数理解析研究所講究録
第 840 巻 1993 年 1-19

1Tunnel numbers of knots and Jones-Witten invariants

TOSHITAKE KOHNO
Department of Mathematical Sciences
University of Tokyo
Komaba, Tokyo 153, Japan河野俊文 (東大数理)1. Introduction and main results

The purpose of this paper is to discuss some topological consequences derived from
Jones-Witten invariants for links in 3-manifolds. In [Jo], Jones obtained upper bounds
for the values of the Jones polynomial at roots of unity in terms of the braid index. In
the present paper we focus on classical invariants called the tunnel numbers of knots.
Let $K$ be a knot in . We are going to describe alower bound for the tunnel number$t(K)$ by means of special values of the Jones polynomial and its generalizations.
The tunnel number is the minimal number of arcs
, so that the closure of thecomplementary space of the regular neighborhood of
is homeomorphic
to a handlebody. In Figure 1 we indicate that the tunnel number of a trefoil knot isequal. to one. It is clear that the tunnel number is bounded from above the minimal
number of crossings of the knot diagram and that the minimal number of generatorsof the fundamental group of the knot complement is at most $t(K)+1$ . This definitionof the tunnel number is also generalized in a natural way to a knot in a closed oriented
3-manifold. In the preceding article [Ko3], we developed a general framework of giving
an estimate for the tunnel number in terms of the Jones-Witten invariants, based on
the monodromy of conformal field theory. The scope of the present paper is to givea refinement of the inequality obtained in [Ko3] and to present a more combinatorialdescription using quantum $6j$ -symbols at roots of unity.
To state our main results let us recall some basic construction for invariants offramed links associated with the quantum group $U_{q}(sl(2, C))$ in the sense of Drinfel’d[D] and Jimbo [J]. We fix a positive integer called a level and we put $r=l+2$ .
In this paper we discuss the invariants at $q=e^{2\pi\sqrt{-1}/r}$ . Let $P_{+}(l)$ be a set of half
integers defined by$S^{3}$$\gamma_{1},$$\cdots$$\gamma_{t}$$K\cup\gamma_{1}\cup\cdots\cup\gamma_{t}$

$l$

(1-1)

$p_{+}(l)=$ $\{0,1/2,1, \cdots l/2\}$.

Here $j\in P_{+}(l)$ corresponds to an irreducible $U_{q}(sl(2, C))$ module of dimension $2j+1$ .
Let be a framed link inwith $m$ components. As was constructed in KirillovReslzetikhin [KR], for: $\{1, \cdots rn\}arrow p_{+}(l)$ , we can associate an invariant of$L$$S^{3}$$\lambda$

Partially supported by Grant-in-Aid for Scicntific Rcscarch on Priority Arcas 231“Infinite Analysis.”
Typesct by$A_{\mathcal{M}^{S-?EX}}$2oriented framed link denoted by
due to Kirby-Melvin [KM]. If
polynomial given by$J_{r}(L, \lambda)$$\lambda\equiv 1/2$

(1-2)

. In this paper, we follow a normalization, then $J.(L, 1/2)$ is a variant of the Jones$J_{r}(L, 1/2)=(2 \cos\frac{\pi}{r})t^{3L\cdot L}V_{L}(e^{-\frac{2\cdot\sqrt{-1}}{r})}$with $t=e^{\pi\sqrt{-1}/2}$ ‘ whereis the original Jones polynomial normalized as in [Jo]. In
is the sum of the sum of the entries of the linking
the above expression the symbol
is the framing on the component . In thematrix $\sum_{ij}lk(L_{i},L_{j})$ , where$K$
case of a knot assigned with a color $j\in P_{+}(l)$ , we have the following description.
, $J_{K,n}=J_{r}(K,n/2)$ , we have a recursive formulaPutting for an integer $n=0,1,$$V_{L}$$L\cdot L$$lk(L_{i\prime}.L_{i})$$\cdots$(1-3)$L_{i}$$l,$$J_{K,\tau\iota}= \sum_{i=0}^{[n/2]}(-1)^{i}(\begin{array}{ll}n -i i\end{array})J_{K^{\mathfrak{n}-2:},1}$stands for the parallel push offs defined by means of the framing. Let us
where
notice that $|J_{r}(K,j)|$ does not depend on the framing on $K$ . For $i,j\in P_{+}(l)$ we put$K^{\gamma}$$\gamma$$S_{ij}= \sqrt{\frac{2}{f}}\sin\frac{(2i+1)(2j+1)\pi}{r}$.

With this notation we have the following.
Theorem A. Let $K$ be a$l\sigma not$in$S^{3}$. Then, the tunnel numb er $t(K)$ sa fies$\sum_{j\in P_{+}(l)}|J_{r}(K,j)|^{2}\leq 2^{-t(K)}S_{00}^{-2(t(K)+1)}$

$tis$

,

wliich gives a lower estim ate for the unn $elnuJnbel\cdot$.
$t$

Let us recall that we have a symmetry$(1rightarrow 4)$$|J_{r}(K,j)|=|J_{r}(K, l/2-j)|$obtained in [KM]. Based on this, we can deduce from Theorem A that
(1-5)$|J_{r}(K,j)|^{2} \leq(\frac{1}{2S_{00}^{2}})^{t(K)+1}-1$

for any $j\in P_{+}(l)$ such that $j\neq 0,$ $l/2,$ $j\neq l/2-j$ .
In [W], Witten introduced a series of invariants for a link in 3-manifold based
quantum field theory with Chern-Simons action. A Dehn surgery formula of the
Witten invariant for $sl(2, C)$ was studied extensively by Reshetikhin-Turaev [RT] and
$011$

