# Diagrammatic criteria for strong irreducibility of Heegaard splittings and
finiteness of Goeritz groups

Yuya Koda  Department of Mathematics, Hiyoshi Campus, Keio University, 4-1-1,
Hiyoshi, Kohoku, Yokohama, 223-8521, Japan/ International Institute for
Sustainability with Knotted Chiral Meta Matter (WPI-SKCM${}^{\text{2}}$),
Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima, 739-8526, Japan
<EMAIL_ADDRESS>and  Kazuto Takao  Graduate School of Information Sciences,
Tohoku University, 6-3-09 Aoba, Aramaki-aza Aoba-ku, Sendai, 980-8579, Japan
<EMAIL_ADDRESS>
###### Abstract.

We give two criteria for diagrams of Heegaard splittings of $3$-manifolds.
Weaker one of them guarantees that the splitting is strongly irreducible, and
the stronger one guarantees in addition that the Goeritz group is finite. They
are generalizations of the criteria introduced by Casson–Gordon and
Lustig–Moriah. Their original ones require as input the diagram formed by a
pair of maximal disk systems. Our present ones accept as input that of
arbitrary disk systems, including minimal disk systems. In other words, our
criteria accept Heegaard diagrams.

###### Key words and phrases:

3-dimensional manifold, Heegaard splitting, Goeritz group, Heegaard diagram,
rectangle condition

###### 1991 Mathematics Subject Classification:

57K20, 57K30, 57M60

The first author is supported by JSPS KAKENHI Grant Numbers JP20K03588,
JP21H00978, JP23H05437. The second author was supported by JSPS KAKENHI Grant
Number JP18K13409.

##  1\. Introduction

A Heegaard splitting is a decomposition of a $3$-manifold into two
handlebodies. Here, a handlebody is an orientable $3$-manifold obtained by
attaching some $1$-handles to a $3$-ball. We define a Heegaard splitting of a
$3$-manifold $M$ as the triple $\left(\Sigma,H,H^{\ast}\right)$ of a closed
surface $\Sigma$ and two handlebodies $H$ and $H^{\ast}$ in $M$ such that
$H\cup H^{\ast}=M$ and $\Sigma=H\cap H^{\ast}=\partial H=\partial H^{\ast}$.
The genus of $\left(\Sigma,H,H^{\ast}\right)$ is the genus of $\Sigma$. It is
an important fact that every closed, orientable, connected $3$-manifold admits
a Heegaard splitting. Since Heegaard splittings of genera zero and one are
well understood, we restrict our attention to those of genera greater than
one.

A Heegaard diagram is a standard diagrammatic representation of a $3$-manifold
and a Heegaard splitting of it. Let $\left(\Sigma,H,H^{\ast}\right)$ be a
Heegaard splitting of genus $g$ of a $3$-manifold. An essential disk of a
handlebody, say $H$, is a disk $D$ properly embedded in $H$ such that
$\partial D$ bounds no disk in $\partial H$. A disk system of $H$ is a family
${\mathcal{D}}$ of essential disks of $H$ that are mutually disjoint and
mutually non-parallel, and cut $H$ into some $3$-balls. Note that the
cardinality of ${\mathcal{D}}$ is at least $g$ and at most $3g-3$. The disk
system ${\mathcal{D}}$ is said to be minimal if the cardinality attains $g$,
and maximal if $3g-3$. Let ${\mathcal{D}}^{\ast}$ be a disk system of
$H^{\ast}$. The boundary curves of the disks in ${\mathcal{D}}$ and
${\mathcal{D}}^{\ast}$ form a diagram on the surface $\Sigma$, which
determines the manifold and the splitting. It is called a Heegaard diagram if
${\mathcal{D}}$ and ${\mathcal{D}}^{\ast}$ are minimal. Non-minimal disk
systems might also be useful under some circumstances, but are excessive for
representing the manifold and the splitting.

###  1.1. Strong irreducibility

It is a-priori hard to extract geometric information of the manifold from a
Heegaard splitting, but has been done from the strong irreducibility of it. A
Heegaard splitting $\left(\Sigma,H,H^{\ast}\right)$ of a $3$-manifold $M$ is
said to be strongly irreducibile if any essential disk of $H$ and any
essential disk of $H^{\ast}$ have non-empty intersection. If
$\left(\Sigma,H,H^{\ast}\right)$ is strongly irreducible, then it is
irreducible as suggested by the name, and then $M$ is irreducible as shown by
Haken [4]. If $\left(\Sigma,H,H^{\ast}\right)$ is irreducible but not strongly
irreducible, then $M$ has an essential surface as shown by Casson–Gordon [1].
If $\left(\Sigma,H,H^{\ast}\right)$ is strongly irreducible, then $M$ may have
only essential surfaces that the component submanifolds are glued along with
bounded complexities, as shown in [10, 11, 12, 15].

We remark that the idea of strong irreducibility streamed abundant studies of
so-called Hempel distance. It is an integral measure of complexity for
Heegaard splittings, introduced by Hempel [6]. The distance is greater than
zero if and only if the splitting is irreducible, and greater than one if and
only if strongly irreducible. The distance bounds from below the genera of
essential surfaces in the $3$-manifold, as shown in [5].

Casson–Gordon introduced a diagrammatic criterion, called the rectangle
condition, for a Heegaard splitting to be strongly irreducible. It requires as
input the diagram formed by a pair of maximal disk systems of the hadlebodies.

We generalize the rectangle condition to diagrams including Heegaard diagrams.

######  Theorem 1.

A Heegaard splitting is strongly irreducible if the rectangle condition holds
for the diagram formed by a pair of disk systems of the handlebodies.

See Section 2 for definitions, and Subsection 4.1 for a proof. See also [13,
Example 2.7] for an example of a Heegaard splitting with infinitely many
different pairs of disk systems that form diagrams satisfying the rectangle
condition.

###  1.2. Finiteness of Goeritz groups

The Goeritz group of a Heegaard splitting is the group of symmetries of the
splitting. More precisely, it is the group of isotopy classes of orientation-
preserving homeomorphisms of the $3$-manifold preserving each of the
handlebodies setwise. It is known that the Goeritz group is finite if the
Hempel distance is greater than three, and infinite if the distance is less
than two as shown by Johnson [8]. In the case where the distance is equal to
three, Zou [16] recently announced a proof of the finiteness. In the case
where the distance is equal to two, there exist Heegaard splittings with
finite and infinite Goeritz groups as shown in [9] and [7], respectively. So
far, we do not know the nature that separates these two phenomena. Note that
measuring the distance of a given splitting is another difficult problem.

Lustig–Moriah [13] introduced a diagrammatic criterion, called the double
rectangle condition, for a Goeritz group to be finite. In fact, they did not
mention Goeritz group, but their conclusion implies its finiteness. The
condition is based on maximal disk systems as well as Casson–Gordon’s.

We generalize the double rectangle condition to diagrams including Heegaard
diagrams. The key theorem is the following.

######  Theorem 2.

For any Heegaard splitting of any $3$-manifold, there exist at most finitely
many pairs of disk systems of the hadlebodies that form diagrams satisfying
the double rectangle condition, up to isotopies in the respective hadlebodies.

See Section 2 for definitions, and Subsection 4.2 for a proof. See also
Section 3 for a family of examples satisfying the double rectangle condition.
We also observe that smaller disk systems may work better than maximal disk
systems, and that the Heegaard splittings in the examples have Hempel distance
two. As a consequence of Theorem 2, we have the following.

######  Corollary 3.

The Goeritz group of a Heegaard splitting is finite if the double rectangle
condition holds for the diagram formed by a pair of disk systems of the
handlebodies.

###### Proof.

Let $\left(\Sigma,H,H^{\ast}\right)$ denote the Heegaard splitting. Let $X$
denote the quotient, by isotopies in $H$ and $H^{\ast}$, of the set of pairs
of disk systems of $H$ and $H^{\ast}$ that form diagrams satisfying the double
rectangle condition. It is non-empty by the hypothesis, and finite by Theorem
2. Let $f$ be a representative of an element of the Goeritz group, that is, an
orientation-preserving homeomorphism of the $3$-manifold preserving $H$ and
$H^{\ast}$ setwise. The map $f$ induces a permutation of the finite set $X$.
For some integer $a$, the induced permutation by $f^{a}$ preserves an element
of $X$. Let $\left({\mathcal{D}},{\mathcal{D}}^{\ast}\right)$ be a
representative of that element. After isotopies in $H$ and $H^{\ast}$, we may
suppose that $f^{a}$ preserves the union of the disks in ${\mathcal{D}}$ and
${\mathcal{D}}^{\ast}$ setwise. By the definition of the double rectangle
condition, the disks cut off the surface $\Sigma$ into finitely many polygonal
disks. The map $f^{a}$ induces a permutation of the polygonal disks. For some
integer $b$, the map $f^{ab}$ preserves each polygonal disk setwise. It
follows that $f^{ab}$ is isotopic to the identity in each polygonal disk, and
hence in $\Sigma$. By [3, Theorem 3.7], the map $f^{ab}$ is isotopic to the
identity in $H$ and $H^{\ast}$, and hence in the entire manifold. Thus the
Goeritz group is a torsion group. Note that the Goeritz group can be naturally
thought of as a subgroup of the mapping class group of $\Sigma$. By [2,
Theorem 6.9] (see also [14]), the mapping class group has a representation
into a finite group such that the kernel is torsion-free. It follows that the
torsion subgroup is finite. ∎

##  2\. Definitions

In this section, we give preliminary definitions and the generalized
definitions of rectangle condition and double rectangle condition.

We begin with miscellaneous preliminary definitions. Let $\Sigma$ be a closed,
orientable surface. Let ${\mathcal{C}}$ be a family of mutually disjoint,
essential, simple, closed curves on $\Sigma$, and ${\mathcal{C}}^{\ast}$ be
another such family. We say that ${\mathcal{C}}$ and ${\mathcal{C}}^{\ast}$
intersect essentially in $\Sigma$ if

  * •

the curves in ${\mathcal{C}}$ and ${\mathcal{C}}^{\ast}$ intersect
transversely, and

  * •

no bigon in $\Sigma$ is cobounded by subarcs of curves in ${\mathcal{C}}$ and
${\mathcal{C}}^{\ast}$.

Suppose that ${\mathcal{C}}$ and ${\mathcal{C}}^{\ast}$ intersect essentially
in $\Sigma$, and that the curves are oriented. We regard an orientation of a
curve on $\Sigma$ as one of the two ways of labeling the two sides of the
curve with the signs. A rectangle in $\Sigma$ is said to be of type
$\left(\left(C_{1},\varepsilon_{1}\right),\left(C_{2},\varepsilon_{2}\right);\left(C^{\ast}_{1},\nu_{1}\right),\left(C^{\ast}_{2},\nu_{2}\right)\right)$
if

  * •

$C_{1},C_{2}\in{\mathcal{C}}$ and
$C^{\ast}_{1},C^{\ast}_{2}\in{\mathcal{C}}^{\ast}$ and
$\varepsilon_{1},\varepsilon_{2},\nu_{1},\nu_{2}\in\\{-,+\\}$,

  * •

its interior is disjoint from the curves in ${\mathcal{C}}$ and
${\mathcal{C}}^{\ast}$, and

  * •

it is cobounded by subarcs of $C_{1}$, $C_{2}$, $C^{\ast}_{1}$, $C^{\ast}_{2}$
as in the left of Figure 1.

A rectangle in $\Sigma$ is said to be of type
$\left(\left(C_{1},\varepsilon_{1}\right),C_{2},\left(C_{3},\varepsilon_{3}\right);\left(C^{\ast}_{1},\nu_{1}\right),\left(C^{\ast}_{2},\nu_{2}\right)\right)$
if

  * •

$C_{1},C_{2},C_{3}\in{\mathcal{C}}$ and
$C^{\ast}_{1},C^{\ast}_{2}\in{\mathcal{C}}^{\ast}$ and
$\varepsilon_{1},\varepsilon_{3},\nu_{1},\nu_{2}\in\\{-,+\\}$, and

  * •

it is composed of two rectangles of types
$\left(\left(C_{1},\varepsilon_{1}\right),\left(C_{2},-\right);\left(C^{\ast}_{1},\nu_{1}\right),\left(C^{\ast}_{2},\nu_{2}\right)\right)$
and
$\left(\left(C_{2},+\right),\left(C_{3},\varepsilon_{3}\right);\left(C^{\ast}_{1},\nu_{1}\right),\left(C^{\ast}_{2},\nu_{2}\right)\right)$
as in the right of Figure 1.

Let $H$ be a handlebody. Let $D_{1},D_{2},\ldots,D_{n}$ be mutually disjoint,
essential disks of $H$, and ${\mathcal{D}}$ denote the family
$\left\\{D_{1},D_{2},\ldots,D_{n}\right\\}$. By convention, we denote the
union $D_{1}\cup D_{2}\cup\cdots\cup D_{n}$ by $\overline{\mathcal{D}}$, and
the family $\left\\{\partial D_{1},\partial D_{2},\ldots,\partial
D_{n}\right\\}$ by $\partial{\mathcal{D}}$. Let ${\mathcal{E}}$ be another
family of mutually disjoint, essential disks of $H$. The families
${\mathcal{D}}$ and ${\mathcal{E}}$ are said to be isotopic in $H$ if
$\overline{\mathcal{D}}$ and $\overline{\mathcal{E}}$ are isotopic in $H$. We
say that ${\mathcal{D}}$ and ${\mathcal{E}}$ intersect essentially in $H$ if

  * •

the disks in ${\mathcal{D}}$ and ${\mathcal{E}}$ intersect transversely,

  * •

$\partial{\mathcal{D}}$ and $\partial{\mathcal{E}}$ intersect essentially in
$\partial H$, and

  * •

no lens in $H$ is cobounded by round subdisks of $\overline{\mathcal{D}}$ and
$\overline{\mathcal{E}}$.

