数理解析研究所講究録
第 1868 巻 2013 年 32-37

32(1,1)-BRIDGE SPLITTINGS WITH DISTANCE EXECTLY$n$AYAKO IDO, YEONHEE JANG AND TSUYOSHI KOBAYASHI

1. INTRODUCTION
Hempel [4] introduced the concept of distance of a Heegaard splitting by using curve
complex, and showed that there exist arbitrarily high distance Heegaard splittings for
closed 3-manifolds by using a construction of Kobayashi [8]. Abrams and Schleimer [1]gave a sharper estimation for the distance of the Heegaard splitting given in [4] by using theresult of Masur and Minsky [9], and Evans [3] gave a combinatorial method to construct
Heegaard splittings of high distance.
On the other hand, the above concept and results have been extended to bridge splittings for links in closed 3-manifolds (for definitions, see subsection 2.3), and have beenstudied by several authors. For example, Saito [11] showed that for any closed 3-manifold
admitting a Heegaard splitting of genus one, there is a knot in the manifold with $a(1,1)-$
bridge splitting of arbitrary high distance. Recently, Blair, Tomova and Yoshizawa [2], and , there exists a manifold $M$ containing ashowed that for given integers$(M,L)$-component link so that
admits $a(g, b)$ -bridge splitting of distance at least $n.$Moreover, Ichihara and Saito [7] showed that for any given closed orientable 3-manifold$M$ with a Heegaard surface of genus, and for any positive integers and , there exists
$K$
$M$
$a(g,
b)$
-bridge splitting of distance greater than $n.$a knot in which admits
In [6], we showed that there exists a Heegaard splitting of a closed orientable 3-manifoldwith distance exactly for each positive integer . To prove this, we gave a method toextend a geodesic in the curve complex of a closed orientable surface to a geodesic withgiven length, and $co$ nstructed a concrete example (for details, see [6, Section 4]).
In this paper, we apply the idea of [6, Section 4] to construct a geodesic of any given
length incurve complex of a twice-punctured torus, and show the following.
$b,$

$c,$

$g$

$n$

$L$

$c$

$b$

$g$

$n$

$n$

$n$

$\mathfrak{h}he$

Theorem 1.1. For any integer $n>0$ , there eansts $a(1,1)$ -bridge splitting with distanceexactly$n.$

2. DEFINITIONS AND NOTATIONS
boundary com2.1. Curve complexes. Let be an orientable surface with genus
simple closed curve in is essential if it does not boundponents and punctures.
a disk or a once-punctured disk in and is not parallel to a component of . An arc
properly embedded in is essential if it does not co-bound a disk in together with anarc on . We say that is sporadic if $g=0,$ $b+p\leq 4$ or $g=1,$ $b+p\leq 1.$
Except in sporadic cases, the curve complex $C(S)$ is defined as follows: each vertex of$C(S)$ is the isotopy class of an essential simple closed curve on, and a collection of $k+1$$C(S)$
if they can be realized by disjoint curves in . In
vertices forms a -simplex ofsporadic cases, we need to modify the definition of the curve complex slightly, as follows.
We assume that is a torus, a torus with one boundary component, or a sphere with
4 boundary components since, otherwise, there are no essential simple closed curves in
$S$

$g,$

$p$

$b$

$S$

$A$

$S$

$\partial S$

$S$

$S$

$S$

$\partial S$

$S$

$k$

$S$

$S$

33
. Whenis a torus or a torus with one boundary component (resp. a sphere with 4boundary components), a collection of $k+1$ vertices forms a -simplex of $C(S)$ if they
can be realized by curves in which mutually intersect exactly once (resp. twice). The
arc-and-curve complex
is defined similarly, as follows: each vertex ofis theisotopy class of an essential properly embedded arc or an essential simple closed curve on, and a collection of $k+1$ vertices forms a -simplex ofif they can be realized
by disjoint arcs or simple closed curves in $S.$
We can define the distance between two vertices in the curve complex $C(S)$ to be theminimal number of 1-simplexes of a simplicial path in $C(S)$ joining the two vertices. We
denote by $d_{C(S)}(x, y)$ , or $d_{S}(x, y)$ in brief, the distance in $C(S)$ between the vertices and. For subsets $X$ and of the vertices of $C(S)$ , we define diam $s(X, Y)=$ diam $s(X\cup Y)$ .
Similarly, we can define the distanceand $diam_{\mathcal{A}C(S)}(X, Y)$ . We denote by$C(S)$
paththeinwith vertices
such that $a_{i}\cap a_{i+1}=\emptyset(i=$$0,1,$$n-1)$ . We call a patha geodesic if $n=d_{s}(a_{0}, a_{n})$ .
$S$

