TitleMAHLER MEASURE OF THE COLORED JONESPOLYNOMIAL AND THE VOLUME CONJECTURE(Volume Conjecture and Its Related Topics)

Author(s) Murakami, Hitoshi

Citation 数理解析研究所講究録 (2002), 1279: 86-99

Issue Date 2002-08

URL http://hdl.handle.net/2433/42351

Right

Type Departmental Bulletin Paper

Textversion publisher

Kyoto University

MAHLER MEASURE OF THE COLORED JONES POLYNOMIALAND THE VOLUME CONJECTURE

東京工業大学大学院理工学研究科 村上 斉 (HITOSHI MURAKAMI)

ABSTRACT. In this note, Iwill discuss apossible relation between the Mahlermeasure of the colored Jones polynomial and the volume conjecture. In par-ticular, Iwill study the colored Jones polynomial of the figure eight knot onthe unit circle. Iwill also propose amethod to prove the volume conjecturefor satellites of the figure eight knot.

1. MAHLER MEASURE

Let $f(t)$ be a(non-zero) Laurent polynomial in $t$ with coefficient in Z. TheMahler measure $\mathrm{M}(f)$ of $f[5,6, 14]$ is defined to be

$\mathrm{M}(f):=\exp(\int_{0}^{1}\log|f(\exp(2\pi\sqrt{-1}x))|dx)$

It is known that $\mathrm{M}(/)$ is the product of the absolute values of the leading coefficientand all the roots that are greater than one. It is convenient to define its logarithmicversion:

$\mathrm{m}(f):=\int_{0}^{1}\log|f(\exp(2\pi\sqrt{-1}x))|dx$ .Then the logarithmic Mahler measure can be regarded as asort of ‘mean’ of thelogarithms of the values on the unit circle. Visit the web pageshttp: $//\mathrm{n}\mathrm{a}\mathrm{t}\mathrm{h}\mathrm{w}\mathrm{o}\mathrm{r}\mathrm{l}\mathrm{d}$ .wolfram. $\mathrm{c}\mathrm{o}\mathrm{m}/\mathrm{U}\mathrm{a}\mathrm{h}\mathrm{l}\mathrm{e}\mathrm{r}\mathrm{I}\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{e}$. htdfor more about the Mahler measure and alsohttp: $//\mathrm{n}\mathrm{a}\mathrm{t}\mathrm{h}$ . $\mathrm{u}\mathrm{c}\mathrm{r}.\mathrm{e}\mathrm{d}\mathrm{u}/\sim \mathrm{x}\mathrm{l}/\mathrm{k}\mathrm{n}\mathrm{o}\mathrm{t}\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{b}/\mathrm{k}\mathrm{n}\mathrm{o}\mathrm{t}\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{b}$.htmlfor problems on the Mahler measure of the Jones polynomial.

2. MAHLER MEASURE OF THE ALEXANDER POLYNOMIAL

Let $K$ be aknot in the three-sphere $S^{3}$ and $Mn(K)$ be the $N$-fold cyclic branchedcovering over $S^{3}$ branched along $K$ . Then it is well known that the order of the firsthomology group of $Mn(K)$ can be obtained in terms of the Alexander polynomial$\Delta(K;t)$ of $K$ (see for example [4, Corollary 9.8]).

Theorem 2.1.

(2.1) $|H_{1}(M_{N}(K); \mathbb{Z})|=\prod_{d=1}^{N-1}\Delta(K, \exp(2d\pi\sqrt{-1}/N))$ ,

where $|A|$ denotes the cardinality of a set $A$ if $A$ is a finite set and 0if it is infinite.Date: $23\mathrm{r}\mathrm{d}$ June, 2002.2000 Mathematics Subject Classification. Primary $57\mathrm{M}27$; Secondary $57\mathrm{M}25,57\mathrm{M}50,17\mathrm{B}37$,

$81\mathrm{R}50$.This research is partially supported by Grant-in-Aid for Scientific Research (B)

数理解析研究所講究録 1279巻 2002年 86-99

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HITOSHI MURAKAMI

If we take the logarithm of the both side of Equation (2.1) and divide by $N$ , wehave

$\frac{\log|H_{1}(M_{N}(I\mathrm{f}),\mathbb{Z})|}{N}.=\frac{\sum_{d=1}^{N-1}\log|\Delta(I\{’\exp(2\pi d\sqrt{-1}/N))|}{N},\cdot$

When $N$ grows, the right hand side approaches to the ‘mean’ of the logarithms ofthe values of $\Delta(K;t)$ on the unit circle, the logarithmic Mahler measure. In factthe following theorem is known to be true.