Note that if ${\mathcal{D}}$ and ${\mathcal{E}}$ intersect essentially in $H$,
then there is no closed component in
$\overline{\mathcal{D}}\cap\overline{\mathcal{E}}$, by the irreducibility of
$H$. For a set $A$ and two subsets $A_{-}$ and $A_{+}$ of it, we call the pair
$\left(A_{-},A_{+}\right)$ a partition of $A$ if $A_{-}\cup A_{+}=A$ and
$A_{-}\cap A_{+}=\emptyset$. By a graph simply, we mean a finite,
$1$-dimensional, simplicial complex without loop edges or multiple edges. A
graph is said to be $2$-connected if it is connected even after deleting an
arbitrary vertex. Let ${\mathcal{G}}$ be a graph whose vertex set is $A$, and
suppose that $\left(A_{-},A_{+}\right)$ is a partition of $A$. We say that
${\mathcal{G}}$ is doubly $2$-connected with respect to
$\left(A_{-},A_{+}\right)$ if ${\mathcal{G}}$ is connected even after deleting
an arbitrary vertex in $A_{-}$ and an arbitrary vertex in $A_{+}$. For
example, the graph in Figure 2 is not doubly $2$-connected with respect to
$\left(\\{1,2,3\\},\\{4,5,6\\}\right)$ because it is disconnected after
deleting the vertices $2$ and $5$, while it is doubly $2$-connected with
respect to $\left(\\{1,2,4,5\\},\\{3,6\\}\right)$.

$C^{\ast}_{2}$$C^{\ast}_{1}$$C_{1}$$C_{2}$$\nu_{2}$$\nu_{1}$$\varepsilon_{1}$$\varepsilon_{2}$$C^{\ast}_{2}$$C^{\ast}_{1}$$C_{1}$$C_{2}$$C_{3}$$\nu_{2}$$\nu_{1}$$\varepsilon_{1}$$-$$+$$\varepsilon_{3}$
Figure 1. Rectangles. Figure 2. A graph.

###  2.1. Criteria for minimal disk systems

We give definitions of the criteria in the case of minimal disk systems first
for simplicity. Let $g$ be an integer greater than one, and
$\left(\Sigma,H,H^{\ast}\right)$ be a Heegaard splitting of genus $g$ of a
closed, orientable, connected $3$-manifold. Let ${\mathcal{D}}$ and
${\mathcal{D}}^{\ast}$ be minimal disk systems of $H$ and $H^{\ast}$,
respectively, such that $\partial{\mathcal{D}}$ and
$\partial{\mathcal{D}}^{\ast}$ intersect essentially in $\Sigma$. Let
$D_{1},D_{2},\ldots,D_{g}$ denote the disks in ${\mathcal{D}}$, and
$D^{\ast}_{1},D^{\ast}_{2},\ldots,D^{\ast}_{g}$ denote the disks in
${\mathcal{D}}^{\ast}$. Endow the boundary curves of these disks with
orientations. For
$(i,\varepsilon),(j,\delta)\in\\{1,2,\ldots,g\\}\times\\{-,+\\}$, let
${\mathcal{G}}_{i,\varepsilon,j,\delta}$ denote the graph defined as:

  * •

The vertex set is $\\{1,2,\ldots,g\\}\times\\{-,+\\}$;

  * •

Two vertices $(v,\nu)$, $(w,\mu)$ span an edge if there is a rectangle on
$\Sigma$ of type $\left(\left(\partial D_{i},\varepsilon\right),\left(\partial
D_{j},\delta\right);\left(\partial D^{\ast}_{v},\nu\right),\left(\partial
D^{\ast}_{w},\mu\right)\right)$.

Let ${\mathcal{G}}$ denote the graph defined as:

  * •

The vertex set is $\\{1,2,\ldots,g\\}\times\\{-,+\\}$;

  * •

Two vertices $(i,\varepsilon)$, $(j,\delta)$ span an edge if
${\mathcal{G}}_{i,\varepsilon,j,\delta}$ is $2$-connected.

######  Definition 4.

We say that $\left({\mathcal{D}},{\mathcal{D}}^{\ast}\right)$, or the diagram
formed by $\left({\mathcal{D}},{\mathcal{D}}^{\ast}\right)$, satisfies the
rectangle condition if ${\mathcal{G}}$ is $2$-connected.

For $d\in\\{1,2,\ldots,g\\}$ and $\kappa\in\\{-,+\\}$, let
$\varLambda_{d,\kappa}$ denote the set
$\left(\\{1,2,\ldots,g\\}\times\\{-,+\\}\right)\setminus\left\\{(d,\kappa)\right\\}$.
For $d\in\\{1,2,\ldots,g\\}$ and
$\left(i_{-},\varepsilon_{-}\right)\in\varLambda_{d,-}$ and
$\left(i_{+},\varepsilon_{+}\right)\in\varLambda_{d,+}$, let
${\mathcal{H}}_{d,i_{-},\varepsilon_{-},i_{+},\varepsilon_{+}}$ denote the
graph defined as:

  * •

The vertex set is $\\{1,2,\ldots,g\\}\times\\{-,+\\}$;

  * •

Two vertices $(v,\nu)$, $(w,\mu)$ span an edge if there is a rectangle on
$\Sigma$ of type $\left(\left(\partial
D_{i_{-}},\varepsilon_{-}\right),\partial D_{d},\left(\partial
D_{i_{+}},\varepsilon_{+}\right);\left(\partial
D^{\ast}_{v},\nu\right),\left(\partial D^{\ast}_{w},\mu\right)\right)$.

For $d\in\\{1,2,\ldots,g\\}$, let ${\mathcal{H}}_{d}$ denote the graph defined
as:

  * •

The vertex set is
$\left(\\{-\\}\times\varLambda_{d,-}\right)\cup\left(\\{+\\}\times\varLambda_{d,+}\right)$;

  * •

Two vertices $(\kappa,i,\varepsilon)$, $(\kappa,j,\delta)$ span an edge if
${\mathcal{G}}_{i,\varepsilon,j,\delta}$ is $2$-connected;

  * •

Two vertices $\left(-,i_{-},\varepsilon_{-}\right)$,
$\left(+,i_{+},\varepsilon_{+}\right)$ span an edge if
${\mathcal{H}}_{d,i_{-},\varepsilon_{-},i_{+},\varepsilon_{+}}$ is
$2$-connected.

######  Definition 5.

We say that $\left({\mathcal{D}},{\mathcal{D}}^{\ast}\right)$, or the diagram
formed by $\left({\mathcal{D}},{\mathcal{D}}^{\ast}\right)$, satisfies the
double rectangle condition if for every $d\in\\{1,2,\ldots,g\\}$, the graph
${\mathcal{H}}_{d}$ is doubly $2$-connected with respect to
$\left(\\{-\\}\times\varLambda_{d,-},\\{+\\}\times\varLambda_{d,+}\right)$,
and the symmetric condition switching $H$ and $H^{\ast}$ holds.

Note that these criteria does not essentially depend on orientations of the
curves.

###  2.2. Criteria for general disk systems

We now give the general definitions of the criteria. These include both those
in the previous subsection and those by Casson–Gordon and Lustig–Moriah [13].
Let $\left(\Sigma,H,H^{\ast}\right)$ be a Heegaard splitting of genus greater
than one, of a closed, orientable, connected $3$-manifold. Let ${\mathcal{D}}$
and ${\mathcal{D}}^{\ast}$ be disk systems of $H$ and $H^{\ast}$,
respectively, such that $\partial{\mathcal{D}}$ and
$\partial{\mathcal{D}}^{\ast}$ intersect essentially in $\Sigma$. Let
$D_{1},D_{2},\ldots,D_{n}$ denote the disks in ${\mathcal{D}}$, and
$D^{\ast}_{1},D^{\ast}_{2},\ldots,D^{\ast}_{n^{\ast}}$ denote the disks in
${\mathcal{D}}^{\ast}$. Endow the boundary curves of these disks with
orientations. Let $H_{1},H_{2},\ldots,H_{m}$ denote the complementary
components of $\overline{\mathcal{D}}$ in $H$, and
$H^{\ast}_{1},H^{\ast}_{2},\ldots,H^{\ast}_{m^{\ast}}$ denote those of
$\overline{{\mathcal{D}}^{\ast}}$ in $H^{\ast}$. For each
$k\in\\{1,2,\ldots,m\\}$, let $A_{k}$ denote the subset of
$\\{1,2,\ldots,n\\}\times\\{-,+\\}$ such that $(i,\varepsilon)$ belongs to
$A_{k}$ if $\partial H_{k}$ has a path converging to $\partial D_{i}$ from the
$\varepsilon$-side. For example, $A_{1}$ is the set
$\left\\{(1,-),(2,+),(3,-),(3,+)\right\\}$ if $H_{1}$ appears as in Figure 3.
We similarly define $A^{\ast}_{\ell}$ for each
$\ell\in\left\\{1,2,\ldots,m^{\ast}\right\\}$. For $k\in\\{1,2,\ldots,m\\}$
and $\ell\in\\{1,2,\ldots,m^{\ast}\\}$ and $(i,\varepsilon),(j,\delta)\in
A_{k}$, let ${\mathcal{G}}_{k,\ell,i,\varepsilon,j,\delta}$ denote the graph
defined as:

  * •

The vertex set is $A^{\ast}_{\ell}$;

  * •

Two vertices $(v,\nu)$, $(w,\mu)$ span an edge if there is a rectangle on
$\Sigma$ of type $\left(\left(\partial D_{i},\varepsilon\right),\left(\partial
D_{j},\delta\right);\left(\partial D^{\ast}_{v},\nu\right),\left(\partial
D^{\ast}_{w},\mu\right)\right)$.

For $k\in\\{1,2,\ldots,m\\}$, let ${\mathcal{G}}_{k}$ denote the graph defined
as:

  * •

The vertex set is $A_{k}$;

  * •

Two vertices $(i,\varepsilon)$, $(j,\delta)$ span an edge if
${\mathcal{G}}_{k,\ell,i,\varepsilon,j,\delta}$ is $2$-connected for every
$\ell\in\\{1,2,\ldots,m^{\ast}\\}$.

######  Definition 6.

We say that $\left({\mathcal{D}},{\mathcal{D}}^{\ast}\right)$, or the diagram
formed by $\left({\mathcal{D}},{\mathcal{D}}^{\ast}\right)$, satisfies the
rectangle condition if ${\mathcal{G}}_{k}$ is $2$-connected for every
$k\in\\{1,2,\ldots,m\\}$.

For $d\in\\{1,2,\ldots,n\\}$ and $\kappa\in\\{-,+\\}$, let $k_{d,\kappa}$
denote the element in $\\{1,2,\ldots,m\\}$ such that $(d,\kappa)$ belongs to
$A_{k_{d,\kappa}}$, and $\varLambda_{d,\kappa}$ denote the set
$A_{k_{d,\kappa}}\setminus\left\\{(d,\kappa)\right\\}$. For
$\ell\in\\{1,2,\ldots,m^{\ast}\\}$ and $d\in\\{1,2,\ldots,n\\}$ and
$\left(i_{-},\varepsilon_{-}\right)\in\varLambda_{d,-}$ and
$\left(i_{+},\varepsilon_{+}\right)\in\varLambda_{d,+}$, let
${\mathcal{H}}_{\ell,d,i_{-},\varepsilon_{-},i_{+},\varepsilon_{+}}$ denote
the graph defined as:

  * •

The vertex set is $A^{\ast}_{\ell}$;

  * •

Two vertices $(v,\nu)$, $(w,\mu)$ span an edge if there is a rectangle on
$\Sigma$ of type $\left(\left(\partial
D_{i_{-}},\varepsilon_{-}\right),\partial D_{d},\left(\partial
D_{i_{+}},\varepsilon_{+}\right);\left(\partial
D^{\ast}_{v},\nu\right),\left(\partial D^{\ast}_{w},\mu\right)\right)$.

For $d\in\\{1,2,\ldots,n\\}$, let ${\mathcal{H}}_{d}$ denote the graph defined
as:

  * •

The vertex set is
$\left(\\{-\\}\times\varLambda_{d,-}\right)\cup\left(\\{+\\}\times\varLambda_{d,+}\right)$;

  * •

Two vertices $(\kappa,i,\varepsilon)$, $(\kappa,j,\delta)$ span an edge if
${\mathcal{G}}_{k_{d,\kappa},\ell,i,\varepsilon,j,\delta}$ is $2$-connected
for every $\ell\in\\{1,2,\ldots,m^{\ast}\\}$;

  * •

Two vertices $\left(-,i_{-},\varepsilon_{-}\right)$,
$\left(+,i_{+},\varepsilon_{+}\right)$ span an edge if
${\mathcal{H}}_{\ell,d,i_{-},\varepsilon_{-},i_{+},\varepsilon_{+}}$ is
$2$-connected for every $\ell\in\\{1,2,\ldots,m^{\ast}\\}$.

Then we define the double rectangle condition exactly as in Definition 5. Note
that these criteria does not essentially depend on orientations of the curves.

$H_{1}$$D_{1}$$-$$D_{2}$$+$$D_{3}$ Figure 3. A possible location of $H_{1}$.

##  3\. Examples

In this section, we give a family of examples satisfying the rectangle
condition and the double rectangle condition.