$S$

$k$

$S$

$\mathcal{A}C(S)$$\mathcal{A}C(S)$

$S$

$k$

$\mathcal{A}C(S)$

$x$

$Y$

$y$

$d_{\mathcal{A}C(S)}(x, y)$$[a_{0}, a_{1}, \ldots, a_{n}]$$a_{0},$$a_{1},$$\ldots,$$a_{n}$$[a_{0}, a_{1}, \ldots, a_{n}]$$\ldots,$

2.2. Subsurface projections. Let
denote the power set of a set . Suppose thatis an essential subsurface of that contains an essential simple closed curve. We call thecompositionof maps::anda subsurface projection if they satisfy the following (see Figure 1): for a vertex , takea representative so thatis minimal, whereis the number of connectedcomponents. Thenis the set of all isotopy classes of the components ofis the union for all $i=1,$
of the set of all isotopy classes of
the components ofwhich are essential in $X$ , whereis aregular neighborhood of
in $X.$$Y$$\mathcal{P}(Y)$

$X$

$S$

$\pi_{0}\circ\pi_{A}$$\pi_{A}$$C^{0}(S)arrow \mathcal{P}(\mathcal{A}C^{0}(X))$$\pi_{0}$$\mathcal{P}(\mathcal{A}C^{0}(X))arrow \mathcal{P}(C^{0}(X))$$\alpha$.

$\bullet$$|\alpha\cap X|$$\alpha$$|$$|$$\pi_{A}(\alpha)$$\alpha\cap X,$$\pi_{0}(\{\alpha_{1}, \ldots, \alpha_{n}\})$$\ldots n\rangle$$\partial N(\alpha_{i}\cup\partial X)$$N(\alpha_{i}\cup\partial X)$$\alpha_{i}\cup\partial X$

FIGURE 1
2.3. $(g, b)$ -bridge splittings. Let $H$ be a genus-g $(\geq 0)$ handlebody. We say that a setof arcs
properly embedded in $H$ is a set of trivial arc.s ifis$H$
parallel to. Let
be a handlebody anda set of trivial arcs in $H.$
Then can be isotoped in $H$ so that the projection fromto $[0,1)$ has exactly
one critical point in each
It is well known that every closed orientable 3-manifold $M$ has a genus-g Heegaard
splitting for some $g(\geq 0)$ , i.e.,, whereandare genus-g handlebodies$n$$\{t_{1}, \ldots, t_{n}\}$$\eta$$\partial H$$\tau=\{t_{1}, \ldots, t_{n}\}$$n$$\partial H\cross[O, 1)$$\tau$$t_{i}.$$M=V_{1} \bigcup_{P}V_{2}$$V_{1}$$t_{1}\cup\cdots\cup t_{n}$$V_{2}$

34
such that $M=V_{1}\cup V_{2}$ and. Let be a link in $M$ . We say
that
is $a(g, b)$ -bridge splitting $(or (g, b)$ -splitting for short) for the pair$(M, L)$ if
separates $(M, L)$ into two components
and
where
resp. $\tau_{2}=L\cap V_{2})$ is a set of trivial arcs in (resp.
). Then we say thatis a$(g, b)$ -bridge
$(M,
L)$
(or
short).
bndge
for
a
It
is
known
that
each
has a
surfacesurface$(g, b)$ -bridge splitting for someand . (For a detailed discussion, see [5, Lemma 2.1]).
For $i=1$ or 2,
denotes the subset of
consisting of the vertices withrepresentatives bounding disks in. Then the (Hempel) distance ofis defined by, where$V_{1}\cap V_{2}=\partial V_{1}=\partial V_{2}=P$$L$$(V_{1}, \tau_{1})\bigcup_{P}(V_{2}, \tau_{2})$$P$$(V_{1}, \tau_{1})$

$A$

$b$

$($

$\tau_{1}=L\cap V_{1}$$(V_{2}, \tau_{2})$

$B$

$P$

$b$

$g$

$C^{0}(\partial V_{i}-\tau_{i})$$\mathcal{D}(V_{i})$$V_{i}-\tau_{i}$$(V_{1}, \tau_{1})\bigcup_{P}(V_{2}, \tau_{2})$$P’=\partial V_{1}-\tau_{1}=\partial V_{2}-\tau_{2}.$$d_{P’}(\mathcal{D}(V_{1}), \mathcal{D}(V_{2}))$

3. EXTENDING GEODESICS
Let $F$ be a twice-punctured torus. The following two propositions can be shown by
using arguments in the proof of [6, Propositions 4.1 and 4.4].

