数理解析研究所講究録
第 1991 巻 2016 年 148-153

148

THE CANONICAL DECOMPOSITION OF THE TWO-BRIDGE KNOTWITH SLOPE 12/29 AND THE CANNON-THURSTON MAP
ASSOCIATED WITH THE KNOTNAOKI SAKATA

1. INTRODUCTION
This note is a summary of the author’s talk at the workshop “Topology, Geometry andAlgebra of low-dimensional manifolds”’
Let $M$ be a hyperbolic once-punctured torus bundle over . Then $M$ admits a canonical decomposition into ideal polyhedra by Epstein-Penner [6] and Weeks [18]. In fact,rgensen [ $12]$ has constructed the canonical decomposition of $M$ . The canonical decomposition induces a triangulation of the peripheral torus. Thus we have a triangulation of
the complex plane as the lift of the triangulation of the torus to the universal cover.
On the other hand, we have another tesscllation of which nicely reflects the nature of
the Cannon-Thurston map associated with $M$ . The tessellation is called the “CannonDicks-Thurston fractal tessellation” (see Definition 2.4).
Dicks and Sakuma [5] have proved that there exists a nice relation between the twotessellations. In particular, the two tessellations share the same vertex set.
It is natural to expect that similar results hold in a more general setting, and theauthor has been trying to realize this expectation for hyperbolic fibered two-bridge links.
The first task is to generalize the fractal tessellation to more general hyperbolic punctured
surface bundles, and the author announced such a generalization for hyperbolic punctured
surface bundles in [14], provided that all singularities of the stable and unstable foliationsof the monodromy are at punctures of the fiber surface. The condition is satisfied forthe hyperbolic fibered two-bridge knots with slope such that has a continued fraction
expansion, 2, . . . , 2 . The main purpose of this note is to present an idea of the proofof the following theorem.
$S^{1}$$J\emptyset$$\mathbb{C}$$\mathbb{C}$$r$$\pm[2$$r$$]$

Theorem 1.1. For the hyperbolic fibered two-bridge knot $K$ with slope $12/29=[2$ , 2, 2, 2
the triangulation induced by the canonical decomposition and the fractal tessellation sharethe same vertex set.
$],$

After the announcement of this result at the workshop, the author learned the workof Gueritaud [10, 11], which generalizes the fractal tessellation and establishes beautiful
relation between (veering” ideal triangulations (cf. [1]) of hyperbolic punctured surface
bundles and the fractal tessellations, generalizing the result of Dicks and Sakuma. Motivated by this work, the author proved the following theorem.
Theorem 1.2 ([15, Theorem 1.1]). The canonical decomposition of a hyperbolic fibered
two-bridge link $K(r)(0<|r|<1/2)$ is veering if and only if the slope has the continued
fraction expansion , 2, . . . , 2$r$$\pm[2$

Received December 26, 2015.

$].$

149
As a consequence of Gueritaud’s result and the above theorem, the canonical decomposition of such a two-bridge knot and the fractal tessellation are intimately related. In
particular, Theorem 1.1 follows immediately.
2. THE CANNON-DICKS-THURSTON FRACTAL TESSELLATIONIn this section, we will recall some properties of the Cannon-Dicks-Thurston fractal
tessellations. Let be a complete hyperbolic punctured surface of finite area, and let
be the holonomy representation of $F$ . For a pseudo-Anosov:$:=F\cross[O, 1]/(x, 0)\sim(h(x), 1)$ admits
homeomorphism of , the mapping torus
Let : $\pi_{1}(F)arrow PSL(2, \mathbb{C})$ be
work.
a complete hyperbolic structure, by Thurston’s$M$. Generalizing the work of
tothe restriction of the holonomy representation of
Cannon and Thurston [4], Bowditch proved the following theorem.
-equivariani continuous surjection from the
Theorem 2.1. There exists a unique.
to that oftimit set ofThis surjection is called a Cannon-Thurston map. Note that the limit sets ofand the boundary of theand are equal to the boundary of the hyperbolic plane, respectively. Thus the surjection is a sphere3-dimensional hyperbolic spacefilling curve.
Moreover, Bowditch has given a complete description of the combinatorics of the
Cannon-Thurston map.
the stable and unstable foliations of , andTheorem 2.2 ([3, Theorem 9.1]). Let. For any leaf of , the images of its
the universal coverbe the lift ofletendpoints under the Cannon-Thurston map are identical. Furthermore, this generates
all the identifications occuwing under
For the case when is a once-punctured torus, Cannon and Dicks [4] have introduced a, where oc is a parabolic fixedcertain fractal tessellation of the complex plane
point of . The fractal tessellation nicely reflects the behavior of the Cannon-Thurston
map. This result was extended by Gueritaud [11] to a more general setting as follows.
$F$$p_{0}$$\gamma r_{1}(F)arrow PSL(2,\mathbb{R})$
$h$