3
-Melvin [KM]. Let $K$ be a
in a closed oriented 3-manifold $M$ assigned with
$j\in P_{-:}(l)$ .
disjoint from $K$ andWe suppose that there is a framed linkin$(M,K)$ is obtained by the Dehn $sur_{o}ery$ on the framed link
. Let us denote by $m$the number of components of . The invariant we are going to investigate is$kn\overline{o}$$Kirb_{-}\cdot-$$S^{3}$

$L$

$L$

$L$

(1-6)

$Z_{r}(M, K;j)=C^{sig\tau\iota(L)} \sum_{\lambda}S_{0\lambda(1)}\cdots S_{0\lambda(m)}J_{r}(K\cup L,j\cup\lambda)$, $m\}arrow P_{+}(l),$ $C=\exp(-3(r-2)\pi\sqrt{-1}/4r)$ ,where the sum is for any: 1,
anddenotes the signature of the linking matrix for . In particular, in thecase;Le knot $K$ is empty, we obtain an invariant of 3-manifold $Z_{r}(M)$ . Let us notethat
normalize the invariant so that $Z.(S^{3})=1$ , in contrast to the Witten’s
original normalization. Our result for the tunnel numbers of knots in 3-manifolds isthe following.
$\lambda$$\cdots$

$\{$

$L$

$\sim^{\overline{\backslash }:gn(L)}$$v^{-}e$

Theorem B. Let $M$ be a closed oriented 3-man ifold and let $K$ be a knot in M. Tlien,byof the Witten invarian , th tunnel number $t(K)llas$ a lower es $tim$ ate$n3e\infty s$

$t$

$e$

$\sum_{j\in P_{+}(l)}|Z_{r}(JI, K;j)|^{2}\leq S_{00}^{-2(t(K)+1)}$$Le\iota\Sigma$

class

.

its mapping
be a closed oriented surface of genus and we denote by. To prove our main
we describe a projective unitary representation$g$$\mathcal{M}_{g}$$resul\sim*$$\circ roup$$\rho;_{J}\triangleright 1_{g}arrow GL(\mathcal{H}\Sigma)$using an idea due to Moore-Seiberg [MS] and Reshetikhin-Turaev [RT].
By means of this representation we show$constr\iota lcted$

(1-6)
with

$Z_{r}(M, K;j)=S_{00}^{-g}<v_{j}^{*},$ $\rho(h)v_{0}>$$\vee\prec^{-}$} $mev_{0},v_{j}\in \mathcal{H}_{\Sigma}$

if the knot exterior admits a splitting$E(K)=cl(V-N(K)) \bigcup_{h}(-V)$

with a handlebodyof genus . Here $N(K)$ is a tubular neighborhood of $K,$
stands for the closure, and:is a glueing map. The equality holds
when the glueing map is identity. Theoren) is a consequence of the unitarity of ,and Theorem A is shown as a special case $M=S^{3}$ with an improvement using thesymmetry derived from the Dynkin agram automorphism.
Let us observe tlxat in tlie case $K$ is cmpty the unitarity implies
$V$

$cl$

$g$

$h$

$\partial Varrow\partial V$

$B$

$di$

(1-8)

$|Z_{r}(M)|\leq S_{00}^{-g}$$\rho$

4
which gives a lower bound for the Heegaard genus of $M$ . The inequality 1.8 has beenobtained in [Wa], [Ko2] and [G].
The above interpretation of the Jones-Witten invariants in terms of the mappingclass group was studied in [Kol] and independently in [C] and [CLM]. Even in the casethe above 3-dimensionaJ point of view plays an essential role to showof knots in
our inequality, which reveals a close relationship between the special values Jones
polynomial and the quantum field theory.
nomialLet us also recall that the topological meaning of the values ofJonesat certain roots of unity is well studied (see [Jo}). Let
be the 2-fold branchedcovering of
branched along a link L. In the case $r=6$ , it is known by Lipson [Lp]and Lickorish-Millet [LM] that$S^{3}$$t1_{1}e$$P^{O_{\wedge}^{]\underline{\}}}}$$M_{L}$$S^{3}$(1-9)$|V_{L}(e^{\underline{\pi}\sqrt{-1}/6})|=3^{d/2}$

where $d=\dim H_{1}(M_{L}, Z/3Z)$ . Hence the inequality 1.5 implies in this case(1-10)

2 $\cdot 3^{d(K)+1}+|J_{r}(K, 1)|^{2}\leq 2\cdot 6^{t(K)+1}-2$with $d(K)=\dim H_{1}(M_{K}, Z/3Z)$ for $\tau=6$ . A numerical computation in$\dot{\overline{\iota}}he$case$r=5showsthatifaknotKsatisfies$$|V_{K}(e^{2\pi\sqrt{-1}/5})|>2.1489\ldots$,

then we have $t(K)\geq 2$ . Although we could not get an inequality sharper than this,we can derive several interesting consequences (see Remark 4.10).
The paper is organized in the following way. In section 2, we recall basic facts on
quantum $6j$ symbols at roots of unity and the associated invariants of framed links
following Kirillov and Reshetikhin [KR]. In section 3, we construct projective unitary
representations of the mapping classand we prove our main results. In section
4, we discuss a generalization of our construction for a complcx simple Lie algebra
based on the monodromy of the Wess-Zumino-Witten model.
$\circ roups$

Acknowledgement: The author would like to thank T. Kobayashi and M. Sakuma
for helpful conversations on the tunnel number.

Figurc 1

5

Figure 2

2. Quantum $6j$ symbols and tangle operators
Let us recall an explicit form of quantum $6j$ -symbols at $q=e^{2\pi\sqrt{-1}/r}follo\backslash ving$
Killirov and Reshetikhin [KR] (see also [TV]).
We say that a triple $i,j,$ $k\in P_{+}(l)$ is admissible if and only if it satisfies theconditions$|i-j|\leq k\leq i+j$(2-1)

$i+j+k\in Z$$i+j+k\leq l$ .