Proposition 3.1 (cf. [6, Proposition 4.1]). For an integer, let
a path in $C(F)$ satisfying the following.
(Hl)andare geodesics in $C(F)$ ,(H2), where $X_{n-2}w$ the component of Cl
that contains an essential simple closed curve.
Then
is a geodesic in $C(F)$ .
$r\iota(\geq 4)$$[\alpha_{0}, \ldots, \alpha_{n-2}]$$[\alpha_{U}, \alpha_{1}, \ldots, \alpha_{n}]$be$[\alpha_{n-2}, \alpha_{n-1}, \alpha_{n}]$$diam_{X_{n-2}}(\pi_{X_{n-2}}(\alpha_{n-4}), \pi_{X_{n-2}}(\alpha_{n}))\geq 4n$$(F\backslash$$N(\alpha_{n-2}))$$[\alpha_{0}, \alpha_{1}, \ldots, \alpha_{n}]$Proposition 3.2 (cf. [6, Proposition 4.4]). For an integer $n(\geq 3)$ , leta path in $C(F)$ satisfying the following.
(Hl)andare geodesics in $C(F)$ ,(H2’), wheoeis the component of Cl
that contains an essential simple closed curve.
Then
is a geodesic in $C(F)$ .

$[\alpha_{0}, \alpha_{1}, \ldots, \alpha_{n}]$$[\alpha_{0}, \ldots, \alpha_{n-2}]$be$[\alpha_{n-2}, \alpha_{n-1}, \alpha_{n}]$$diam_{X_{n-2}}(\pi_{X_{n-2}}(\alpha_{0}), \pi_{X_{n-2}}(\alpha_{n}))>2n$$X_{n-2}$$(F\backslash$$N(\alpha_{n-2}))$$[\alpha_{0}, \alpha_{1}, \ldots, \alpha_{n}]$By using Propositions 3.1 and 3.2, we construct a geodesic, for a positive integer$d_{F}(\alpha_{0}, \alpha_{n})=\prime r\iota$$[\alpha_{0}, \alpha_{1}, \ldots, \alpha_{n}]$in $C(F)$ , i.e.,$n.$3.1. $A$ construction of a concrete example: the case when is even. We first
assume that is even. Let
be essential non-separating simple closed curves on
which intersect transversely in one point, and let
be an essential simple closed curveon which is disjoint from
. Let $X_{2}=$ Cl. Note that
$C(F)$
is a geodesic of length two in. Choose a homeomorphism: $Farrow F$ such that$f_{2}(N(\alpha_{2}))=N(\alpha_{2})$ and that
. This is possible by [9,Proposition 4.6]. Let. Note thatandis a geodesic oflength two in $C(F)$ andtransversely in one point.
intersects$n$$\prime n$$\alpha_{U},$$F$$\alpha_{2}$$\alpha_{1}$$S$$(F\backslash N(\alpha_{2}))$$\alpha_{0}\cup\alpha_{2}$$[\alpha_{0}, \alpha_{1}, \alpha_{2}]$$f_{2}$$diam_{X_{2}}(\pi_{X_{2}}(\alpha_{0}), \pi_{X_{2}}(f_{2}(\alpha_{0})))\geq 4r\iota$$\alpha_{3}=f_{2}(\alpha_{1})$$\alpha_{2}$$\alpha_{4}=f_{2}(\alpha_{0})$$\alpha_{4}$FIGURE 2$[\alpha_{2}, \alpha_{3}, \alpha_{4}]$35
We repeat this process to construct a pathinductively as follows. Suppose that we have constructed a pathwithfor each even$i(<n)$ . Then let $X_{i}=$ Cl. Choose a homeomorphism: $Farrow F$ such thatand that$[a_{0}, a_{1}, \ldots, a_{n}]$$[a_{0}, a_{1)}\ldots, a_{i}]$$|\alpha_{i-2}\cap\alpha_{i}|=1$$(F\backslash N(\alpha_{i}))$$f_{i}$$f_{i}(N(\alpha_{i}))=N(\alpha_{i})$(1)$diam_{X_{i}}(\pi_{X_{i}}(\alpha_{i-2}), \pi_{X_{i}}(f_{i}(\alpha_{i-2})))\geq 4/r\iota.$

Then we let
and. Note thatof length two in $C(F)$ , and we have obtained a path$\alpha_{i+1}=f_{i}(\alpha_{i-1})$$\alpha_{i+2}=f_{i}(\alpha_{i-2})$$[\alpha_{i}, \alpha_{i+1}, \alpha_{i+2}]$$[a_{0}, a_{1)}\ldots, a_{i+2}]$