$F$

$M_{h}$$\rho$$\pi_{1}(F)$$\omega$$\pi_{1}(F)$$\Gamma=\rho(\pi_{1}(F))$$\Gamma_{0}=p_{0}(\pi_{1}(F))$$\Gamma_{0}$$\omega$$\partial \mathbb{H}^{2}$$\Gamma$$\partial \mathbb{H}^{3}=\hat{\mathbb{C}}$$\omega$$h$$\mathcal{F}\pm be$$\tilde{\mathcal{F}}_{\star}$$\mathbb{B}^{2}$$ノ^{}=\sim\pm$$\overline{J^{\sim}}\pm to$$\omega$$\omega.$$F$$\mathbb{C}=\hat{\mathbb{C}}\backslash \{\infty\}$$\Gamma$

Proposition 2.3 (cf. [11, Theorem 1.2]). Suppose that all singularities of the invariant
foliations of the pseudo-Anosov homeomorphism are at punctures of F. Then there, bounding domains , with the following
exists a -family of Jordan curvesof$h$$\mathcal{F}_{\pm}$$\partial \mathbb{H}^{3}$$D_{i}$$J_{i}$$\mathbb{Z}$properties:(1) For each$i\in \mathbb{Z}$, the Jordan curve$J_{i}$

passes through$\infty$, a parabolic fixed point of$\Gamma.$(2) For each(3)(4) For(5) For$i\in \mathbb{Z},$$\bigcap_{i\in \mathbb{Z}}D_{i}=\emptyset$$D_{i}\supset D_{i+1}.$

and$\bigcup_{i\epsilon \mathbb{Z}}D_{i}=\mathbb{C}.$$i,$ $i’\in \mathbb{Z},$$ij|i-i’|>1$ , then $J_{i}\cap J_{i’}=\{\infty\}.$$i,$ $i’\in \mathbb{Z},$$if|i-i^{J}|=1$ , then $J_{i}\cap J_{i’}$ is a discrete set which accumulates atfrom both directions.
(6) The closure of each component of$\partial \mathbb{H}^{3}\backslash \bigcup_{i\in \mathbb{Z}}J_{i}$$\infty$

is homeomorphic to the disk.

fills
Moreover, Gu\’eritaud describes the way in which the Cannon-Thurston map
a
have
we
proposition,
above
By
the
(see [11, Section 1.2], and see also [4]).
CW-decomposition of the complex plane$\omega$$\hat{\mathbb{C}}=\partial{\} 83$$\mathbb{C}=\partial\Re^{3}\backslash \{\infty\}.$150
Definition 2.4. The symbol$C$denotes the CW-decomposition of$\mathbb{C}$defined as follows:$C^{(0)}= \bigcup_{i\in Z}(J_{i}\cap J_{i+1})\backslash \{\infty\},$$C^{(1)}=${$C^{(2)}=${$c1(\gamma)|\gamma$

is a component of$c1(\delta)|\delta$

is a component of$J_{i}\backslash (J_{i-1}\cup J_{i+1})$$\partial \mathbb{H}^{3}\backslash \bigcup_{i\in Z}\sqrt{}i,$,$i\in \mathbb{Z}$$i\in \mathbb{Z}$},}.

