# Unbraided wiring diagrams for Stein fillings of lens spaces

Mohan Bhupal and Burak Ozbagci  Department of Mathematics, METU, Ankara,
Turkey,
<EMAIL_ADDRESS>Department of Mathematics, Koç University, Istanbul,
Turkey
<EMAIL_ADDRESS>
###### Abstract.

In a previous work [2], we constructed a planar Lefschetz fibration on each
Stein filling of any lens space equipped with its canonical contact structure.
Here we describe an algorithm to draw an unbraided wiring diagram whose
associated planar Lefschetz fibration obtained by the method of Plamenevskaya
and Starkston [10], where the lens space with its canonical contact structure
is viewed as the contact link of a cyclic quotient singularity, is equivalent
to the Lefschetz fibration we constructed on each Stein filling of the lens
space at hand. Coupled with the work of Plamenevskaya and Starkston, we obtain
the following result as a corollary: The wiring diagram we describe can be
extended to an arrangement of symplectic graphical disks in ${\mathbb{C}}^{2}$
with marked points, including all the intersection points of these disks, so
that by removing the proper transforms of these disks from the blowup of
${\mathbb{C}}^{2}$ along those marked points one recovers the Stein filling
along with the Lefschetz fibration. Moreover, the arrangement is related to
the decorated plane curve germ representing the cyclic quotient singularity by
a smooth graphical homotopy.

As another application, we set up an explicit bijection between the Stein
fillings of any lens space with its canonical contact structure, and the
Milnor fibers of the corresponding cyclic quotient singularity, which was
first obtained by Némethi and Popescu-Pampu [8], using different methods.

##  1\. Introduction

In their recent work, Plamenevskaya and Starkston [10] showed that every Stein
filling of the link of a rational surface singularity with reduced fundamental
cycle, equipped with its canonical contact structure, can be obtained from a
configuration of symplectic graphical disks in $\mathbb{C}^{2}$ with marked
points including all the intersection points of these disks, by removing the
union of the proper transforms of these disks from the blowup of
${\mathbb{C}}^{2}$ at the marked points. Their purely topological proof relies
on a theorem of Wendl [11], which implies that each Stein filling of the
contact singularity link of the type above admits a planar Lefschetz fibration
over $D^{2}$, since the contact link itself is supported by a planar open book
([1],[9]), and moreover, the Lefschetz fibration corresponds to a positive
factorization of the monodromy of this open book. In their proof,
Plamenevskaya and Starkston developed a method to reverse-engineer a braided
wiring diagram producing any such factorization, and then extended this
diagram to an arrangement of symplectic graphical disks which, in turn, gives
the Stein filling along with the Lefschetz fibration via the method described
above (see Section 2 for more details of their construction).

In the present paper, we focus on cyclic quotient singularities—a subclass of
rational surface singularities with reduced fundamental cycle. It is well-
known that the oriented link of any cyclic quotient singularity is orientation
preserving diffeomorphic to a lens space $L(p,q)$. Let $\xi_{can}$ denote the
canonical contact structure on the singularity link $L(p,q)$. In [7], Lisca
classified the minimal symplectic fillings of the contact $3$-manifold
$(L(p,q),\xi_{can})$, up to diffeomorphism, and also showed that any such
filling of $(L(p,q),\xi_{can})$ is in fact a Stein filling. In [2], we
constructed a planar Lefschetz fibration $W\to D^{2}$ on each Stein filling
$W$ of $(L(p,q),\xi_{can})$, by explicitly describing the ordered set of
vanishing cycles on a disk with holes. In this paper, our primary goal is to
describe an algorithm to draw a wiring diagram, which turns out to be
unbraided, that corresponds to each of these planar Lefschetz fibrations via
the work of Plamenevskaya and Starkston.

######  Theorem 1.

There is an algorithm to draw an explicit unbraided wiring diagram whose
associated planar Lefschetz fibration obtained by the method of Plamenevskaya
and Starkston [10] is equivalent to the planar Lefschetz fibration $W\to
D^{2}$ constructed by the authors [2] on each Stein filling $W$ of the contact
$3$-manifold $(L(p,q),\xi_{can})$.

As we mentioned in the first paragraph, Plamenevskaya and Starkston described
an algorithm (see [10, Section 5.4]) to obtain a braided wiring diagram from
the ordered set of vanishing cycles of a planar Lefschetz fibration, by
reverse-engineering. Their algorithm involves many choices (see [10, Remark
5.7]) and although we do not rely on their reverse-engineering algorithm here,
we show indirectly that by appropriate choices, one can obtain “unbraided”
wiring diagrams, which means that all the braids in the diagram can be chosen
to be the identity.

The article of Plamenevskaya and Starkston was admittedly inspired by the work
of de Jong and van Straten [5], who studied the Milnor fibers and deformation
theory of sandwiched singularities—which includes rational surface
singularities with reduced fundamental cycle. In their work, deformation
theory of a surface singularity in the given class is reduced to deformations
of the germ of a reducible plane curve representing the singularity.

In particular, to any cyclic quotient singularity germ, de Jong and van
Straten associate a decorated germ of a reduced plane curve singularity
$\mathcal{C}=C_{1}\cup\cdots\cup C_{n}\subset({\mathbb{C}}^{2},0)$ with smooth
irreducible branches, where the decoration on each $C_{i}$ is a certain
positive integer, which we omit here from the notation for simplicity. The
outcome of de Jong and van Straten’s construction is that there is a bijection
between one-parameter deformations of the cyclic quotient singularity and
“picture deformations” of $\mathcal{C}$ representing that singularity (see
Section 2 for more details of their construction).

Moreover, Plamenevskaya and Starkston [10, Proposition 5.5], extends any given
braided wiring diagram (viewed as a collection of intersecting curves in
${\mathbb{R}}\times{\mathbb{C}}$) to a collection of symplectic disks in
${\mathbb{C}}\times{\mathbb{C}}$. Consequently, as a corollary to Theorem 1,
we obtain the following result coupled with [10, Proposition 5.8].

######  Corollary 2.

For each Stein filling $W$ of the contact $3$-manifold $(L(p,q),\xi_{can})$,
there is an explicit collection of symplectic graphical disks
$\Gamma_{1},\ldots,\Gamma_{n}$ in ${\mathbb{C}}^{2}$, with marked points
$p_{1},\ldots,p_{m}\subset\bigcup_{i}\Gamma_{i}$, which include all the
intersection points of these disks, so that by removing the union of the
proper transforms $\widetilde{\Gamma}_{1},\ldots,\widetilde{\Gamma}_{n}$ of
$\Gamma_{1},\ldots,\Gamma_{n}$, from the blowup of ${\mathbb{C}}^{2}$ at these
marked points we obtain $W$ along with the Lefchetz fibration mentioned in
Theorem 1. Moreover, the collection of symplectic graphical disks
$\Gamma_{1},\ldots,\Gamma_{n}$ is related to the decorated plane curve germ
$\mathcal{C}=C_{1}\cup\cdots\cup C_{n}\subset({\mathbb{C}}^{2},0)$
representing the cyclic quotient surface singularity at hand by a smooth
graphical homotopy.

For each Stein filling $W$ of $(L(p,q),\xi_{can})$, the symplectic graphical
disk arrangement $\Gamma:=\Gamma_{1}\cup\cdots\cup\Gamma_{n}$ with marked
points $p_{1},\ldots,p_{m}\subset\Gamma$ in ${\mathbb{C}}^{2}$ described in
Corollary 2 determines immediately the $m\times n$ incidence matrix
$\mathcal{I}(\Gamma,\\{p_{j}\\})$, defined so that its $(i,j)$-entry is $1$ if
$p_{j}\in\Gamma_{i}$, and $0$ otherwise. Since there is no canonical labelling
of the points $p_{j}$, in general, the incidence matrix is only defined up to
permutation of the columns.

######  Corollary 3.

For each Stein filling $W$ of $(L(p,q),\xi_{can})$, there is an iterative
algorithm to obtain the incidence matrix $\mathcal{I}(\Gamma,\\{p_{j}\\})$ for
the symplectic graphical disk arrangement
$\Gamma:=\Gamma_{1}\cup\cdots\cup\Gamma_{n}$ with marked points
$p_{1},\ldots,p_{m}\subset\Gamma$ in ${\mathbb{C}}^{2}$ described in Corollary
2.

As a matter of fact, one can read off the incidence matrix directly from the
wiring diagram from which the symplectic disk configuration arises. As
explained in [10, Section 6] and [5, Section 5], the incidence matrix
$\mathcal{I}(\Gamma,\\{p_{j}\\})$ determines the fundamental group, the
integral homology and the intersection form of $W$, as well as the first Chern
class of the Stein structure on $W$.

Note that each Milnor fiber of the cyclic quotient singularity is a Stein
filling of its boundary—which is the link $L(p,q)$ of the singularity,
equipped with its canonical contact structure $\xi_{can}$. In [8], Némethi and
Popescu-Pampu showed that there is an explicit one-to-one correspondence
between Stein fillings of $(L(p,q),\xi_{can})$ and Milnor fibers of the
corresponding cyclic quotient singularity, proving in particular a conjecture
of Lisca [7]. As another application of Theorem 1, we obtain an alternate
proof of their result formulated as Corollary 4. We say that two (smooth) disk
arrangements $(\Gamma,\\{p_{j}\\})$ and
$(\Gamma^{\prime},\\{p^{\prime}_{j}\\})$ in $\mathbb{C}^{2}$ are
combinatorially equivalent if their incidence matrices coincide up to
permutation of columns, i.e. up to relabelling of the marked points.

######  Corollary 4.

For each Stein filling $W$ of $(L(p,q),\xi_{can})$, the arrangement of
symplectic graphical disks $(\Gamma,\\{p_{j}\\})$, described in Corollary 2,
is combinatorially equivalent to the arrangement of the smooth branches of a
picture deformation of the decorated plane curve germ $\mathcal{C}$
representing the corresponding cyclic quotient surface singularity, described
by de Jong and van Straten [5]. This equivalence gives an explicit bijection
between Stein fillings of $(L(p,q),\xi_{can})$ and Milnor fibers of the
corresponding cyclic quotient singularity.

In other words, the wiring diagrams we obtain via Theorem 1 can be viewed as
picture deformations of the decorated plane curve germ representing the
associated cyclic quotient singularity. In Figure 1 below, we summarized the
correspondence that takes each Stein filling of $(L(p,q),\xi_{can})$ given by
Lisca to the Milnor fiber of the associated cyclic quotient singularity
described by de Jong and van Straten via a picture deformation of the
decorated plane curve representing the singularity.

Stein filling Lefschetz fibrationWiring diagramConfiguration of symp.graphical
disks in $\mathbb{C}^{2}$Incidence matrixpicture deformationMilnor fiber

Figure 1. From Lisca to de Jong and van Straten

It is well-known that for any contact singularity link $(L(p,q),\xi_{can})$,
the Milnor fiber of the Artin smoothing component of the corresponding cyclic
quotient singularity gives a Stein filling which is Stein deformation
equivalent to the one obtained by deforming the symplectic structure on the
minimal resolution (see [3]) of the singularity. In addition, according to
[5], there is a canonical picture deformation, called the Scott deformation,
of the decorated plane curve germ which corresponds to the Artin smoothing. As
a final application of Theorem 1, we obtain a wiring diagram that represents
the combinatorial equivalence class of the Scott deformation.

######  Corollary 5.

For the Milnor fiber of the Artin smoothing component of the cyclic quotient
singularity, the arrangement of symplectic disks given in Corollary 2, arising
from the wiring diagram described in Theorem 1, is combinatorially equivalent
to the Scott deformation of a decorated plane curve representing the
singularity.