Claim 3.3. For each $k\in\{2,4, \ldots, n\}$ , the path$[\alpha_{0}, \alpha_{1}, \ldots, \alpha_{k}]$geodesic in $C(F)$ .

with

is a geodesic$|\alpha_{i}\cap\alpha_{i+2}|=1.$

constructed above$w$a

Proof. We prove the claim by mathematical induction on . It is clear thatisa geodesic in $C(F)$ . Hence, Claim 3.1 holds for $k=2$ . Assume thatis ageodesic in $C(F)$ for some $k\in\{2,4, \ldots, n-2\}$ . We note thatis a geodesic
in $C(F)$ . Furthermore, by the inequality (1), we have$4n>4k$ . Hence, by Proposition 3.1, the path
is a geodesic in $C(F)$ ,which shows that Claim 3.3 holds for $k+2$ . This completes the proof of Claim 3.3.
$k$$[\alpha_{U}, \alpha_{1}, \alpha_{2}]$$[\alpha_{0}, \alpha_{1}, \ldots, \alpha_{k}]$$[\alpha_{k}, \alpha_{k+1}, \alpha_{k+2}]$$diam_{X_{k}}(\pi_{X_{k}}(\alpha_{k-2}), \pi_{X_{k}}(\alpha_{k+2}))\geq$$[\alpha_{0}, \alpha_{1}, \ldots, \alpha_{k+2}]$$\square$3.2. $A$ construction of a concrete example: the case when is odd. Supposethat is odd. Let
be essential non-separating simple closed curves on which aremutually disjoint. Let be an essential simple closed curve which intersects
andtransversely in one point, respectively. Choose an essential simple closed curveonthat is disjoint fromand . Let $X_{2}=$ Cl. By [10, Proposition 4.6], there
$n$

$n$

$\alpha_{2},$$F$$\alpha_{3}$$x$$\alpha_{2}$$\alpha_{3}$$y_{1}$

exists homeomorphism
(2)

Let

$f$

$F$

$(F\backslash N(\alpha_{2}))$$x$$\alpha_{2}$: $Farrow F$ such that $f_{2}(N(\alpha_{2}))=N(\alpha_{2})$ and that$diam_{X_{2}}(\pi_{X_{2}}(\alpha_{3}), \pi_{X_{2}}(f_{2}(x)))>2n.$and. Note that
andintersects
transversely in one point, which implies thatis a geodesic in $C(F)$ . On
the other hand, choose an essential simple closed curve
on $F$ that is disjoint from
$X_{3}=$
and . LetCl
. By [10, Proposition 4.6], there exists a homeomorphism: $Farrow F$ such that $f_{3}(N(\alpha_{3}))=N(\alpha_{3})$ and that(3)$\alpha_{0}=f_{2}(x)$$\alpha_{1}=f_{2}(y_{1})$$\alpha_{0}\cap\alpha_{1}=\emptyset,$$\alpha_{2}$$\alpha_{1}\cap\alpha_{2}=\emptyset$$\alpha_{0}$$[\alpha_{0}, \alpha_{1}, \alpha_{2}]$$y_{2}$$x$$\alpha_{3}$$(F\backslash N(\alpha_{3}))$$f_{3}$$diam_{X_{3}}(\pi_{X_{3}}(\alpha_{0}), \pi_{X_{3}}(f_{3}(x)))>2n.$Letand. Note thattransversely in one point, which implies that$\alpha_{4}=f_{3}(y_{2})$$\alpha_{6}=f_{3}(x)$$\alpha_{3}\cap\alpha_{4}=\emptyset,$$[\alpha_{3}, \alpha_{4}, \alpha_{5}]$andintersects
is a geodesic in $C(F)$ .

$\alpha_{4}\cap\alpha_{5}=\emptyset$$\alpha_{3}$FIGURE 3
Claim 3.4. The path$[\alpha_{0}, \alpha_{1}, \ldots, \alpha_{5}]$constructed above is a geodesic in $C(F)$ ,$\alpha_{5}$

36
is a
geodesic. By Proposition 3.2 again together with the inequality (3), we see that the pathis also a geodesic in $C(F)$ .