We call the Cannon-Dicks-Thurston fractal tessellation (a fractal tessellation, in brief)
associated with the Cannon-Thurston map$C$$\omega.$

In order to state the main result, we introduce the notion of a “spider

Definition 2.5. Suppose that all singularities of the invariant foliationsof the pseudo
Anosov homeomorphism are at punctures of $F.$
(1) Let be a parabolic fixed point of . We denote by $s(p)$ the union of the leavesofwhich have as an endpoint, and we call it the spider with head . We call an
endpoint of a leaf of.
a foot of(2) A parabolic fixed point ofwithis called a neighbor of if the numberof points ofis equal to 2, where is the parabolic fixed point of such that$\mathcal{F}_{\pm}$$h$$\Gamma_{0}$$p$$\tilde{\mathcal{F}}_{\pm}$

$p$

$p$

$\mathfrak{s}(p)$$\mathfrak{s}(p)$$p$$\Gamma_{0}$$\mathfrak{s}(\infty)\cap s(p)$$p\neq\infty$$\infty$$\infty$$\Gamma_{0}$$\omega(\infty)=\infty.$

Note that, by Theorem 2.2, the preimage of, under the Cannon-Thurston
map , is equal to the set of all feet of. By the construction of , we have thefollowing lemma.
$\omega(p)\in\partial \mathbb{H}^{3}$$C$$\mathfrak{s}(p)$$\omega$Lemma 2.6. Suppose thatunique parabolic fixed point

$v$

$p$

is a vertex of the fractal tessellation. Then there exists aof such that $v=\omega(p)$ . Furthermore, is a neighbor of$\Gamma_{0}$$p$$\infty.$

3. SINGULAR EUCLIDEAN STRUCTURE ON THE FIBER SURFACE OF THE TWO-BRIDGEKNOT WITH SLOPE 12/29

In this section, we describe the stable and unstable foliations of the monodromy,of the hyperbolic fibered two-bridge knot $K$ with slope 12/29. To this end, we give asingular Euclidean structure on the fiber surface, , with respect to which acts by
aﬃne transformation, following the construction of Thurston [17] (cf. [7]).
First of all, we recall the fiber surface of $K$ and the monodromy . The rational
number 12/29 has the following continued fraction expansion:
$h,$

$F$

$h$

$F$

$h$

$12/29=[2, 2, 2, 2 ] = \frac{1}{2+\frac{}{2+\frac{11}{1}}}.$$2+_{\overline{2}}$Thus the two-bridge knot $K$ is the boundary of the fiber surface obtained by successively plumbing the unknotted four positive and negative Hopf-bands (see Figure 1). The
symbolsanddenote the simple loops in as shown in Figure 1. Each ofthem is a core curve of a Hopf-band. For a simple loop
denotes the Dehn twist(resp.
along . Let) be the product of(resp.
). Then theandandmonodromy is a pseudo-Anosov homeomorphism$F$$\alpha_{1},$$\alpha_{2},$$\beta_{1}$$F$$\beta_{2}$$\gamma,$$\gamma$$T_{A}$$h$$T_{B}$$T_{\alpha_{1}}$$T_{\gamma}$$T_{\alpha_{2}}$$T_{A}\circ(T_{B})^{-1}.$$T_{\beta_{1}}$$T_{\beta_{2}}$151

FIGURE 1. The fiber surface of the hyperbolic fibered two-bridge knot $K$
with slope 12/29.
Next, we give a singular Euclidean structure on the fiber surface $F$ . We cut $F$ intobe the rectangle, as illustrated in Figure 2(a). Letandthree octagons,$F_{1},$$E_{i}$$F_{3}$$F_{2}$$F_{1} F_{2} F_{3}$$E_{1}$$\alpha_{1}$$\beta_{1}$$\beta_{1} \beta_{2}$(b)FIGURE 2. (a) The octagons$E_{1},$$E_{2}$

