q 2 2 γn = L(−2 n − 1,χ12). 24n n! k=−∞ 2 2 n +···+n +n1+···+nm−1 L q 1 m−1 By use of the relationship (3–10) between the -series q · , =( )∞ q ... q q and the T -numbers, we find ( )nm−1−nm−2 ( )n2−n1 ( )n1 nm−1≥···≥n1≥0 (3,2) Tn γn = −2 , where the last equality is the Gordon–Andrews identity, 24n n! a generalization of the Rogers–Ramanujan identity (m = which proves Equation (3–25). 2).

Remark 3.12. From Equation (3–29), the right-hand side Conjecture 3.10 suggests that there should be a of Equation (3–25) is regarded as a “half-differential” of q-series identity as a generalization of Zagier’s iden- η the Dedekind -function [Zagier 01]. tity (3–31), which we hope to report in a future pub- lication [Hikami 02]. Remark 3.13. Conjecture 3.10 is formally derived as follows: Equation (3–25), i.e., a proof of Conjecture 3.10 Remark 3.14. We consider an expansion of the q-series in the case m = 1, suggests that we may apply a naive (2m+1) n with q → 1 − x, and define an as coefficients of x : analytic continuation ∞ 2 π (2m+1,2) (2m+1) n N ←→ , F (1 − x)= a x . (3–38) t (3–33) n i n=0 in the integral (3–7), i.e., we may set (2m+1) (2m+1) (2m+1) To calculate an from Tn , we also define bn F (2m+1,2)(e−t) following [Zagier 01] by 3/2 2 2(2m +1)π (2m−1) t i e 8(2m+1) ∞ b(2m+1) t F (2m+1,2) −t n tn. (e )= (3–39) 2 n! 2(2m+1)π w2 sh(2 πw) n=0 × dw e t w . (3–34) ch((2 m +1)πw) (2m+1) C It is easy to see that a series bn is written in terms (2m+1,2) In fact, substituting the expansion (3–8) with (m, p) → of Tn as m , (2 +1 2), 2 n n n (2m+1,2) (2 m − 1) Tn ∞ b(2m+1) . x n = 2k sh(2 ) χ n −nx 8(2m +1) k (2 m − 1) m x = 8m+4( )e k=0 ch((2 +1) ) n=0 (3–40) ∞ (2m+1,2) Tn Some of the computed terms are given in Table 2. =2 (−1)n x2n+1, (3–35) (2 n + 1)! n=0 We use the nonnegative Stirling numbers of the first n mod (8 m +4) 2 m − 12m +3 6m +1 6m +5 others kind [Goldberg et al. 72] defined by χ8m+4(n) 1 −1 −110 n−1 n we obtain the right-hand side of Equation (3–24). Using m (x + j)= s(n, m) x . (3–41) the Mellin transformation, we also see that j=0 m=0 (2m+1,2) −t F (e ) Itisknownthatwehave ∞ 1 − t (n2−(2m−1)2) m ∞ −t n ∼− nχ8m+4(n)e 8(2m+1) . (3–36) t (1 − e ) = s(n, m) . (3–42) 2 n=0 m n ! n=m ! 328 Experimental Mathematics, Vol. 12 (2003), No. 3

n 01 2 3 4 5 6 7 (3) bn 1 1 3 19 207 3451 81663 2602699 (5) bn 1 2 10 104 1870 51632 2027470 107354144 (7) bn 1 3 21 303 7581 291903 16004541 1184112303 (9) bn 1 4 36 664 21276 1050664 73939356 7024817944 (11) bn 1 5 55 1235 48235 2906315 249689275 28969703915 (13) bn 1 6 78 2064 95082 6762216 686010858 94007233704 (15) bn 1 7 105 3199 169785 13919647 1628324985 257347060159 (17) bn 1 8 136 4688 281656 26150768 3465278776 620465295248 (19) bn 1 9 171 6579 441351 45771579 6776104311 1355621381739 (21) bn 1 10 210 8920 660870 75714880 12384774150 2737845857680

TABLE 2.

n 01 2 3 4 5 6 7 (3) an 1 1 2 5 15 53 217 1014 (5) an 1 2 6 23 109 621 4149 31851 (7) an 1 3 12 62 402 3162 29308 312975 (9) an 1 4 20 130 1070 10738 127316 1741705 (11) an 1 5 30 235 2345 28623 413441 6896695 (13) an 1 6 42 385 4515 64911 1105573 21759966 (15) an 1 7 56 588 7924 131124 2572640 58354762 (17) an 1 8 72 852 12972 242820 5392464 138497502 (19) an 1 9 90 1185 20115 420201 10419057 298862100 (21) an 1 10 110 1595 29865 688721 18859357 597554925