Theorem 2.2 (D. Silver and S. Williams [15]).

$\lim_{Narrow\infty}\frac{\log|H_{1}(M_{N}(I\mathrm{f}),\mathbb{Z})|}{N}.=\mathrm{m}$ (.A (K;t))

See [2, 1, 13] for other topics of the homology of the branched cyclic cover overaknot. See also [16] for the Mahler measure of the Alexander polynomial of alink.

3. MAHLER MEASURE $\mathrm{o}\mathrm{F}$ THE COLORED JONES POLYNOMIALS

Let $J_{N}(K;t)$ be the $N$-dimensional colored Jones polynomial of aknot $K$ nor-malized so that $J_{N}(O;t)=1$ for the unknot $O$ . We want to know the asymptoticbehavior of $J_{N}(K;t)$ for large $N$ .

Since

$\mathrm{m}(J_{N}(Kjt))=\int_{0}^{1}\log|J_{N}(K;\exp(2\pi\sqrt{-1}x))|dx$

$= \int_{0}^{N}\frac{\log|J_{N}(I\acute{\mathrm{c}},\exp(2r\pi\sqrt{-1}/N))|}{N}\cdot dr$,

it is helpful to study the asymptotic behavior of $\log|J_{N}(I\acute{\mathrm{s}};\exp(2\pi\sqrt{-1}r/N))|$ forafixed $r$ . Note that for $r=1$ , this problem is nothing but the volume conjecture[11, 8, 10, 9, 19, 18, 17, 12].

In the following sections Iwill discuss the colored Jones polynomials of thefigure-eight knot evaluated on the unit circle.

4. SOME CALCULATIONS ABOUT THE FIGURE-EIGHT KNOT

Let $E$ denote the figure-eight knot $4_{1}$ . Due to K. Habiro and T. Le, the followingformula is known.

(4.1) $J_{N}(E;t)= \sum_{k=0}^{N-1}\prod_{j=1}^{k}(t^{(N+j)/2}-t^{-(N+j)/2})(t^{(N-j)/2}-t^{-(N-j)/2})$ .

Using this formula we can prove the following result.

Theorem 4.1. Let $r$ be a positive integer or a real number satisfying $5/6<r$ $<7/6$ .Then

$\lim_{Narrow\infty}2\pi\frac{\log|J_{N}(E,\exp(2r\pi\sqrt{-1}))|}{N}.=\frac{2\Lambda(r\pi+\theta(r)/2)-2\Lambda(r\pi-\theta(r)/2)}{r}$ ,

where $\mathrm{A}(\mathrm{z}):=-\int_{0}^{z}\log|\sin x|dx$ is the Lobachevski function and $\mathrm{o}(\mathrm{r})$ is the smallest

posrtive number satisfying $\cos \mathrm{O}(\mathrm{r})=\cos(2r\pi)-1/2$ .In particular, if $r$ is a positive integer then

$2 \pi\lim_{Narrow\infty}\frac{\log|J_{N}(E,\exp(2r\pi\sqrt{-1}/N))|}{N}.=\frac{\mathrm{V}\mathrm{o}\mathrm{l}(S^{3}\backslash E)}{r}$.

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MAHLER MEASURE AND THE VOLUME CONJECTURE

Proof of Theorem 4.1 when $r$ is a positive integer. Replacing $t$ with $\exp(2\mathrm{r}\pi\sqrt{-1}/f$

in Equation (4.1), we have

$J_{N}(E; \exp(2\mathrm{r}\pi\sqrt{-1}/N))=\sum_{k=0}^{N-1}\prod_{j=1}^{k}\{2\sin(jr\pi/N)\}^{2}$

If we put $f(k):= \prod_{j=1}^{k}\{2\sin(j\mathrm{r}\pi/N)\}^{2}$ , then $f$ takes its maximum at $k\mathrm{r}\pi/N=$