We adopt the following convention of a q-integer.
$[n]= \frac{q^{n/2}-q^{-n/2}}{q^{1/2}-q-1/2}=\frac{\sin n\pi/r}{\sin\pi/r}$

For an admissible triple $(a, b, c)$ we put$\triangle(abc)=(\frac{[a+b-c]![a+c-b]![b+c-a]!}{[a+b+c+1]!})^{1/2}$

An ordered 6-tuple$a,$ $b,$ $c,$ $d,$ $e,$ $f\in

P_{+}(l)$is said to be admissible if$t1_{1}e$triples$(a, b, c),$ $(c, d, e),$ $(a, e, f),$ $(b, d, f)$

are admissible in the above sense. For an admissible 6-tuple the Rakah-Wigner symbol
is given by$\{\begin{array}{lll}a b cd e f\end{array}\}=\triangle(abc)\triangle(aef)\triangle(bdf)\triangle(cde)\sum_{z}(-1)^{z}[z+1]!\cross$$-${ $[z-a-b-c]$ ! $[z-a-e-f]$ ! $[z-b-d-f]$ ! $[z-c-d-e]$ !
$\cross[a+b+d+e-z]$ ! $[a+c+d+f-z]$ ! $[b+c+e+f-z]!\}^{-1}$where the sum is for any non-negative integer such that all expressions in thebrackets are non-negative. Normalizing tlxe Rakah-Wigner symbol, the quantm 6jsymbol is given by$z$.
$\{\begin{array}{lll}a b cd e f\end{array}\}=(-1)^{d+c+2c-a-b}\sqrt{\frac{S_{0c}}{S_{00}}\frac{S_{0f}}{S_{00}}}\{\begin{array}{lll}a b cd e f\end{array}\}$6
For an admissible triple $(i,j, k)$ and a trivalent graph depicted as in Figure 2, we. If
with basis denoted byassociate a one dimensional complex vector space
to be zero. When we reverse the orientation$(i,j, k)$ is not admissible, we defineof the edge corresponding to we use the convention that the same space is denoted. We adopt a similar convention for and .
by the symbol
By means of the quantum $6j$ -symbols we first define the fusing operators$e_{ij}^{k}$$\prime H_{ij}^{k}$$\mathcal{H}_{ij}^{k}$

$i$

$j$

$\mathcal{H}_{i\dagger j}^{k}$$F_{+}$$k$$:\oplus_{j}\mathcal{H}_{ab}^{j}\otimes \mathcal{H}_{jc}^{d}arrow\oplus_{i}\mathcal{H}_{bc}^{i}\otimes \mathcal{H}_{ai}^{d}$$(2- 2)$$F_{-}:$$\oplus_{j}\mathcal{H}_{bc}^{j}\otimes \mathcal{H}_{aj}^{d}arrow\oplus_{i}\mathcal{H}_{ab}^{i}\otimes H_{ic}^{d}$in the following way (see Figure 3)$F_{+}(e_{ab}^{j} \otimes e_{jc}^{d})=\sum_{i}\{\begin{array}{lll}a b jc d i\end{array}\}e_{bc}^{i} \otimes e_{ai}^{d}$.

$F_{-}(e_{bc}^{j} \otimes e_{aj}^{d})=\sum_{i}\{\begin{array}{lll}c b ja d i\end{array}\}e_{ab}^{i} \otimes e_{ic}^{d}$$\ulcorner_{+}$$arrow^{F_{-}}$$arrow$

Figure 3

Let us introduce linear maps$\Omega_{+}$

and$\Omega_{-}$$\Omega\pm:\oplus_{j}\mathcal{H}_{j_{1}j_{2}}^{j}arrow\oplus_{j}\mathcal{H}_{j_{2}j_{1}}^{j}$given by
(2-3)$\Omega_{\pm}(e_{j_{1}j_{2}}^{j})=(-1)^{j_{1}+j_{2}-j}\exp(\pm\pi\sqrt{-1}(\triangle_{i}-\triangle_{i\iota}-\triangle_{j_{2}}))e_{j_{2}j_{1}}^{j}$

We define the braiding operators$B\pm:\oplus_{j}\mathcal{H}_{ab}^{j}\otimes \mathcal{H}_{jc}^{d}arrow\oplus_{j}\mathcal{H}_{ac}^{j}\otimes \mathcal{H}_{jb}^{d}$

by

(2-4)

$E\pm=F_{-}(\Omega\pm\otimes id)F_{+}$

.

.

7

$\oplus_{j}\mathcal{H}_{ab}^{j}\otimes \mathcal{H}_{jc}^{d}arrow^{F_{+}}\oplus_{i}H_{bc}^{i}\otimes \mathcal{H}_{ai}^{d}$$\Omega\pm\otimes id\downarrow$$B\pm\downarrow$$\oplus_{j}\mathcal{H}_{ac}^{j}\otimes’kt_{jb}^{d}arrow^{F-}\oplus_{i}H_{cb}^{j}\otimes \mathcal{H}_{ai}^{d}$

Associated with the diagram (a) in Figure 4, the fusing operators give the linearmaps$Y_{jj}^{0},$$:\oplus_{k}\cdot \mathcal{H}_{ij}^{k}\otimes \mathcal{H}_{k\cdot j}^{i},$$Y_{j^{1}j}^{0}$$:\oplus_{k}\mathcal{H}_{ij}^{k},$$arrow \mathcal{H}_{jj\dagger}^{0}\otimes Tt_{i0}^{i}\cong C$$\otimes Tt_{k\cdot j}^{i}arrow \mathcal{H}_{j\dagger j}^{0}\otimes \mathcal{H}_{i0}^{i}\cong$

C.