Proof. By Proposition 3.2 together with the inequality (2), the path$[\alpha_{0}, \alpha_{1}, \alpha_{2}, \alpha_{3}]$$\square$$[\alpha_{0}, \alpha_{1}, \ldots, \alpha_{5}]$as follows. Suppose that we have
We extend the above geodesic
for an odd integer with $5\leq i<n$ . Let $X_{i}=$ Cl
constructed a path
. Then there exists a homeomorphism : $Farrow F$ such that $f_{i}(N(\alpha_{i}))=N(\alpha_{i})$
.
. Let
and
and thattransverselyintersects
is a geodesic in $C(F)$ and that
Note thatis a geodesic in $C(F)$ . We
in one point. By Proposition 3.1, the pathof length . Note that
repeat this process until we obtain a geodesic
transversely in one point.
intersects$[\alpha_{0}, \alpha_{1}, \alpha_{2}, \alpha_{3}, \alpha_{4}, \alpha_{5}]$$(F\backslash$$i$$[\alpha_{0}, \alpha_{1}, \ldots, \alpha_{i}]$$N(\alpha_{i}))$$f_{i}$$\alpha_{i+2}=f_{i}(\alpha_{i-2})$$\alpha_{i+1}=f_{i}(\alpha_{i-1})$$diam_{X_{i}}(\pi_{X_{i}}(\alpha_{i-2}), \pi_{X_{t}}(f_{i}(\alpha_{i-2})))\geq 4n$$[\alpha_{i}, \alpha_{i+1}, \alpha_{i+2}]$$\alpha_{i}$$\alpha_{i+2}$$[\alpha_{1}, \alpha_{2}, \ldots, \alpha_{i+2}]$$[\alpha_{1}, \alpha_{2}, \ldots, \alpha_{n}]$$n$$\alpha_{n-2}$$\alpha_{n}$4. PROOF OF THEOREM 1.1
be a solid torus
Basically we mimic the proof of [6, Theorem 1.1]. For $i=1,2$ , letand a trivial arc properly embedded in . The following assertion is proved by Saito[11, Proposition 3.8].
$V_{i}$$V_{i}$$t_{i}$

be an essential disk inAssertion 4.1. Let$dwk$
is isotopic toseparating essential
incan be otoped to be disjoint from
$D_{i}$$V_{i}-t_{i}$$V_{i}-t_{i}$as in Figure 4. Then any nonand any sepamting essential disk in$V_{i}-t_{i}$$D_{i}$$D_{i}.$$u\sigma$

FIGURE 4
in $C(P-t_{1})$ with. Then starting with a geodesic$|\alpha_{n}\cap\alpha_{n+2}|=1$
as in
with
, we construct a geodesic
homeomorphism
by
such
a
glue
and
We
Subsections 3.1 and 3.2.
. Then the argument in the proof of [6, Theorem
andthat1.1], together with Assertion 4.1, enables us to show that the distance of the (1,1)-bridge
is exactly
splittingLet$P=\partial V_{1}$$[\alpha_{0}(=\partial D_{1}), \alpha_{1}, \alpha_{2}]$$|\alpha_{0}\cap\alpha_{2}|=1$$[\alpha_{U}, \alpha_{1}, \ldots, \alpha_{n+2}]$$\partial V_{2}$$\partial V_{1}$$h(\partial t_{1})=\partial t_{2}$$h:\partial V_{1}arrow\partial V_{2}$$h(\alpha_{n+2})=\partial D_{2}$$(V_{1}, t_{1}) \bigcup_{P}(V_{2}, t_{2})$$n.$

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[3] T. Evans, High distance Heegaard splittings of 3-manifolds, Topology and its Apphcations, 153 (14)(20oe), 2631-2647.
[4] J. Hempel, 3-manifolds as mewed from the curve complex, Topology 40 (2001), no. 3, 631-657.
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197 (2001), no. 2, 301-324.
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37[7] K. Ichihara, T. Saito, Knots with arbitrary high d\’istance bndge decomposiitons, preprint,

arXiv: 1209.0097v2.

[8] T. Kobayashi, Heights of simple loops and pseudo-anosov homeomorp hisms, Contemporary Math.
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[9] H. Masur and Y. Minsky, Geometry of the complex of curves. I. Hyperbolicity, Invent. Math.,$138(1):103-149$ , 1999.

[10] H. Masur and Y. Minsky, Geometry of the complex of curves. . Hierarchical structure, Geom. Ifunct.
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$\Pi$

DEPARTMENT OF MATHEMATICS, NARA WOMEN’S UNIVERSITY-mail address: eaa. ido@cc.nara-wu.ac.jp$E$

DEPARTMENT OF MATHEMATICS, NARA WOMEN’S UNIVERSITY-mail address: yeonheej ang@cc. nara-wu. ac. jp$E$

DEPARTMENT OF MATHEMATICS, NARA WOMEN’S UNIVERSITY-mail address: tsuyoshi@cc. nara-wu. ac. jp$E$