TABLE 3. Using this identity, we obtain the a-series from the b- (2m+1) 1 4 3 a6 = m (m +1) 2360 m + 6544 m series as 180 + 6841 m2 + 3209 m + 576 , n 1 (2m+1) (2m+1) 1 5 4 a(2m+1) = s(n, k) b . (3–43) a = m (m +1) 99136 m + 330440 m n n k 7 2520 ! k=1 + 440960 m3 + 294775 m2 + 98919 m + 13410 . (3) Some of the a-series are given in Table 3. We should note that the series an coincides with the (2m+1) Table 3 indicates that an is given by the n-th upper bound of the number of linearly independent Vas- order polynomial of m, e.g., siliev invariants of degree n [Stoimenow 98].

(2m+1) Remark 3.15. It would be interesting to construct the ex- a =1, 0 plicit form of Kashaev’s invariant for the arbitrary torus (2m+1) knot K =Trs(m, p), and to study an asymptotic expan- a1 = m, sion as a q-series based on Equation (3–4). (2m+1) a = m (m +1), 2 Remark 3.16. The Rogers-Ramanujan identities are the (2m+1) 1 following set of equations: a3 = m (m +1)(8m +7), 6 ∞ qn2 S q 1 , 0( )= = 4 5 (3–44a) (2m+1) 1 2 (q)n (q,q ; q )∞ a = m (m + 1) (14 m +22m +9), n=0 4 6 ∞ qn2+n S q 1 . (2m+1) 1 2 1( )= = 2 3 5 (3–44b) a5 = m (m +1)(8m + 7) (19 m +25m +9), (q)n (q ,q ; q )∞ 30 n=0 Hikami: Volume Conjecture and Asymptotic Expansion of q-Series 329

With these functions, we have the modular property, Conjecture 3.18. The q-series F˜(5,2)(q) with q → ω ≡ e2πi/N has an asymptotic expansion in N →∞as c − 1 2π π c τ 0( τ ) 2 sin 5 sin 5 0( ) N √ , 1 = (5,2) −ab c − π 2π c τ F˜ ω ω ω a+b−1 1( τ ) 5 sin 5 − sin 5 1( ) ( )= ( ) a,b=0 (3–45) 1≤a+b≤N where we have set q =exp(2π i τ), and 2 3 πi − πi 2 π − Nπi √ N 2 e 4 20N 2sin e 20 5 5 1/40 9/40 ∞ c0(q)=q S0(q),c1(q)=q S1(q). (5,2) n π − 9Nπi − πi Tñ π 20 20N . +sin e +e n N 5 n=0 ! 20 i Conjecture 3.10 for m = 2 is an identity for a half dif- (3–50) ferential of S1(q). We expect that there should exist an identity for S0(q). For this purpose, we define another (5,2) With the above conjecture and Equation (3–15b), the series Tñ : transformation property (3–17) should be reformulated as a variant of Equation (3–45); we set ∞ (5,2) ∞ sh(4 x) n Tñ 2n+1 −nx =2 (−1) x = χ˜20(n)e . (5) 9 πiα (5,2) 2πiα x n Φ (α)=e20 F (e ), (3–51) ch(5 ) n=0 (2 + 1)! n=0 (3–46) (5) 1 πiα (5,2) 2πiα Ψ (α)=e20 F˜ (e ). (3–52) Here, we have Using the fact that we have n mod 20 1 9 11 19 others Φ(5)(0) = 1, Ψ(5)(0) = 2, χ˜20(n) 1 −1 −11 0 and a recursion relation, (5,2) (5,2) 1 n+1 and some of the Tñ , Tñ = (−1) L(−2 n − (5) 9 πi (5) 2 Φ (α +1)=e20 · Φ (α), 1, χ˜20) are as follows: (5) 1 πi (5) Ψ (α +1)=e20 · Ψ (α),

n 01 2 3 4 5 we get for n ∈ Z (5,2) . T˜n 2 118 23762 10104358 7370639522 8214744720598 (5) 9 πin (5) 1 πin Φ (n)=e20 , Ψ (n)=2e20 .