Normalizing these, we define the annihilation operators by(2-5)$\sqrt{S_{0j}/S_{00}}Y_{jj}^{0}$,and$(-1)^{2j\sqrt{S_{0j}/S_{00}}}Y_{j^{1}j}^{0}$respectively. In a similar way, the fusion operators induce$Y_{0^{jj}}$:$C\cong \mathcal{H}_{jj}^{0},$$Y_{0}^{j^{t}j}$:$C\cong \mathcal{H}_{jj}^{0_{\dagger}}\otimes \mathcal{H}_{i0}^{i}arrow\oplus_{k}\mathcal{H}_{ij}^{k^{\backslash }},$$\otimes H_{i0}^{i}arrow\oplus_{k}\mathcal{H}_{ij}^{k}\otimes \mathcal{H}_{k\cdot j}^{i}$

associated with the diagram (b) in Figure 4. Using
operators by(2-6)$\sqrt{S_{0j}/S_{00}}Y_{0}^{jj^{\uparrow}}$and,$\otimes \mathcal{H}_{k\cdot j}^{i}$$t1_{1}is$, we define the creation$(-1)^{2j\sqrt{S_{0j}/S_{00}}}Y_{0}^{jj}$respectively.
We put$\mathcal{H}_{j_{1}\cdots j_{\mathfrak{n}}}=\oplus_{k_{1},\cdots,k_{\mathfrak{n}-l}}\cdot\in P_{+}(l)\mathcal{H}_{0j^{1}l}^{k}\otimes \mathcal{H}_{k_{1}^{2}j_{2}}^{k}\otimes\cdots \mathcal{H}_{k\cdot j}^{0,-1},.\cdot$8
Let be an oriented framed $(m, n)$ tangle, by which we mean a union of oriented arcs
with boundary points contained inas shown in:inisrepresented]. By
byframinthethe blackboard framing in the sense of
5. Heredecomposing the tangle diagram into elementary diagrams and by associating Io eachelementary diagram the braiding, creation or annihilation operator defined, weobtain a linear map.
$T$

$R^{2}\cross[0,1]$$R^{2}\cross\{0\}\cup R^{2}\cross\{1\}$$i_{o}ure$$I\llcornerarrow$$g$$abo_{1}^{arrow-}e$$J_{f}(T):7\{j_{1}\cdots j_{n}arrow \mathcal{H}_{i_{1}\cdots i_{m}}$.

In particular, for an oriented framed link with $m$ components assigned with coior :$\{1, \cdots m\}arrow P_{+}(l)$ we obtain an invariant of colored oriented framed link.
It turns out that this invariant does not depend on the orientation on the link . In
, the invariant $J_{L}=J_{r}(L, 1/2)$ is characterized by the skein
the case$L$$\lambda$$J_{r}(L, \lambda)$$L$$re_{\wedge}^{\backslash }ation$$\lambda\equiv 1/2$

$tJ_{L+}-t^{-1}J_{L_{-}}=(t^{2}-t^{-2})J_{L_{0}}$$(2- 7)$$J_{O}= \{2]=2\cos\frac{\pi}{r}$,where we use the usual llotation $L+,$
as in [Jo].
$R$ be the representation$sl(2,
C)$
Let
ring of. The above invariant defines a-nearmap$L_{-},$ $L_{0}$(2-8)$J_{r}(L, \cdot):R^{\otimes m}arrow C$.

Moreover, it factors through the fusion algebra
for the $SU(2)$ Wess-Zumino-Witten
model at level $l=r+2$ (see [V]). This linear map is compatible with the product$j\in P_{+}(l)$ with the productstructure as shown in Figure 6. Herehas generators$R_{r}$$R_{r}$$v_{j},$

structure$v_{i} \cdot v_{j}=\sum_{k}N_{ij}^{k}v_{k}$(see also $[MoS1]$ and $[MoS2]$ ). As the notation indicates it is
whereclear from the above consideration that the invariant $J.(K,j)$ for a framed knot $K$
coincides with the one defined in the introduction by means of the recursive formula$N_{ij}^{k}=\dim \mathcal{H}_{ij}^{k}$

1-3.

Figure 6

9
Let us recaU that we have the following (see [KM]).

2.9 Proposition. Let
be a colored oriented framed tangle. We write $J(T)$ for$J_{r}(T)$ . Then, with respect to the following local modffi cations of the tangle we have$T$$J)= \frac{S_{0j}}{S_{00}}J$$J_{C^{/_{\backslash }}}j=e^{\sqrt{-1}}xp2\pi\Delta_{j}J_{(}j$$J rightarrow^{i_{t}}j=\frac{S_{ij}}{S_{0i}}J)i$

Here$\Delta_{j}$,.

is given by$\Delta_{j}=\frac{j(j+1)}{r}$.