$5\mathrm{t}\mathrm{t}/6$ if $N$ is large. Therefore

$N^{\cdot} arrow\infty \mathrm{h}\mathrm{m}\frac{\log|J_{N}(E\exp(2\mathrm{r}\pi\sqrt{-1}/N))|}{N}=2\lim_{Narrow\infty}\sum_{\mathrm{j}=1}^{5N/6r}\frac{\log(2\sin(j\mathrm{r}\pi/N))}{N}$

$= \frac{2}{r\pi}\lim_{Narrow\infty}\int_{0}^{5\pi/6}\log(2\sin x)dx$

$=- \frac{2}{\mathrm{r}\pi}\Lambda(5\pi/6)$

$= \frac{\mathrm{V}\mathrm{o}\mathrm{l}(S^{3}\backslash E)}{2\mathrm{r}\pi}$ .See [8, Theorem 4.2] for details.

$\square$

Remark 4.2. The case where $r$ $=1$ is due to R. Kashaev [3] and T. Ekholm [8].Proof of Theorem 4.1 when $5/6<f$ $<1$ . We will assume $N$ is sufficiently large sothat $j/N$ can behave as if it is acontinuous parameter.

Put $\omega$ $:=\exp(2\pi\sqrt{-1}/N)$ . Since$\omega^{r(N+j)/2}-\omega^{-r(N+j)/2}=2\sqrt{-1}\sin(f(N+j)\pi/N)$

and

$\omega^{r(N-j)/2}-\omega^{-r(N-j)/2}=2\sqrt{-1}\sin(\mathrm{r}(N-j)\pi/N)$ ,we have

$\prod_{\mathrm{j}=1}^{k}(\omega^{r(N+\mathrm{j})/2}-\omega^{-r(N+j)/2})(\omega^{r(N-\mathrm{j})/2}-\omega^{-r(N-j)/2})$

Put

$= \prod_{j=1}^{k}4$ si$\mathrm{n}(rj\pi/N +\mathrm{r}\pi)$ $\sin(\gamma j\pi/N-r\pi)$ .

$g(j)$ : $=4\sin(rj\pi/N+\mathrm{r}\pi)$ $\sin(\mathrm{r}j\pi/N-\mathrm{r}\pi)$

$=2\cos(2r\pi)-2\cos(2\mathrm{r}j\pi/N)$

and

$f(k)$ $:= \prod_{j=1}^{k}g(j)$

so that $J_{N}(E; \omega^{r})=\sum_{k=0}^{N-1}f(k)$ . We also put

$A:= \frac{N(1-\mathrm{r})}{f}$ , $B:= \frac{N\theta(\mathrm{r})}{2\mathrm{r}\pi}$ , and $C:= \frac{N(2\pi-\theta(\mathrm{r}))}{2r\pi}$

where $\theta(\mathrm{r})$ is the smallest positive number satisfying $\cos\theta(r)=\cos(2\mathrm{r}\pi)-1/2$ asbefore. Note that since $5/6<f$ $<1,1/2<\mathrm{c}\mathrm{o}\mathrm{e}(2\mathrm{r}\pi)$ $<1$ and so the equation$\cos\theta(r)=\cos(2\mathrm{r}\pi)-1/2$ has asolution.

Note that $0<A<B<C<N$ (see Figure 1)

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HITOSHI MURAKAMI

FIGURE 1. Graph of $g(j)$ when $5/6<r<1$

Since we have(1) $\mathrm{g}(\mathrm{j})<0$ for $j<A$ , and $g(j)>0$ for $j>A$ , and(2) $f_{j}>1$ for $B<j<C$ ,

we see(3) If $j<A$ then the signs of $f(j)$ alternate, that is, $f(j-1)\mathrm{f}(\mathrm{j})<0$ , and if

$j>A$ then the signs of $f(j)$ are constant, and(4) $|f(\mathrm{O})|>|f(1)|>\cdots>|f(B)|$ and $|f(B+1)|<\cdots<|f(C)|$ .

Let $f_{\mathrm{M}\mathrm{A}\mathrm{X}}1$ be the maximum of $\{|f_{j}|\}$ for $0\leq j\leq N-1$ . Note that $f_{\mathrm{M}\mathrm{A}\mathrm{X}}=\mathrm{f}(\mathrm{j})$ .We can show the following inequality.