##  2\. Rational singularities with reduced fundamental cycle, picture
deformations, and braided wiring diagrams

In [8], Némethi and Popescu-Pampu showed that there is an explicit bijective
correspondence between Stein fillings of the link of a cyclic quotient
singularity and Milnor fibres of smoothing components of the given
singularity, as conjectured by Lisca [7]. As cyclic quotient singularities are
examples of sandwiched singularites, de Jong and van Straten’s picture
deformation construction (see [5]) can be used to describe these Milnor
fibres. We give a brief description of the construction of de Jong and van
Straten here. As the theory is easier to describe in the case of rational
singularities with reduced fundamental cycle, a class which contains cyclic
quotient singularities, we will restrict attention to these.

Let $(X,x)$ be the germ of a rational singularity with reduced fundamental
cycle. De Jong and van Straten associate to $(X,x)$ a, possibly nonunique,
decorated germ of a reduced plane curve singularity
$\mathcal{C}=C_{1}\cup\cdots\cup C_{n}\subset({\mathbb{C}}^{2},0)$ with
_smooth_ irreducible branches. Each such singularity can be resolved by a
finite sequence of blowups. For each branch $C_{i}$, let $m_{i}$ denote the
number of times $C_{i}$, or its proper transform, is blown up in the minimal
resolution. For example, if $\mathcal{C}$ consists of a collection of curves
intersecting (pairwise) transversally at $0$, then $m_{i}=1$ for all $i$. The
decoration on $\mathcal{C}$ consists of an $n$-tuple $l=(l_{1},\dots,l_{n})$
of positive integers such that $l_{i}\geq m_{i}$ for each $i$. Given such a
decorated curve $(\mathcal{C},l)$, one can recover the corresponding surface
singularity as follows: Take the minimal embedded resolution of $\mathcal{C}$
and iteratively blow up the proper transform of $C_{i}$ $(l_{i}-m_{i})$-times
on the preimage of $0$ for each $i$. Under the condition that $l_{i}>m_{i}$
for each $i$, the set of exceptional curves that do not meet the proper
transform of $\mathcal{C}$ will be connected. Collapsing them then gives the
corresponding surface singularity.

Given a decorated curve $(\mathcal{C},l)$, let
$\tilde{\mathcal{C}}=\tilde{C}_{1}\cup\cdots\cup\tilde{C}_{n}$ denote the
normalization of $\mathcal{C}$ (which in our present situation is just the
disjoint union of the irreducible components of $\mathcal{C}$). Geometrically,
one may think of the decoration $l$ as a collection of $l_{i}$ marked points
on $\tilde{C}_{i}$, for each $i$, all concentrated on the preimage of the
singular point. The outcome of de Jong and van Straten’s construction is that
there is a one-to-one correspondence between one-parameter deformations of
$(X,x)$ and “picture deformations” of $(\mathcal{C},l)$. Roughly speaking, a
_picture deformation_ of $(\mathcal{C},l)$ consists of a $\delta$-constant
deformation $\mathcal{C}^{s}=C_{1}^{s}\cup\cdots\cup C_{n}^{s}$ of
$\mathcal{C}$, which in the present situation means that the branches of
$\mathcal{C}$ are deformed separately and not allowed to merge, together with
a redistribution $l^{s}$ of the marked points so that we have exactly $l_{i}$
marked points on $\tilde{C}^{s}_{i}$ for each $i$, where
$\tilde{C}^{s}_{1},\ldots,\tilde{C}^{s}_{n}$ denote the irreducible components
of the normalization $\tilde{\mathcal{C}}^{s}$ of $\mathcal{C}^{s}$. Here
$\mathcal{C}^{0}=\mathcal{C}$ and we require that for $s\neq 0$ the only
singularities of $\mathcal{C}^{s}$ are ordinary $k$-tuple points, for various
$k$, that is, transversal intersections of $k$ smooth branches, and that each
such multiple point is marked. There may be additional “free” marked points on
the branches of $\mathcal{C}^{s}$. The Milnor fibre of the smoothing
associated to $(\mathcal{C}^{s},l^{s})$ can then be constructed by blowing up
all the marked points, taking the complement of the proper transforms of
$C_{1}^{s},\ldots,C_{n}^{s}$ and smoothing corners. Here the Milnor fiber will
be noncompact, but by working in a small ball centered at the origin in
${\mathbb{C}}^{2}$ we can obtain compact Milnor fibers.

The topological information from picture deformations can be conveniently
extracted by using the notion of braided wiring diagrams. These were
introduced by Cohen and Suciu [4] in their study of complex hyperplane
arrangements and have been used fruitfully by Plamenevskaya and Starkston [10]
in their investigation of unexpected Stein fillings in the case of rational
surface singularities with reduced fundamental cycle. We briefly describe
these next.

A _braided wiring diagram_ is a collection of curves
$f_{i}\colon[0,1]\to{\mathbb{R}}\times{\mathbb{C}}$ for $1\leq i\leq n$,
called _wires_ , such that $f_{i}(t)\in\\{t\\}\times{\mathbb{C}}$. At finitely
many interior points $t_{1},\ldots,t_{m}$, a subcollection of the wires may
intersect with the remaining being disjoint, but at each such point the wires
intersecting are assumed to have distinct tangent lines. We will make the
further assumption that there is a number $\varepsilon>0$ such that the
positions of the wires above the points $0,1$ and $t_{i}\pm\varepsilon$ take
the same given values in ${\mathbb{R}}\subset{\mathbb{C}}$ and the restriction
of each wire $f_{j}$ to $(t_{i}-\varepsilon,t_{i}+\varepsilon)$ is linear. Any
braided wiring diagram can be made to satisfy this assumption by a homotopy of
braided wiring diagrams. Then the portions of the braided wiring diagram
between $t_{i}-\varepsilon$ and $t_{i}+\varepsilon$ can be specified by
declaring which adjacent wires intersect and on the complementary intervals
the wires may be braided. Moreover, any wiring diagram will be presented by
its projection onto
${\mathbb{R}}\times{\mathbb{R}}\subset{\mathbb{R}}\times{\mathbb{C}}$, where
the second ${\mathbb{R}}$ is the real part of ${\mathbb{C}}$.

We now describe how to obtain a braided wiring diagram from a picture
deformation $(\mathcal{C}^{s},l^{s})$. By choosing coordinates of
${\mathbb{C}}^{2}$ generically, we may assume that each $C_{i}^{0}$ is
graphical, that is, $C_{i}^{0}=\\{(x,y)\in{\mathbb{C}}^{2}\,|\,x\in
D,y=f_{i}(x)\\}$ for some complex function $f_{i}$, where $D$ is a small disk
in ${\mathbb{C}}$ centered at $0$. For $s>0$ sufficiently small, it follows
that each $C_{i}^{s}$ is graphical. Let $\eta_{1},\ldots,\eta_{m}$ denote the
images of the intersection points of $C_{1}^{s},\ldots,C_{n}^{s}$ under the
map $\pi_{x}\colon{\mathbb{C}}^{2}\to{\mathbb{C}}$ given by projecting onto
the first coordinate and choose a smooth curve $\gamma\colon[0,1]\to D$ whose
interior passes through these points such that $\gamma^{\prime}(t)$ has
nonpositive real part for all $t$. Then $(C_{1}^{s}\cup\cdots\cup
C_{n}^{s})\cap\pi_{x}^{-1}(\gamma)$ is a braided wiring diagram.

Next we review how Plamenevskaya and Starkston constructed planar Lefschetz
fibrations based on a configuration of smooth disks in ${\mathbb{C}}^{2}$; see
[10, Lemma 3.2]. Let $\Gamma_{1},\ldots,\Gamma_{n}$ be smooth disks in
${\mathbb{C}}^{2}$ which are graphical with respect to the projection
$\pi_{x}$. Assume that whenever two or more of these disks meet at a point,
they intersect transversally and positively with respect to the orientation on
the graph $\Gamma_{i}$ induced from the natural orientation on ${\mathbb{C}}$.
Let $p_{1},\ldots,p_{m}$ be the marked points on $\bigcup_{i}\Gamma_{i}$ which
include all the intersection points, and let
$\Pi\colon{\mathbb{C}}^{2}\\#m\hbox{$\overline{{\mathbb{C}}P^{2}}$}\to{\mathbb{C}}^{2}$
be the blow-up at the points $p_{1},\ldots,p_{m}$. If
$\widetilde{\Gamma}_{1},\ldots,\widetilde{\Gamma}_{n}$ denote the proper
transforms of $\Gamma_{1},\ldots,\Gamma_{n}$, then

$\pi_{x}\circ\Pi\colon{\mathbb{C}}^{2}\\#m\hbox{$\overline{{\mathbb{C}}P^{2}}$}\setminus(\widetilde{\Gamma}_{1},\ldots,\widetilde{\Gamma}_{n})\to{\mathbb{C}}$

is a Lefschetz fibration whose regular fibers are punctured planes, where each
puncture corresponds to a component $\widetilde{\Gamma_{i}}$. There is one
vanishing cycle for each point $p_{j}$, which is a curve in the fiber
enclosing the punctures that correspond to the components $\Gamma_{i}$ passing
through $p_{j}$.

Moreover, restricting to an appropriate Milnor ball in ${\mathbb{C}}^{2}$ that
contains all the points $p_{1},\ldots,p_{m}$ one obtains a Lefschetz fibration
whose fiber is a disk with holes, where the holes correspond to the components
$\Gamma_{i}$ and the vanishing cycles correspond to the points $p_{j}$ in the
same way as described above. Furthermore, if the curvettas
$C_{1}^{s},\ldots,C_{n}^{s}$ with marked points are the result of a picture
deformation of a germ associated to a surface singularity, the Lefschetz
fibration constructed as above is compatible with the complex structure on the
Milnor fiber of the corresponding smoothing.

###  2.1. From wiring diagrams to planar Lefschetz fibrations

Here we outline the method of Plamenevskaya and Starkston that gives a set of
ordered vanishing cycles associated to any braided wiring diagram, which in
turn determines a planar Lefschetz fibration on the associated Stein fillings;
see [10, Section 5.2]. In this paper, we will only deal with wiring diagrams
without any braids and we will call them unbraided wiring diagrams. In the
following, we describe their method for the case of unbraided wiring diagrams.
We should emphasize that our conventions will be different from those of [10],
for the purposes of this paper. We denote the marked points (consisting of
intersection points and free points) by $x_{i}$, and enumerate them according
to their geometric position from right to left, as illustrated in Figure 2.

$x_{1}$$x_{3}$$x_{4}$$x_{2}$

Figure 2. An example of an unbraided wiring diagram without any free points.

For each marked point $x_{s}$ in the wiring diagram, there is a convex curve
$\delta(x_{s})$ in $D_{k}$ enclosing a certain set of adjacent holes, which is
determined as follows.

######  Definition 6.

(Convex curve assigned to a marked point) Suppose that the marked point
$x_{s}$ is a simultaneous intersection point of some geometrically consecutive
wires in a given wiring diagram. The convex curve $\delta(x_{s})$ encircling
the adjacent holes whose geometric order from the top in $D_{k}$ coincides
with the local geometric order of the wires simultaneously intersecting at
that marked point is called the convex curve assigned to $x_{s}$. If $x_{s}$
is a free marked point on a single wire, then the convex curve $\delta(x_{s})$
assigned to $x_{s}$ is the curve which is parallel to a single interior
boundary component of $D_{k}$ whose order from the top coincides with the
local geometric order of the wire.

For example, in Figure 2, the geometrically top four wires intersect at the
marked point $x_{4}$; the geometrically top two wires intersect at the marked
point $x_{3}$; the geometrically bottom two wires intersect at the marked
point $x_{2}$ and the geometrically second and third wires intersect at the
marked point $x_{1}$. It follows that the convex curves
$\delta(x_{4}),\delta(x_{3}),\delta(x_{2}),\delta(x_{1})$ depicted in Figure 3
are assigned to the marked points $x_{4},x_{3},x_{2},x_{1}$, respectively, in
Figure 2.