and$F_{1},$$F_{2}$

and$F_{3}.$(b) The Euclidean rectangles$E_{3}.$

into a point, forby collapsing each of the four components ofobtained from$\mu=(1+\sqrt{5})/2$
the Euclidean
withrectangle. Identify each
each $i\in\{1$ , 2, 3 . Set
rectangle as illustrated in Figure 2(b). For any Euclidean rectangles , the length ofor ) is equal to 1 (resp. ). This gives a singular(resp.
orthe edges crossed$F.$
Euclidean structure onand , respectively, act by aﬃne transformaWe can see that the homeomorphisms
can be described by the following matrices
andtions on , and their derivative:and$F_{i}\cap\partial F$$F_{i}$$E_{i}$$\}$$E_{i}$$\alpha_{1}$$\beta_{2}$$\beta_{1}$$T_{A}$$F$$D_{A}$$\mu$$\alpha_{2}$$dT_{A}$$T_{B}$$dT_{B}$$D_{B}$$D_{A}=(\begin{array}{ll}1 \mu 0 1\end{array})$$D_{B}=(\begin{array}{ll}1 0-\mu 1\end{array}).$152
Since the monodromy, it acts on by aﬃne transformation withis equal torespect to the singular Euclidean structure, such that its derivative $dh=dT_{A}o(dT_{B})^{-1}$
is described by the following matrix
:
$h$

$F$

$T_{A}o(T_{B})^{-1}$$D_{h}$$D_{h}=D_{A}(D_{B})^{-1}=(\begin{array}{lll}1+ \mu^{2} \mu\mu 1\end{array}).$The stable and unstable foliations of the monodromy are described as follows. Set$\lambda=(\mu+3+\sqrt{7\mu+6})/2$ . Let
be the vector $(1, (-\mu+\sqrt{\mu+5})/2)^{T}$ Then the vector
is an eigenvector of, and is the eigenvalue associated with
. Hence the set
consisting of the straight lines on $F$ with the same slope $(-\mu+\sqrt{\mu+5})/2$ forms a foliation,, which is invariant under . In fact, the invariant foliationis the stable foliationof with dilatation . Similarly, the vector $v_{-}:=(1, (-\mu-\sqrt{\mu+5})/2)^{T}$ is the othereigenvector of, and it gives the unstable foliation,
, of
$h$

$v_{+}$$\lambda$$D_{h}$$v_{+}$$v_{+}$$h$$\mathcal{F}_{+}$$h$$\mathcal{F}_{+}$$\lambda$$D_{h}$$\mathcal{F}_{-}$$h.$

4. IDEA OF THE PROOF OF THEOREM 1.1

First of all, we recall the edges of the canonical decomposition, , of
, whereis the hyperbolic two-bridge knot with slope 12/29. By the work of Gu\’eritaud [8]$(cf. [2,9,16 we can see$ that $the$ edges
Figure $3(a)$ . Moreover, byshown
the author’s previous work [13], the canonical decomposition is “layered” with respect
to the fiber structure of. In particular, each edge of can be isotoped into thefiber surface (see Figure $3(b)$ ).
$\mathcal{D}$$S^{3}\backslash K$

$K$

$in$

$of \mathcal{D} are as$$\mathcal{D}$$S^{3}\backslash K$$\mathcal{D}$

$F$

(b)

FIGURE 3. The edges of the canonical decomposition$\mathcal{D}$of$S^{3}\backslash K.$

By Lemma 2.6, each vertex of the fractal tessellation corresponds a unique neighborof . On the other hand, each vertex of the cusp triangulation induced by the canonicaldecomposition is contained in a unique edge of . Since the edges of are isotoped
into the fiber surface $F$ , what we need to show is the following identity:
$C$

$p$

$\infty$$\mathcal{D}$$\mathcal{D}${ $\pi(g_{p})|p$ is a neighbor of$\infty$$\mathcal{D}$}$=\mathcal{D}^{(1)},$

whereis the vertical geodesic, in, above, and :
is thenatural projection. By using the singular Euclidean structure introduced in Section 3, we
can describe each neighbor of . Hence we can prove the above identity.
$\mathbb{H}^{2}$$g_{p}$$\infty$$p\in\partial \mathbb{H}^{2}$$\pi$$\mathbb{H}^{2}arrow F=\mathbb{H}^{2}/\Gamma_{0}$153
REFERENCES[1] 1. Agol, Ideal triangulations of pseudo-Anosov mapping tori, Topology and geometry in dimens\’ion three, Contemp. Math., vol. 560, Amer. Math. Soc., Providence, RI, 2011, pp. 1-17, DOI