2.10 Remark. Let us note that the sign convention adopted here is different fromthe one in [KR] and $[MoS1]$ , where the creation and annihilation operators are defined without the sign $(-1)^{2j}$ and consequently the effect of local modifications in
Proposition 2.9 appear with signs.
3. Proof of main results

Figure 7
Let be a closed oriented surface of genus . The mapping class group ofby
is by definition$\Sigma$$g$$\Sigma$denoted$\mathcal{A}4_{g}$$\mathcal{M}_{g}=Diff^{+}(\Sigma)/\sim$

where $Diff^{+}(\Sigma)$ denotes the group of orientation preserving diffeomorphisms ofandstands for the equivalence
isotopy. We are going to investigate an
action of the mapping class groupdefined byon a vector space$\Sigma$$\sim$$mod \iota\iota 10$$\mathcal{M}_{g}$(3-1)$\mathcal{H}_{\Sigma}=\oplus_{j_{1},\cdots,j_{9}\in P_{+}(l)}\mathcal{H}_{j_{1}j:\cdots j_{9}j_{g}^{1}}$$\mathcal{H}_{\Sigma}$

10$arrow$

Figure 8(see Figure 7).

is a diffeomorphim of
Let us recal that the Dehn twist along a circle
indefined by cutting the surface along and by reglueing after the rotation ofgeneratedisthe positive direction (see Figure 8). It is known by Lickorish [L] thatby
, $1\leq i\leq g$ ;, $1\leq i\leq g$, $2\leq i\leq g$ ;$C\subset\Sigma$$\Sigma$$2\pi$$C$$\Sigma$$\mathcal{M}_{g}$$\delta_{i}$$\beta_{i}$$\alpha_{i}$which are Dehn twists along circles shown as in Figure 8.

. ..

$(b)$

$\overline{\backslash _{/}/c\prec\backslash }$$\backslash CJ_{]}r\neg\backslash \prec r^{C\backslash }$$(a)$$T_{O}$$T(\beta_{d})$$T(\alpha_{d^{\prime\cdot)}}$$T(f_{j})$Figure 9

we associate a framed $(2g, 2g)$ tangle $T(x)$ in the following way.
To eachwebe a $(2g, 2g)$ tangle illustrated in Figure $9a$ . To eachLet
$9b$$T(x_{i})$we associate its
and fordepicted as in Figure
associate a $(2g, 2g)$ tangle
$(2g,
2g)$
. Let
be a
mirror image
tangle shown in Figure 9. Given
and we define a tangle
we express using Lickorish generators as$T(x)$ by the composition$x\in \mathcal{M}_{g}$$T_{0}$$x_{i},$$(x=\alpha, \beta, \delta)$$x_{i}^{-1}$$T(x_{i}^{-1})$$x\in \mathcal{M}_{g}$$T_{0}$

$x$

(3-2)

$x_{i_{1}}^{\epsilon_{1}}\cdots x_{i_{k}^{k}}^{\epsilon},$$\epsilon_{i}=\pm 1$$T(x)=T_{0}oT(x_{i_{1}^{1}}^{\epsilon})0\cdots oT(x_{i_{k}}^{\epsilon_{k}})$.

in the previous
Let $T(x)$ be an oriented framed $(2g, 2g)$ tangle defined forparagraph by fixing a way of writing in terms of Lickorish generators. We divide
the set of arcs in $T(x)$ into $E^{0}(x),$ $E^{1}(x)$ and $L(x)$ defined by$x\in \mathcal{M}_{g}$$x$$E^{0}(x)=\{\gamma;\partial\gamma\subset R^{2}\cross\{0\}\}$,$E^{1}(x)=\{\gamma;\partial\gamma\subset R^{2}\cross\{1\}\}$,$L(x)=\{\gamma;\partial\gamma=\phi\}$

11

Figure 10(see Figure 10).
We define.a linear map$\rho(x):\mathcal{H}\Sigmaarrow Tt\Sigma$

in the following way. For$u\in \mathcal{H}_{j_{1}j:\cdots j_{g}j_{9}^{1}}$,$v^{*}\in(\mathcal{H}_{i_{1}i_{1}^{1}\cdots i_{9}i:})^{*}$

we associate the colors to the arcs in $E^{0}(x)$ and $E^{1}(x)$ as $sl_{1}own$ in Figure 10.
As is explained above, each time we associate a color$\lambda:\{1,2, \cdots m\}arrow P_{+}(l)$to the components of the oriented framed link L(x) $=L_{1}\cup\cdots\cup L_{m}$ we havea linear
map. Let us denote by$J_{r}(T(x), \lambda):\mathcal{H}\Sigmaarrow \mathcal{H}_{\Sigma}$$<$$>:\mathcal{H}_{\Sigma}^{*}\cross \mathcal{H}\Sigmaarrow C$

the canonical pairing with the dual space. We define$\rho(x)$by$<v^{*},$ $\rho(x)u>=\sqrt{S_{0j_{1}}S_{0j_{9}}}\sqrt{S_{0i_{1}}S_{0i_{9}}}$(3-3)$\sum_{\lambda}C^{sign(L(x))}S_{0\lambda(1)}\cdots S_{0\lambda(m)}<v^{*},$$J_{r}(T(x), \lambda)u>$

where runs over all colors on the components of the link $L(x)$ .
For $x,$
we fix a way of writing and in tcrms of Lickorislx generators andwe define $(2g, 2g)$ tangles $T(x),$ $T(y)$ and $T(xy)$ inabove way. Corresponding to$L(y)$$L(xy)$
oriented framed links $L(x),$. We put
this weand$\lambda$$y\in \mathcal{M}_{g}$

$x$

$y$

$t1_{1}e$$]_{1}ave$(3-4)$\sigma(x, y)=sign(L(xy))-sign(L(x))-sign(L(y))$

and we define $\xi(x, y)$ by$\zeta(x, y)=C^{\sigma(x,y)}$.