Claim 4.3.$0<fMAX$ $-1\leq|J_{N}(E;\omega^{r})|\leq N$ fMAX

Proof of the Claim 4.3. We only show the second inequality for the case where $A$

is even. In this case since $f(0)=1$ , $f(2j-1)+f(2j)<0$ for $2j<A$ , and $f(j)<0$for $j\geq A-1$ , we have

$|J_{N}(E;\omega^{r})|$

$=|f(0)+\{f(1)+f(2)\}+\{f(3)+f(4)\}+\cdots+\{f(A-3)+f(A-2)\}$

$+f(A-1)+\mathrm{f}(\mathrm{j})+f(A+1)+\cdots+f(N-1)|$

$=|f(1)+f(2)|+|f(3)+f(4)|+\cdots+|f(A-3)+f(A-2)|$

$+|f(A-1)|+|f(A)|+\cdots+|f(N-1)|$

-1$>f_{\mathrm{M}\mathrm{A}\mathrm{X}}-1$

and the second equality follows. 0

IMAX are temporarily Nana, Reina, and Lina

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MAHLER MEASURE AND THE VOLUME CONJECTURE

Therefore we have

$\lim_{Narrow\infty}\frac{\log|J_{N}(E_{j}\omega^{r})|}{N}$

$= \lim_{Narrow\infty}\frac{\log(f_{\mathrm{M}\mathrm{A}\mathrm{X}})}{N}$

$= \lim_{Narrow\infty}\frac{1}{N}\sum_{\mathrm{j}=0}^{C}\{\log($2 $\sin(\frac{rj\pi}{N}+r\pi-\pi))+\log($2 $\sin(-\frac{\mathrm{r}j\pi}{N}+\mathrm{r}\pi))\}$

$= \frac{1}{\mathrm{r}\pi}\int_{r\pi-\pi}^{r\pi-\theta(r)/2}\log(2\sin x)dx$$+ \frac{1}{\mathrm{r}\pi}\int_{r\pi-\pi+\theta(r)/2}^{r\pi}\log(2\sin x)dx$

$= \frac{1}{r\pi}(\mathrm{A}(\mathrm{r}\pi-\pi)-\Lambda(\mathrm{r}\pi-\theta(\mathrm{r})/2)+\Lambda(r\pi-\pi+\theta(\mathrm{r})/2)-\Lambda(r\pi))$

$= \frac{1}{r\pi}(\Lambda(r\pi+\theta(\mathrm{r})/2)-\Lambda(\mathrm{r}\pi-\theta(r)/2))$ .

Here we use the $\pi$-periodicity of the Lobachevski function. (See [7].) $\square$

Proof when $1<r<7/6$ . The proof is similar to the case where $5/6<r<1$ . SeeFigure 2. $\square$

FIGURE 2. Graph of $g(j)$ when $1<f$ $<7/6$

As acorollary we have

Corollary 4.4.

$\lim_{rarrow 1}\{\lim_{Narrow\infty}\frac{\log|J_{N}(E,\exp(2r\pi\sqrt{-1}))|}{N}.\}=\lim_{Narrow\infty}\frac{\log|J_{N}(E,\exp(2\pi\sqrt{-1}))|}{N}$

.

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HITOSHI MURAKAMI

By some calculation using PARI-GP 2 and MAPLE $\mathrm{V}$ , it seems that the followingequality holds.

(4.2) $2r \pi\lim_{Narrow\infty}\frac{\log|J_{N}(E\omega^{r})|}{N}=\{$$V(r)$ if $0\leq r\leq 1$ ,

$W(r-[r])$ if $r>1$ ,

where $[r]$ denotes the greatest integer which does not exceed $r$ , and

$V(x):=$

’

0 if $0\leq x<1/6$ ,

$\Lambda(x\pi+\theta(x)/2-\pi/2)-\Lambda(x\pi-\theta(x)/2-\pi/2)$ if $1/6\leq x<3/4$ ,

$\backslash \Lambda(x\pi+\theta(x)/2)-\Lambda(x\pi-\theta(x)/2)$ if $3/4\leq x\leq 1$ ,

and

$W(x):=\{$

$\Lambda(x\pi)+\theta(x)/2-\Lambda(x\pi-\theta(x)/2)$ if $0\leq x<1/4$ ,

$\Lambda(x\pi)+\theta(x)/2-\pi/2)-\Lambda(x\pi-\theta(x)/2-\pi/2)$ if $1/4\leq x<3/4$ ,

$\Lambda(x\pi+\theta(x)/2)-\Lambda(x\pi-\theta(x)/2)$ if $3/4\leq x\leq 1$ .

