arXiv:math/0508138v4 [math.GT] 4 Mar 2006 where ipiilvlm ftecmlmn of complement the of volume simplicial ioaih nain n h yeblcvlm ftecmlmn of complement the of volume hyperbolic the and invariant dilogarithm tetrahedron. yeblcpee nteJJdcmoiino h opeetof complement knot the the of form, JSJ-decomposition original the in pieces hyperbolic ojcue11(ahe 2,Mrkm-uaai[5]). Murakami-Murakami [2], (Kashaev 1.1 Conjecture reformulate follows. and as Kashaev Murakami by and Murakami proposed was conjecture volume The Introduction 1 ealthat Recall ahmtc ujc lsicto 00 72,57N10 57M25, 2000: Classification knot Subject torus Mathematics double, Whitehead conjecture, volume Keywords: ro ftevlm ojcuefrWhitehead for conjecture volume the of Proof h ouecnetr o htha obe fafml fto of family a observations. of interesting doubles some presented Whitehead show is for knots, conjecture of volume doubles the Whitehead the by illustrated J K,N ehiu ocluaeteclrdJnsplnmaso sa of polynomials Jones colored the calculate to technique A ste(omlzd ooe oe oyoilof polynomial Jones colored (normalized) the is obe fafml ftrsknots torus of family a of doubles v 3 eateto ahmtc,ZoghnUniversity Zhongshan Mathematics, of Department k S 3 2 π \ K N K -al [email protected] E-mail: lim →∞ shproi n h qaini ntrso h quantum the of terms in is equation the and hyperbolic is k sntigbttesmo h yeblcvlmsof volumes hyperbolic the of sum the but nothing is unzo 125 China 510275, Guangzhou log
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v = 3 stevlm fteielregular ideal the of volume the is v 3 k S 3 \ K k ,Wieedlink Whitehead s, hnw prove we Then . K u nt and knots rus , o n knot any For elt knots, tellite K n endby refined and d k S nKashaev’s In . 3 \ K K . k sthe is (1.1) K , The conjecture is marvellous in the sense that it reveals the topological meaning of the quantum invariants of knots which is quite unobvious from definition. How- ever, it also turns out to be rather hard to be proved. Till now, besides positive numerical evidences (ref. [1, 6]) for some hyperbolic knots, only the cases of torus knots (Kashaev-Tirkkonen [3]) and the simplest hyperbolic knot, the figure 8 knot (ref. [2]) have been verified. In view of the compatible behavior of both sides of the conjectured equation (1.1) under connect sum
J = J J , (1.2) K1♯K2,N K1,N · K2,N S3 K ♯K = S3 K + S3 K , (1.3) k \ 1 2k k \ 1k k \ 2k the volume conjecture, in fact, may be reduced to the consideration of prime knots. By Thurston’s Hyperbolization Theorem (ref. [7]), the prime knots further fall into three families: torus knots, hyperbolic knots and satellite knots. In this article, we deal with the conjecture by examining a special case of the third family, the Whitehead doubles of torus knots. The approach is emphasized on the relation between the colored Jones polynomial of a satellite knot and those of the associated companion knot and pattern link. In particular, we show a technique to calculate the colored Jones polynomial of satellite knots by cutting and gluing method.