The Jacobi triple identity gives (5) (5) As a result, we find that the functions Ψ and Φ can be regarded as a set of “nearly” modular functions [Zagier ∞ 2 3 5 5 1 (n2−1) 01] satisfying (q ,q ,q ; q )∞ =(q)∞ · S0(q)= χ˜20(n) q 40 . n=0 (5) 1 (3–47) Ψ ( N ) Φ(5)( 1 ) N Conjecture 3.17. We define 2π π (5) −N 3 2 sin 5 sin 5 Ψ ( ) +(−i N) 2 · √ 5 sin π − sin 2π Φ(5)(−N) ∞ 5   5 F˜(5,2) q q−ab q . ∞ (5,2) ( )= ( )a+b−1 (3–48) 1 Tñ π n a,b=0 =   . (3–53) (a,b) =(0,0) n (5,2) N n=0 ! Tn 20 i

Note that the transformation matrix coincides with that Then, we have of Equation (3–45). ∞ ˜(5,2) n (5,2) −t t/40 Tn t As a generalization of the previous re- F˜ (e )=e . (3–49) Remark 3.19. n m> q n=0 ! 40 mark to the case 2, we deﬁne a formal -series 330 Experimental Mathematics, Vol. 12 (2003), No. 3

(2m+1,2;a) (2m+1) F˜ (q)fora =0, 1,...,m− 2by We deﬁne functions Ψa (α)fora =0, 1,...,m−2 by F˜(2m+1,2;a)(q) 2 2m−2 (2m−2a−3) πiα cj (2m+1) 4(2m+1) (2m+1,2;a) 2πiα = (−1) j=3 Ψa (α)=e F˜ (e ), (3–58)

0≤c1<∞ 0≤c ≤···≤c <∞ 2m−2 2 (2m+1) 0

m n 01 2 3 4 5 6 7 (5) 2 an 1 2 6 23 109 621 4149 31851 (5;0) a˜n 2 3 9 35 168 966 6496 50103

(7) 3 an 1 3 12 62 402 3162 29308 312975 (7;0) a˜n 2 5 20 105 690 5478 51102 548244 (7;1) a˜n 3 6 24 127 840 6699 62689 674091

(9) 4 an 1 4 20 130 1070 10738 127316 1741705 (9;0) añ 2 7 35 231 1925 19481 232309 3191199 (9;1) añ 3 9 45 300 2520 25641 306915 4227525 (9;2) añ 4 10 50 335 2825 28821 345618 4767048

(11) 5 an 1 5 30 235 2345 28623 413441 6896695 (11;0) añ 2 9 54 429 4329 53235 772863 12939498 (11;1) añ 3 12 72 578 5880 72702 1059436 17785437 (11;2) añ 4 14 84 679 6944 86163 1258684 21168134 (11;3) añ 5 15 90 730 7485 93039 1360788 22905630

(13) 6 an 1 6 42 385 4515 64911 1105573 21759966 (13;0) añ 2 11 77 715 8470 122584 2097326 41414087 (13;1) añ 3 15 105 985 11760 171084 2937544 58154346 (13;2) añ 4 18 126 1191 14301 208845 3595347 71312841 (13;3) añ 5 20 140 1330 16030 234682 4047162 80376063 (13;4) añ 6 21 147 1400 16905 247800 4277077 84995664

(15) 7 an 1 7 56 588 7924 131124 2572640 58354762 (15;0) añ 2 13 104 1105 15028 250172 4928300 112114184 (15;1) añ 3 18 144 1545 21168 354105 6998985 159603426 (15;2) añ 4 22 176 1903 26224 440363 8726795 199383701 (15;3) añ 5 25 200 2175 30100 506880 10064600 230275675 (15;4) añ 6 27 216 2358 32724 552096 10976580 251378289 (15;5) añ 7 28 224 2450 34048 574966 11438630 262082935

TABLE 4.

we obtain a relationship between the T -series and the A table of these series is given in Table 4. For conven- a-series as a result of Equation (3–42): tion, we have also included the a-series deﬁned by Equa- tion (3–38).