$\delta(x_{3})$$\delta(x_{1})$$\delta(x_{2})$$\delta(x_{4})$

Figure 3. Convex curves in $D_{5}$ assigned to the marked points in Figure 2.

For each marked point $x_{s}$ in the wiring diagram, there is a
counterclockwise half-twist $\Delta(x_{s})\colon D_{k}\to D_{k}$, which is
determined as follows.

######  Definition 7.

(Counterclockwise half-twist corresponding to a marked point) The
counterclockwise half-twist $\Delta(x_{s})$ along the subdisk in $D_{k}$
enclosed by the convex curve $\delta(x_{s})$ is called the counterclockwise
half-twist corresponding to $x_{s}$.

Suppose that a wiring diagram has $k$ wires and $r$ marked points
$x_{r},x_{r-1},\ldots,x_{1}$, reading from left to right. According to [10],
for each $1\leq s\leq r$, there is a vanishing cycle $V(x_{s})$ in $D_{k}$
associated to the marked point $x_{s}$, which is determined as follows.

######  Definition 8.

(Vanishing cycle associated to a marked point) For each $2\leq s\leq r$, the
vanishing cycle $V(x_{s})$ associated to the marked point $x_{s}$ is the curve
in $D_{k}$ given as

$\Delta(x_{1})\circ\cdots\circ\Delta(x_{s-1})(\delta(x_{s})),$

and $V(x_{1})=\delta(x_{1})$.

For example, the vanishing cycles for the marked points in Figure 2 are
calculated as follows. The curve
$V(x_{4})=\Delta(x_{1})\circ\Delta(x_{2})\circ\Delta(x_{3})(\delta(x_{4}))$ is
illustrated in Figure 4. Similarly,
$V(x_{3})=\Delta(x_{1})\circ\Delta(x_{2})(\delta(x_{3}))$ is illustrated in
Figure 5. Finally, the vanishing cycle
$V(x_{2})=\Delta(x_{1})(\delta(x_{2}))=\delta(x_{2})$ and
$V(x_{1})=\delta(x_{1})$ by definition.

$\delta(x_{4})$$V(x_{4})$$\Delta(x_{3})$$\Delta(x_{2})$$\Delta(x_{1})$

Figure 4. Starting from $\delta(x_{4})$, we apply a counterclockwise half-
twist on the subdisk enclosed by the dotted curve, at each step, going from
left to right.

$\delta(x_{3})$$V(x_{3})$$\Delta(x_{1})$$\Delta(x_{2})$

Figure 5. Starting from $\delta(x_{3})$, we apply a counterclockwise half-
twist on the subdisk enclosed by the dotted curve, at each step, going from
left to right.

##  3\. Planar Lefschetz fibrations on Stein fillings of lens spaces

###  3.1. Symplectic fillings of lens spaces

In [7], Lisca classified the minimal symplectic fillings of the contact
$3$-manifold $(L(p,q),\xi_{can})$, up to diffeomorphism. It turns out any
minimal symplectic filling of $(L(p,q),\xi_{can})$ is in fact a Stein filling.
We first briefly review Lisca’s classification [7] of Stein fillings of
$(L(p,q),\xi_{can})$, up to diffeomorphism.

######  Definition 9.

(Blowup of a tuple of positive integers) For any integer $r\geq 2$, a blowup
of an $r$-tuple of positive integers at the $i$th term is a map
$\varphi_{i}\colon\mathbb{Z}^{r}_{+}\to\mathbb{Z}^{r+1}_{+}$ defined by

$\displaystyle(n_{1},\ldots,n_{i},n_{i+1},\ldots,n_{r})\mapsto(n_{1},\ldots,n_{i-1},n_{i}+1,1,n_{i+1}+1,n_{i+2},\ldots,n_{r})$

for any $1\leq i\leq r-1$ and by

$\displaystyle(n_{1},\ldots,n_{r})\mapsto(n_{1},\ldots,n_{r-1},n_{r}+1,1)$

when $i=r$. The case when $1\leq i\leq r-1$ is called an interior blowup,
whereas the case $i=r$ is called an exterior blowup. We also say that
$(0)\to(1,1)$ is the initial blowup.

Suppose that $p>q\geq 1$ are coprime integers and let

$\frac{p}{p-q}=[b_{1},b_{2},\ldots,b_{k}]=b_{1}-\cfrac{1}{b_{2}-\cfrac{1}{\ddots-\cfrac{1}{b_{k}}}}$

be the Hirzebruch-Jung continued fraction, where $b_{i}\geq 2$ for $1\leq
i\leq k$. Note that the sequence of integers $\\{b_{1},b_{2}\ldots,b_{k}\\}$
is uniquely determined by the pair $(p,q)$.

For any $k\geq 2$, a $k$-tuple of positive integers $(n_{1},\ldots,n_{k})$ is
called admissible if each of the denominators in the continued fraction
$[n_{1},\ldots,n_{k}]$ is positive, where we do not assume that $n_{i}\geq 2$.
For any $k\geq 2$, let $\mathcal{Z}_{k}\subset\mathbb{Z}^{k}$ denote the set
of admissible $k$-tuples of positive integers
$\textbf{n}=(n_{1},\ldots,n_{k})$ such that $[n_{1},\ldots,n_{k}]=0$ and let
$\mathcal{Z}_{1}=\\{(0)\\}$. As a matter of fact, any $k$-tuple of positive
integers in $\mathcal{Z}_{k}$ can be obtained from $(0)$ by a sequence of
blowups as observed by Lisca [7, Lemma 2]. Note that the only possible blowup
of $(0)$ is the initial blowup $(0)\to(1,1)$. Let

$\mathcal{Z}_{k}(\textstyle{\frac{p}{p-q}})=\\{(n_{1},\ldots,n_{k})\in\mathcal{Z}_{k}\,|\,0\leq
n_{i}\leq b_{i}\;\mbox{for}\;i=1,\ldots,k\\}.$

Next, for every $k$-tuple
$\textbf{n}=(n_{1},\ldots,n_{k})\in\mathcal{Z}_{k}(\textstyle{\frac{p}{p-q}})$,
we describe a $4$-manifold $W_{p,q}(\textbf{n})$ whose boundary is
orientation-preserving diffeomorphic to $L(p,q)$. We start with a chain of
unknots in $S^{3}$ with framings $n_{1},n_{2},\ldots,n_{k}$, respectively. It
can be easily verified that the result of Dehn surgery on this framed link,
which we denote $N(\textbf{n})$, is diffeomorphic to $S^{1}\times S^{2}$. Let
$\textbf{L}=\bigcup_{i=1}^{k}L_{i}$ denote the framed link in $N(\textbf{n})$
depicted in red in Figure 6, where each $L_{i}$ has $b_{i}-n_{i}$ components.

$n_{1}$$n_{2}$$n_{k-1}$$n_{k}$$b_{1}-n_{1}$$b_{2}-n_{2}$$b_{k-1}-n_{k-1}$$b_{k}-n_{k}$$-1$$-1$$-1$$-1$$-1$$-1$$-1$$-1$$-1$$-1$$-1$$-1$

Figure 6. The relative handlebody decomposition of the $4$-manifold
$W_{(p,q)}(\textbf{n})$.

Since $N(\textbf{n})$ is diffeomorphic to $S^{1}\times S^{2}$, one can fix a
diffeomorphism $\phi:N(\textbf{n})\to S^{1}\times S^{2}$. By attaching
$2$-handles to $S^{1}\times D^{3}$ along the framed link
$\phi(\textbf{L})\subset S^{1}\times S^{2}$, we obtain a smooth $4$-manifold
$W_{p,q}(\textbf{n})$ whose boundary is orientation-preserving diffeomorphic
to $L(p,q)$. As noted by Lisca, the diffeomorphism type of
$W_{p,q}(\textbf{n})$ is independent of the choice of $\phi$ since any self-
diffeomorphism of $S^{1}\times S^{2}$ extends to $S^{1}\times D^{3}$.

According to Lisca, any minimal symplectic filling (in fact Stein filling) of
($L(p,q),\xi_{can})$ is orientation-preserving diffeomorphic to
$W_{p,q}(\textbf{n})$ for some $\textbf{n}\in\mathcal{Z}_{k}(\frac{p}{p-q})$.

###  3.2. Planar Lefschetz fibrations on Stein fillings

In [2], we described an algorithm to construct a planar Lefschetz fibration
$W_{p,q}(\textbf{n})\to D^{2}$, based on any given blowup sequence

$(0)\to(1,1)\to\cdots\to\textbf{n}=(n_{1},\ldots,n_{k})\in\mathcal{Z}_{k}(\frac{p}{p-q}).$

Here we briefly review our algorithm, which consists of two parts,
stabilization and surgery, that gives an ordered set of vanishing cycles on a
disk with $k$ holes which is the fiber of our Lefschetz fibration
$W_{p,q}(\textbf{n})\to D^{2}$. We begin by describing the first part of our
algorithm which we call the stabilization algorithm.

####  3.2.1. The stabilization algorithm

For any positive integer $r$, let $D_{r}$ denote the disk with $r$ holes. We
assume that the holes are aligned horizontally on $D_{r}$ and we enumerate the
holes on $D_{r}$ from left to right as $H_{1},H_{2},\ldots,H_{r}.$

The initial step of the algorithm corresponding to $(0)$ is the disk $D_{1}$
with no vanishing cycle, as depicted on the top in Figure 7. Recall that the
only blowup starting from $(0)$ is the initial blowup $(0)\to(1,1)$. The
corresponding fiber is the disk $D_{2}$ with one vanishing cycle $\alpha_{1}$,
which is parallel to the boundary of $H_{1}$, as depicted in the middle in
Figure 7. This is a stabilization of the previous step, where we had the
annulus $D_{1}$ with no vanishing cycle. Depending on the type of the next
blowup, we proceed as follows.

$1$$2$$1$$2$$3$$1$$2$$3$$1$$(1,1)$$(2,1,2)$$(1,2,1)$interior blowupexterior
blowupinitial
blowup$\alpha_{1}$$\alpha_{1}$$\alpha_{1}$$\alpha_{2}$$\alpha_{2}$$(0)$

Figure 7. Stabilizations depending on the type of the blowup.

If we have an interior blowup at the first term $(1,1)\to(2,1,2)$, then
$H_{2}$ “splits” into two holes, where the new hole $H_{3}$ is placed to the
right of $H_{2}$. The curve $\alpha_{1}$ becomes a convex curve enclosing
$H_{2}$ and $H_{3}$ in $D_{3}$. We introduce a new vanishing cycle
$\alpha_{2}$ which encloses $H_{1}$ and $H_{3}$ in $D_{3}$ as shown at the
bottom left in Figure 7. We can view the introduction of $\alpha_{2}$ as a
stabilization of the previous step.

On the other hand, if we have an exterior blowup $(1,1)\to(1,2,1)$, then we
simply introduce a new hole $H_{3}$ to the right, and the new vanishing cycle
$\alpha_{2}$ is parallel to the boundary of $H_{3}$ in $D_{3}$ as shown at the
bottom right in Figure 7. Again, we can view the introduction of $\alpha_{2}$
as a stabilization of the previous step.

Now suppose that we have a set of $r-1$ vanishing cycles
$\alpha_{1},\alpha_{2},\ldots,\alpha_{r-1}$ on a disk with $r$ holes
corresponding to some blowup sequence

$(0)\to(1,1)\to\cdots\to(n_{1},\ldots,n_{r}).$

Depending on the type of the next blowup we insert a new hole and introduce a
new vanishing cycle $\alpha_{r}$ as follows.

If we have an interior blowup at the $i$th term, for $1\leq i\leq r-1$, then
the hole $H_{i+1}$ “splits” into two holes, where the new hole $H_{i+2}$ is
placed to the right of $H_{i+1}$ in the resulting disk $D_{r+1}$. We introduce
a new vanishing cycle $\alpha_{r}$ which encloses the holes
$H_{1},H_{2},\ldots,H_{i}$ and the new hole $H_{i+2}$ in $D_{r+1}$. We can
view the introduction of $\alpha_{r}$ as a stabilization of the previous step.