See Figures 3 and 4 for graphs of $V$ and $W$ . See also Figures 6, 7, 8, 9, 10, and 11

FIGURE 3. Graph of $V$ , where 2.029883213... is the volume of thefigure-eight knot complement.

for some results of calculations supporting Equation 4.2.

2 $\mathrm{G}\mathrm{P}/\mathrm{P}\mathrm{A}\mathrm{R}\mathrm{I}$ CALCULATOR Version 2.0.20 (beta)

i586 running $\mathrm{c}\mathrm{y}\mathrm{g}\mathrm{w}\mathrm{i}\mathrm{n}_{-}98-4$.10 (ix86 kernel) $32$ -bit version(readline v1.O enabled, extended helP not $\mathrm{a}\mathrm{v}\mathrm{a}\mathrm{i}\mathrm{l}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e}\rangle$

Copyright (C) by 1989-1999 byC. Batut, K. Belabas, D. Bernardi, H. Cohen and $\mathrm{n}$ . Olivier.

The program is available at http://www.parigp-home.de

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MAHLER MEASURE AND THB VOLUME CONJECTURE

FIGURE 4. Graph of W.

lf Equation (4.2) is true, one could have the following result on the asymptoticbehavior of the logarithmic Mahler measure of the colored Jones polynomials of thefigure-eight knot.

Remark 4.5. Caution! There are fake calculations in the following.

$\lim_{Narrow\infty}\frac{\mathrm{m}(J_{N}(E_{j}t))}{1\mathrm{o}\mathrm{g}N}=\lim_{Narrow\infty}\frac{1}{\log N}\int_{0}^{1}\log|J_{N}(E;\exp(2\pi\sqrt{-1}x))|dx$

$= \lim_{N?arrow\infty}\frac{1}{\log N}\int_{0}^{N}\frac{\log|J_{N}(E_{j}\exp(2\pi\sqrt{-1}r/N))|}{N}d$ ,

$= \lim_{N?arrow\infty}\frac{\mathrm{l}}{2\pi 1\mathrm{o}\mathrm{g}N}\{\int_{0}^{1}\frac{V(\mathrm{r})}{f}d\mathrm{r}+\sum_{k=1}^{N-1}\int_{k}^{k+1}\frac{W(\mathrm{r}-[\mathrm{r}])}{f}dt\}$

$= \lim_{Narrow\infty}\frac{\mathrm{l}}{2\pi 1\mathrm{o}\mathrm{g}N}\{\int_{0}^{1}\frac{V(f)}{f}dr+\sum_{k=1}^{N-1}\int_{0}^{1}\frac{W(r)}{f+k}d\mathrm{r}\}$ ,

where $=$?means that there is adoubt in the equality. At the first Iuse $N$ in theintegral, which should be independent of $N$ , and at the second Iassume (4.2).

Now since$\frac{1}{k+1}\leq\frac{1}{\mathrm{r}+k}$. $\leq\frac{1}{k}$

for $0\leq f$ $\leq 1$ , we have

$\int_{0}^{1}\frac{W(\mathrm{r})}{k+1}d\mathrm{r}\leq\int_{0}^{1}\frac{W(\mathrm{r})}{\mathrm{r}+k}d\mathrm{r}\leq\int_{0}^{1}\frac{W(\mathrm{r})}{k}d\mathrm{r}$ .Therefore we have

$\sum_{k=1}^{N-1}\int_{0}^{1}\frac{W(\mathrm{r})}{k+1}d\mathrm{r}\leq\sum_{k=1}^{N-1}\int_{0}^{1}\frac{W(\mathrm{r})}{\mathrm{r}+k}dr\leq\sum_{k=1}^{N-1}\int_{0}^{1}\frac{W(\mathrm{r})}{k}dr$.