m − a − 2 n ˜b(2m+1;a) (2 2 3) n = m 4. HYPERBOLIC KNOTS 8(2 +1) n n T˜(2m+1,2;a) In the previous section, we analytically studied an × k , (3–63) asymptotic behavior of Kashaev’s invariant of the torus k (2 m − 2 a − 3)2k k=0 knot. Here, we consider numerically an asymptotic for- n mula for the invariant of the hyperbolic knots and links: 1 (2m+1;a) a˜(2m+1;a) = s(n, k) ˜b . (3–64) knots up to 6-crossing, the Whitehead link, and Bor- n n! k k=1 romean rings. Our conjecture based on both analytic 332 Experimental Mathematics, Vol. 12 (2003), No. 3 results for the torus knot (Corollary 3.9) and numerical This computation is essentially the same as that of the results for hyperbolic knots is summarized as follows. central charge from the character [Richmond and Szek- eres 81]. The invariant may be represented by the integral Conjecture 4.1. Kashaev’s invariant behaves in a large of the potential VK(x), N limit as N i N 3 3 0 KN ∼ dxi exp VK(x) . (4–4) log KN ∼ v3 · S \K · + #(K) · log N + O(N ), π 2 π 2 i 2 (4–1) where #(K) is the number of prime factors of a knot as In the large N limit, we apply a stationary phase approx- a connected-sum of prime knots. imation, and obtain a saddle point x0 as a solution of the set of equations, Numerical computation is performed with the help of PARI/GP [PARI 00]. We compute Kashaev’s invariant ∂ VK(x) =0. (4–5) K K ∂xi N for the hyperbolic knot numerically (see Fig- x=x0 2 π ures 2–8). We plot N logKN as a function of N, and numerical data is given as • in those ﬁgures. The With this solution, we may obtain solid line denotes a result of the least-squares method 2 π with a trial function, lim logKN =iVK(x0), (4–6) N→∞ N π v N 2 K K( )= N log N whose real part is expected to coincide with the hyper- 2 π c3(K) c4(K) bolic volume (Conjecture 2.3). = c1(K)+c2(K) · log N + + . N N N 2 In the following, for several hyperbolic knots and links (4–2) we give a list of numerical data c1(K),...,c4(K), poten- V x x This trial function is motivated from an analytic re- tial K( ), and a saddle point 0 of the potential. We sult (3–4) of the torus knot. will see that We also give some computations which support nu- 3 c1(K)= i VK(x0) =Vol(S \K), (4–7) merical results of c1(K). Although there is no mathematically rigorous proof of the asymptotics of each invariant, 3 c2(K)= , (4–8) it is known [Kashaev 95] that a semirigorous proof works 2 well to obtain an asymptotic limit of Kashaev’s invariant. c K < , In the limit N →∞,wemayreplacetheω-series with 3( ) 0 (4–9) the dilogarithm function, which supports Equation (4–1) (Conjecture 4.1) for K n 2 π i 2 π i a prime knot. Note that Equation (3–21) proves this log(ω)n = log(1 − exp(2 π i j/N)) N N K m, p j=1 conjecture for =Trs( ). x ∼ dt log(1 − et) 0 Figure-Eight Knot 41. 2 π x = − Li2(e ). 6 Thus, we formally obtain a potential from Kashaev’s 2πi invariant KN by the following steps (we set N ai = log xi): N aiaj i ω → exp − log xi log xj , 2 π i N π2 (ω)a → exp Li2(xi) − , (4–3) i 2 π 6 N−1 2 2 ω . ∗ i N −1 π 41 N = ( )a (4–10) (ω) → exp − Li2(x )+ . ai 2 π i 6 a=0 Hikami: Volume Conjecture and Asymptotic Expansion of q-Series 333

v(N) v(N)

2.2 3.2 2.175 3.15 2.15 3.1 2.125 3.05 2.1 3 2.075 2.95 2.05 2.9 N N 100 1000 10000 100000. 100 150 200 300 500 700 1000 15002000

FIGURE 2. Figure-eight knot. FIGURE 3.Knot52.

• Numerical result (Figure 2): 3 πi/3 • Potential and saddle point: Vol(S \ 41)=2D(e )=2.029883212819307... c . ± . × −9 1 =2029883193056962 7 77 10 π2 −6 V x, y y x−1 x y − , c . ± . × 52 ( )=2Li2( )+Li2( ) + log log 2 =150002685 2 42 10 2 c3 = −1.7269321 ± 0.000095 (4–13) c . ± . . 4 =3575981 0 0027 x0 0.122561 + 0.744862 i = . y0 0.337641 − 0.56228 i • Potential and saddle point: V x x − x−1 , 41 ( )=Li2( ) Li2( ) (4–11) 61 Knot. x0 =exp(−π i/3). Note that asymptotic behavior of this ω-series is proved rigorously (see, e.g., [Murakami 00]).

52 Knot.

N−1 2 (ω)c (c−a−b)(c−a+1) 61N = ∗ ω . (4–14) ω a ω a,b,c=0 ( ) ( )b a+b≤c 2 (ω)b −(b+1)a 52N = ω . (4–12) ω ∗ • Numerical result (Figure 4): 0≤a≤b≤N−1 ( )a

• Numerical result (Figure 3): 3 Vol(S \ 61)=3.16396322888... 3 Vol(S \ 52)=2.828122088330783... −8 c1 =3.1639628602 ± 3.04 × 10 −8 c1 =2.8281219744 ± 1.5571 × 10 −6 c2 =1.5000356 ± 1.88 × 10 −6 c2 =1.5000269858 ± 2.01 × 10 c3 = −4.0343627 ± 0.0000611 c3 = −2.648116951 ± 0.0000732 c4 =3.971777 ± 0.000970. c4 =4.22788 ± 0.00169. 334 Experimental Mathematics, Vol. 12 (2003), No. 3

v(N) v(N)

5.2

3.6 5.1 N 50 100 150 200 250 300 3.5 4.9

4.8 3.4 4.7 N 100 200 300 400 500 4.6

FIGURE 4.Knot61. FIGURE 5.Knot62.