We construct $3$-manifolds, Heegaard splittings, and pairs of minimal disk
systems as follows. Let $g$ be an integer greater than one, and let $H$ and
$H^{\ast}$ be handlebodies with boundary surfaces of genus $g$. Let
${\mathcal{D}}$ and ${\mathcal{D}}^{\ast}$ be minimal disk systems of $H$ and
$H^{\ast}$, respectively, and let $D_{1},D_{2},\ldots,D_{g}$ and
$D^{\ast}_{1},D^{\ast}_{2},\ldots,D^{\ast}_{g}$ denote their disks,
respectively. Let $h\colon\partial H^{\ast}\to\partial H$ be a homeomorphism
that maps $\partial D_{i}^{\ast}$ to $\partial D_{i}$ for each
$i\in\\{1,2,\ldots,g\\}$. Let $\gamma$ denote the simple, closed curve on
$\partial H$ as in Figure 4, and $\tau\colon\partial H\to\partial H$ denote
the Dehn twist along $\gamma$. Let $l$ be an integer such that $|l|\geq 2$,
and $M$ denote the $3$-manifold obtained by gluing $H$ and $H^{\ast}$ by
$\tau^{l}\circ h$. Let $\Sigma$ denote the common boundary of $H$ and
$H^{\ast}$ in $M$. Note that $\left(\Sigma,H,H^{\ast}\right)$ is a Heegaard
splitting of $M$ of genus $g$. We restrict to the case where $g=3$ and $l=2$
for simplicity in the following paragraphs, and leave the other cases to the
reader.

$D_{1}$$D_{2}$$D_{3}$$\cdots$$D_{g-1}$$D_{g}$$-$$+$$-$$+$$-$$+$$-$$+$$-$$+$$\gamma$
Figure 4. Essential disks of $H$, orientations of their boundaries, and a
simple, closed curve on $\partial H$.

We verify the criteria as follows. Endow $\partial D_{1}$, $\partial D_{2}$,
$\partial D_{3}$ with orientations as in Figure 4, and $\partial
D_{1}^{\ast}$, $\partial D_{2}^{\ast}$, $\partial D_{3}^{\ast}$ with
orientations to agree with those of $\partial D_{1}$, $\partial D_{2}$,
$\partial D_{3}$ via $h$. Then the curves of $\partial{\mathcal{D}}^{\ast}$
appear on $\partial H$ as in Figure 5. Let
${\mathcal{G}}_{i,\varepsilon,j,\delta}$, ${\mathcal{G}}$,
$\varLambda_{d,\kappa}$,
${\mathcal{H}}_{d,i_{-},\varepsilon_{-},i_{+},\varepsilon_{+}}$ and
${\mathcal{H}}_{d}$ be as in Subsection 2.1. Note that, if $\gamma$ has a
subarc that leaves $D_{i}$ to the $\varepsilon$-side and goes without
intersecting $\overline{\mathcal{D}}$ and arrives $D_{j}$ from the
$\delta$-side, then a bunch of subarcs of
$\partial\overline{{\mathcal{D}}^{\ast}}$ goes the same way, and the
$\varepsilon$-side of $\partial D_{i}^{\ast}$ faces the $\delta$-side of
$\partial D_{j}^{\ast}$ in the bunch. These show that the graph
${\mathcal{G}}_{i,\varepsilon,j,\delta}$ contains the subgraph as in the left
of Figure 6, which shows that ${\mathcal{G}}_{i,\varepsilon,j,\delta}$ is
$2$-connected. Similarly for the graph
${\mathcal{H}}_{d,i_{-},\varepsilon_{-},i_{+},\varepsilon_{+}}$. They show
that ${\mathcal{G}}$ coincides with the graph in the left of Figure 6, and
${\mathcal{H}}_{1}$ is as shown in the right. We can see that ${\mathcal{G}}$
is $2$-connected, and ${\mathcal{H}}_{1}$ is doubly $2$-connected with respect
to $\left(\\{-\\}\times\varLambda_{1,-},\\{+\\}\times\varLambda_{1,+}\right)$.
Similarly for the graphs ${\mathcal{H}}_{2}$ and ${\mathcal{H}}_{3}$. We can
also see that the symmetric condition switching $H$ and $H^{\ast}$ holds.
Thus, $\left({\mathcal{D}},{\mathcal{D}}^{\ast}\right)$ satisfies the
rectangle condition and the double rectangle condition. By Theorem 1 and
Corollary 3, the Heegaard splitting $\left(\Sigma,H,H^{\ast}\right)$ is
strongly irreducible, and its Goeritz group is finite.

$\partial D_{1}^{\ast}$$\partial D_{2}^{\ast}$$\partial D_{3}^{\ast}$ Figure
5. The three curves of $\partial{\mathcal{D}}^{\ast}$ on $\partial H$. Figure
6. A subgraph of ${\mathcal{G}}_{i,\varepsilon,j,\delta}$ and the graph
${\mathcal{H}}_{1}$.

We remark that the double rectangle condition fails for the pairs of maximal
disk systems obtained by naively adding disks. Let $D_{4}$, $D_{5}$, $D_{6}$
denote the essential disks of $H$ as in Figure 7, and $D^{\ast}_{4}$,
$D^{\ast}_{5}$, $D^{\ast}_{6}$ be essential disks of $H^{\ast}$ such that
$h\left(\partial D_{4}^{\ast}\right)=\partial D_{4}$, $h\left(\partial
D_{5}^{\ast}\right)=\partial D_{5}$, $h\left(\partial
D_{6}^{\ast}\right)=\partial D_{6}$. Let ${\mathcal{E}}$ and
${\mathcal{E}}^{\ast}$ denote the maximal disk systems
$\left\\{D_{1},D_{2},\ldots,D_{6}\right\\}$ of $H$ and
$\left\\{D^{\ast}_{1},D^{\ast}_{2},\ldots,D^{\ast}_{6}\right\\}$ of
$H^{\ast}$, respectively. Then the pair
$\left({\mathcal{E}},{\mathcal{E}}^{\ast}\right)$ fails the double rectangle
condition, in the absence of rectangles of the type illustrated in dark gray
in Figure 7. This supports that our generalization has some advantage over the
original by Lustig–Moriah [13].

$\gamma$$D_{1}$$D_{2}$$D_{3}$$D_{4}$$D_{5}$$D_{6}$ Figure 7. Additional
essential disks of $H$, and an example of missing types of rectangles.

We also remark that their Hempel distance is equal to two. On one hand, by the
rectangle condition, the distance is greater than or equal to two. On the
other hand, by the definition, the sequence from $\partial D_{1}$ to $\partial
D_{1}^{\ast}$ through the curve $C$ as in Figure 8 shows that the distance is
less than or equal to two.

$D_{1}$$\gamma$$C$ Figure 8. A simple, closed curve on $\Sigma$ disjoint from
$\partial D_{1}$ and $\partial D_{1}^{\ast}$.

##  4\. Proofs

In this section, we give proofs of the main theorems.

The following notions and facts play important roles throughout the proofs.
Let $H$ be a handlebody, and ${\mathcal{D}}$ be a disk system of $H$. A wave
disk with respect to ${\mathcal{D}}$ is a bigon $\Omega$ in $H$ such that

  * •

$\Omega\cap\partial H$ and $\Omega\cap\overline{\mathcal{D}}$ are the two
edges of the bigon $\Omega$, and

  * •

no bigon in $\partial H$ is cobounded by $\Omega\cap\partial H$ and any subarc
of $\partial\overline{\mathcal{D}}$.

The arc $\Omega\cap\partial H$ is called a wave with respect to
${\mathcal{D}}$. This generalizes the classical notion in the case where
${\mathcal{D}}$ is maximal. If $E$ is an essential disk of $H$ such that
$\\{E\\}$ and ${\mathcal{D}}$ intersect essentially in $H$, then $E$ is
disjoint from $\overline{\mathcal{D}}$ or contains a wave disk with respect to
${\mathcal{D}}$. This can be seen by recalling that there is no closed
component in $E\cap\overline{\mathcal{D}}$, and considering an outermost
subdisk of $E$ cut off by the arcs of $E\cap\overline{\mathcal{D}}$.

We use the following convention. For a handlebody $H$, and a disk system
${\mathcal{D}}$ of $H$, we would like to consider “components cut off from $H$
by $\overline{\mathcal{D}}$” and regard them as closed $3$-balls. Note that
the closure of a complementary component of $\overline{\mathcal{D}}$ in $H$
is, however, possibly a handlebody with one or more $1$-handles as in Figure
3. To make things clear, we take the universal covering space of $H$, denoted
by $\widetilde{H}$, and the covering map, denoted by $\pi_{H}$. Let
$\widetilde{\mathcal{D}}$ denote the family of lifts in $\widetilde{H}$ of
disks of ${\mathcal{D}}$. Let ${\mathcal{B}}({\mathcal{D}})$ denote the family
of closures of complementary components in $\widetilde{H}$ of the lifted
disks. Note that ${\mathcal{B}}({\mathcal{D}})$ consists of infinitely many
closed $3$-balls. For $B\in{\mathcal{B}}({\mathcal{D}})$, let
$\widetilde{\mathcal{D}}_{B}$ denote the family of disks in
$\widetilde{\mathcal{D}}$ contained in $\partial B$. When the curves of
$\partial{\mathcal{D}}$ are endowed with orientations, we regard the disks of
${\mathcal{D}}$ and $\widetilde{\mathcal{D}}$ as endowed with the induced
orientations. A disk $\widetilde{D}$ in $\widetilde{\mathcal{D}}_{B}$ is
called the $\varepsilon$-side lift of $\pi_{H}\bigl{(}\widetilde{D}\bigr{)}$
in $B$, if $B$ is adjacent to $\widetilde{D}$ from the $\varepsilon$-side, for
$\varepsilon\in\\{-,+\\}$.

###  4.1. Strong irreducibility

We prove Theorem 1. Let $\left(\Sigma,H,H^{\ast}\right)$, ${\mathcal{D}}$,
${\mathcal{D}}^{\ast}$ be as in Subsection 2.2. Assume that
$\left({\mathcal{D}},{\mathcal{D}}^{\ast}\right)$ satisfies the rectangle
condition but $\left(\Sigma,H,H^{\ast}\right)$ is not strongly irreducible.
Then there exist an essential disk $E$ of $H$ and an essential disk $E^{\ast}$
of $H^{\ast}$ such that $E$ and $E^{\ast}$ are disjoint. We can arrange that
$\left\\{E\right\\}$ and ${\mathcal{D}}$ intersect essentially in $H$, and
similarly for $E^{\ast}$, after some isotopies if necessary. Then $E$ is
disjoint from $\overline{\mathcal{D}}$ or contains a wave disk with respect to
${\mathcal{D}}$, and similarly for $E^{\ast}$. By the following lemma, we have
a contradiction to the disjointness of $E$ and $E^{\ast}$.

######  Lemma 7.

Let $\left(\Sigma,H,H^{\ast}\right)$, ${\mathcal{D}}$, ${\mathcal{D}}^{\ast}$
be as in Subsection 2.2. Let $O$ be either an essential disk of $H$ disjoint
from $\overline{\mathcal{D}}$ or a wave disk with respect to ${\mathcal{D}}$.
Define $O^{\ast}$ similarly in $H^{\ast}$. If
$\left({\mathcal{D}},{\mathcal{D}}^{\ast}\right)$ satisfies the rectangle
condition, then $O$ and $O^{\ast}$ have non-empty intersection.

###### Proof.

We use the notation in Subsection 2.2. Let $k_{0}$ denote the element in
$\\{1,2,\ldots,m\\}$ such that the closure of $H_{k_{0}}$ contains $O$, and
$\ell_{0}$ denote that in $\\{1,2,\ldots,m^{\ast}\\}$ for $O^{\ast}$. Let
$B_{k_{0}}$ denote the closure of a lift of $H_{k_{0}}$ in $\widetilde{H}$.
Define $B^{\ast}_{\ell_{0}}$ similarly over $H^{\ast}_{\ell_{0}}$. Let
$\widetilde{O}$ denote the lift of $O$ in $B_{k_{0}}$, and
$\widetilde{O^{\ast}}$ denote the lift of $O^{\ast}$ in $B^{\ast}_{\ell_{0}}$.
Let $\widetilde{D}_{i,\varepsilon}$ denote the $\varepsilon$-side lift of
$D_{i}$ in $B_{k_{0}}$ for $(i,\varepsilon)\in A_{k_{0}}$. Define
$\widetilde{D^{\ast}}_{v,\nu}$ similarly for $(v,\nu)\in A^{\ast}_{\ell_{0}}$.
Let $(I,J)$ denote the partition of $\bigl{\\{}(i,\varepsilon)\in
A_{k_{0}}\bigm{|}\widetilde{D}_{i,\varepsilon}\cap\widetilde{O}=\emptyset\bigr{\\}}$
such that $\widetilde{O}$ separates
$\bigl{\\{}\widetilde{D}_{i,\varepsilon}\bigm{|}(i,\varepsilon)\in
I\bigr{\\}}$ and $\bigl{\\{}\widetilde{D}_{j,\delta}\bigm{|}(j,\delta)\in
J\bigr{\\}}$ in $B_{k_{0}}$. Since the graph ${\mathcal{G}}_{k_{0}}$ is
$2$-connected, there exist a member $\left(i_{0},\varepsilon_{0}\right)$ in
$I$ and a member $\left(j_{0},\delta_{0}\right)$ in $J$ that span an edge in
${\mathcal{G}}_{k_{0}}$. By the definition, the graph
${\mathcal{G}}_{k_{0},\ell_{0},i_{0},\varepsilon_{0},j_{0},\delta_{0}}$ is
$2$-connected. Let $(V,W)$ denote the partition of $\bigl{\\{}(v,\nu)\in
A^{\ast}_{\ell_{0}}\bigm{|}\;\widetilde{D^{\ast}}_{v,\nu}\cap\widetilde{O^{\ast}}=\emptyset\bigr{\\}}$
such that $\widetilde{O^{\ast}}$ separates
$\bigl{\\{}\widetilde{D^{\ast}}_{v,\nu}\bigm{|}(v,\nu)\in V\bigr{\\}}$ and
$\bigl{\\{}\widetilde{D^{\ast}}_{w,\mu}\bigm{|}(w,\mu)\in W\bigr{\\}}$ in
$B^{\ast}_{\ell_{0}}$. There exist a member $\left(v_{0},\nu_{0}\right)$ in
$V$ and a member $\left(w_{0},\mu_{0}\right)$ in $W$ that span an edge in
${\mathcal{G}}_{k_{0},\ell_{0},i_{0},\varepsilon_{0},j_{0},\delta_{0}}$. By
the definition, there exists a rectangle $R$ on $\Sigma$ of type
$\left(\left(\partial D_{i_{0}},\varepsilon_{0}\right),\left(\partial
D_{j_{0}},\delta_{0}\right);\left(\partial
D^{\ast}_{v_{0}},\nu_{0}\right),\left(\partial
D^{\ast}_{w_{0}},\mu_{0}\right)\right)$, as illustrated in Figure 9. Since
$\widetilde{O}$ separates $\widetilde{D}_{i_{0},\varepsilon_{0}}$ and
$\widetilde{D}_{j_{0},\delta_{0}}$ in $B_{k_{0}}$, the rectangle $R$ contains
a subarc of $O\cap\Sigma$ that separates the two edges in $\partial D_{i_{0}}$
and $\partial D_{j_{0}}$. Similarly $R$ also contains a subarc of
$O^{\ast}\cap\Sigma$ that separates the two edges in $\partial
D^{\ast}_{v_{0}}$ and $\partial D^{\ast}_{w_{0}}$. It follows that
$O\cap\Sigma$ and $O^{\ast}\cap\Sigma$ intersect in $R$. ∎

$O$$D_{i_{0}}$$D_{j_{0}}$$\partial D^{\ast}_{v_{0}}$$\partial
D^{\ast}_{w_{0}}$$R$ Figure 9. The rectangle $R$ in a possible case where $O$
is a wave disk.