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HITOSHI MURAKAMI

Since$\lim_{Narrow\infty}\frac{\sum_{k=1}^{N-1}\frac{1}{k+1}}{1\mathrm{o}\mathrm{g}N}=\lim_{Narrow\infty}\frac{\sum_{k=1}^{N-1}\frac{1}{k}}{1\mathrm{o}\mathrm{g}N}=1$

and $V(r)=0$ for $0\leq r\leq 1/6$ , we finally have

$2 \pi\lim_{Narrow\infty}\frac{\mathrm{m}(J_{N}(E,t))}{1\mathrm{o}\mathrm{g}N}.=?\int_{0}^{1}W(r)dr=1.450191516\ldots$.

5. SATELLITES OF THE FIGURE-EIGHT KNOT

In this section, Iwould like to study the volume conjecture for the $(2, 1)$-cableand the Whitehead double of the figure-eight knot. Linear skein method givesus formulas to describe the colored Jones polynomials of such knots but one ofthe difficulties is that the value of the unknot is not 1but $(t^{N/2}-t^{-N/2})/(t^{1/2}-$

$t^{-1/2})$ (see for example [4, Chapter 14]), and so they vanish if we evaluate themat the $N$-th root of unity. To avoid this Iwill use Corollary 4.4 to analyze theasymptotic behaviors of the colored Jones polynomials. Unfortunately, Icannotgive arigorous result here but Ihope that this method gives an insight to solve thevolume conjecture for satellite knots.

Remark 5.1. Caution! There are many fake arguments in this section.

Let $E^{2}$ be the $(2, 1)$-cable of the figure-eight knot. By using techniques in [4,Chapter 14], we see

$J_{N}(E^{2}; t)(t^{N/2}-t^{-N/2})/(t^{1/2}-t^{-1/2})=$ $c \leq 2N-,1\sum_{c\mathrm{o}\mathrm{d}\mathrm{d}}u(c;t^{1/4})J_{c}(E;t)$

,

where $u(c;t^{1/4})$ is amonomial in $t^{1/4}$ . Replacing $t$ with $\omega^{r}$ with $5/6<r<7/6$$(r\neq 1)$ , we have

$J_{N}(E^{2}; \omega^{r})=\frac{\sin(r\pi/N)}{\sin(r\pi)}$$1 \leq c\leq 2N’-1\sum_{c\mathrm{o}\mathrm{d}\mathrm{d}}u(c;\omega^{r/4})J_{c}(E;\omega^{r})$

.

Note that $\sin(r\pi)\neq 0$ . If one could show that the maximum of the terms in thesummation dominates the limit, which is akind of saddle point method, we couldhave

$\lim_{Narrow\infty}\frac{\log|J_{N}(E^{2}\cdot\omega)|}{N},=\lim_{r?arrow 1}\{\lim_{Narrow\infty}\frac{\log|J_{N}(E^{2}\cdot\omega^{r})|}{N},\}$

$= \lim_{r?arrow 1}\{\frac{\log|_{1\leq}\max_{c\leq 2N-1}J_{\mathrm{c}}(E,\omega^{r})|}{N}\}$

$= \lim_{r?arrow 1}\{\frac{\log|J_{N}(E,\omega^{r})|}{N}\}$

$= \frac{\log|J_{N}(E,\omega)|}{N}$ ,

proving the volume conjecture for the $(2, 1)$ -cable of the figure-eight knot. Here $=$

?

indicates that there is adoubt in the equality; at the fist equality, Ichange theorder of the limits, at the second, Iassume the maximum dominates the limit, andat the third, Iassume that $J_{c}(E,\omega^{r})$ takes its maximum at $c=N$ , which can beobserved by calculation using PARI. See Figure 5.

Ibelieve that the gaps here are not so big

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MAHLER MEASURE AND THB VOLUME CONJECTURE

FIGURE 5. Plot of (c,$2\pi\log|J_{\mathrm{c}}(E;\omega)|/N)$ with N $=800$ .