• • Potential and saddle point: Potential and saddle point: V x, y, z y xz − x x−1 −1 −1 62 ( )=2Li2( )+Li2( ) Li2( )+Li2( ) V6 (x, y, z)=Li2(z) − Li2(z ) − Li2(x)+Li2(y ) 1 π2 z − y/x x yz − , − z/x π x/z , Li2( ) + log( ) log( ) log xy log( )+2 i log( ) 3 (4–17) (4–15)         x0 0.09043267 + 1.60288 i x . .     0 0 17385 + 1 06907 i y0 = −0.232705 − 1.09381 i . y   . .  . 0 = 0 322042 + 0 15778 i z0 −0.964913 − 0.621896 i z0 0.278726 − 0.48342 i

63 Knot. 62 Knot.

N−1 2 −a(b+c+1) (ω)b (ω)a+c 2N ω . 6 = N−1 (ω)a (ω)b−a 2 a,b,c=0 (ω)a+b+c ∗ (a+1)(b−c) a≤b 63N = (ω)a+b (ω)a+c ω . ω b ω c 0≤a+c≤N−1 a,b,c=0 ( ) ( ) (4–16) a+b+c≤N−1 (4–18) • Numerical result (Figure 5): • Numerical result (Figure 6):

S3 \ . ... 3 Vol( 62)=440083251 Vol(S \ 63)=5.69302109... c . ± . × −7 1 =4400828513 2 97 10 c1 =5.69289987 ± 0.0000124 c . ± . × −6 2 =1500213389 9 83 10 c2 =1.50411 ± 0.00026 c − . ± . 3 = 4 685095 0 00028 c3 = −5.6162 ± 0.0066 c . ± . . 4 =602178 0 00266 c4 =10.315 ± 0.0397. Hikami: Volume Conjecture and Asymptotic Expansion of q-Series 335

v(N) v(N)

6.8 5

4.8 6.6

4.6 6.4 4.4 6.2 4.2 N 50 100 150 200 250 N 25 50 75 100 125 150

FIGURE 6.63 Knot. FIGURE 7. Whitehead link.

• • Potential and saddle point: Potential and saddle point: −1 −1 V52 (x, y, z)=− Li2((xz) )+Li2(x) − Li2(y) V6 (x, y, z)=Li2(xyz) − Li2((xyz) ) − Li2(y) 1 3 −1 −1 −1 +Li2(z ) − log(z) log(x/y), +Li2(y ) − Li2(z)+Li2(z ) (4–21) − xy −1 xz − x y/z , Li2(( ) )+Li2( ) log( ) log( )     (4–19) x0 −i         y0 = i . x0 0.204323 − 0.978904 i z 1     0 2 1+i y0 = 1.60838 + 0.558752 i . z0 0.554788 + 0.19273 i 3 Borromean Rings 62.

2 Whitehead Link 51.

2 3 (ω)a+c (ω)b+d 62N = ω d ω a+c−b N−1 ∗ a,b,c,d=0 ( ) ( ) (ω) (ω)a 2 a+c ωc(a−b), a≤b≤a+c≤N−1 51 N = ∗ (4–20) b+d≤N−1 ω b ω a,b,c=0 ( ) ( )c b≤a 1 (b+1)(c−d+a−b) a+c≤N−1 × ∗ ω . (4–22) (ω)a (ω)b−a • Numerical result (Figure 7): • Numerical result (Figure 8):

S3 \ 2 . ... 3 3 Vol( 51)=366386237 Vol(S \ 62)=7.32772475... c . ± . −6 1 =3663960 0 000113 c1 =7.3276812 ± 4.1 × 10 c . ± . 2 =149978 0 00190 c2 =1.50176 ± 0.00011 c − . ± . 3 = 3 2729 0 0461 c3 = −8.76447 ± 0.00296 c . ± . . 4 =61846 0 2549 c4 =11.116 ± 0.025. 336 Experimental Mathematics, Vol. 12 (2003), No. 3

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Kazuhiro Hikami, Department of Physics, Graduate School of Science, University of Tokyo, Hongo 7-3-1, Bunkyo, Tokyo 113-0033, Japan ([email protected])

Received August 10, 2002; accepted in revised form March 20, 2003