###  4.2. Finiteness of disk systems

We prove Theorem 2 in this subsection.

We have the following relation between our criteria as their names suggest.

######  Lemma 8.

Let ${\mathcal{D}}$ and ${\mathcal{D}}^{\ast}$ be as in Subsection 2.2. If
$\left({\mathcal{D}},{\mathcal{D}}^{\ast}\right)$ satisfies the double
rectangle condition, then $\left({\mathcal{D}},{\mathcal{D}}^{\ast}\right)$
satisfies the rectangle condition.

###### Proof.

We use the notation in Subsection 2.2. Let $k$ be any element in
$\\{1,2,\ldots,m\\}$. We consider deleting an arbitrary vertex
$(i,\varepsilon)$ from the graph ${\mathcal{G}}_{k}$. Since
${\mathcal{G}}_{k}$ has at least three vertices, we can take another vertex
$(d,\kappa)$ than $(i,\varepsilon)$. By the double rectangle condition, the
graph ${\mathcal{H}}_{d}$ is doubly $2$-connected with respect to
$\left(\\{-\\}\times\varLambda_{d,-},\\{+\\}\times\varLambda_{d,+}\right)$.
Hence, ${\mathcal{H}}_{d}$ is connected even after deleting the vertex
$(\kappa,i,\varepsilon)$ and any vertex in
$\\{-\kappa\\}\times\varLambda_{d,-\kappa}$. There is a natural simplicial map
from ${\mathcal{H}}_{d}$ to ${\mathcal{G}}_{k}$ whereby
$\\{\kappa\\}\times\varLambda_{d,\kappa}$ injects to
$A_{k}\setminus\\{(d,\kappa)\\}$ and
$\\{-\kappa\\}\times\varLambda_{d,-\kappa}$ collapses to $(d,\kappa)$. These
show that ${\mathcal{G}}_{k}$ is connected even after deleting
$(i,\varepsilon)$, to conclude that ${\mathcal{G}}_{k}$ is $2$-connected. ∎

We use the following notation. Let $g$ be an integer greater than one, and
$\left(\Sigma,H,H^{\ast}\right)$ be a Heegaard splitting of genus $g$ of a
closed, orientable, connected $3$-manifold. We fix maximal disk systems
${\mathcal{E}}$ and ${\mathcal{E}}^{\ast}$ of $H$ and $H^{\ast}$,
respectively, such that $\partial{\mathcal{E}}$ and
$\partial{\mathcal{E}}^{\ast}$ intersect essentially in $\Sigma$. Let
$\mathfrak{DR}$ denote the set of pairs of disk systems of $H$ and $H^{\ast}$
such that $\left({\mathcal{D}},{\mathcal{D}}^{\ast}\right)$ belongs to
$\mathfrak{DR}$ if

  * •

$\partial{\mathcal{D}}$ and $\partial{\mathcal{D}}^{\ast}$ intersect
essentially in $\Sigma$,

  * •

$\left({\mathcal{D}},{\mathcal{D}}^{\ast}\right)$ satisfies the double
rectangle condition,

  * •

${\mathcal{E}}$ and ${\mathcal{D}}$ intersect essentially in $H$,

  * •

${\mathcal{E}}^{\ast}$ and ${\mathcal{D}}^{\ast}$ intersect essentially in
$H^{\ast}$,

  * •

$\partial{\mathcal{E}}$ and $\partial{\mathcal{D}}^{\ast}$ intersect
essentially in $\Sigma$,

  * •

$\partial{\mathcal{E}}^{\ast}$ and $\partial{\mathcal{D}}$ intersect
essentially in $\Sigma$, and

  * •

$\overline{\mathcal{D}}\cap\overline{{\mathcal{D}}^{\ast}}\cap\overline{\mathcal{E}}=\overline{\mathcal{D}}\cap\overline{{\mathcal{D}}^{\ast}}\cap\overline{{\mathcal{E}}^{\ast}}=\overline{\mathcal{D}}\cap\overline{\mathcal{E}}\cap\overline{{\mathcal{E}}^{\ast}}=\overline{{\mathcal{D}}^{\ast}}\cap\overline{\mathcal{E}}\cap\overline{{\mathcal{E}}^{\ast}}=\emptyset$.

We also define ${\mathfrak{R}}$ by dropping “double” in the second condition.
Note that ${\mathfrak{R}}$ contains $\mathfrak{DR}$ by Lemma 8. Note also that
if $\left({\mathcal{D}},{\mathcal{D}}^{\ast}\right)$ satisfies the top two
conditions, then we can arrange the remaining conditions by isotopies of
${\mathcal{D}}$ and ${\mathcal{D}}^{\ast}$ in $H$ and $H^{\ast}$,
respectively. The task in this subsection is, therefore, to show the
finiteness of $\mathfrak{DR}$ modulo isotopies in the respective hadlebodies.

First, we consider some $1$-dimensional objects, and show the finiteness of
their types. Let ${\mathcal{D}}$ and ${\mathcal{D}}^{\prime}$ be disk systems
of $H$ intersecting essentially in $H$. An intersection arc of ${\mathcal{D}}$
with respect to ${\mathcal{D}}^{\prime}$ is a component arc of
$\overline{\mathcal{D}}\cap\overline{{\mathcal{D}}^{\prime}}$. A lift of it in
$\widetilde{H}$ is called an intersection arc of $\widetilde{\mathcal{D}}$
with respect to $\widetilde{{\mathcal{D}}^{\prime}}$. An isolated circle of
${\mathcal{D}}$ with respect to ${\mathcal{D}}^{\prime}$ is a component circle
of $\partial\overline{\mathcal{D}}$ disjoint from
$\overline{{\mathcal{D}}^{\prime}}$. A connecting arc of ${\mathcal{D}}$ with
respect to ${\mathcal{D}}^{\prime}$ is the closure of a component arc of
$\partial\overline{\mathcal{D}}\setminus\overline{{\mathcal{D}}^{\prime}}$. A
lift of it in $\widetilde{H}$ is called a connecting arc of
$\widetilde{\mathcal{D}}$ with respect to
$\widetilde{{\mathcal{D}}^{\prime}}$. Let
$\left({\mathcal{D}}_{1},{\mathcal{D}}_{1}^{\ast}\right)$ and
$\left({\mathcal{D}}_{2},{\mathcal{D}}_{2}^{\ast}\right)$ be pairs in
${\mathfrak{R}}$. Intersection arcs $\alpha_{1}$ and $\alpha_{2}$ of
${\mathcal{D}}_{1}$ and ${\mathcal{D}}_{2}$, respectively, with respect to
${\mathcal{E}}$ are said to be
$\left({\mathcal{E}},{\mathcal{E}}^{\ast}\right)$-parallel if
$\left(\alpha_{1},\partial\alpha_{1}\right)$ and
$\left(\alpha_{2},\partial\alpha_{2}\right)$ are isotopic in
$\left(\overline{\mathcal{E}},\partial\overline{\mathcal{E}}\setminus\overline{{\mathcal{E}}^{\ast}}\right)$.
Isolated circles $C_{1}$ and $C_{2}$ of ${\mathcal{D}}_{1}$ and
${\mathcal{D}}_{2}$, respectively, with respect to ${\mathcal{E}}$ are said to
be $\left({\mathcal{E}},{\mathcal{E}}^{\ast}\right)$-parallel if $C_{1}$ and
$C_{2}$ are isotopic in $\Sigma\setminus\overline{\mathcal{E}}$. Connecting
arcs $\beta_{1}$ and $\beta_{2}$ of ${\mathcal{D}}_{1}$ and
${\mathcal{D}}_{2}$, respectively, with respect to ${\mathcal{E}}$ are said to
be $\left({\mathcal{E}},{\mathcal{E}}^{\ast}\right)$-parallel if
$\left(\beta_{1},\partial\beta_{1}\right)$ and
$\left(\beta_{2},\partial\beta_{2}\right)$ are isotopic in
$\left(\Sigma,\partial\overline{\mathcal{E}}\setminus\overline{{\mathcal{E}}^{\ast}}\right)$.
Let $\mathcal{IA}$, $\mathcal{IC}$, $\mathcal{CA}$ denote the sets

$\displaystyle\bigl{\\{}\,\text{intersection arcs of ${\mathcal{D}}$ with
respect to
${\mathcal{E}}$}\bigm{|}\left({\mathcal{D}},{\mathcal{D}}^{\ast}\right)\in{\mathfrak{R}}\bigr{\\}}\big{/}\text{$\left({\mathcal{E}},{\mathcal{E}}^{\ast}\right)$-parallel},$
$\displaystyle\bigl{\\{}\,\text{isolated circles of ${\mathcal{D}}$ with
respect to
${\mathcal{E}}$}\bigm{|}\left({\mathcal{D}},{\mathcal{D}}^{\ast}\right)\in{\mathfrak{R}}\bigr{\\}}\big{/}\text{$\left({\mathcal{E}},{\mathcal{E}}^{\ast}\right)$-parallel},$
$\displaystyle\bigl{\\{}\,\text{connecting arcs of ${\mathcal{D}}$ with
respect to
${\mathcal{E}}$}\bigm{|}\left({\mathcal{D}},{\mathcal{D}}^{\ast}\right)\in{\mathfrak{R}}\bigr{\\}}\big{/}\text{$\left({\mathcal{E}},{\mathcal{E}}^{\ast}\right)$-parallel},$

respectively.

######  Lemma 9.

The sets $\mathcal{IA}$, $\mathcal{IC}$, $\mathcal{CA}$ are finite.

###### Proof.

We can see the finiteness of $\mathcal{IA}$ similarly to [13, Lemma 3.3], and
that of $\mathcal{IC}$ more easily. We show that of $\mathcal{CA}$ as follows,
similarly to [13, Lemma 3.4]. Let $Y_{1},Y_{2},\ldots,Y_{2g-2}$ be $3$-balls
in ${\mathcal{B}}\left({\mathcal{E}}\right)$ whose images under $\pi_{H}$
cover the whole handlebody $H$. Let $\widetilde{E}_{p,1}$,
$\widetilde{E}_{p,2}$, $\widetilde{E}_{p,3}$ denote the disks of
$\widetilde{\mathcal{E}}_{Y_{p}}$ for each $p\in\\{1,2,\ldots,2g-2\\}$. We may
suppose that ${\mathfrak{R}}$ is not empty. It follows by Theorem 1 that
$\pi_{H}\bigl{(}\widetilde{E}_{p,q}\bigr{)}$ and any disk $E^{\ast}$ in
${\mathcal{E}}^{\ast}$ have non-empty intersection for each $q\in\\{1,2,3\\}$.
The lift in $\partial\widetilde{E}_{p,q}$ of each of the intersection points
is adjacent to a closed arc properly embedded in $\partial\widetilde{H}\cap
Y_{p}$ whose image under $\pi_{H}$ lies in $\partial E^{\ast}$. Fix such a
closed arc $\widetilde{\theta}_{p,q}$ for $p\in\\{1,2,\ldots,2g-2\\}$ and
$q\in\\{1,2,3\\}$. Let $s_{p,q}$ denote the twisting number of
$\widetilde{\theta}_{p,q}$ along $\partial\widetilde{E}_{p,q}$, that is, the
number of twists relative to a reference arc on $\partial\widetilde{H}\cap
Y_{p}$ and an orientation of $\partial\widetilde{E}_{p,q}$. The ambiguities in
those choices do not affect our conclusion below. Let $S$ denote the finite
set

$\bigl{\\{}s\in{\mathbb{Z}}\bigm{|}p\in\\{1,2,\ldots,2g-2\\},\
q\in\\{1,2,3\\},\ \left|s-s_{p,q}\right|\leq 2\bigr{\\}}.$

Now, we take a pair $\left({\mathcal{D}},{\mathcal{D}}^{\ast}\right)$ in
${\mathfrak{R}}$ and a connecting arc $\beta$ of ${\mathcal{D}}$ with respect
to ${\mathcal{E}}$. Suppose that $p$ is the index in $\\{1,2,\ldots,2g-2\\}$
such that $\pi_{H}\left(Y_{p}\right)$ contains $\beta$. Let
$\widetilde{\beta}$ denote the lift of $\beta$ in $Y_{p}$. Suppose that $q$ is
one of the indices in $\\{1,2,3\\}$ such that $\widetilde{\beta}$ has an
endpoint in $\partial\widetilde{E}_{p,q}$. Let $t$ denote the twisting number
of $\widetilde{\beta}$ along $\partial\widetilde{E}_{p,q}$. The following
claim guarantees that $t$ belongs to $S$.

######  Claim.

$\left|t-s_{p,q}\right|\leq 2$.