Let $D(E)$ be the Whitehead double of the figure-eight knot (with any framing).Then using similar techniques we have

$J_{N}(D(E)_{j}t)(t^{N/2}-t^{-N/2})/(t^{1/2}-t^{-1/2})= \sum_{\mathrm{c}\cdot \mathrm{o}\mathrm{d}\underline{\mathrm{d}}c\leq 2N1}v(c;t)J_{c}(E;t)$

,

with

$v(c;t)=$$d \leq 2N-,1\sum_{d\mathrm{o}\mathrm{d}\mathrm{d}}\frac{\Delta_{d}\theta(N-1,N-1,c-1)}{\Delta_{c}\theta(N-1,N-1,d-1)}$

$\{\begin{array}{lllll}N -\mathrm{l} N -1 cN -1 N -\mathrm{l} d\end{array}\}$

where $\Delta_{x}$ , $\theta(x, y, z)$ and $\{\begin{array}{lll}x y zu v w\end{array}\}$ are defined in [4, Chapter 14]. Similar calcu-lation shows that for the Whitehead link $W$ we have

$J_{N}(W;t)(t^{N/2}-t^{-N/2})/(t^{1/2}-t^{-1/2})=. \sum_{c\cdot \mathrm{o}\mathrm{d}\underline{\mathrm{d}}c\leq 2N1}v(.c; t)J_{N,c}(H;t)$,

where $J_{N,c}(H;t)$ is the colored Jones polynomial of the Hopf link $H$ colored with$N$ and $c$ , which is equal to $\Delta_{(N-1)(c-1)}$ .

Now we have the following fake calculations with doubtful equalities:

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HITOSHI MURAKAMI

$= \lim_{r?arrow 1}\{\frac{\log|v(N,\omega^{r})J_{N}(E,\omega^{r})|}{N}..\}$

$= \lim_{rarrow 1}\frac{\log|v(N,\omega^{r})|}{N}.+\lim_{rarrow 1}\frac{\log|J_{N}(e\cdot\omega^{r})|}{N}$

,

$= \lim_{rarrow 1}\frac{\log|v(N,\omega^{r})|}{N}.+\frac{\log|J_{N}(E,\omega)|}{N}$ ,

On the other hand

$\lim_{Narrow\infty}\frac{\log|J_{N}(W\omega)|}{N}=\lim_{r?arrow 1}\{\lim_{Narrow\infty}\frac{\log|J_{N}(W,\omega^{r})|}{N}.\}$

$= \lim_{r?arrow 1}\{\frac{\log|v(N_{}\omega^{r})J_{N,N}(H\omega^{r})|}{N}\}$

$= \lim_{rarrow 1}\frac{\log|v(N\cdot\omega^{r})|}{N}$,

since $J_{N,N}(W, \omega^{r})$ can be expressed in terms of sine of $1/N$ . Therefore if we acceptthese calculations, we could prove

$\lim_{narrow\infty}\frac{\log|J_{N}(D(E),\omega)|}{N}=\lim_{narrow\infty}\frac{\log|J_{N,N}(W,\omega)|}{N}+\lim_{narrow\infty}\frac{\log|J_{N}(E,\omega)|}{N}$.

Noting that the complement of $D(E)$ is the union of those of the figure-eight knotand the Whitehead link, which is the volume conjecture for the Whitehead doubleof the figure-eight knot.

Acknowledgments. This article is prepared for the proceedings of the workshop‘Volume Conjecture and Its Related Topics’ held at the International Institutefor Advanced Studies from 5th to 8th March, 2002. It was also supported bythe Research Institute for Mathematical Sciences, Kyoto University. Iwould liketo thank the institutes for their hospitality. Ialso thank Kazuhiro Hikami forintroducing the computer program PARI.

Part of this work was done when Iwas visiting Warwick University to attendthe workshop Quantum Topology’ from 18th to $22\mathrm{n}\mathrm{d}$ March, 2002, and Universitedu Quebec \‘a Montr\’eal to attend the workshop ‘Knots in Montreal $\mathrm{I}\mathrm{I}$ ’from 20th to21st April, 2002. Thanks are due to the universities and to the organizers of theworkshops, Stavros Garoufalidis, Colin Rourke, Steven Boyer, and Adam Sikora.