On the other hand, if we have an exterior blowup, then we simply insert a new
hole $H_{r+1}$ to the right, which is the last hole in the geometric order
from the left in the resulting disk $D_{r+1}$ and the new vanishing cycle
$\alpha_{r}$ is parallel to the boundary of $H_{r+1}$. Again, we can view the
introduction of $\alpha_{r}$ as a stabilization of the previous step.

Next, we describe the second part of our algorithm which we call the surgery
algorithm.

####  3.2.2. The surgery algorithm

The surgery algorithm is based on the link
$\textbf{L}=\bigcup_{i=1}^{k}L_{i}$, which is used to define
$W_{p,q}(\textbf{n})$. The vanishing cycles in this subsection will be
mutually disjoint and hence their order does not matter. So we can describe
all the vanishing cycles as a set of curves on the disk $D_{k}$ with $k$
holes.

######  Definition 10.

(The $\gamma$-curves) For each $1\leq i\leq k$, let $\gamma_{i}$ be the convex
curve on $D_{k}$ enclosing the holes $H_{1},H_{2},\ldots,H_{i}$.

Then the set of vanishing cycles in this part of the algorithm is

$\\{\underbrace{\gamma_{1},\ldots,\gamma_{1}}_{b_{1}-n_{1}},\underbrace{\gamma_{2},\ldots,\gamma_{2}}_{b_{2}-n_{2}},\ldots,\underbrace{\gamma_{k},\ldots\gamma_{k}}_{b_{k}-n_{k}}\\},$

where each $\gamma_{i}$ appears $b_{i}-n_{i}$ times in the set. In particular,
if $b_{i}=n_{i}$, then $\gamma_{i}$ is not in the set of vanishing cycles.

####  3.2.3. Total monodromy

The fiber of the planar Lefschetz fibration $W_{p,q}(\textbf{n})\to D^{2}$ is
the disk $D_{k}$ with $k$ holes, where $k$ is the length of the continued
fraction $\frac{p}{p-q}=[b_{1},b_{2}\ldots,b_{k}]$. The set of vanishing
cycles consists of the curves $\alpha_{1},\alpha_{2},\ldots,\alpha_{k-1}$
coming from the stabilization algorithm and
$\gamma_{1},\gamma_{2},\ldots,\gamma_{k}$ (each with a multiplicity) coming
from the surgery algorithm. Let $D(\alpha)$ denote the right-handed twist
along a simple closed curve $\alpha$ on a surface. The total monodromy of the
planar Lefschetz fibration $W_{p,q}(\textbf{n})\to D^{2}$ is given as the
following composition of Dehn twists along the vanishing cycles

$D(\alpha_{1})D(\alpha_{2})\cdots
D(\alpha_{k-1})D^{b_{1}-n_{1}}(\gamma_{1})D^{b_{2}-n_{2}}(\gamma_{2})\cdots
D^{b_{k}-n_{k}}(\gamma_{k}).$

In Lemma 11 below, we describe another planar Lefschetz fibration on
$W_{p,q}(\textbf{n})$.

######  Lemma 11.

Let $f:W_{p,q}(\textbf{n})\to D^{2}$ be the planar Lefschetz fibration we
constructed in [2]. The total space of the planar Lefschetz fibration obtained
by reversing the order of the vanishing cycles of $f$, while taking their
mirror images is diffeomorphic to $W_{p,q}(\textbf{n})$.

###### Proof.

The result follows from the fact that such a transformation of the vanishing
cycles can be achieved by rotating the absolute handlebody diagram inducing
the planar Lefschetz fibration constructed in [2]. To see this, consider for
example the handlebody diagram in [2, Figure 7], which is depicted on the
left-hand side in Figure 8.

Rotate by $180^{\circ}$

Figure 8. By rotating the handlebody diagram $180^{\circ}$ in a direction
normal to the page, we obtain the mirror images of the vanishing cycles in
reverse order.

By rotating this handlebody diagram $180^{\circ}$ in a direction normal to the
page, we get the handlebody diagram on the right-hand side whose total space
is still the same. But this new handlebody diagram corresponds to a planar
Lefschetz fibration, where the mirror images of the vanishing cycles appear in
reverse order. Note that here we view the base disk $D_{k}$ “horizontally” and
the mirror image $\overline{\alpha}$ of a curve $\alpha\subset D_{k}$ is
defined to be the reflection of $\alpha$ along the $x$-axis, once the holes in
$D_{k}$ are aligned horizontally along the $x$-axis. This definition of mirror
image, of course, coincides with the mirror image in a vertical $D_{k}$ by
rotating the horizontal $D_{k}$ clockwise by $90^{\circ}$. ∎

###  3.3. An example

For $p=56$ and $q=17$, we have $\dfrac{56}{56-17}=[2,2,5,2,3]$. The $5$-tuple
$\textbf{n}=(2,1,4,1,2)$ belongs to $\mathcal{Z}_{5}(\frac{56}{56-17})$ since
we have the blowup sequence

$(0)\to(1,1)\to(1,2,1)\to(2,1,3,1)\to(2,1,4,1,2)$

and hence we conclude that $W_{(56,17)}((2,1,4,1,2))$ is a Stein filling of
the contact $3$-manifold $(L(56,17),\xi_{can})$. The fiber of the planar
Lefschetz fibration

$W_{(56,17)}(2,1,4,1,2)\to D^{2}$

is the disk $D_{5}$ with $5$ holes, and to obtain the vanishing cycles
$\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}$ coming from the stabilization
algorithm, we start from the step $(1,2,1)$ which is already shown at the
bottom right in Figure 7 and apply the stabilization algorithm to the interior
blowups $(1,2,1)\to(2,1,3,1)\to(2,1,4,1,2)$ as depicted in Figure 9.

$5$$4$$1$$1$$2$$2$$3$$3$$1$$2$$3$$4$$(2,1,4,1,2)$$(2,1,3,1)$$(1,2,1)$$\alpha_{2}$$\alpha_{1}$$\alpha_{1}$$\alpha_{1}$$\alpha_{2}$$\alpha_{2}$$\alpha_{4}$$\alpha_{3}$$\alpha_{3}$

Figure 9. The vanishing cycles $\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}$
coming from the stabilization algorithm.

Note that $b_{1}-n_{1}=0$, whereas
$b_{2}-n_{2}=b_{3}-n_{3}=b_{4}-n_{4}=b_{5}-n_{5}=1$, which implies that the
set of vanishing cycles coming from the surgery algorithm in this case is
$\gamma_{2},\gamma_{3},\gamma_{4}$ and $\gamma_{5}$ as shown in Figure 10.

$5$$1$$2$$3$$4$$D_{5}$$\gamma_{2}$$\gamma_{3}$$\gamma_{5}$$\gamma_{4}$

Figure 10. The vanishing cycles $\gamma_{2},\gamma_{3},\gamma_{4},\gamma_{5}$
coming from the surgery algorithm.

Consequently, the total monodromy is given as the follows

$D(\alpha_{1})D(\alpha_{2})D(\alpha_{3})D(\alpha_{4})D(\gamma_{2})D(\gamma_{3})D(\gamma_{4})D(\gamma_{5}).$

######  Remark 12.

By Lemma 11, there is a planar Lefschetz fibration $W_{(56,17)}(2,1,4,1,2)\to
D^{2}$ whose monodromy factorization is given by

$D(\gamma_{5})D(\gamma_{4})D(\gamma_{3})D(\gamma_{2})D(\overline{\alpha}_{4})D(\overline{\alpha}_{3})D(\overline{\alpha}_{2})D(\overline{\alpha}_{1}).$

##  4\. Unbraided wiring diagrams

###  4.1. The blowup algorithm

In this subsection, we describe an algorithm to construct an unbraided wiring
diagram corresponding to a blowup sequence starting from the initial blowup
$(0)\to(1,1)$. The wiring diagram corresponding to $(0)$ is a single wire
$w_{1}$ without any marked points and the wiring diagram corresponding to
$(1,1)$, consists of two parallel wires $\\{w_{1},w_{2}\\}$ so that $w_{1}$ is
on top without any marked points, and $w_{2}$ has a single marked point
$x_{1}$. The next step in the algorithm depends on whether we have an interior
or exterior blowup that follows the initial blowup $(0)\to(1,1)$.

If we have an interior blowup at the first term $(1,1)\to(2,1,2)$ we introduce
a new wire $w_{3}$, which is initially below $w_{2}$ on the right-hand side of
the diagram and as it moves to the left, it goes through the marked point
$x_{1}$ on $w_{2}$, but otherwise remains parallel to $w_{2}$ and then
intersects $w_{1}$ at a new marked point $x_{2}$, which is to the left of
$x_{1}$. This diagram with three wires $\\{w_{1},w_{2},w_{3}\\}$ corresponds
to $(2,1,2)$, which we depicted in Figure 11.

$(0)$$(1,1)$$(2,1,2)$$x_{1}$$x_{2}$$x_{1}$$w_{1}$$w_{1}$$w_{1}$$w_{2}$$w_{2}$$w_{3}$

Figure 11. Wiring diagrams corresponding to the blowup sequence
$(0)\to(1,1)\to(2,1,2)$.

On the other hand, if we have an exterior blowup $(1,1)\to(1,2,1)$, we insert
in the diagram a new wire $w_{3}$ which is right below $w_{2}$ and parallel to
it. We place a marked point $x_{2}$ on $w_{3}$ so that $x_{2}$ is to the left
of $x_{1}$. This diagram with three wires $\\{w_{1},w_{2},w_{3}\\}$
corresponds to $(1,2,1)$, which we depicted in Figure 12.

$(0)$$(1,1)$$(1,2,1)$$x_{1}$$x_{1}$$x_{2}$$w_{1}$$w_{1}$$w_{1}$$w_{2}$$w_{2}$$w_{3}$

Figure 12. Wiring diagrams corresponding to the blowup sequence
$(0)\to(1,1)\to(1,2,1)$.

Now suppose that we have an unbraided wiring diagram $\mathcal{W}$ consisting
of $r$ wires $\\{w_{1},w_{2}$, $\ldots,w_{r}\\}$ corresponding to some blowup
sequence starting from the initial blowup $(0)\to(1,1)$ and ending with some
$r$-tuple of positive integers. We would like to emphasize that the indices of
the wires in the set $\mathcal{W}$ above indicate the order in which the wires
are introduced into the diagram. Depending on the type of the next blowup we
insert a new wire in $\mathcal{W}$ and adjust the diagram accordingly as
follows.

Suppose that we have an interior blowup at the $i$th term, for some $1\leq
i\leq r-1$. Let $w_{j}\in\mathcal{W}$ be the $(i+1)$st wire with respect to
the geometric ordering of the wires on the right-hand side of the diagram, and
let $\mathcal{W}_{i}$ denote the subset of $\mathcal{W}$ consisting of all the
wires which appears before $w_{j}$ in this ordering. In other words,
$\mathcal{W}_{i}$ is the set of the top $i$ wires in the geometric ordering of
the wires on the right-hand side of the diagram. Now we introduce a new wire,
named $w_{r+1}$, into the diagram, which is initially right below $w_{j}$ on
the right-hand side of the diagram and as it moves to the left, goes through
all the marked points on $w_{j}$ but otherwise remains parallel to $w_{j}$,
and then we insert a new marked point $x_{r}$ on $w_{r+1}$ which is the
simultaneous intersection of $w_{r+1}$ and all the wires in $\mathcal{W}_{i}$.
We place the marked point $x_{r}$ to the left of $x_{r-1}$. For this to work,
we need to know that the set $\mathcal{W}_{i}\cup\\{w_{r+1}\\}$ of wires is
geometrically consecutive on the left-hand side, which we verify in Lemma 13
below, where we refer to this step in the algorithm as the last twist.