###### Proof.

Assume that $\left|t-s_{p,q}\right|>2$. Note that $\widetilde{E}_{p,q}$
contains one or more intersection arcs of $\widetilde{\mathcal{D}}$ with
respect to $\widetilde{\mathcal{E}}$, since $\widetilde{\beta}$ has an
endpoint there. Let $\widetilde{\alpha}$ be outermost one of them. Let
$\widetilde{\beta}_{1}$ and $\widetilde{\beta}_{2}$ denote the connecting arcs
of $\widetilde{\mathcal{D}}$ with respect to $\widetilde{\mathcal{E}}$ in
$Y_{p}$ adjacent to $\partial\widetilde{\alpha}$. Let $t_{1}$ and $t_{2}$
denote the twisting numbers of $\widetilde{\beta}_{1}$ and
$\widetilde{\beta}_{2}$, respectively, along $\partial\widetilde{E}_{p,q}$. We
have either that $\widetilde{\beta}=\widetilde{\beta}_{1}$ or that
$\widetilde{\beta}=\widetilde{\beta}_{2}$, or that $\widetilde{\beta}$ is
disjoint from $\widetilde{\beta}_{1}$ and $\widetilde{\beta}_{2}$. It follows
that $\left|t_{1}-t\right|\leq 1$ and $\left|t_{2}-t\right|\leq 1$, and hence
that $\left|t_{1}-s_{p,q}\right|>1$ and $\left|t_{2}-s_{p,q}\right|>1$. Then
$\pi_{H}\bigl{(}\widetilde{\theta}_{p,q}\bigr{)}$ contains a wave $\omega$
with respect to ${\mathcal{D}}$ as illustrated in Figure 10. By shifting
$\omega$ slightly, we obtain a wave with respect to ${\mathcal{D}}$ disjoint
from $E^{\ast}$. This contradicts the rectangle condition by Lemma 7. ∎

Hence, we obtain an injection from $\mathcal{CA}$ to the finite set

$\\{1,2,\ldots,2g-2\\}\times\pi_{0}\left(\partial\overline{{\mathcal{E}}}\setminus\overline{{\mathcal{E}}^{\ast}}\right)\times\pi_{0}\left(\partial\overline{{\mathcal{E}}}\setminus\overline{{\mathcal{E}}^{\ast}}\right)\times
S\times S$

by assigning $\left(p,e,e^{\prime},t,t^{\prime}\right)$ to $[\beta]$ if
$\beta$ has the endpoints in $e$ and $e^{\prime}$, and has a lift in $Y_{p}$
with twisting numbers $t$ and $t^{\prime}$, respectively. ∎

$Y_{p}$$\widetilde{\beta}_{1}$$\widetilde{\beta}_{2}$$\widetilde{\theta}_{p,q}$$\widetilde{\omega}$$\widetilde{\alpha}$$\widetilde{E}_{p,q}$$\widetilde{\theta}_{p,q}$$\widetilde{\alpha}$$\widetilde{\beta}_{1}$$\widetilde{\beta}_{2}$$\widetilde{\omega}$
Figure 10. The top shows arcs on $\partial Y_{p}$ and a subarc
$\widetilde{\omega}$ whose image under $\pi_{H}$ is a wave $\omega$ with
respect to ${\mathcal{D}}$. The gray band represents a part of a disk in
$\widetilde{\mathcal{D}}$, and the dots represent an arc whose image and
$\omega$ cobound a wave disk. The bottom shows them so that the disk in
$\widetilde{\mathcal{D}}$ looks plain.

Second, we consider some $2$-dimensional objects, and show the finiteness of
their types. For disk systems ${\mathcal{D}}$ and ${\mathcal{D}}^{\prime}$ of
$H$ intersecting essentially in $H$, a disk piece of ${\mathcal{D}}$ with
respect to ${\mathcal{D}}^{\prime}$ is the closure of a component of
$\overline{\mathcal{D}}\setminus\overline{{\mathcal{D}}^{\prime}}$. A lift of
it in $\widetilde{H}$ is called a disk piece of $\widetilde{\mathcal{D}}$ with
respect to $\widetilde{{\mathcal{D}}^{\prime}}$. For pairs
$\left({\mathcal{D}}_{1},{\mathcal{D}}_{1}^{\ast}\right)$ and
$\left({\mathcal{D}}_{2},{\mathcal{D}}_{2}^{\ast}\right)$ in ${\mathfrak{R}}$,
disk pieces $\Delta_{1}$ and $\Delta_{2}$ of ${\mathcal{D}}_{1}$ and
${\mathcal{D}}_{2}$, respectively, with respect to ${\mathcal{E}}$ are said to
be $\left({\mathcal{E}},{\mathcal{E}}^{\ast}\right)$-parallel if
$\left(\Delta_{1},\Delta_{1}\cap\overline{\mathcal{E}},\Delta_{1}\cap\Sigma,\Delta_{1}\cap\overline{{\mathcal{E}}^{\ast}}\right)$
and
$\left(\Delta_{2},\Delta_{2}\cap\overline{\mathcal{E}},\Delta_{2}\cap\Sigma,\Delta_{2}\cap\overline{{\mathcal{E}}^{\ast}}\right)$
are isotopic in
$\left(H,\overline{\mathcal{E}},\Sigma,\partial\overline{{\mathcal{E}}^{\ast}}\right)$.
We have the following, similarly to [13, Proposition 3.6].

######  Lemma 10.

The set

$\bigl{\\{}\,\text{disk pieces of ${\mathcal{D}}$ with respect to
${\mathcal{E}}$}\bigm{|}\left({\mathcal{D}},{\mathcal{D}}^{\ast}\right)\in{\mathfrak{R}}\bigr{\\}}\big{/}\text{$\left({\mathcal{E}},{\mathcal{E}}^{\ast}\right)$-parallel}$

is finite.

###### Proof.

Note that a disk piece $\Delta$ with respect to ${\mathcal{E}}$ is a disk, and
each lift of it is properly embedded in a $3$-ball in
${\mathcal{B}}\left({\mathcal{E}}\right)$. It follows from the irreducibility
of the $3$-ball that the parallelism class of $\Delta$ is determined by those
of the $1$-dimensional objects in $\partial\Delta$. The present lemma,
therefore, almost follows from Lemma 9. It remains to rule out the possibility
that infinitely many non-parallel disk pieces are generated by composing
arbitrarily many connecting arcs and intersection arcs in the same parallelism
classes.

######  Claim.

At most one connecting arc in $\partial\Delta$ belongs to one
$\left({\mathcal{E}},{\mathcal{E}}^{\ast}\right)$-parallelism class.

###### Proof.

Let $\left({\mathcal{D}},{\mathcal{D}}^{\ast}\right)$ denote the pair in
${\mathfrak{R}}$ and $Y$ denote the $3$-ball in
${\mathcal{B}}\left({\mathcal{E}}\right)$ such that $\Delta$ is a disk piece
of ${\mathcal{D}}$ with respect to ${\mathcal{E}}$ in $\pi_{H}(Y)$. Assume
that distinct connecting arcs $\beta_{1}$ and $\beta_{2}$ in $\partial\Delta$
are $\left({\mathcal{E}},{\mathcal{E}}^{\ast}\right)$-parallel. There is a
subarc $\omega$ of
$\partial\overline{\mathcal{E}}\setminus\overline{{\mathcal{E}}^{\ast}}$
connecting $\beta_{1}$ and $\beta_{2}$. Let $\omega^{\circ}$ denote the
interior of $\omega$. If $\omega^{\circ}$ intersects $\partial\Delta$ then
there are more $\left({\mathcal{E}},{\mathcal{E}}^{\ast}\right)$-parallel
connecting arcs. By rechoosing $\beta_{1}$ and $\beta_{2}$ to be innermost, we
may suppose that $\omega^{\circ}$ is disjoint from $\partial\Delta$. Assume
that $\omega^{\circ}$ still intersects $\partial\overline{\mathcal{D}}$. There
is another disk piece $\Delta^{\prime}$ of ${\mathcal{D}}$ with respect to
${\mathcal{E}}$ in $\pi_{H}(Y)$ intersecting $\omega^{\circ}$. Let
$\widetilde{\Delta}$ denote the lift of $\Delta$ in $Y$, and
$\widetilde{\omega}$ denote the lift of $\omega$ in $Y$ connecting the lifts
of $\beta_{1}$ and $\beta_{2}$. Note that $\widetilde{\omega}$ and a subarc of
$\partial\widetilde{\Delta}$ form a simple, closed curve, which is separating
in the sphere $\partial Y$, and that $\partial\Delta$ and
$\partial\Delta^{\prime}$ are disjoint. It follows that $\omega^{\circ}$ and
$\partial\Delta^{\prime}$ have even number of intersection points, and hence
$\partial\Delta^{\prime}$ has distinct two connecting arcs between $\beta_{1}$
and $\beta_{2}$. Note that they are
$\left({\mathcal{E}},{\mathcal{E}}^{\ast}\right)$-parallel as well as
$\beta_{1}$ and $\beta_{2}$. By repeating this process, we can rechoose
$\beta_{1}$ and $\beta_{2}$ to be innermost in such pairs of arcs. Then
$\omega^{\circ}$ is disjoint from $\partial\overline{\mathcal{D}}$, and hence
$\omega$ is a wave with respect to ${\mathcal{D}}$. Recall that $\omega$ is
disjoint from $\overline{{\mathcal{E}}^{\ast}}$. This contradicts the
rectangle condition by Lemma 7. ∎

######  Claim.

At most two intersection arcs in $\partial\Delta$ belong to one
$\left({\mathcal{E}},{\mathcal{E}}^{\ast}\right)$-parallelism class.

###### Proof.

Assume that distinct three intersection arcs in $\partial\Delta$ are
$\left({\mathcal{E}},{\mathcal{E}}^{\ast}\right)$-parallel. Let
$\widetilde{\Delta}$ be a lift of $\Delta$ in $\widetilde{H}$. The lifts in
$\partial\widetilde{\Delta}$ of at least two of the three arcs are contained
in the same disk in $\widetilde{\mathcal{E}}$. Similarly to the proof of the
previous claim, we obtain a contradiction. ∎

These claims complete the proof of Lemma 10. ∎

Third, we consider the $3$-dimensional objects called thin parts and thick
parts. For a pair $\left({\mathcal{D}},{\mathcal{D}}^{\ast}\right)$ in
${\mathfrak{R}}$, a part of $\widetilde{\mathcal{D}}$ with respect to
$\widetilde{\mathcal{E}}$ is a component $\widetilde{P}$ of the intersection
of a $3$-ball in ${\mathcal{B}}\left({\mathcal{E}}\right)$ and a $3$-ball in
${\mathcal{B}}\left({\mathcal{D}}\right)$. We also call $\widetilde{P}$ a part
of $B$ with respect to $\widetilde{\mathcal{E}}$ if $B$ is the $3$-ball in
${\mathcal{B}}\left({\mathcal{D}}\right)$ containing $\widetilde{P}$. The
image $\pi_{H}\bigl{(}\widetilde{P}\bigr{)}$ is denoted by $P$, and called a
part of ${\mathcal{D}}$ with respect to ${\mathcal{E}}$. Note that
$\widetilde{P}$ is a $3$-ball, but $P$ is possibly not. The part
$\widetilde{P}$ and the part $P$ are said to be thin if $P$ is the region of
$\left({\mathcal{E}},{\mathcal{E}}^{\ast}\right)$-parallelism between two disk
pieces of ${\mathcal{D}}$ with respect to ${\mathcal{E}}$, or thick otherwise.
For pairs $\left({\mathcal{D}}_{1},{\mathcal{D}}_{1}^{\ast}\right)$ and
$\left({\mathcal{D}}_{2},{\mathcal{D}}_{2}^{\ast}\right)$ in ${\mathfrak{R}}$,
parts $P_{1}$ and $P_{2}$ of ${\mathcal{D}}_{1}$ and ${\mathcal{D}}_{2}$,
respectively, with respect to ${\mathcal{E}}$ are said to be
$\left({\mathcal{E}},{\mathcal{E}}^{\ast}\right)$-parallel if
$\left(P_{1},P_{1}\cap\overline{\mathcal{E}},P_{1}\cap\Sigma,P_{1}\cap\overline{{\mathcal{E}}^{\ast}}\right)$
and
$\left(P_{2},P_{2}\cap\overline{\mathcal{E}},P_{2}\cap\Sigma,P_{2}\cap\overline{{\mathcal{E}}^{\ast}}\right)$
are isotopic in
$\left(H,\overline{\mathcal{E}},\Sigma,H\cap\overline{{\mathcal{E}}^{\ast}}\right)$.

We have the following structure lemma of thin parts.

######  Lemma 11.

Let $\left({\mathcal{D}},{\mathcal{D}}^{\ast}\right)$ be a pair in
${\mathfrak{R}}$, and $\widetilde{P}$ be a thin part of
$\widetilde{\mathcal{D}}$ with respect to $\widetilde{\mathcal{E}}$. Then we
have the following:

  * (1)

Each disk piece of $\widetilde{\mathcal{E}}$ with respect to
$\widetilde{\mathcal{D}}$ adjacent to $\widetilde{P}$ is a rectangle cobounded
by two intersection arcs and two connecting arcs of $\widetilde{\mathcal{E}}$
with respect to $\widetilde{\mathcal{D}}$;

  * (2)

$\widetilde{P}$ intersects exactly two disks in $\widetilde{\mathcal{D}}$.