A Whitehead double of a knot K is a knot obtained as follows. Remove the regular neighborhood of one component of the Whitehead link from S3 thus get a knot inside a torus, then knot the torus in the shape of a knot K. Note that, when K is nontrivial, a Whitehead double K′ of K is a satellite knot whose complement contains an obvious essential torus T 2. Cutting along the torus,
2 we get 3 2 3 3 (S K′) T = (S Whitehead link) (S K) (1.4) \ \ ∼ \ ∪ \ thus 3 3 3 S K′ = S Whitehead link + S K . (1.5) k \ k k \ k k \ k In particular, if K is a nontrivial torus knot, the complement of K is Seifert fibred and the complement of the Whitehead link is hyperbolic, hence 3 3 v S K′ = vol(S Whitehead link). (1.6) 3k \ k \ The article proceeds as follows. First, we calculate the colored Jones polynomi- als of the twisted Whitehead links and the Whitehead doubles of knots in section 2. Next, as a warming-up we prove in the next two consecutive sections the following two theorems, of which the former one is, in fact, the volume conjecture for twisted Whitehead links and both extends the estimation (1.1) to the second order. Theorem 1.2. For every twisted Whitehead link L, we have
2π√ 1 3 2π log J (e N− ) = vol(S L) N +3π log N + O(1) (1.7) L,N \ · as N . →∞ Theorem 1.3. For every nontrivial torus knot T (p, q) with q =2, we have
2π√ 1 − 2π log JT (p,q),N (e N ) =3π log N + O(1) (1.8) as N . →∞ Then we prove the main theorem in Section 5 and show some observations in the final section. Theorem 1.4. If K is a Whitehead double of a nontrivial torus knot T (p, q) with q =2, then
2π√ 1 − 3 2π log J (e N ) = v S K N +4π log N + O(1) (1.9) K,N 3k \ k· as N . In particular, the volume conjecture is true for K. →∞ Remark 1.5. In their proof of the volume conjecture for torus knots, Kashaev and Tirkkonen [3] derived the following estimation
2π√ 1 − 2π log JT (p,q),N (e N ) = O(log N). (1.10)
But improving the estimation to (1.8) requires a nonvanishing proposition on num- ber theory (see Proposition 4.1) to which both Theorem 1.3 and Theorem 1.4 are reduced in this article. With a technical condition q = 2 we proved the nonvanish- ing proposition in section 4. A complete proof has been beyond the scope of the article. We only mention here that our technique can be sharpened to prove the nonvanishing proposition, hence both theorems, at least for the cases that both p, q are odd or one of them is a power of 2.
3 Remark 1.6. It is noteworthy that the coefficient “4π” of the second term in the asymptotic expansion (1.9) disagrees with the observation due to Hikami [1]
2π√ 1 3 2π log J (e N− ) = v S K N +3π log N + O(1) (1.11) K,N 3k \ k·
for many prime knots K.
2 Calculation of colored Jones polynomial
In this section, we calculate the colored Jones polynomials of the twisted Whitehead link W L(r) and the Whitehead double WD(K,r) of a knot K.
double(K) r twists r twists PSfrag replacements
WL(r) WD(K,r) In the figure, double(K) denotes the (2,2)-tangle obtained by doubling the knot K to a link with zero linking number and then removing a pair of parallel segments. Our trick is cutting the link diagrams into (2,2)-tangles and gluing the tangle invariants together.
PSfrag replacements
twist belt double(K) clasp Colored Jones polynomial is also defined for tangles, but, instead of a Lau- rent polynomial of t, it is in general a module homomorphism of Uq(sl2) (choose t = q2). Especially, the colored Jones polynomial of a (2,2)-tangle is a module homomorphism V V V V (2.1) N ⊗ N → N ⊗ N 4 where VN is the N dimensional irreducible representation of Uq(sl2). Note that the tensor product admits the decomposition
N 1 − V V = V . (2.2) N ⊗ N 2n+1 Mn=0 A straightforward calculation shows that the (framing independent, unnormalized) colored Jones polynomials of the tangles are
N 1 − J˜ = tn(n+1) id , (2.3) twist,N · V2n+1 Mn=0 N 1 N(2n+1)/2 N(2n+1)/2 − t t− J˜ = − id , (2.4) belt,N t(2n+1)/2 t (2n+1)/2 · V2n+1 Mn=0 − − N 1 − J˜ = J id , (2.5) double(K),N K,2n+1 · V2n+1 Mn=0 N 1 − J˜ = ξ id , (2.6) clasp,N N,n · V2n+1 Mn=0 where
N 1 n n − − N i j i+j (N 2 1)/2+N(N 1)/2 N(i+n) (1 t − − )(1 t ) ξ = t − − t− − − . (2.7) N,n 1 tj Xi=0 Yj=1 − Combining the tangle invariants together, one has
N 1 (2n+1)/2 (2n+1)/2 N(2n+1)/2 N(2n+1)/2 − t t− t t− J = − trn(n+1) ξ − , (2.8) WL(r),N tN/2 t N/2 · · N,n · t(2n+1)/2 t (2n+1)/2 Xn=0 − − − − and N 1 (2n+1)/2 (2n+1)/2 − t t− J = − trn(n+1) ξ J . (2.9) WD(K,r),N tN/2 t N/2 · · N,n · K,2n+1 Xn=0 − − Note that, in the expression of JWD(K,r),N , the factor JK,2n+1 is contributed by the companion knot K and the other part is precisely obtained from the expression of JWL(r),N by removing the factor contributed by the belt tangle.