On the other hand, if we have an exterior blowup, we insert a new wire
$w_{r+1}$ below all the wires in $\mathcal{W}$ with no intersection points
with the other wires, and place a single marked point $x_{r}$ on $w_{r+1}$,
which is to the left of $x_{r-1}$.

We call this procedure the blowup algorithm for wiring diagrams. Note that in
the resulting wiring diagram, the wires are indexed in the order they are
introduced into the diagram but their geometric ordering on the right-hand
side (or the left-hand side) of the diagram as viewed on the page, might be
different from the index ordering. Moreover, by our algorithm, $w_{1}$ will
always be at the top on the right-hand side of the diagram.

######  Lemma 13.

If $\mathcal{W}$ is an unbraided wiring diagram consisting of wires
$\\{w_{1},w_{2},\ldots,w_{r}\\}$, which is obtained by the blowup algorithm
with respect to some blowup sequence starting from the initial blowup
$(0)\to(1,1)$, then any set of wires including $w_{1}$, which is consecutive
with respect to the geometric ordering on the right-hand side of the diagram,
is also geometrically consecutive (perhaps with a different geometric
ordering) on the left-hand side of the diagram. Moreover, if any wire other
than $w_{1}$ carries an even (resp. odd) number of marked points, then on the
left-hand side it is above (resp. below) all the wires which appear before it
in the geometric ordering of the wires on the right-hand side of the diagram.

###### Proof.

We prove the lemma by induction on the number of wires. The two wiring
diagrams we described above corresponding to the blowup sequences
$(0)\to(1,1)\to(2,1,2)$ and $(0)\to(1,1)\to(1,2,1)$, respectively, can be
taken to be the initial step of our induction argument. The properties stated
in Lemma 13 hold for these wiring diagrams.

Suppose that both properties stated in Lemma 13 hold when there are up to
$r\geq 3$ wires in any unbraided wiring diagram obtained as a result of the
blowup algorithm with respect to some blowup sequence starting from the
initial blowup $(0)\to(1,1)$. We will prove that these properties continue to
hold when a new wire is inserted into the diagram corresponding to a new
blowup. If the new wire inserted corresponds to an exterior blowup, it is
clear that both properties stated in Lemma 13 continue to hold in the new
diagram with $r+1$ wires. This is because in this case, the new wire will be
inserted at the bottom of the diagram with a single marked point on it and
without any intersections with the other wires.

Suppose that a new wire $w_{r+1}$ is inserted into $\mathcal{W}$ with respect
to an interior blowup at the $i$th term, for some $1\leq i\leq r-1$. Let
$\mathcal{W}_{s}$ be the subset of $\mathcal{W}$ consisting of the top $s$
wires in the geometric ordering of the wires on the right-hand side of the
diagram. Note that $\mathcal{W}_{i+1}=\mathcal{W}_{i}\cup\\{w_{j}\\}$, since
by definition, $w_{j}\in\mathcal{W}$ is the $(i+1)$st wire with respect to the
geometric ordering of the wires on the right-hand side of the diagram.

Assume that $w_{j}$ has an odd number of marked points. By the induction
hypotheses, before we insert $w_{r+1}$, the wires in the set
$\mathcal{W}_{i+1}$ are geometrically consecutive (perhaps with a different
geometric ordering) on the left-hand side, while $w_{j}$ is at the bottom of
these geometrically consecutive wires. The new wire $w_{r+1}$ will be
initially right below the wire $w_{j}$ on the right-hand side of the diagram
and $w_{r+1}$ will go through all the marked points on $w_{j}$, and otherwise
it will remain parallel to $w_{j}$, before the last twist in the algorithm.
But since $w_{j}$ has an odd number of marked points, and $w_{r+1}$ is
initially right below $w_{j}$, the wire $w_{r+1}$ will be right above $w_{j}$
on the left-hand side before the last twist. Therefore, before the last twist,
the wires in the set $\mathcal{W}_{i+1}\cup\\{w_{r+1}\\}$ will be
geometrically consecutive on the left-hand side, and moreover $w_{r+1},w_{j}$
will be the bottom two wires in that order. Finally, when we twist once all
the wires in the set $\mathcal{W}_{i}\cup\\{w_{r+1}\\}$ (to create a
simultaneous intersection point of these $i+1$ wires) as part of the blowup
algorithm, the wires in the set $\mathcal{W}_{i+1}\cup\\{w_{r+1}\\}$ will
remain geometrically consecutive on the left-hand side, where $w_{r+1}$ will
appear at the top, and $w_{j}$ will appear at the bottom of this consecutive
set of wires.

Assume that $w_{j}$ has an even number of marked points. By the induction
hypotheses, before we insert $w_{r+1}$, the wires in the set
$\mathcal{W}_{i+1}$ is geometrically consecutive (perhaps with a different
geometric ordering) on the left-hand side, while $w_{j}$ is at the top of
these geometrically consecutive wires. The new wire $w_{r+1}$ will be
initially right below the wire $w_{j}$ on the right-hand side of the diagram
and $w_{r+1}$ will go through all the marked points on $w_{j}$, and otherwise
it will remain parallel to $w_{j}$, before the last twist in the algorithm.
But since $w_{j}$ has an even number of marked points, and $w_{r+1}$ is
initially right below $w_{j}$, the wire $w_{r+1}$ will be right below $w_{j}$
on the left-hand side before the last twist. Therefore, before the last twist,
the wires in the set $\mathcal{W}_{i+1}\cup\\{w_{r+1}\\}$ will be
geometrically consecutive on the left-hand side and, moreover, $w_{j},w_{r+1}$
will be the top two wires in that order. Finally, when we twist once all the
wires in the set $\mathcal{W}_{i}\cup\\{w_{r+1}\\}$ (to create a simultaneous
intersection point of these $i+1$ wires) as part of the blowup algorithm, the
wires in the set $\mathcal{W}_{i+1}\cup\\{w_{r+1}\\}$ will remain
geometrically consecutive on the left-hand side, where $w_{j}$ will appear at
the top, and $w_{r+1}$ will appear at the bottom of this consecutive set of
wires.

The discussion above proves that, after we insert $w_{r+1}$, any set of wires
in $\mathcal{W}\cup\\{w_{r+1}\\}$ including $w_{1}$, which is consecutive with
respect to the geometric ordering on the right-hand side of the diagram, is
also geometrically consecutive (perhaps with a different geometric ordering)
on the left-hand side of the diagram.

Moreover, if $w_{j}$ has an odd (resp. even) number of marked points, then
$w_{r+1}$ will have even (resp.odd) number of marked points by the blowup
algorithm and it will be above (resp. below) all the wires in
$\mathcal{W}_{i+1}$ on the left-hand side of the diagram. The upshot is that
both properties stated in Lemma 13 hold true for the unbraided wiring diagram
$\mathcal{W}\cup\\{w_{r+1}\\}$. ∎

###  4.2. An example

Consider the blowup sequence

$(0)\to(1,1)\to(1,2,1)\to(2,1,3,1)\to(2,1,4,1,2).$

In Figure 13 below we depict the diagrams corresponding to

$(1,2,1)\to(2,1,3,1)\to(2,1,4,1,2)$

starting from the diagram of $(1,2,1)$ already depicted in Figure 12.

$(1,2,1)$$(2,1,3,1)$$(2,1,4,1,2)$$x_{1}$$x_{1}$$x_{1}$$x_{3}$$x_{3}$$x_{4}$$x_{2}$$x_{2}$$x_{2}$$w_{1}$$w_{1}$$w_{1}$$w_{3}$$w_{3}$$w_{3}$$w_{2}$$w_{2}$$w_{2}$$w_{4}$$w_{4}$$w_{5}$

Figure 13. Wiring diagrams corresponding to the blowup sequence
$(1,2,1)\to(2,1,3,1)\to(2,1,4,1,2)$.

###  4.3. The twisting algorithm

Suppose that $\mathcal{W}$ is an unbraided wiring diagram consisting of wires
$\\{w_{1},w_{2},\ldots,w_{k}\\}$, which is obtained by the blowup algorithm
with respect to some blowup sequence, starting from the initial blowup
$(0)\to(1,1)$ and ending with some $k$-tuple of positive integers. Let
$\mathcal{W}_{s}$ be the subset of $\mathcal{W}$ consisting of the top $s$
wires in the geometric ordering of the wires on the right-hand side of the
diagram, as in Section 4.1. Based on any $k$-tuple
$\textbf{m}=(m_{1},\ldots,m_{k})$, where $m_{i}$ is a nonnegative integer, we
describe a procedure called the twisting algorithm to extend the unbraided
wiring diagram $\mathcal{W}$ to another unbraided wiring diagram
$\mathcal{W}({\textbf{m})}$ with the same number of wires but with more marked
points obtained by extra twists inserted to the left.

If $m_{1}=0$, then we do not modify $\mathcal{W}$ and move onto the next step.
If $m_{1}>0$, then we simply add $m_{1}$ extra marked points
$\underbrace{y_{1},y_{1},\ldots,y_{1}}_{m_{1}}$ on $w_{1}$ to the left of
$x_{k-1}$. If $m_{2}=0$, then we do not modify the diagram any further and
move onto the next step. If $m_{2}>0$, then by Lemma 13, we know that the
wires in $\mathcal{W}_{2}$ are geometrically consecutive on the left-hand side
of the diagram $\mathcal{W}$. We extend $\mathcal{W}$ by twisting
$m_{2}$-times the wires in $\mathcal{W}_{2}$, creating consecutive
simultaneous intersection points
$\underbrace{y_{2},y_{2},\ldots,y_{2}}_{m_{2}}$ to the left of the last, if
any, $y_{1}$. If $m_{3}=0$, then we do not modify the diagram any further and
move onto the next step. Now suppose that $m_{3}>0$. Since the wires in
$\mathcal{W}_{3}$ are geometrically consecutive on the left-hand side of the
diagram $\mathcal{W}$ by Lemma 13 these wires will remain geometrically
consecutive after the first additional twists we possibly put into the diagram
corresponding to $m_{2}$. We extend the diagram further by twisting
$m_{3}$-times the wires in $\mathcal{W}_{3}$, creating simultaneous
intersection point $\underbrace{y_{3},y_{3},\ldots,y_{3}}_{m_{3}}$ to the left
of the last, if any, $y_{2}$. By iterating this procedure, we extend
$\mathcal{W}$ to $\mathcal{W}({\textbf{m})}$ with additional marked points
corresponding to m.

######  Remark 14.

Here, we think of $m_{i}$ as the “multiplicity” of the point $y_{i}$. If
$m_{i}=0$, then $y_{i}$ does not appear in the diagram, and if $m_{i}>1$, then
$y_{i}$ is repeated $m_{i}$-times. To avoid cumbersome notation, we do not put
an extra index to distinguish between different $y_{i}$ type points.

###  4.4. An example

Here we give an example where we extend the wiring diagram $\mathcal{W}$
corresponding to the blowup sequence

$(0)\to(1,1)\to(1,2,1)\to(2,1,3,1)\to(2,1,4,1,2)$

depicted in Figure 13 to $\mathcal{W}({\textbf{m}})$ applying the twisting
algorithm based on $\textbf{m}=(0,1,1,1,1)$. Note that inside the dotted
square in Figure 14, there is a copy of $\mathcal{W}$ from Figure 13.

$\mathcal{W}$$y_{3}$$x_{1}$$y_{2}$$x_{3}$$x_{4}$$y_{5}$$y_{4}$$x_{2}$$w_{1}$$w_{3}$$w_{2}$$w_{4}$$w_{5}$

Figure 14. Extending $\mathcal{W}$ to $\mathcal{W}((0,1,1,1,1))$ by applying
the surgery algorithm based on $\textbf{m}=(0,1,1,1,1)$.