10. $1090/conm/560/11087.$[2] H. Akiyoshi, M. Sakuma, M. Wada, and Y. Yamashita, Punctured torus groups and 2-bridge knot
groups. , Lecture Notes in Mathematics, vol. 1909, Springer, Berlin, 2007.
[3] B. H. Bowditch, The Cannon-Thurston map for punctured-surface groups, Math. Z. 255 (2007),
no. 1, 35-76, DOI $10.1007/s00209-O\langle$) $6-0012-4.$
J. W. Cannon and W. P. Thurston, Group invariant Peano curves, Geom. Topol. 11 (2007), 13151355, DOI 10. $2140/gt.2007.11.1315.$[5] W. Dicks and M. Sakuma, On hyperbolic once-punctured-torus bundles III: compamng two tessellations of the complex plane, Topology Appl. 157 (2010), no. 12, 1873-1899.
[6] I3. B. A. Epstein and R. C. Penner, Euclidean decompositions of noncompact hyperbolic manifolds,
J. Diﬀerential Geom. 27 (1988), no. 1, 67-80.
[7] B. Farb and D. Margalit, A primer on mapping class groups, Princeton Mathematical Series, vol. 49,
Princeton University Press, Princeton, NJ, 2012.
[8] I . Gu\’eritaud, Hyperbolic geometry and canonical triangutations in dimension three (2006), 164.
Thesis (Ph.D.)-University of Southern California.
[9] –, On canonical triangulations of once-punctured torus bundles and two-bridge link compleDOI 10. $2140/gt.2006.10.1239$ . With an appendix byments, Geom. Topol. 10 (2006), 1239
David Futer.
$|10]$ –, Veering mangutations and the Cannon-Thurston Map, Oberwolfach Rep. 9 (2012), 14191421, DOI $1\zeta\rangle.4171/OWR/2012/24.$$|11]$ –,triangulation and Cannon-Thurston maps, available at arXiv : 1506.03387 [math.
$I$

$|4]$

$\dagger$$Vee\prime\dot{\eta}ng$

GTI.

[12] T. Jrge sen, On pairs of once-punctured tori, Kleinian groups and hyperbolic 3-manifolds (Warw\’ick,2001), London Math. Soc. Lecture Note Ser., vol. 299, Cambridge Univ. Press, Cambridge, 2003,$\mathfrak{n}$pp. 183-207, DOI 10. $1017/CB09780511542817.010.$[13] N. Sakata, Canonical decompositions of hyperbolic fibered two-bridge link complements, Topology
Appl. 196 (2015), 821-845, OOI 10. $1016/j.$ topol. $2015.05.054.$[14] –, A generalization of the Cannon-Dicks fractal tessellation, Talk at “The Tenth East Asian
School of Knots and Related Topics”, Shanghai, China, Jan. 26–29, 2015.
[15] –, Veering structures of the canonical decompositions of hyperbolic fibered two-bridge link complements, in preparation.
[16] M. Sakuma and J. Weeks, Examples of canonical decompositions of hyperbolic link complements,
Japan. J. Math. (N.S.) 21 (1995), no. 2, 393-439.
[17] W. P. Thurston, On the geometry and dynamics of diﬀeomorphisms of surfaces, Bull. Amer. Math.
Soc. (N.S.) 19 (1988), no. 2, 417-431, DOI 10.1090/S0273-0979-l988-l5685-6.
[18] J. R. Weeks, Convex hulls and isometries of cusped hyperbolic 3-manifolds, Topology Appl. 52 (1993),
no. 2, 127-149, DOI $10.1016/0\lambda 66-8641(93)90032-9.$
OF MATHEMATICS, GRADUATE SCHOOL OF SCIENCE, HIROSHIMA UNIVERSITY,$HIGASBI-H\ddagger$
ROSHiMA, $739arrow 8526$ , JAPAN (RESEARCH FELLOW o\S JAPAN SOCiETY FOR
KAGAMIYAMA,SCIENCE)OF
THE PROMOTION$DEi)AR’fMEN\Upsilon$-mail address: $sakata202988e-05Wiroshima-u$ . ac. jp$\mathcal{B}$$1arrow 3-1$