5 3 Proof of Theorem 1.2
2π√ 1 Let L denote the twisted Whitehead link W L(r). Setting t = e N− , we have
N 1 N 1 n n i j i+j 2π√ 1 − − − − 1/2 rn(n+1) (1 t− − )(1 t ) J (e N )= t− (2n + 1)t − − L,N − 1 tj Xn=0 Xi=0 Yj=1 − (3.1) N 1 N 1 n π√ 1 − − − − 4r 1 = e− N (2n + 1)a − S − n n,i Xn=0 Xi=0 where n(n+1) n n(n+1 N) π√ 1 π√ 1 − π√ 1 an = e 2N − − 2 − = e 2N − (3.2) and n 2 (i+j)π 4 sin N Sn,i = . (3.3) 2 sin jπ Yj=1 N
First, we prepare a lemma to estimate the norm factor Sn,i. Put
n jπ sn = log 2 sin (3.4) − N Xj=1 and let x L(x)= log 2 sin u du (3.5) − Z0 | | be the Lobachevsky function.
Lemma 3.1. For 0 <α< 1 we have uniform estimations
N mπ N nπ 1 s s = L( ) L( )+ O(N − )(m n) (3.6) m − n π N − π N − on α N 6 Proof. We have jπ jπ N N log 2 sin + log 2 sin u du (j 1)π − N π Z − | | N π jπ N N jπ = log 2 sin + log 2 sin( u) du (3.9) − N π Z0 N − π jπ N N sin( N u) = log jπ− du. π Z0 sin N Since sin(x u) log − = O(u) (3.10) sin x α α uniformly on x [ π, (1 )π] as u 0, the first estimation follows as ∈ 2 − 2 → N mπ N nπ s s L( )+ L( ) m − n − π N π N m π jπ N sin( u) N N (3.11) = log jπ− du π Z0 sin j=Xn+1 N 1 = O(N − )(m n). − Note that sin(x u) x − =1+ O(u) (3.12) x u · sin x − thus sin(x u) x u log − = log − + O(u) (3.13) sin x x uniformly on x [ απ,απ ] 0,u as u 0. It follows that ∈ − \{ } → n π jπ N N nπ N sin( N u) sn L( )= log jπ− du − π N π Z0 sin Xj=1 N n π jπ N N N u 1 = log jπ− du + nO(N − ) π Z0 Xj=1 N (3.14) n j 1 = (j 1) log − 1 + O(1) − − j − Xj=1 n! = log n + O(1) − nn − uniformly on 0 L(x)+ L(π x) = 0 (3.16) − and sn 1 + sN n = sN 1. (3.17) − − − In particular, we have π L( ) = 0 (3.18) 2 and, by the second estimation, sN 1 = s[ N 1 ] + s[ N ] = log N + O(1). (3.19) − 2− 2 − Therefore, nπ sn = sN 1 sN n log 2 sin − − − − N N nπ 1 2(N n)π = log N + L( )+ log( N n) log − + O(1) (3.20) − π N 2 − − N N nπ 1 = L( ) log(N n)+ O(1) π N − 2 − uniformly on (1 α)N