###### Proof.

The assertion (1) can be seen from the definitions of thinness and
$\left({\mathcal{E}},{\mathcal{E}}^{\ast}\right)$-parallelism for disk pieces.
Also by the thinness, $\widetilde{P}$ is adjacent to exactly two disk pieces
of $\widetilde{\mathcal{D}}$ with respect to $\widetilde{\mathcal{E}}$. Assume
that they are contained in one disk in $\widetilde{\mathcal{D}}$. Then a
connecting arc of $\widetilde{\mathcal{E}}$ with respect to
$\widetilde{\mathcal{D}}$ adjacent to $\widetilde{P}$ maps by $\pi_{H}$ to a
wave with respect to ${\mathcal{D}}$. Since the part $\widetilde{P}$ is thin,
the wave is disjoint from $\overline{{\mathcal{E}}^{\ast}}$. This contradicts
the rectangle condition by Lemma 7, to conclude the assertion (2). ∎

We have the following finiteness lemma of thick parts.

######  Lemma 12.

The set

$\bigl{\\{}\,\text{thick parts of ${\mathcal{D}}$ with respect to
${\mathcal{E}}$}\bigm{|}\left({\mathcal{D}},{\mathcal{D}}^{\ast}\right)\in{\mathfrak{R}}\bigr{\\}}\big{/}\text{$\left({\mathcal{E}},{\mathcal{E}}^{\ast}\right)$-parallel}$

is finite, and there exists the maximum in the set

$\bigl{\\{}\,\text{the number of thick parts of ${\mathcal{D}}$ with respect
to
${\mathcal{E}}$}\bigm{|}\left({\mathcal{D}},{\mathcal{D}}^{\ast}\right)\in{\mathfrak{R}}\bigr{\\}}.$

###### Proof.

Note that a thick part $\widetilde{P}$ with respect to
$\widetilde{\mathcal{E}}$ is cut off from a $3$-ball in
${\mathcal{B}}\left({\mathcal{E}}\right)$ by some disk pieces. The parallelism
class of $\pi_{H}\bigl{(}\widetilde{P}\bigr{)}$ is, therefore, almost
determined by those of the images of the disk pieces. To examine more
carefully, note that only one stratum of parallel disk pieces possibly cuts
off a $3$-ball in ${\mathcal{B}}\left({\mathcal{E}}\right)$ into two thick
parts, as illustrated in Figure 11. Still at most two
$\left({\mathcal{E}},{\mathcal{E}}^{\ast}\right)$-parallelism classes of thick
parts correspond to one collection of
$\left({\mathcal{E}},{\mathcal{E}}^{\ast}\right)$-parallelism classes of disk
pieces. Note also that if a thick part has two
$\left({\mathcal{E}},{\mathcal{E}}^{\ast}\right)$-parallel disk pieces, then
that part must be the region of parallelism, to contradict the thickness.
Taking these into account, the first assertion follows from Lemma 10. Note
also that for a pair $\left({\mathcal{D}},{\mathcal{D}}^{\ast}\right)$ in
${\mathfrak{R}}$, two distinct thick parts of ${\mathcal{D}}$ with respect to
${\mathcal{E}}$ cannot be
$\left({\mathcal{E}},{\mathcal{E}}^{\ast}\right)$-parallel. The number of
thick parts is, therefore, bounded from above by the finite cardinality in the
first assertion, to obtain the second assertion. ∎

thick

thick

thin Figure 11. Two thick parts separated by a stack of thin parts.

We focus on thin and thick parts lying peripherally in the handlebody. Let
$\left({\mathcal{D}},{\mathcal{D}}^{\ast}\right)$ be a pair in
${\mathfrak{R}}$, and $B$ be a $3$-ball in
${\mathcal{B}}\left({\mathcal{D}}\right)$. Let $\widetilde{P}$ be a thin part
of $B$ with respect to $\widetilde{\mathcal{E}}$, and $\widetilde{\nabla}$ be
a disk piece of $\widetilde{\mathcal{E}}$ with respect to
$\widetilde{\mathcal{D}}$ adjacent to $\widetilde{P}$. Note that
$\widetilde{\nabla}$ is a properly embedded disk in the $3$-ball $B$. Let
${\mathbf{P}}$ denote the closure of the complementary component of
$\widetilde{\nabla}$ in $B$ such that ${\mathbf{P}}$ contains $\widetilde{P}$.
Suppose that $B\setminus{\mathbf{P}}$ intersects all the disks in
$\widetilde{\mathcal{D}}_{B}$. Then $\widetilde{P}$ is said to be peripheral,
and ${\mathbf{P}}$ is called the peripheral component by $\widetilde{P}$, or a
peripheral component of $B$ with respect to $\widetilde{\mathcal{E}}$. We
imported this terminology from [13], but modified the definition. Still
similarly to [13, Lemma 4.5], we have the following structure lemma.

######  Lemma 13.

Let $\left({\mathcal{D}},{\mathcal{D}}^{\ast}\right)$ be a pair in
${\mathfrak{R}}$, and $B$ be a $3$-ball in
${\mathcal{B}}\left({\mathcal{D}}\right)$, and ${\mathbf{P}}$ be a peripheral
component of $B$ with respect to $\widetilde{\mathcal{E}}$. Then we have the
following:

  * (1)

Any thin part in ${\mathbf{P}}$ is peripheral, and the peripheral component by
it is contained in ${\mathbf{P}}$;

  * (2)

${\mathbf{P}}$ intersects exactly two disks $\widetilde{D}_{1}$ and
$\widetilde{D}_{2}$ in $\widetilde{\mathcal{D}}_{B}$;

  * (3)

${\mathbf{P}}\cap\widetilde{D}_{1}$ and ${\mathbf{P}}\cap\widetilde{D}_{2}$
are bigons composed of the same number of disk pieces of
$\widetilde{\mathcal{D}}$ with respect to $\widetilde{\mathcal{E}}$.

As the assertion (1), there is possibly nesting among peripheral components.
We call a peripheral component maximal if it is not contained in another.

###### Proof of Lemma 13.

The assertion (1) can be seen from the construction of ${\mathbf{P}}$. Let
$\widetilde{P}$ and $\widetilde{\nabla}$ as above. By Lemma 11 (2), the thin
part $\widetilde{P}$ intersects exactly two disks in
$\widetilde{\mathcal{D}}_{B}$. Since ${\mathbf{P}}$ contains $\widetilde{P}$,
the peripheral component ${\mathbf{P}}$ intersects those two disks. Assume
that ${\mathbf{P}}$ also intersects another disk $\widetilde{D}_{3}$ in
$\widetilde{\mathcal{D}}_{B}$ than the two. Since $B\setminus{\mathbf{P}}$
intersects all the disks, $\widetilde{D}_{3}$ has non-empty intersections with
both ${\mathbf{P}}$ and $B\setminus{\mathbf{P}}$. By the construction, the
border $\widetilde{\nabla}$, and hence $\widetilde{P}$, intersects
$\widetilde{D}_{3}$. This is a contradiction, and thus the two disks are
$\widetilde{D}_{1}$ and $\widetilde{D}_{2}$ in the assertion (2) of the
present lemma. By Lemma 11 (1), the disk $\widetilde{\nabla}$ is a rectangle
cobounded by two intersection arcs $\widetilde{\alpha}_{1}$ and
$\widetilde{\alpha}_{2}$ and two connecting arcs $\widetilde{\xi}_{1}$ and
$\widetilde{\xi}_{2}$ of $\widetilde{\mathcal{E}}$ with respect to
$\widetilde{\mathcal{D}}$. The surface ${\mathbf{P}}\cap\partial B$ is cut off
from the $2$-sphere $\partial B$ by the circle $\partial\widetilde{\nabla}$,
and hence is a disk. The surfaces ${\mathbf{P}}\cap\widetilde{D}_{1}$ and
${\mathbf{P}}\cap\widetilde{D}_{2}$ are cut off from $\widetilde{D}_{1}$ and
$\widetilde{D}_{2}$, respectively, by $\widetilde{\alpha}_{1}$ and
$\widetilde{\alpha}_{2}$, and hence are bigons. Since
${\mathbf{P}}\cap\partial B$ is disjoint from other disks in
$\widetilde{\mathcal{D}}_{B}$, the surface
${\mathbf{P}}\cap\partial\widetilde{H}$ is the closure of the complement of
the two bigons in the disk ${\mathbf{P}}\cap\partial B$, and hence is a
rectangle cobounded by $\widetilde{\xi}_{1}$ and $\widetilde{\xi}_{2}$ and two
subarcs of $\partial\widetilde{D}_{1}$ and $\partial\widetilde{D}_{2}$. Since
${\mathcal{E}}$ and ${\mathcal{D}}$ intersect essentially in $H$, the
connecting arcs in the rectangle ${\mathbf{P}}\cap\partial\widetilde{H}$ are
parallel, and hence have the same number of endpoints in
$\partial\widetilde{D}_{1}$ and $\partial\widetilde{D}_{2}$. They are the
endpoints of the same number of intersection arcs in
${\mathbf{P}}\cap\widetilde{D}_{1}$ and ${\mathbf{P}}\cap\widetilde{D}_{2}$.
They cut off ${\mathbf{P}}\cap\widetilde{D}_{1}$ and
${\mathbf{P}}\cap\widetilde{D}_{2}$ into the same number of disk pieces, as
asserted in (3). ∎

Based on the above observation, we gather some of the $2$-dimensional objects
into what we call peripheral subdisks. Let
$\left({\mathcal{D}},{\mathcal{D}}^{\ast}\right)$ be a pair in
${\mathfrak{R}}$, and $\widetilde{D}$ be a disk in $\widetilde{\mathcal{D}}$.
Let $B_{-}$ and $B_{+}$ denote the $3$-balls in
${\mathcal{B}}\left({\mathcal{D}}\right)$ such that $B_{-}\cap
B_{+}=\widetilde{D}$. Let ${\mathbf{P}}_{-}$ and ${\mathbf{P}}_{+}$ be maximal
peripheral components of $B_{-}$ and $B_{+}$, respectively, with respect to
$\widetilde{\mathcal{E}}$ intersecting $\widetilde{D}$. Figure 12 shows their
possible relative positions. We consider the left case, where one of the
bigons ${\mathbf{P}}_{-}\cap\widetilde{D}$ and
${\mathbf{P}}_{+}\cap\widetilde{D}$ contains the other, that is, the smaller
bigon is the intersection ${\mathbf{P}}_{-}\cap{\mathbf{P}}_{+}$. The image
$\pi_{H}\left({\mathbf{P}}_{-}\cap{\mathbf{P}}_{+}\right)$ is called a
peripheral subdisk of ${\mathcal{D}}$ with respect to ${\mathcal{E}}$.

${\mathbf{P}}_{-}$${\mathbf{P}}_{+}$$B_{-}$$B_{+}$$\widetilde{D}$ Figure 12.
Relative positions of peripheral components of $B_{-}$ and $B_{+}$.

We divide the argument into outside and inside of the peripheral subdisks in
the next two paragraphs, respectively, to show the relevant uniform
finiteness. By uniform finiteness, we mean the existence of the maximum in the
set

$\bigl{\\{}\,\text{the number of disk pieces of ${\mathcal{D}}$ with respect
to
${\mathcal{E}}$}\bigm{|}\left({\mathcal{D}},{\mathcal{D}}^{\ast}\right)\in\mathfrak{DR}\bigr{\\}}.$

This and Lemma 10 imply that we have only finitely many possible constructions
for a disk system ${\mathcal{D}}$ up to isotopy in $H$ so that ${\mathcal{D}}$
takes part in a pair in $\mathfrak{DR}$. Whole symmetric arguments hold for
disk systems of the other handlebody $H^{\ast}$, to complete the proof of
Theorem 2.

The double rectangle condition guarantees the uniform finiteness outside of
the peripheral subdisks. To see this we consider thickness of disk pieces as
follows. A disk piece of ${\mathcal{D}}$ with respect to ${\mathcal{E}}$ is
said to be thick if it is adjacent to a thick part of ${\mathcal{D}}$ with
respect to ${\mathcal{E}}$. We also define thickness for disk pieces of
$\widetilde{\mathcal{D}}$ with respect to $\widetilde{\mathcal{E}}$ similarly.
By Lemma 12, there exists the maximum, denoted by $T$, in the set

$\bigl{\\{}\,\text{the number of thick disk pieces of ${\mathcal{D}}$ with
respect to
${\mathcal{E}}$}\bigm{|}\left({\mathcal{D}},{\mathcal{D}}^{\ast}\right)\in\mathfrak{DR}\bigr{\\}}.$

The following lemma implies that $T$ is also a uniform upper bound for the
number of disk pieces outside of the peripheral subdisks.

######  Lemma 14.

For a pair $\left({\mathcal{D}},{\mathcal{D}}^{\ast}\right)$ in
$\mathfrak{DR}$, any disk piece of ${\mathcal{D}}$ with respect to
${\mathcal{E}}$ is either thick or contained in a peripheral subdisk.