In this example $m_{1}=0$ and $m_{2}=1$ and the wires
$\mathcal{W}_{2}=\\{w_{1},w_{2}\\}$ are geometrically consecutive on the left-
hand side of $\mathcal{W}$. Now we twist them together once to obtain the
marked point $y_{2}$, which is to the left of $x_{4}$. Since $m_{3}=1$, next
we twist the wires in $\mathcal{W}_{3}=\\{w_{1},w_{2},w_{4}\\}$ (which are
geometrically consecutive) together once to obtain the marked point $y_{3}$,
which is to the left of $y_{2}$. Since $m_{4}=1$, we twist the wires in
$\mathcal{W}_{4}=\\{w_{1},w_{2},w_{4},w_{3}\\}$ (which are geometrically
consecutive) together once to obtain the marked point $y_{4}$, which is to the
left of $y_{3}$. Finally, since $m_{5}=1$, we twist all the wires in
$\mathcal{W}_{5}=\mathcal{W}$ together once to obtain the marked point
$y_{5}$, which is to the left of $y_{4}$, as illustrated in Figure 14.

######  Remark 15.

We will also speak about $\delta(y_{s})$, $\Delta(y_{s})$ and $V(y_{s})$ for
each marked point $y_{s}$ in the rest of the paper, as described in
Definitions 6, 7, and 8.

##  5\. From vanishing cycles to unbraided wiring diagrams

We recall the main theorem from the introduction, where we have replaced $W$
with $W_{p,q}(\textbf{n})$ below, to be more precise.

Theorem 1. There is an algorithm to draw an explicit unbraided wiring diagram
whose associated planar Lefschetz fibration obtained by the method of
Plamenevskaya and Starkston [10] is equivalent to the planar Lefschetz
fibration $W_{p,q}(\textbf{n})\to D^{2}$ constructed by the authors in [2].

Before we give the proof of Theorem 1 below, we illustrate the statement and
its proof on an example. First we introduce some notation that will be used in
the following discussion. The disk $D_{k}$ with $k$ holes will be viewed in
two different but equivalent ways as follows: (i) the holes are aligned
horizontally in $D_{k}$ and enumerated from left to right or (ii) the holes
are aligned vertically in $D_{k}$ and enumerated from top to bottom. Here we
identify the “horizontal” $D_{k}$ in (i) with the “vertical” $D_{k}$ in (ii)
by rotating the “horizontal” $D_{k}$ clockwise by $90^{\circ}$. The reason why
we consider these two embeddings of a disk with holes is that the vanishing
cycles in [2] are described on a horizontal $D_{k}$, while the vanishing
cycles in [10] are described on a vertical $D_{k}$. Here we compare them on a
vertical $D_{k}$ via the identification given above. When we view $D_{k}$
vertically, the mirror image $\overline{\alpha}$ of a curve $\alpha\subset
D_{k}$ is defined to be the reflection of $\alpha$ along the $y$-axis, once
the holes in $D_{k}$ are aligned vertically along the $y$-axis.

###  5.1. An example

In Section 3.3, we constructed a planar Lefchetz fibration

$W_{(56,17)}((2,1,4,1,2))\to D^{2}$

whose fiber is the disk $D_{5}$ with $5$ holes and whose vanishing cycles are
the curves

$\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4},\gamma_{2},\gamma_{3},\gamma_{4},\gamma_{5}$

in $D_{5}$, which are depicted in Figures 9 and 10. We claim that the planar
Lefschetz fibration obtained by using the method of Plamenevskaya and
Starkston associated to the unbraided wiring diagram $W((0,1,1,1,1))$ in
Figure 14 has exactly the same set of vanishing cycles (viewed in a vertical
$D_{5}$), except that we have to take “mirror images” of all the curves and
reverse the order of the vanishing cycles in the total monodromy. In other
words, first we rotate the disks in Figures 9 and 10 clockwise by $90^{\circ}$
and then take the mirror images of the curves. This modification of the
vanishing cycles is not an issue by Lemma 11. Note that the mirror image of a
$\gamma$-curve is equal to itself, and hence we only need to take the mirror
images of the $\alpha$-curves. As a matter of fact, we claim that
$V(y_{j})=\gamma_{j}$, for $2\leq j\leq 5$ and
$V(x_{i})=\overline{\alpha}_{i}$, for $1\leq i\leq 4$ (see Remark 15 for
notation). To verify our claim, we apply the method of Plamenevskaya and
Starkston (see Section 2), to describe a set of ordered vanishing cycles
associated to the marked points in Figure 14, where we depicted the convex
curves assigned to the marked points in Figure 15.

$\delta(y_{2})$$\delta(y_{3})$$\delta(y_{5})$$\delta(x_{3})$$\delta(x_{1})$$\delta(x_{2})$$\delta(x_{4})$$\delta(y_{4})$

Figure 15. Convex curves in $D_{5}$ assigned to the marked points in Figure
14.

Note that

$V(y_{5})=\Delta(x_{1})\circ\cdots\circ\Delta(x_{4})\circ\Delta(y_{2})\circ\Delta(y_{3})\circ\Delta(y_{4})(\delta(y_{5}))=\delta(y_{5})=\gamma_{5},$
$V(y_{4})=\Delta(x_{1})\circ\cdots\circ\Delta(x_{4})\circ\Delta(y_{2})\circ\Delta(y_{3})(\delta(y_{4}))=\gamma_{4},\;\mbox{as
illustrated in Figure~{}\ref{fig: ex5c}},$
$V(y_{3})=\Delta(x_{1})\circ\cdots\circ\Delta(x_{4})\circ\Delta(y_{2})(\delta(y_{3}))=\gamma_{3},\;\mbox{as
illustrated in Figure~{}\ref{fig: ex5}, and}$
$V(y_{2})=\Delta(x_{1})\circ\cdots\circ\Delta(x_{4})(\delta(y_{2}))=\gamma_{2},\;\mbox{as
illustrated in Figure~{}\ref{fig: ex6}}.$

$\delta(y_{4})$$\gamma_{4}$

Figure 16. Starting from $\delta(y_{4})$, we apply a counterclockwise half-
twist on the subdisk enclosed by the dotted curve, at each step from left to
right.

$\delta(y_{3})$$\gamma_{3}$

Figure 17. Starting from $\delta(y_{3})$, we apply a counterclockwise half-
twist on the subdisk enclosed by the dotted curve, at each step from left to
right.

$\delta(y_{2})$$\gamma_{2}$

Figure 18. Starting from $\delta(y_{2})$, we apply a counterclockwise half-
twist on the subdisk enclosed by the dotted curve, at each step from left to
right.

######  Remark 16.

In Figure 16, we have not included $\delta(y_{2})$ and $\delta(y_{3})$ as
dotted curves since $\Delta(y_{2})\circ\Delta(y_{3})$ would not have any
effect on $\delta(y_{4})$. Similarly, we have not included $\delta(y_{2})$ as
a dotted curve in Figure 17 since $\Delta(y_{2})$ would not have any effect on
$\delta(y_{3})$. We will generalize this observation as Lemma 19 in Section
5.2.1.

Moreover, $V(x_{4})=\overline{\alpha}_{4}$ by comparing Figure 9 and Figure 4;
$V(x_{3})=\overline{\alpha}_{3}$ by comparing Figure 9 and Figure 5, and
finally $V(x_{2})=\delta(x_{2})=\overline{\alpha}_{2}=\alpha_{2}$ and
$V(x_{1})=\delta(x_{1})=\overline{\alpha}_{1}=\alpha_{1}$, by comparing Figure
9 and Figure 3. Note that the total monodromy of the planar Lefschetz
fibration is

$D(\gamma_{5})D(\gamma_{4})D(\gamma_{3})D(\gamma_{2})D(\overline{\alpha}_{4})D(\overline{\alpha}_{3})D(\overline{\alpha}_{2})D(\overline{\alpha}_{1}),$

which coincides with the monodromy in Remark 12.

Now we are ready to give a proof of Theorem 1.

###  5.2. Proof of the main result

Suppose that $p>q\geq 1$ are coprime integers and let

$\dfrac{p}{p-q}=[b_{1},b_{2},\ldots,b_{k}]$

be the Hirzebruch-Jung continued fraction, where $b_{i}\geq 2$ for $1\leq
i\leq k$. We set

$\textbf{b}=(b_{1},b_{2},\ldots,b_{k}).$

######  Definition 17.

(The wiring diagrams $\mathcal{W}_{\textbf{n}}$, and
$\mathcal{W}_{\textbf{n}}({\textbf{m}})$) For any

$\textbf{n}=(n_{1},n_{2},\ldots,n_{k})\in\mathcal{Z}_{k}(\textstyle{\frac{p}{p-q}})$

let $(0)\to(1,1)\to\cdots\to\textbf{n}$ be a blowup sequence, and let
$\textbf{m}=\textbf{b}-\textbf{n}$. We denote by $\mathcal{W}_{\textbf{n}}$,
the unbraided wiring diagram with $k$ wires $\\{w_{1},w_{2},\ldots,w_{k}\\}$
and $k-1$ marked points $x_{k-1},x_{k-2},\ldots,x_{1}$ (reading from left to
right) constructed by applying the blowup algorithm in Section 4.1 to the
given blowup sequence. We denote by $\mathcal{W}_{\textbf{n}}({\textbf{m}})$
the extension of $\mathcal{W}_{\textbf{n}}$ to the left obtained by applying
the twisting algorithm in Section 4.3 based on the $k$-tuple m. Note that
$\mathcal{W}_{\textbf{n}}({\textbf{m}})$ is obtained from
$\mathcal{W}_{\textbf{n}}$ by inserting additional marked points

$\underbrace{y_{k},\ldots,y_{k}}_{m_{k}},\underbrace{y_{k-1},\ldots,y_{k-1}}_{m_{k-1}},\dots,\underbrace{y_{1},\ldots,y_{1}}_{m_{1}},$

reading from left to right.

Vanishing cycles associated to $\mathcal{W}_{\textbf{n}}({\textbf{m}})$: Now
we can apply the method of Plamenevskaya and Starkston (see Section 2) to the
wiring diagram $\mathcal{W}_{\textbf{n}}({\textbf{m}})$, to obtain the
associated planar Lefschetz fibration by describing a set of ordered vanishing
cycles on the disk $D_{k}$ with $k$ holes. According to their algorithm, there
is a vanishing cycle associated to each marked point in
$\mathcal{W}_{\textbf{n}}({\textbf{m}})$. So, for each $1\leq t\leq k-1$,
there is a vanishing cycle $V(x_{t})$ associated to the marked point $x_{t}$
in $\mathcal{W}_{\textbf{n}}({\textbf{m}})$, and for each $1\leq s\leq k$,
there is a vanishing cycle $V(y_{s})$ associated to the marked point $y_{s}$
in $\mathcal{W}_{\textbf{n}}({\textbf{m}})$. Note that there are $k-1$
vanishing cycles associated to type $x$ marked points, and since each $y_{t}$
is repeated $m_{t}=b_{t}-n_{t}$ times, there are

$m_{1}+m_{2}+\cdots+m_{k}=(b_{1}-n_{1})+(b_{2}-n_{2})+\cdots+(b_{k}-n_{k})$

vanishing cycles in total associated to type $y$ marked points.

Planar Lefschetz fibration $W_{p,q}(\textbf{n})\to D^{2}$: Let
$W_{p,q}(\textbf{n})$ be the minimal symplectic filling of
$(L(p,q),\xi_{can})$ as in Section 3.1. As we described in Section 3.2, there
is a planar Lefschetz fibration $W_{p,q}(\textbf{n})\to D^{2}$ with fiber
$D_{k}$, which is obtained by applying the stabilization algorithm and the
surgery algorithm. Note that there are $k-1$ vanishing cycles
$\alpha_{1},\alpha_{2},\ldots,\alpha_{k-1}$ coming from the stabilization
algorithm and

$(b_{1}-n_{1})+(b_{2}-n_{2})+\cdots+(b_{k}-n_{k})$

vanishing cycles

$\\{\underbrace{\gamma_{1},\ldots,\gamma_{1}}_{b_{1}-n_{1}},\underbrace{\gamma_{2},\ldots,\gamma_{2}}_{b_{2}-n_{2}},\ldots,\underbrace{\gamma_{k},\ldots\gamma_{k}}_{b_{k}-n_{k}}\\}$

coming from the surgery algorithm.