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[9] –, Optimistic calculations about the Witten-Reshetikhin-Tu faev invariants ofclosed three-manifolds obtained from the figure-eight knot by integral Dehn surgeries,

S\={u}rikaisekikenky\={u}sho K\={o}ky\={u}roku (2000), no. 1172, 70-79. MR 1805 729

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[10] –, Kashaev’s invariant and the volume of a hyperbolic knot after Y. Yokota, Physicsand combinatorics 1999 (Nagoya), World Sci. Publishing, River Edge, NJ, 2001, pp. 244-272.MR 1865040

[11] H. Murakami and J. Murakami, The colored Jones polynomials and the simplicial volume ofa knot, Acta Math. 186 (2001), no. 1, 85-104. MR $2\propto 12\mathrm{b}:5705$

[12] H. Murakami, J. Murakami, M. Okamoto, T. Takata, and Y. Yokota, Kashaev’s conjectureand the Chem-Simons invariants of knots and links, $\mathrm{a}\mathrm{r}\mathrm{X}\mathrm{i}\mathrm{v}:\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{h}.\mathrm{G}\mathrm{T}/0203119$.

[13] R. Riley, Growth of order of homology of cyclic branched covers of knots, Bull. London Math.Soc. 22 (1990), no. 3, 287-297. MR92g:570l7

[14] K. Schmidt, Dynamical systems of algebraic origin, Birkhauser Verlag, Basel, 1995. MR97c:2804l

[15] D. S. Silver and S. G. Williams, Mahler measure, links and homology growth,$\mathrm{a}\mathrm{r}\mathrm{X}\mathrm{i}\mathrm{v}:\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{h}.\mathrm{G}\mathrm{T}/003127$.

[16] –, Mahler measure of Alexander polynomials, $\mathrm{a}\mathrm{r}\mathrm{X}\mathrm{i}\mathrm{v}\mathrm{m}\mathrm{a}t\mathrm{h}.\mathrm{G}\mathrm{T}/0105234$ .[17] Y. Yokota, On the volume conjecture for hyperbolic knots, $\mathrm{a}\mathrm{r}\mathrm{X}\mathrm{i}\mathrm{v}.\cdot \mathrm{m}\mathrm{a}\mathrm{t}\mathrm{h}.\mathrm{Q}\mathrm{A}/0009165$ .[18] –, On the volume conjecture for hyperbolic knots, Proceedings of the 47th All Japan

Topology Symposium (Inamori Hall, Kagoshima University), July 2000, pp. 38-44.[19] –, On the volume conjecture of hyperbolic knots, Knot Theory -dedicated to Professor

Kunio Murasugi for his 70th birthday (M. Sakuma, ed.), March 2000, pp. 362-367.

FIGURE 6. Graph of W (gray) and $2r\pi\log|J_{N}(E;\omega^{r})|/N$ withN $=2000$ (black) for $0\leq f$ $\leq 1$ .

DBPARTMENT OF MATHEMATICS, Tokyo $\mathrm{I}\mathrm{N}\mathrm{S}\mathrm{T}\Pi.\cup \mathrm{T}\mathrm{E}$ OF TECHNOLOGY, OH-OKAYAMA, MBGURO,TOkyo 152-8551, iApAN

$E$-mail address: starsheabky3.3web.ne.jp

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FIGURE 7. Graph of $W$ (gray) and $2r\pi\log|J_{N}(E;\omega^{r})|/N$ with$N=2000$ (black) for $1\leq r\leq 2$ .

FiGURE 8. Graph of $W$ (gray) and $2r\pi\log|J_{N}(E;\omega^{r})|/N$ with$N=2000$ (black) for $2\leq r\leq 3$ .

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FIGURE 9. Graph of W (gray) and $2r\pi\log|J_{N}(E;\omega^{r})|/N$ withN $=2000$ (black) for $3\leq f$ $\leq 4$ .

FIGURE 10. Graph of W (gray) and $2\mathrm{r}\pi\log|J_{N}(E;\omega^{r})|/N$ withN $=2000$ (black) for $4\leq f$ $\leq 5$ .

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FIGURE 11. Graph of $W$ (gray) and $2r\pi\log|J_{N}(E;\omega^{r})|/N$ with$N=8000$ (black) for $4\leq r\leq 5$ .

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