###### Proof.

We use the notation in Subsection 2.2. Note that any disk $E^{\ast}$ in
${\mathcal{E}}^{\ast}$ is disjoint from $\overline{{\mathcal{D}}^{\ast}}$ or
contains a wave disk with respect to ${\mathcal{D}}^{\ast}$. Let $O^{\ast}$
denote $E^{\ast}$ in the former case, or the wave disk in the latter case. Let
$\ell_{0}$ denote the element in $\\{1,2,\ldots,m^{\ast}\\}$ such that the
closure of $H^{\ast}_{\ell_{0}}$ contains $O^{\ast}$. Let
$B^{\ast}_{\ell_{0}}$ denote the closure of a lift of $H^{\ast}_{\ell_{0}}$ in
$\widetilde{H^{\ast}}$. Let $\widetilde{O^{\ast}}$ denote the lift of
$O^{\ast}$ in $B^{\ast}_{\ell_{0}}$, and $\widetilde{D^{\ast}}_{v,\nu}$ denote
the $\nu$-side lift of $D_{v}$ in $B^{\ast}_{\ell_{0}}$ for $(v,\nu)\in
A^{\ast}_{\ell_{0}}$. Let $(V,W)$ denote the partition of
$\bigl{\\{}(v,\nu)\in
A^{\ast}_{\ell_{0}}\bigm{|}\widetilde{D^{\ast}}_{v,\nu}\cap\widetilde{O^{\ast}}=\emptyset\bigr{\\}}$
such that $\widetilde{O^{\ast}}$ separates
$\bigl{\\{}\widetilde{D^{\ast}}_{v,\nu}\bigm{|}(v,\nu)\in V\bigr{\\}}$ and
$\bigl{\\{}\widetilde{D^{\ast}}_{w,\mu}\bigm{|}(w,\mu)\in W\bigr{\\}}$ in
$B^{\ast}_{\ell_{0}}$. Now, assume that a disk piece $\Delta$ of
${\mathcal{D}}$ with respect to ${\mathcal{E}}$ is neither thick nor contained
in any peripheral subdisk. Let $d$ denote the element in $\\{1,2,\ldots,n\\}$
such that $D_{d}$ contains $\Delta$, and $\widetilde{D}_{d}$ be a lift of
$D_{d}$ in $\widetilde{H}$. Let $B_{\kappa}$ denote the $3$-ball in
${\mathcal{B}}\left({\mathcal{D}}\right)$ adjacent to $\widetilde{D}_{d}$ from
the $\kappa$-side for $\kappa\in\\{-,+\\}$. Let
$\widetilde{D}_{\kappa,i,\varepsilon}$ denote the $\varepsilon$-side lift of
$D_{i}$ in $B_{\kappa}$ for $\kappa\in\\{-,+\\}$ and
$(i,\varepsilon)\in\varLambda_{d,\kappa}$. Let $\widetilde{P}_{-}$ and
$\widetilde{P}_{+}$ denote the parts of $B_{-}$ and $B_{+}$ with respect to
$\widetilde{\mathcal{E}}$ adjacent to the lift of $\Delta$ in
$\widetilde{D}_{d}$. Since $\Delta$ is not thick, $\widetilde{P}_{-}$ and
$\widetilde{P}_{+}$ are thin parts. By Lemma 11 (2), there exists a unique
member $\left(i_{\kappa},\varepsilon_{\kappa}\right)$ in
$\varLambda_{d,\kappa}$ such that $\widetilde{P}_{\kappa}$ intersects only
$\widetilde{D}_{d}$ and
$\widetilde{D}_{\kappa,i_{\kappa},\varepsilon_{\kappa}}$ in
$\widetilde{\mathcal{D}}_{B_{\kappa}}$, for $\kappa\in\\{-,+\\}$. Since
$\Delta$ is not contained in any peripheral subdisk, one of the following
holds:

  * •

Either $\widetilde{P}_{-}$ or $\widetilde{P}_{+}$ is not peripheral;

  * •

Both $\widetilde{P}_{-}$ and $\widetilde{P}_{+}$ are peripheral, and the
peripheral components by them lie as in the right of Figure 12.

In either case, $\widetilde{P}_{-}\cup\widetilde{P}_{+}$ separates
$\bigl{(}\widetilde{\mathcal{D}}_{B_{-}}\cup\widetilde{\mathcal{D}}_{B_{+}}\bigr{)}\setminus\bigl{\\{}\widetilde{D}_{d},\widetilde{D}_{-,i_{-},\varepsilon_{-}},\widetilde{D}_{+,i_{+},\varepsilon_{+}}\bigr{\\}}$
into two or more non-empty subsets in $B_{-}\cup B_{+}$. Let
$\left(I_{\triangleleft},I_{\triangleright}\right)$ be a partition of
$\left(\\{-\\}\times\varLambda_{d,-}\right)\cup\left(\\{+\\}\times\varLambda_{d,+}\right)\setminus\left\\{\left(-,i_{-},\varepsilon_{-}\right),\left(+,i_{+},\varepsilon_{+}\right)\right\\}$
such that $\widetilde{P}_{-}\cup\widetilde{P}_{+}$ separates
$\bigl{\\{}\widetilde{D}_{\kappa,i,\varepsilon}\bigm{|}(\kappa,i,\varepsilon)\in
I_{\triangleleft}\bigr{\\}}$ and
$\bigl{\\{}\widetilde{D}_{\kappa,i,\varepsilon}\bigm{|}(\kappa,i,\varepsilon)\in
I_{\triangleright}\bigr{\\}}$ in $B_{-}\cup B_{+}$. Since the graph
${\mathcal{H}}_{d}$ is doubly $2$-connected with respect to
$\left(\\{-\\}\times\varLambda_{d,-},\\{+\\}\times\varLambda_{d,+}\right)$,
there exist a member
$\left(\kappa_{\triangleleft},i_{\triangleleft},\varepsilon_{\triangleleft}\right)$
in $I_{\triangleleft}$ and a member
$\left(\kappa_{\triangleright},i_{\triangleright},\varepsilon_{\triangleright}\right)$
in $I_{\triangleright}$ that span an edge in ${\mathcal{H}}_{d}$. We divide
the argument into two cases:

  * •

Suppose that $\kappa_{\triangleleft}$ and $\kappa_{\triangleright}$ coincide.
Since
${\mathcal{G}}_{k_{d,\kappa_{\triangleleft}},\ell_{0},i_{\triangleleft},\varepsilon_{\triangleleft},i_{\triangleright},\varepsilon_{\triangleright}}$
is $2$-connected, there exist a member $\left(v_{0},\nu_{0}\right)$ in $V$ and
a member $\left(w_{0},\mu_{0}\right)$ in $W$ that span an edge in
${\mathcal{G}}_{k_{d,\kappa_{\triangleleft}},\ell_{0},i_{\triangleleft},\varepsilon_{\triangleleft},i_{\triangleright},\varepsilon_{\triangleright}}$.
By the definition, there exists a rectangle $R$ on $\Sigma$ of type
$\left(\left(\partial
D_{i_{\triangleleft}},\varepsilon_{\triangleleft}\right),\left(\partial
D_{i_{\triangleright}},\varepsilon_{\triangleright}\right);\left(\partial
D^{\ast}_{v_{0}},\nu_{0}\right),\left(\partial
D^{\ast}_{w_{0}},\mu_{0}\right)\right)$;

  * •

Suppose that $\kappa_{\triangleleft}$ and $\kappa_{\triangleright}$ are
opposite. Since
${\mathcal{H}}_{\ell_{0},d,i_{\triangleleft},\varepsilon_{\triangleleft},i_{\triangleright},\varepsilon_{\triangleright}}$
is $2$-connected, there exist a member $\left(v_{0},\nu_{0}\right)$ in $V$ and
a member $\left(w_{0},\mu_{0}\right)$ in $W$ that span an edge in
${\mathcal{H}}_{\ell_{0},d,i_{\triangleleft},\varepsilon_{\triangleleft},i_{\triangleright},\varepsilon_{\triangleright}}$.
By the definition, there exists a rectangle $R$ on $\Sigma$ of type
$\left(\left(\partial
D_{i_{\triangleleft}},\varepsilon_{\triangleleft}\right),\partial
D_{d},\left(\partial
D_{i_{\triangleright}},\varepsilon_{\triangleright}\right);\left(\partial
D^{\ast}_{v_{0}},\nu_{0}\right),\left(\partial
D^{\ast}_{w_{0}},\mu_{0}\right)\right)$, as illustrated in Figure 13.

In either case, since $\widetilde{O^{\ast}}$ separates
$\widetilde{D^{\ast}}_{v_{0},\nu_{0}}$ and
$\widetilde{D^{\ast}}_{w_{0},\mu_{0}}$ in $B^{\ast}_{\ell_{0}}$, the rectangle
$R$ contains a subarc of $O^{\ast}\cap\Sigma$ that separates the two edges in
$\partial D^{\ast}_{v_{0}}$ and $\partial D^{\ast}_{w_{0}}$. It follows that
the lift of $O^{\ast}\cap R$ in $B_{-}\cup B_{+}$ connects the disks
$\widetilde{D}_{\kappa_{\triangleleft},i_{\triangleleft},\varepsilon_{\triangleleft}}$
and
$\widetilde{D}_{\kappa_{\triangleright},i_{\triangleright},\varepsilon_{\triangleright}}$,
and hence goes across $\widetilde{P}_{-}\cup\widetilde{P}_{+}$. This
contradicts that $\widetilde{P}_{-}$ and $\widetilde{P}_{+}$ are thin parts. ∎

Lemma 14 slightly extends as follows.

######  Lemma 15.

Let $\left({\mathcal{D}},{\mathcal{D}}^{\ast}\right)$ be a pair in
$\mathfrak{DR}$, and $\widetilde{D}$ be a disk in $\widetilde{\mathcal{D}}$.
Let $B_{-}$ and $B_{+}$ denote the $3$-balls in
${\mathcal{B}}\left({\mathcal{D}}\right)$ such that $B_{-}\cap
B_{+}=\widetilde{D}$. Let ${\mathbf{P}}_{-}$ be a peripheral component of
$B_{-}$ with respect to $\widetilde{\mathcal{E}}$. Let $\widetilde{\Delta}$ be
any disk piece in ${\mathbf{P}}_{-}\cap\widetilde{D}$, and $\widetilde{P}_{+}$
denote the part of $B_{+}$ adjacent to $\widetilde{\Delta}$. Then one of the
following holds:

  * •

The disk piece $\widetilde{\Delta}$ is thick;

  * •

The part $\widetilde{P}_{+}$ is thin and peripheral, and the peripheral
component by $\widetilde{P}_{+}$ is adjacent to a subdisk of
${\mathbf{P}}_{-}\cap\widetilde{D}$.

###### Proof.

Assume the failure of the first assertion. Then $\widetilde{P}_{+}$ is thin by
the definition, and $\pi_{H}\bigl{(}\widetilde{\Delta}\bigr{)}$ is contained
in a peripheral subdisk by Lemma 14. The lift of the peripheral subdisk in
$\widetilde{D}$ is the intersection of maximal peripheral components of
$B_{-}$ and $B_{+}$. Those components contain ${\mathbf{P}}_{-}$ and
$\widetilde{P}_{+}$. By Lemma 13 (1), the thin part $\widetilde{P}_{+}$ is
peripheral, and the peripheral component by $\widetilde{P}_{+}$ is contained
in the maximal one. Thus the relevant subdisks are well nested, to satisfy the
last assertion. ∎

$\pi_{H}\bigl{(}\widetilde{P}_{-}\bigr{)}$$\pi_{H}\bigl{(}\widetilde{P}_{+}\bigr{)}$$D_{d}$$D_{i_{\triangleleft}}$$D_{i_{\triangleright}}$$\partial
D^{\ast}_{v_{0}}$$\partial D^{\ast}_{w_{0}}$$R$ Figure 13. The rectangle $R$
in a possible case where both $\widetilde{P}_{-}$ and $\widetilde{P}_{+}$ are
peripheral.

It remains to show the uniform finiteness inside of the peripheral subdisks.
We can divide it into the following two lemmas.

######  Lemma 16.

There exists the maximum in the set

$\bigl{\\{}\,\text{the number of peripheral subdisks of ${\mathcal{D}}$ with
respect to
${\mathcal{E}}$}\bigm{|}\left({\mathcal{D}},{\mathcal{D}}^{\ast}\right)\in\mathfrak{DR}\bigr{\\}}.$

###### Proof.

Let $\left({\mathcal{D}},{\mathcal{D}}^{\ast}\right)$ be a pair in
$\mathfrak{DR}$. By Lemma 14, the number of peripheral subdisks of
${\mathcal{D}}$ with respect to ${\mathcal{E}}$ is less than or equal to the
number of complementary components in $\overline{\mathcal{D}}$ of the thick
disk pieces. Recall that ${\mathcal{D}}$ consists of at most $3g-3$ disks, and
they contains at most $T$ thick disk pieces. By Lemma 10, there exists the
maximum, denoted by $c$, in the set

$\left\\{\,\begin{minipage}{130.0pt} the number of complementary components in
$D$ of $\Delta$ \end{minipage}\;\middle|\;\begin{minipage}{195.0pt}
$\left({\mathcal{D}},{\mathcal{D}}^{\ast}\right)\in{\mathfrak{R}}$,\\\
$D\in{\mathcal{D}}$,\\\ $\Delta$ is a disk piece of ${\mathcal{D}}$ with
respect to ${\mathcal{E}}$ in $D$ \end{minipage}\right\\}.$

The number at issue is, therefore, coarsely bounded from above by
$3g-3+T(c-1)$. ∎

######  Lemma 17.

There exists the maximum in the set

$\left\\{\,\mathop{\mathrm{Area}}\left({\mathbf{\Delta}}\right)\;\middle|\;\begin{minipage}{210.0pt}
$\left({\mathcal{D}},{\mathcal{D}}^{\ast}\right)\in\mathfrak{DR}$,\\\
${\mathbf{\Delta}}$ is a peripheral subdisk of ${\mathcal{D}}$ with respect to
${\mathcal{E}}$ \end{minipage}\right\\}.$

Here, $\mathop{\mathrm{Area}}\left({\mathbf{\Delta}}\right)$ denotes the
number of disk pieces in ${\mathbf{\Delta}}$. We use this symbol not only for
a peripheral subdisk ${\mathbf{\Delta}}$, but also for a general disk composed
of some disk pieces of ${\mathcal{D}}$ with respect to ${\mathcal{E}}$, or of
$\widetilde{\mathcal{D}}$ with respect to $\widetilde{\mathcal{E}}$.