Theorem 1 is in fact equivalent to Proposition 18 coupled with Lemma 11.

######  Proposition 18.

Let $\mathcal{W}_{\textbf{n}}({\textbf{m}})$ be an unbraided wiring diagram
with $k$ wires as described in Definition 17. Then

  1. (a)

for any $1\leq s\leq k$, the vanishing cycle $V(y_{s})$ associated to the
marked point $y_{s}\in\mathcal{W}_{\textbf{n}}({\textbf{m}})$ is isotopic to
$\gamma_{s}$ in $D_{k}$, and

  2. (b)

for any $1\leq t\leq k-1$, the vanishing cycle $V(x_{t})$ associated to the
marked point $x_{t}\in\mathcal{W}_{\textbf{n}}({\textbf{m}})$ is isotopic to
$\overline{\alpha}_{t}$ (the mirror image of $\alpha_{t}$) in $D_{k}$.

In the rest of the article we will provide a proof of Proposition 18. In
Section 5.2.1, we will first formulate Proposition 24 (a necessarily very
technical result) and Lemma 25 will show that it implies Proposition 18(a).
Then we will turn our attention to Proposition 18(b) in Section 5.2.2, where
we will formulate the result as Proposition 26.

####  5.2.1. The case of $\gamma$-curves:

To prove our claim in Proposition 18, we will verify that for $1\leq s\leq k$,
the vanishing cycle $V(y_{s})$ is isotopic to the curve $\gamma_{s}$ in
$D_{k}$. We begin with a simple but crucial observation.

######  Lemma 19.

For any wiring diagram $\mathcal{W}_{\textbf{n}}({\textbf{m}})$ with $k$ wires
as in Definition 17, and for any $1\leq s\leq k$, we have

$V(y_{s})=\Delta(x_{1})\circ\cdots\circ\Delta(x_{k-1})(\delta(y_{s})).$

###### Proof.

By Definition 8 we have

$V(y_{s})=\Delta(x_{1})\circ\cdots\circ\Delta(x_{k-1})\circ(\Delta(y_{1}))^{b_{1}-n_{1}}\circ\cdots\circ(\Delta(y_{s-1}))^{b_{s-1}-n_{s-1}}(\delta(y_{s})).$

But

$(\Delta(y_{1}))^{b_{1}-n_{1}}\circ\cdots\circ(\Delta(y_{s-1}))^{b_{s-1}-n_{s-1}}(\delta(y_{s}))=\delta(y_{s}),$

since the convex curves associated to the type $y$ marked points are nested,
due to the construction and the order of the type $y$ marked points in the
wiring diagram. ∎

Therefore, to prove our claim in Proposition 18 (a), for each $1\leq s\leq k$,
we need to verify that

$\Delta(x_{1})\circ\cdots\circ\Delta(x_{k-1})(\delta(y_{s}))=\gamma_{s}$

by Definition 8 and Lemma 19. Equivalently, we need to verify that for each
$1\leq s\leq k$,

$(\Delta(x_{k-1}))^{-1}\circ\cdots\circ(\Delta(x_{1}))^{-1}(\gamma_{s})=\delta(y_{s}).$

For technical reasons, we will prove a more refined statement in Proposition
24 from which our claim will follow by Lemma 25. Before giving the statement
we make the following definition.

######  Definition 20.

(Right/Left-convexity) A curve in a disk with holes enclosing two distinct
sets of adjacent holes as illustrated in Figure 19 is called right-convex, and
the mirror image of a right-convex curve is called left-convex. By definition
any convex curve enclosing a set of adjacent holes is both right-convex and
left-convex.

$0\leq$

Figure 19. A right-convex curve in a disk with holes.

Notation for the rest of the paper: During the proof, it will be convenient to
keep track of the number of wires in our wiring diagrams when talking about
marked points. Thus we will write $x_{i}^{k}$ (resp. $y_{j}^{k}$) when talking
about the marked point $x_{i}$ (resp. $y_{j}$) in a wiring diagram with $k$
wires. Although this decoration will make the notation cumbersome, it is
necessary for the accuracy of the arguments, but the reader can safely ignore
this superscript for the most part in the text below. Similarly, when talking
about curves and half-twists in a disk with holes, it will be convenient to
keep track of the number of holes. For example, we will write $\gamma_{s}^{k}$
when talking about the convex curve $\gamma_{s}$ in the disk $D_{k}$ with $k$
holes. Moreover, the counterclockwise half-twist $\Delta(x_{i})$ in $D_{k}$
will be abbreviated by $\Delta^{k}_{i}$ and its inverse, the clockwise half-
twist, by $(\Delta^{k}_{i})^{-1}$. (Fortunately, we will not need to use
$\Delta(y_{j})$ in our discussion below by Lemma 19, and hence the notation
$\Delta^{k}_{i}$ will not lead to any confusion.) Furthermore, we will denote
the $i$th hole (with respect to the geometric order from top to bottom) in
$D_{k}$ by $H_{i}^{k}$. We will also need the following definitions.

######  Definition 21.

For $2\leq s\leq k$, we denote by $\Gamma_{s}^{k}$ the collection of red arcs
in $D_{k}$ shown in Figure 20, where we set $\Gamma_{1}^{k}:=\emptyset$.

$\gamma^{k}_{s}$$H^{k}_{s}$$D_{k}$

Figure 20.  $\Gamma_{s}^{k}$ is the collection of red arcs.

Fix any wiring diagram $\mathcal{W}_{\textbf{n}}$ with $k$ wires and $k-1$
marked points $x^{k}_{k-1},\ldots,x^{k}_{1}$ (reading from left to right), as
in Definition 17.

######  Definition 22.

For $2\leq s\leq k$, let $\rho_{s}^{k}$ be the smallest
$t\in\\{1,\ldots,k-1\\}$ such that the convex curve $\delta(x_{t}^{k})\subset
D_{k}$ assigned to $x_{t}^{k}$ contains $H_{s}^{k}$. For $1\leq s\leq k$ and
$1\leq r\leq k-1$, we define

$\displaystyle\gamma_{s,r}^{k}$
$\displaystyle:=(\Delta_{r}^{k})^{-1}\circ(\Delta_{r-1}^{k})^{-1}\circ\cdots\circ(\Delta_{1}^{k})^{-1}(\gamma_{s}^{k}),$
$\displaystyle\Gamma_{s,r}^{k}$
$\displaystyle:=\begin{cases}(\Delta_{r}^{k})^{-1}\circ(\Delta_{r-1}^{k})^{-1}\circ\cdots\circ(\Delta_{\rho_{s}^{k}}^{k})^{-1}(\Gamma_{s}^{k})&\text{if
}\;s\geq 2\;\text{and }\;r\geq\rho_{s}^{k}\\\
\Gamma_{s}^{k}&\text{otherwise,}\end{cases}$

and set $\gamma_{s,0}^{k}:=\gamma_{s}^{k}$ and
$\Gamma_{s,0}^{k}:=\Gamma_{s}^{k}$.

Note that $\Gamma_{1,r}^{k}=\emptyset$ for any $0\leq r\leq k-1$, by
Definition 22.

######  Definition 23.

We abbreviate

$\Psi_{r}^{k}:=(\Delta_{r}^{k})^{-1}\circ(\Delta_{r-1}^{k})^{-1}\circ\cdots\circ(\Delta_{1}^{k})^{-1}$

for $r\geq 1$ and set $\Psi_{0}^{k}=\operatorname{id}$.

######  Proposition 24.

Fix any wiring diagram $\mathcal{W}_{\textbf{n}}$ with $k$ wires and $k-1$
marked points $x^{k}_{k-1}$, $\ldots,$ $x^{k}_{1}$ (reading from left to
right), as in Definition 17. Then the following three statements hold for any
$1\leq s\leq k$, and $1\leq r\leq k-1$:

  1. (L1)

The curve $\gamma_{s,r}^{k}$ is left or right-convex.

  2. (L2)

The sidedness of the convexity of $\gamma_{s,r}^{k}$ is opposite to that of
$\gamma_{s,r-1}^{k}$ if and only if the convex curve assigned to $x_{r}^{k}$
contains $\Psi_{r-1}^{k}(H_{s}^{k})$.

  3. (L3)

If $s\geq 2$ and $r\geq\rho_{s}^{k}$, then $\Gamma_{s,r}^{k}$ has one of the
two forms shown in Figure 21. In particular, the holes enclosed by
$\gamma_{s,r}^{k}$ can be split into two collections of adjacent holes, which
we will call “lobes” with the lobe containing $\Psi_{r}^{k}(H_{s}^{k})$ being
called the “primary lobe” and the other lobe the “secondary lobe”. We require
$\Psi_{r}^{k}(H_{s}^{k})$ to be the innermost hole of the primary lobe and the
intersection of $\Gamma_{s,r}^{k}$ with a half-plane containing the primary
lobe to have precisely one of the two forms illustrated in Figure 21. When
$s=1$, we have $\Gamma_{1,r}^{k}=\emptyset$, for any $0\leq r\leq k-1$.

$D_{k}$$D_{k}$$\gamma_{s,r}^{k}$$\gamma_{s,r}^{k}$$0\leq$$\geq
0$$\Psi_{r}^{k}(H_{s}^{k})$$\Psi_{r}^{k}(H_{s}^{k})$

Figure 21. $\Gamma_{s,r}^{k}$ is the collection of red arcs in both forms in
(L3).

######  Lemma 25.

Proposition 24 implies Proposition 18 (a).

###### Proof.

Lemma 13 implies that, for each $1\leq s\leq k$, the top $s$ wires according
to their geometric order on the right-hand side of a wiring diagram as
described in Definition 17 will be consecutive (perhaps with a different
geometric order) on the left-hand side as well. Therefore, by definition, the
convex curve $\delta(y^{k}_{s})$ encloses the set of adjacent holes in $D_{k}$
each of whose order is the same as the local geometric order of one of these
$s$ wires on the left-hand side of the diagram.

On the other hand, by definition, the convex curve $\gamma_{s}^{k}$ encloses
the top $s$ holes in $D_{k}$ and the set of images of these holes under
$(\Delta_{k-1}^{k})^{-1}\circ\cdots\circ(\Delta_{1}^{k})^{-1}$ will be the
same as the set of adjacent holes enclosed by $\delta(y^{k}_{s})$. To see
this, imagine that each wire has a colour and that each hole in the initial
copy of $D_{k}$ has a colour so that the $i$th hole from the top has the same
colour as the $i$th wire from the top on the right hand side. As the wires
move from right to left, they will be locally reordered each time a marked
points appears in the diagram. Similarly, the clockwise half-twist
corresponding to that marked point will reorder the holes on the disk $D_{k}$.
We set up our algorithm so that at each step the colour of each wire remains
the same as the colour of the corresponding hole.

Moreover, for each $1\leq s\leq k$, we know by Proposition 24 that the curve

$\gamma_{s,k-1}^{k}:=(\Delta_{k-1}^{k})^{-1}\circ\cdots\circ(\Delta_{1}^{k})^{-1}(\gamma_{s}^{k})$

is right or left-convex, but since it encloses a set of adjacent holes, it
must be convex. Therefore we conclude that for each $1\leq s\leq k$, the
convex curve $\gamma_{s,k-1}^{k}$ is isotopic to the convex curve
$\delta(y^{k}_{s})$. ∎

###### Proof of Proposition 24..

We will prove Proposition 24 by induction on the number of wires in the wiring
diagram. These three statements are vacuously true for a wiring diagram with
one wire and no marked points. Now suppose that $k\geq 2$ and these statements
hold for any wiring diagram constructed using the blowup algorithm above with
$k-1$ wires and $k-2$ marked points. We will prove that they hold for any
wiring diagram with $k$ wires and $k-1$ marked points
$x^{k}_{k-1},\ldots,x^{k}_{1}$ (reading from left to right), constructed as in
Definition 17. Our induction argument naturally splits into several cases.