12Let us recall that a closed oriented 3-manifold $M$ is represented as$M=V \bigcup_{h}(-V)$. This description is called awith a handlebody $V$ and a glueing map$M$
Heegaard splitting. The minimal such that
admits a Heegaard splitting with ahandlebody of genus is called the Heegaard genus and is denoted by $g(M)$ .
$h:\partial Varrow\partial V$

$g$

$g$

3.5 Proposition. The above constructi on defnes a projectively linear representa tion$\rho:\mathcal{M}_{g}arrow GL(\mathcal{H}\Sigma)$

rvith a 2-cocycle$\xi.$

Namely,. we have$\rho(xy)=\xi(x, y)\rho(x)\rho(y)$for any $x,$$y\in \mathcal{M}_{g}$. Moreo$e_{0j_{1}^{1}}^{k}\otimes e_{k_{1}^{2}j:}^{k}$$\rho(x)$\copyright, with respect to a asis of$b$$1^{7}er$$\cdots\otimes e_{k\cdot j_{9}^{\not\in}}^{0_{2g-1}}$,$j_{1},$$\cdots j_{g},$is represented as a unit $ary$ matrix for any$k_{1},$$x\in \mathcal{M}_{g}$$\mathcal{H}_{\Sigma}$$\cdots k_{2g-1}\in P_{+}(l)$,.

let us first recall thatfor
Proof: To show the well-definedness of$sl_{1}ow$
invariancethe topologicalof the expresstion 1.6 in [RT] is thea key step tofollowing equality for the tangle operators corresponding to Kirby moves (see [K] and[FR]). For the tangle diagrams and ‘ as shown in Figure 11, we have$\rho(h)$$\Gamma$(3-6)$h\in \mathcal{M}_{g}$$\Gamma$$C^{-1} \sum_{j}S_{0j}J_{r}(\Gamma, i_{1}, \cdots i_{\tau\iota},j)=J_{r}(\Gamma’, i_{1} , \cdots i_{n})$.

In our situation this equality can be applied to any local modification of the above
type, allowing also the arcs in $E^{0}(h)$ and $E^{1}(h)$ as strands passing through the circle
in the angle diagram F.
$t$$<arrow$$\ulcorner$$\ulcorner’$Figure 11

13
Let us also recaU that as sociated with a decomposition of into a product of
Lickorish generators we can construct a link inso that the Dehn surgery on thislink gives a 3-manifold obtained as a Heegaard splitting with a glueing map (see[Ro]). In our case the Dehn surgery on the link $L(h)$ gives a 3-manifold.
$E^{i}(h),$The arcs in$i=0,1$ appear as a spine of the handlebody. Now we apply arelative version of Kirby’s theorem (see [K] and [RT]) and we can conclude that if
and are isotopic the as sociated tangle diagrams $T(h)$ and $T(h’)$ are obtainable one
from the other by a finitlely many steps of Kirby moves shown in Figure 11. Here weallow any arcs in the tangle diagram as strands passing through the circle. Hence theequality 3.6 implies $\rho(h)=\rho(h’)$ .
Now we describe the relation between $\rho(xy)$ and $\rho(x)\rho(y)$ . First, let us notice thatas sociated with the tangle
the linear operator
is the identity operator. Thiscan be shown by considering the compositionin the following way. Applying
Kirby moves, one sees that
idempotent.
isis non-singular, thisSince
implies that
is identity. Combining with the definition 3.3, it is easy to deduce
the assertion $\rho(xy)=\xi(x,y)\rho(x)\rho(y)$ .
Noticing that the entries of the fusing matrices are real at $q=e^{2\pi\sqrt{-1}/r}$ , it follows
from the orthogonality of the quantum $6j$ symbol (see [KR] and [RT]) that the braiding operator is represented by a unitary matrix with respect to the above basis. The
unitarity offor the Lickorish generators follows immediately by looking at theabove tangle representation. This completes the proof.
$h$$S^{3}$$h$$V_{g} \bigcup_{h}(-V_{g})$

$h$

$h’$

$\rho(T_{0})$$T_{0}$$T_{0}\cdot T_{0}$$\rho(T_{0})$$\rho(T_{0})$$\rho(T_{0})$$\rho(x_{i})$Now we are in a position to prove our main results. Let $M$ be a closed oriented
3-manifold and let $K$ be a knot in $M$ . We denote by $E(K)$ the knot exterior. Namely,$E(K)$ is the closure of the complementary space of the tubular neighborhood $N(K)$ .
If the tunnel number of the knot $K$ is $t(K)$ , then wea splitting of the knot
exterior$1_{1}ave$$E(K)=cl(V-N(K)) \bigcup_{h}(-V)$

with a handlebody of genus $g=t(K)+1$ and a glueing map. Herethe knot $K$ is embedded in a simple position as shown in Figure 12. More precisely,
there exists a separating disc $D$ properly embedded in $V$ withsuch that $K$
is embedded as a core of the solid torus obtained by cutting the handlebody along$D$ . Let us notethat $cl(V-N(K))$ is a special case of a so-called compression body.
$h:\partial Varrow\partial V$$V$$\partial D\subset V$$V$

Figure 12

14
We put$v_{j}=e_{0j}^{j}\otimes e_{jj}^{0},$$\otimes e_{00}^{0}\otimes\cdots\otimes e_{00}^{0}$which is a basis of$\mathcal{H}_{0j}^{j}\otimes H_{jj^{2}}^{0}\otimes H_{00}^{0}\otimes\cdots\otimes \mathcal{H}_{00}^{0}\cong$

C.

With this notation we obtain the following.