###### Proof of Lemma 17.

This proof is based on the idea of “disk pushing procedure” by Lustig–Moriah
[13, Section 5]. Let $\left({\mathcal{D}},{\mathcal{D}}^{\ast}\right)$ be a
pair in $\mathfrak{DR}$, and ${\mathbf{\Delta}}$ be a peripheral subdisk of
${\mathcal{D}}$ with respect to ${\mathcal{E}}$. We construct a sequence of
disks as illustrated schematically in Figure 14, and derive from it a uniform
upper bound for $\mathop{\mathrm{Area}}\left({\mathbf{\Delta}}\right)$ as
follows. Let $\widetilde{\mathbf{\Delta}}_{1}$ be a lift of
${\mathbf{\Delta}}$ in $\widetilde{H}$, and $\widetilde{D}_{1}$ denote the
disk in $\widetilde{\mathcal{D}}$ containing
$\widetilde{\mathbf{\Delta}}_{1}$. Let $B_{0}$ and $B_{1}$ denote the
$3$-balls in ${\mathcal{B}}\left({\mathcal{D}}\right)$ such that $B_{0}\cap
B_{1}=\widetilde{D}_{1}$. Let ${\mathbf{P}}_{0}$ and ${\mathbf{P}}_{1}$ denote
the maximal peripheral components of $B_{0}$ and $B_{1}$, respectively, with
respect to $\widetilde{\mathcal{E}}$ containing
$\widetilde{\mathbf{\Delta}}_{1}$. We may suppose that
$\widetilde{\mathbf{\Delta}}_{1}={\mathbf{P}}_{1}\cap\widetilde{D}_{1}$
without loss of generality. For an inductive construction, assume that
$\widetilde{\mathbf{\Delta}}_{\iota}$, $\widetilde{D}_{\iota}$, $B_{\iota}$,
${\mathbf{P}}_{\iota}$ have been defined for a positive integer $\iota$. If
${\mathbf{P}}_{\iota}$ is empty, then set
$\widetilde{\mathbf{\Delta}}_{\iota+1}$, $\widetilde{D}_{\iota+1}$,
$B_{\iota+1}$, ${\mathbf{P}}_{\iota+1}$ to be empty. Assume
${\mathbf{P}}_{\iota}$ to be non-empty. By Lemma 13 (2), it intersects only
$\widetilde{D}_{\iota}$ and another disk $\widetilde{D}_{\iota+1}$ in
$\widetilde{\mathcal{D}}_{B_{\iota}}$. Let $B_{\iota+1}$ denote the $3$-ball
in ${\mathcal{B}}\left({\mathcal{D}}\right)$ adjacent to
$\widetilde{D}_{\iota+1}$ other than $B_{\iota}$, and
$\widetilde{\mathbf{\Delta}}^{\prime}_{\iota+1}$ denote the disk
${\mathbf{P}}_{\iota}\cap\widetilde{D}_{\iota+1}$. If there exist peripheral
components of $B_{\iota+1}$ with respect to $\widetilde{\mathcal{E}}$ that
intersect $\widetilde{D}_{\iota+1}$ in
$\widetilde{\mathbf{\Delta}}^{\prime}_{\iota+1}$, then let
${\mathbf{P}}_{\iota+1}$ be one of those peripheral components such that
$\mathop{\mathrm{Area}}\bigl{(}{\mathbf{P}}_{\iota+1}\cap\widetilde{D}_{\iota+1}\bigr{)}$
is maximal. If not exist, then set ${\mathbf{P}}_{\iota+1}$ to be empty. Let
$\widetilde{\mathbf{\Delta}}_{\iota+1}$ denote the set
${\mathbf{P}}_{\iota+1}\cap\widetilde{D}_{\iota+1}$. In this way, we obtain
the sequence
$\bigl{(}\widetilde{\mathbf{\Delta}}_{\iota}\bigr{)}_{\iota\in{\mathbb{N}}}$
together with $\bigl{(}\widetilde{D}_{\iota}\bigr{)}_{\iota\in{\mathbb{N}}}$,
$\left(B_{\iota}\right)_{\iota\in{\mathbb{N}}}$,
$\left({\mathbf{P}}_{\iota}\right)_{\iota\in{\mathbb{N}}}$ and
$\bigl{(}\widetilde{\mathbf{\Delta}}^{\prime}_{\iota+1}\bigr{)}_{\iota\in{\mathbb{N}}}$.
Note that
$\mathop{\mathrm{Area}}\bigl{(}\widetilde{\mathbf{\Delta}}^{\prime}_{\iota+1}\bigr{)}\geq\mathop{\mathrm{Area}}\bigl{(}\widetilde{\mathbf{\Delta}}_{\iota+1}\bigr{)}$
since
$\widetilde{\mathbf{\Delta}}^{\prime}_{\iota+1}\supset\widetilde{\mathbf{\Delta}}_{\iota+1}$,
and that
$\mathop{\mathrm{Area}}\bigl{(}\widetilde{\mathbf{\Delta}}_{\iota}\bigr{)}=\mathop{\mathrm{Area}}\bigl{(}\widetilde{\mathbf{\Delta}}^{\prime}_{\iota+1}\bigr{)}$
by Lemma 13 (3). The sequence
$\bigl{(}\mathop{\mathrm{Area}}\bigl{(}\widetilde{\mathbf{\Delta}}_{\iota}\bigr{)}\bigr{)}_{\iota\in{\mathbb{N}}}$
is, therefore, monotonically decreasing. We analyze how it decreases in the
following claims.

######  Claim.

There exists a positive integer such that
$\mathop{\mathrm{Area}}\bigl{(}\widetilde{\mathbf{\Delta}}_{\iota}\bigr{)}=0$
for every greater integer $\iota$.

###### Proof.

Since
$\bigl{(}\mathop{\mathrm{Area}}\bigl{(}\widetilde{\mathbf{\Delta}}_{\iota}\bigr{)}\bigr{)}_{\iota\in{\mathbb{N}}}$
is a decreasing sequence of non-negative integers, it stays constant after
$\iota$ exceeds some integer $L$. It remains to show that the constant area is
zero. If not, $\left({\mathbf{P}}_{\iota}\right)_{\iota>L}$ is a sequence of
non-empty peripheral components with constant cross-section area. Then
${\mathbf{P}}_{\iota+1}$ is uniquely determined by ${\mathbf{P}}_{\iota}$ to
be adjacent to the same subdisk in $\widetilde{D}_{\iota+1}$, for each integer
$\iota$ greater than $L$. Note that there are only finitely many parts of
${\mathcal{D}}$ with respect to ${\mathcal{E}}$, and hence finitely many
images of peripheral components. It follows that the sequence
$\left(\pi_{H}\left({\mathbf{P}}_{\iota}\right)\right)_{\iota>L}$ is cyclic.
Note that ${\mathbf{P}}_{\iota}$ is cut off from $B_{\iota}$ by a rectangular
disk piece of $\widetilde{\mathcal{E}}$ with respect to
$\widetilde{\mathcal{D}}$. The images of such disk pieces form an annulus in a
disk in ${\mathcal{E}}$, to contradict that ${\mathcal{E}}$ and
${\mathcal{D}}$ intersect essentially. ∎

This says that the length of the sequence
$\bigl{(}\widetilde{\mathbf{\Delta}}_{\iota}\bigr{)}_{\iota\in{\mathbb{N}}}$
is essentially finite, but does not mean the uniform boundedness of that
length. Instead we consider the length of
$\bigl{(}\mathop{\mathrm{Area}}\bigl{(}\widetilde{\mathbf{\Delta}}_{\iota}\bigr{)}\bigr{)}_{\iota\in{\mathbb{N}}}$
after reducing repetitions of the same areas. Let $N$ denote the set
$\bigl{\\{}\iota\in{\mathbb{N}}\bigm{|}\mathop{\mathrm{Area}}\bigl{(}\widetilde{\mathbf{\Delta}}_{\iota}\bigr{)}>\mathop{\mathrm{Area}}\bigl{(}\widetilde{\mathbf{\Delta}}_{\iota+1}\bigr{)}\bigr{\\}}$.

######  Claim.

The cardinality of $N$ is less than or equal to the maximum number $T$ of
thick disk pieces.

###### Proof.

Suppose that $\iota$ belongs to $N$, that is,
$\mathop{\mathrm{Area}}\bigl{(}\widetilde{\mathbf{\Delta}}_{\iota}\bigr{)}>\mathop{\mathrm{Area}}\bigl{(}\widetilde{\mathbf{\Delta}}_{\iota+1}\bigr{)}$.
Since
$\mathop{\mathrm{Area}}\bigl{(}\widetilde{\mathbf{\Delta}}_{\iota}\bigr{)}$ is
positive, ${\mathbf{P}}_{\iota}$ is not empty. Let $\widetilde{P}_{\iota}$
denote the peripheral thin part such that ${\mathbf{P}}_{\iota}$ is the
peripheral component by $\widetilde{P}_{\iota}$. Let
$\widetilde{\Delta}_{\iota+1}$ denote the disk piece in
$\widetilde{D}_{\iota+1}$ adjacent to $\widetilde{P}_{\iota}$. Assume that
$\widetilde{\Delta}_{\iota+1}$ is not thick. Then by Lemma 15, there is a
peripheral component of $B_{\iota+1}$ adjacent to
$\widetilde{\mathbf{\Delta}}^{\prime}_{\iota+1}$. By the construction, that
peripheral component must be ${\mathbf{P}}_{\iota+1}$. It follows that
$\mathop{\mathrm{Area}}\bigl{(}\widetilde{\mathbf{\Delta}}_{\iota}\bigr{)}=\mathop{\mathrm{Area}}\bigl{(}\widetilde{\mathbf{\Delta}}^{\prime}_{\iota+1}\bigr{)}=\mathop{\mathrm{Area}}\bigl{(}\widetilde{\mathbf{\Delta}}_{\iota+1}\bigr{)}$,
a contradiction. Thus $\widetilde{\Delta}_{\iota+1}$, and hence
$\pi_{H}\bigl{(}\widetilde{\Delta}_{\iota+1}\bigr{)}$, is a thick disk piece.
Note that the sequence
$\bigl{(}\pi_{H}\bigl{(}\widetilde{\Delta}_{\iota+1}\bigr{)}\bigr{)}_{\iota\in
N}$ does not recur the same term, since
$\bigl{(}\mathop{\mathrm{Area}}\bigl{(}\widetilde{\mathbf{\Delta}}_{\iota}\bigr{)}\bigr{)}_{\iota\in
N}$ is strictly decreasing. This gives the claimed bound. ∎

Next we estimate how sharply
$\mathop{\mathrm{Area}}\bigl{(}\widetilde{\mathbf{\Delta}}_{\iota}\bigr{)}$
decreases at each step of $\iota$.

######  Claim.

There exists a constant $r$, independent from the choice of
$\left({\mathcal{D}},{\mathcal{D}}^{\ast}\right)$ in $\mathfrak{DR}$, such
that
$\mathop{\mathrm{Area}}\bigl{(}\widetilde{\mathbf{\Delta}}_{\iota}\bigr{)}<r\bigl{(}\mathop{\mathrm{Area}}\bigl{(}\widetilde{\mathbf{\Delta}}_{\iota+1}\bigr{)}+1\bigr{)}$
for every positive integer $\iota$.

###### Proof.

We may suppose that $\iota$ belongs to $N$. There are some peripheral
components of $B_{\iota+1}$ with respect to $\widetilde{\mathcal{E}}$ whose
intersections with $\widetilde{D}_{\iota+1}$ are proper subdisks of
$\widetilde{\mathbf{\Delta}}^{\prime}_{\iota+1}$. Let
$\widetilde{\mathbf{\Delta}}_{\iota+1}^{\flat}$ denote the closure of the
exterior in $\widetilde{\mathbf{\Delta}}^{\prime}_{\iota+1}$ of those
peripheral components. It follows from Lemma 15 that
$\widetilde{\mathbf{\Delta}}_{\iota+1}^{\flat}$ is composed of thick disk
pieces, which are no more than $T$. As mentioned in the proof of Lemma 16,
there exists a uniform upper bound $c$ for the number of complementary
components of any disk piece. The complementary components of
$\widetilde{\mathbf{\Delta}}_{\iota+1}^{\flat}$ in
$\widetilde{\mathbf{\Delta}}^{\prime}_{\iota+1}$ are no more than $T(c-1)$,
and have areas at most
$\mathop{\mathrm{Area}}\bigl{(}\widetilde{\mathbf{\Delta}}_{\iota+1}\bigr{)}$.
Thus
$\mathop{\mathrm{Area}}\bigl{(}\widetilde{\mathbf{\Delta}}^{\prime}_{\iota+1}\bigr{)}$,
and hence
$\mathop{\mathrm{Area}}\bigl{(}\widetilde{\mathbf{\Delta}}_{\iota}\bigr{)}$,
is bounded from above by
$T+T(c-1)\mathop{\mathrm{Area}}\bigl{(}\widetilde{\mathbf{\Delta}}_{\iota+1}\bigr{)}$,
and hence by
$T(c+1)\bigl{(}\mathop{\mathrm{Area}}\bigl{(}\widetilde{\mathbf{\Delta}}_{\iota+1}\bigr{)}+1\bigr{)}$.
∎

These show that
$\bigl{(}\mathop{\mathrm{Area}}\bigl{(}\widetilde{\mathbf{\Delta}}_{\iota}\bigr{)}\bigr{)}_{\iota\in
N}$ is bounded from above by a geometric sequence, and that
$\mathop{\mathrm{Area}}\left({\mathbf{\Delta}}\right)=\mathop{\mathrm{Area}}\bigl{(}\widetilde{\mathbf{\Delta}}_{1}\bigr{)}<r^{T}+r^{T-1}+r^{T-2}+\cdots+r+1$.
∎

$\widetilde{D}_{\iota-1}$$\widetilde{\mathbf{\Delta}}_{\iota-1}$$B_{\iota-1}$${\mathbf{P}}_{\iota-1}$$\widetilde{D}_{\iota}$$\widetilde{\mathbf{\Delta}}^{\prime}_{\iota}\supset\widetilde{\mathbf{\Delta}}_{\iota}$$B_{\iota}$${\mathbf{P}}_{\iota}$$\widetilde{D}_{\iota+1}$$\widetilde{\mathbf{\Delta}}_{\iota+1}\subset\widetilde{\mathbf{\Delta}}^{\prime}_{\iota+1}$$B_{\iota+1}$${\mathbf{P}}_{\iota+1}$$\widetilde{D}_{\iota+2}$$\widetilde{\mathbf{\Delta}}^{\prime}_{\iota+2}$
Figure 14. Construction of a sequence of disks.

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