3.7 Proposition. In the ab $ove$ situation we have$Z_{r}(M, K;j)=S_{00}^{-g}<v_{j}^{*},$ $\rho(h)v_{0}>$.

Proof: Let us close the $(2g, 2g)$ tangle diagram $T(h)$ to get a plat diagram as
contains $2g$
in Figure 13. This is a link diagram expressed as $L(h)\cup L$ whereis equal to and the color on the other,. The color oncomponents
components is zero. Let us recall that by the Dehn surgery on the framed link $L(h)$we get our 3-manifold $M$ with link in it.
$L$$L_{1},$$\cdots$$j$$L_{1}$$L_{2g}$$L$

Figure 13

Let us consider the above situation with $K=L_{1}$ . Noticing that the fusion operator$Y_{jj^{t}}^{0}$:$C\cong \mathcal{H}_{0j}^{j}\otimes \mathcal{H}_{jj}^{0},$$arrow \mathcal{H}_{jj^{\uparrow}}^{0}\otimes \mathcal{H}_{00}^{0}\cong C$

is an identity map, we obtain the assertion from definition 1.5 and 3.3.
Now the proof of our main results is completed in tlxe following way. We obtain$\rho(h)v_{0}>$ is an entry of a unitary lnatrix.
Hence itfrom Proposition 3.5 that
follows that$\rho(h)v_{0}>|^{2}\leq 1$ ,$<v_{j}^{*},$$\sum_{j\in P_{+}(\iota)}|<v_{j}^{*},$

which shows Theorem B. In the case $M=S^{3}$ we liave$Z_{l}.(S^{3}, K;j)=J_{r}(K,j)$.

15
Moreover, by means of the symmetry principle due to Kirby and Melvin [KM], we

have$|J_{r}(K,j)=J,.(L_{1}\cup L_{2}\cup\cdots L_{g}, \lambda)|$for $\lambda(1)=j$ and$\lambda(k)=0$

or $l/2$ ,Combining this symmetry with Theorem
the proof.

$B$$2\leq k\leq g$.

, we obtain Theorem A. This completes

4. Comments on generalizations and discussions

We discuss briefly how our results can be generalized to an arbitrary complex simple
Lie algebra . We refer the readers to [Ko3] for a more detailed description. We fix
a positive integer and we adopt as the set of colors$\mathcal{G}$

$l$

(4-1)

$P_{+}(l)=\{\lambda\in P+:0\leq(\lambda, \theta)\leq l\}$is the Cartan-Killing form normalized so that
where
for the longest root.
Let be an oriented framed link in . By looking at the monodromy representationof the Knizhinik-Zamolodchikov equation in Wess-Zumino-Witten model at level(see [TK]), for a color : $\{1, \cdots m\}arrow P_{+}(l)$ , one can define an invariant of colored
framed link
where we put $r=l+h^{*}$ with the dual Coxeter number .
The phase factor derived from the $c1_{1}ange$ of framing in Proposition 2.9 is expressedby the conformal weight$(\theta, \theta)=2$$(\cdot, \cdot)$$S^{3}$$L$$l$$\lambda$$h^{*}$$J_{\mathcal{G},r}(L, \lambda)$$\triangle_{\lambda}=\frac{(\lambda,\lambda+2\rho)}{2(l+h^{*})}$

where is the half sum of the positive roots.
Now in the situation of Theorem , the Witten invariant for$\rho$$B$$\mathcal{G}$at level is expressed
$l$

as
(4-2)

$Z_{\mathcal{G},r}(M, K, \lambda)=C^{sign(L)}\sum_{\mu}S_{0\mu(1)}\cdots S_{0\mu(m)}J_{Q,r}(K\cup L, \lambda\cup\mu)$where the sum is for any
are given by

$\mu$

: 1,
$\{$

$\cdots$, $m\}arrow P_{+}(l)$ . In

$t1_{1}e$above expression

$C$

and

$S_{0\lambda}$$C= \exp(-\frac{l\dim \mathcal{G}\pi\sqrt{-1}}{4(l+l\iota^{*})})$(4-3)$S_{0\lambda}= \frac{1}{(l+h^{*})^{rank\cdot(\mathcal{G})/2}}(\frac{vol\Lambda^{W}}{vol\Lambda^{R}})^{1/2}\prod_{\alpha\in\triangle_{+}}2$

siu$\frac{\pi(\lambda+\rho,\alpha)}{l+h^{*}}$

Hereis the volume of tlte wciglit lattice,is the volume oflattice, andstands for the set of positive roots. Let us notice that$vol\Lambda^{W}$$vol\Lambda^{R}$$\Delta_{+}$$t1_{1}e$$C$corootcan be16
with the central charge of the Virasoro algebra actingasthe integrable highest weight module of highest weight . We also recaU thatxppears as an entry of the modular transformation matrix on the characters of level: integrable highest weight modules determined in [KP] and [KW].
Several approaches have been developed to establish a more combinatorial descripof such invariants for classical Lie algebras (see [M], [TW], [We] and [KT]). The-sociated representation of the mapping class group is related to the monodromy:epresentation of the conformal field theory on Riemann surfaces (see [TUY]).
Let $K$ be a knot in a closed oriented 3-manifold $M$ . Generalizing Theorem , we
: ave a lower estimate for the tunnel number $t(K)$$\overline{\sim}\tau itten$$( \exp 2\pi\sqrt{-1}\frac{c}{24})^{-3}$$c$$\lambda$$S_{0\lambda}$$\gammaarrow n$$S$$\underline{\neg}on$$\simeq^{\backslash }\backslash$$B$$|Z_{\mathcal{G},r}(M, K;\lambda)|^{2}\leq S_{00}^{-2(t(K)+1)}$$\sum$$\underline{\dagger}4)$.

$\lambda\in P+(l)$also notice that considering the case $K$ is empty, we have the following lower‘:ound for the Heegaard genus (see [Wa], [Ko2] and [G]).
$-\prime t^{-}e$$|Z_{\mathcal{G},r}(M)|\leq S_{00}^{-g(M)}$$-’-5)$$1\leq i\leq n-1$ , a set of simple
Let us discuss the case $\mathcal{G}=sl(n, C)$ . We denote by
the reflection on the weight lattice defined by:oots and.
$P_{+}(l)$-ne set of weights
is written as$\alpha_{i},$$s_{\alpha 1}.(v)=v-(v, \alpha_{i})\alpha_{i}$$s_{\alpha:}$$\neg$$P_{+}(l)= \{\sum_{i=1}^{\tau\iota-1}a_{i}\Lambda_{i};a_{i}\in Z, a_{i}\geq 0,\sum_{i=1}^{n-1}a_{i}\leq l\}$$1\leq i\leq n-1$ , which is naturally identified with thethe fundamental weights
at levelof dominant integral weights for the corresponding affine Lie algebra:- Hence we have a Dynkin diagram automorphism for the affine Lie algebra, which
duces an automorphism on the set $P_{+}(l)$ . This automorphism is isomorphic to aclic group of order generated by defined by$\overline{\sim_{\wedge}}th$$\Lambda_{i},$$\mathcal{G}\wedge$$-\simeq_{\vee}b$

$=-$

$n$

$—$

$\sigma$$\sigma(\lambda)=C(\lambda)+l\Lambda_{1}$is the Coxeter transformation. In [KT] we described the phase
we havecoming from this action, which implies that for a knot $K$ in$\overline{\sim}-;lereC$$\simeq-\vee\backslash ctor$$s_{\alpha_{1}}\cdots s_{\alpha_{n-1}}$$S^{3}$$|J_{\mathcal{G}^{f}},(K, \sigma(\lambda))|=|J_{\mathcal{G},r}(K, \lambda)|$$\underline{\prime}6)$.

Let us llotice tltat has a fixed point
if $1\equiv 0_{1}nodn$ and is fixed point
otherwise. We suppose thatis not a fixed point of and that is not-Iltained in the orbit of $0\in P_{+}(l)$ . Thcn, we have an inequality$l/n \sum_{i^{\iota}=^{-}1^{1}}\Lambda_{i}$$\sigma$$—\yen_{-}$$\lambda\in P_{+}(l)$$\lambda$$\sigma$$\sim-$$\underline{\prime}\tau)$$\sum$$\lambda\in P_{+}(l)$$|J_{\mathcal{G},r}(K, \lambda)|^{2}\leq n^{-t(K)}S_{00}^{-2(t(K)+1)}$.

17
We recaU that the skein (or HOMFLY) polynomial $P_{L}(a, z)$ for an oriented link is
a two-variable generalization of the Jones polynomial (see [Jo]) characterized by theskein relation:$P_{O}=1$(4-8)$a^{-1}P_{L_{+}}-aP_{L-}=zP_{L_{O}}$.

Let us suppose that $n\geq 3$ and $l\geq 2$ . Considering the case whento the vector representation, we have$\lambda\in P_{+}(l)$corresponds$P_{K}(s^{n}, s-s^{-1})= \frac{\sin\pi/r}{\sin n\pi/r}J_{\mathcal{G},r}(K, \lambda)$

for $s=e^{\pi\sqrt{-1}/r}$ . Combining with 4.6 and 4.7, we obtain a upper bound for the above
special values of the skein polynomial. A similar estimate can be formulated for special
values of the Kauffman polynomial by examiningcase $\mathcal{G}=o(n, C),$ $sp(n, C)$ .
Our construction may also be generalized to certain links. Let $L=L_{1}\cup L_{2}$ be alink
with 2 components in a closed oriented 3-manifold $M$ . We define its tunnel number$t(L)$ to be the minimal number $g+1$ such that the link exterior $E(L)=cl(M-N(L))$
admits a splitting$t1_{1}e$$E(L)=(V^{1}-N(L_{1}))\cup(V^{2}-N(L_{2}))$

with a reversed orientation.
with a handlebodyof genus and its second copy
Here we assume thatin a sinlple position as shown in$i=1,2$ , is embedded in
Figure 12. In this situation we obtain for$V^{1}$$V^{2}$$g$$V^{i}$$L_{i},$$\lambda_{1}\in P_{+}(l)$

(4-9)

where

$\sum_{\lambda_{2}\in P_{+}(l)}|Z_{\mathcal{G},r}(M, L;\lambda_{1}, \lambda_{2})|^{2}\leq S_{00}^{-2(t(L)+1)}$$\lambda_{i}$is a color on$L_{i},$$i=1,2$ .

4.10 Concluding Remarks.
(i) It was shown by Jones [ $Jo|$ that if an oriented link

(4-11)$|V_{L}(e^{2\pi\sqrt{-1}/r})| \leq(2\cos\frac{\pi}{r})^{n-1}$,$L$

is a closed$r=3,4,$$n$braid, then$\cdots$

The unitarity of the monodromy representation enables us to generalize this inequality
in the form$|J_{\mathcal{G},r}(L, \lambda)|\leq(\frac{S_{0\lambda}}{S_{00}})^{n}$using the framed hnk invariant with color
for eaclt component of . Combining with our estimate of the tunnel number, we can deduce several consequenccs.
$\lambda\in P_{+}(l)$

$L$

18
For example, if a knot $K$ satisfies the equality for $r=5$ and $n=3$ in 4.11 (see [Jo]),
then we have $t(K)\geq 2$ .
(1i) For a connected sum of knots we have
(4-12)$J_{\mathcal{G},r}(K_{1} \# K_{2}, \lambda)=J_{\mathcal{G},\prime}(K_{1}, \lambda)J_{\mathcal{G},\prime}(K_{2}, \lambda)\frac{S_{00}}{S_{0\lambda}}$.

For a knot $K,$
us consider $K\#\cdots\# K$ ( times). Our result can be used to give alower estimate for the tunnel number of such knot in terms of a function of which
is of linear order with respect to .
$\ddagger et$

$n$

$n$

$n$

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E-mail address: t-kohno@tansei.cc.u-tokyo.ac.jp