Case I (Exterior blowup): This is the easiest case. Suppose that the last wire
$w_{k}$ is inserted into the diagram as a consequence of an exterior blowup so
that $w_{k}$ lies below all the wires and has no “interaction” with the other
wires. Recall that $w_{k}$ carries a free marked point $x^{k}_{k-1}$ which is
placed geometrically to the left of all the previous marked points in the
diagram.

Consider a fixed embedding of $D_{k-1}\subset D_{k}$, where $D_{k-1}$ includes
the top $k-1$ holes in $D_{k}$. In other words $D_{k}$ is obtained from
$D_{k-1}$ by inserting an extra hole, named $H^{k}_{k}$ by our conventions, at
the bottom. Under this embedding, for any $1\leq t\leq k-2$, the convex curve
$\delta(x^{k-1}_{t})$ in $D_{k-1}$ can be identified with the convex curve
$\delta(x^{k}_{t})$ in $D_{k}$, since $x^{k}_{t}=x^{k-1}_{t}$ in the new
diagram. Similarly, $\Delta^{k}_{t}=\Delta^{k-1}_{t}$ for any $1\leq t\leq
k-2$, under this embedding and hence it follows that for $1\leq s\leq k-1$ and
$1\leq r\leq k-2$ we have $\gamma_{s,r}^{k}=\gamma_{s,r}^{k-1}$ and
$\Gamma_{s,r}^{k}=\Gamma_{s,r}^{k-1}$, which proves by induction, that
statements (L1), (L2) and (L3) hold for the new wiring diagram with $k$ wires,
for these cases.

Note that the convex curve $\delta(x^{k}_{k-1})$ is the curve that encloses
the last hole $H^{k}_{k}\subset D_{k}$, by definition. Therefore, for $1\leq
s\leq k-1$, the clockwise half-twist $(\Delta_{k-1}^{k})^{-1}$ has no effect
on the _convex_ curve $\gamma_{s,k-2}^{k}$ nor on the collection of arcs
$\Gamma_{s,k-2}^{k}$. Hence statements (L1), (L2) and (L3) hold for $1\leq
s\leq k-1$ and $r=k-1$ as well in the new wiring diagram with $k$ wires.

Finally, we observe that $\gamma_{k,r}^{k}=\gamma_{k}^{k}$ is convex for each
$r$, hence (L1) and (L2) automatically hold for $s=k$. Also,
$\rho_{k}^{k}=k-1$ and
$\Gamma_{k,k-1}^{k}=(\Delta_{k-1}^{k})^{-1}(\Gamma_{k}^{k})$ has the form
shown in Figure 22, thus (L3) also holds for $s=k$.

$\gamma_{k,k-1}^{k}$$D_{k}$

Figure 22. $\Gamma_{k,k-1}^{k}$ is the collection of red arcs.

Case II (Interior blowup): Suppose that the last wire $w_{k}$ is introduced
into the diagram as a consequence of an interior blowup at the $i$th term so
that $w_{k}$ is initially right below the $(i+1)$st wire with respect to the
geometric ordering of the wires on the right-hand side of the diagram. Suppose
that this $(i+1)$st wire is $w_{j}$. Now imagine that we take a step back in
our blowup algorithm. In other words, we delete the last wire $w_{k}$ (and the
last marked point $x^{k}_{k-1}$ and the associated last twisting) from the
diagram. At the same time we remove the corresponding hole from $D_{k}$ as
follows. First we remove the $(i+2)$nd hole from $D_{k}$ to obtain the
rightmost copy of $D_{k-1}$. As we move from right to left in the wiring
diagram, every time we pass through a marked point, we have a new copy of
$D_{k-1}$ obtained by removing from $D_{k}$ the hole whose order is the same
as the local geometric order of the wire $w_{k}$. All these copies of
$D_{k-1}$ can of course be identified with the rightmost copy of $D_{k-1}$ and
we use this observation in our induction argument below, where we proceed
according to three possible cases.

Case II.A (Interior blowup, $1\leq s\leq i$): Suppose that $1\leq s\leq i$. By
hypothesis, $\gamma_{s,r}^{k-1}$ and $\Gamma_{s,r}^{k-1}$ satisfy statements
(L1), (L2) and (L3) on $D_{k-1}$ for $1\leq r\leq k-2$. Now since
$\gamma^{k-1}_{s}$ does not contain the hole $H^{k-1}_{i+1}$, by the
assumption that $1\leq s\leq i$, the image $\Psi_{r-1}^{k}(H_{i+1}^{k-1})$ is
not contained in $\gamma_{s,r}^{k-1}$ for $1\leq r\leq k-2$. Therefore, we can
insert back the hole we deleted by splitting the image
$\Psi_{r-1}^{k-1}(H_{i+1}^{k-1})$ into two adjacent holes. The new hole will
be inserted right below $H^{k-1}_{i+1}$ in the rightmost copy of $D_{k-1}$ and
it will be inserted right below or above $\Psi_{r-1}^{k-1}(H_{i+1}^{k-1})$ in
an alternating fashion every time we pass a marked point that belongs to the
intersection $w_{j}\cap w_{k}$. As a result, the superscript $k-1$ can be
promoted to $k$, meaning that the curve $\gamma_{s,r}^{k-1}$ can be viewed as
$\gamma_{s,r}^{k}$ and $\Gamma_{s,r}^{k-1}$ can be viewed as
$\Gamma_{s,r}^{k}$, since we have not modified them by the insertion of the
new hole. Hence $\gamma_{s,r}^{k}$ and $\Gamma_{s,r}^{k}$ satisfy the
statements (L1), (L2) and (L3) on $D_{k}$, for $1\leq r\leq k-2$, as well.

To finish the proof of this case, we only need to argue that
$\gamma_{s,k-1}^{k}$ and $\Gamma_{s,k-1}^{k}$ satisfy the statements (L1),
(L2) and (L3). But by the discussion above $\gamma_{s,k-2}^{k}$ is right or
left-convex and belongs to the subdisk in $D_{k}$ along which we apply
$\Delta^{-1}_{k-1}$ corresponding to the new marked point $x_{k-1}$, by our
algorithm. Therefore, it is easy to see that
$\gamma_{s,k-1}^{k}=(\Delta^{k}_{k-1})^{-1}(\gamma_{s,k-2}^{k})$ and
$\Gamma_{s,k-1}^{k}=(\Delta^{k}_{k-1})^{-1}(\Gamma_{s,k-2}^{k})$ satisfy (L1),
(L2) and (L3) as well.

Case II.B (Interior blowup, $i+2\leq s\leq k$): Suppose that $i+2\leq s\leq
k$. We check that $\gamma_{s,r}^{k}$ and $\Gamma_{s,r}^{k}$ satisfy statements
(L1), (L2) and (L3) for $1\leq r\leq k-1$. We will proceed by induction on
$r$. In the case $r<\rho_{s}^{k}$, the statements are trivial since
$\gamma_{s,r}^{k}=\gamma_{s}^{k}$ and $\Gamma_{s,r}^{k}=\Gamma_{s}^{k}$ in
this case. If $r=\rho_{s}^{k}$, then, after applying the clockwise half-twist
$(\Delta_{r}^{k})^{-1}$ to $\gamma_{s,r-1}^{k}=\gamma_{s}^{k}$ and
$\Gamma_{s,r-1}^{k}=\Gamma_{s}^{k}$, we see that $\gamma_{s,r}^{k}$ and
$\Gamma_{s,r}^{k}$ have the form given in Figure 23. Thus statements (L1),
(L2) and (L3) hold in this case also.

$\gamma_{s,\rho_{s}^{k}}^{k}$$D_{k}$

Figure 23. $\Gamma_{s,\rho_{s}^{k}}^{k}$ is the collection of red arcs.

Now suppose that statements (L1), (L2) and (L3) hold for $k-3\geq
r=p\geq\rho_{s}^{k}$. We check that the statements continue to hold for
$r=p+1$. For this first note that the convex curve $\delta(x_{p+1}^{k})$
encloses the hole $\Psi_{p}^{k}(H_{s}^{k})$ if and only if the convex curve
$\delta(x_{p+1}^{k-1})$ encloses the hole $\Psi_{p}^{k-1}(H_{s-1}^{k-1})$.
Indeed, if $s>i+2$, then the wire in geometric position $s$ on the right hand
side of the wiring diagram is wire $w_{l}$ for some $l<k$, since wire $w_{k}$
is in geometric position $i+2$ on the right hand side. If we take a step back
in our blowup algorithm, then wire $w_{l}$ will have geometric position $s-1$
on the right hand side of the wiring diagram, now with $k-1$ wires. For
$s>i+2$, the result claimed now follows from the fact that
$\delta(x_{p+1}^{m})$ encloses the hole $\Psi_{p}^{m}(H_{t}^{m})$ if and only
if the wire in geometric position $t$ on the right hand side of the diagram
passes through the marked point $x_{p+1}^{m}$. If $s=i+2$, then the wire in
geometric position $s$ on the right hand side of the wiring diagram will be
wire $w_{k}$. Since wire $w_{k}$ passes through each marked point $x_{q}^{k}$
that wire $w_{j}$ passes through for $1\leq q\leq k-2$, otherwise remaining
parallel to $w_{j}$, arguing as above, we obtain the same result for $s=i+2$.

By hypothesis of the induction on $r$, it now follows that the curve
$\gamma_{s,p}^{k}$ will be right or left-convex according to whether
$\gamma_{s-1,p}^{k-1}$ is right or left-convex. Furthermore, the holes that
$\gamma_{s,p}^{k}$ encloses can be obtained from the holes that
$\gamma_{s-1,p}^{k-1}$ encloses by splitting the hole
$\Psi_{p}^{k-1}(H_{i+1}^{k-1})$ into two adjacent holes.

In a similar way, the holes that $\delta(x_{p+1}^{k})$ encloses can be
obtained from the holes that $\delta(x_{p+1}^{k-1})$ encloses by splitting the
hole $\Psi_{p}^{k-1}(H_{i+1}^{k-1})$ into two adjacent holes. As a
consequence, the curve
$\gamma_{s,p+1}^{k}=(\Delta^{k}_{p+1})^{-1}(\gamma_{s,p}^{k})$ satisfies (L1)
and (L2) and the holes that $\gamma_{s,p+1}^{k}$ encloses can be obtained from
the holes that $\gamma_{s-1,p+1}^{k-1}$ encloses by splitting the hole
$\Psi_{p+1}^{k-1}(H_{i+1}^{k-1})$ into two adjacent holes.

We now show that $\Gamma_{s,p+1}^{k}$ satisfies (L3) by considering the cases
that $\delta(x_{p+1}^{k})$ encloses and does not enclose the hole
$\Psi_{p}^{k}(H_{s}^{k})$ separately. First suppose that $\delta(x_{p+1}^{k})$
does not enclose the hole $\Psi_{p}^{k}(H_{s}^{k})$. Then
$\delta(x_{p+1}^{k-1})$ does not enclose the hole
$\Psi_{p}^{k-1}(H_{s-1}^{k-1})$. Assume that $\gamma_{s-1,p}^{k-1}$ is right-
convex; the case that $\gamma_{s-1,p}^{k-1}$ is left-convex is similar. Then
$\gamma_{s-1,p+1}^{k-1}$ is also right-convex and we have the following
possibilities for $\delta(x_{p+1}^{k-1})$:

(i) $\delta(x_{p+1}^{k-1})$ is disjoint from $\gamma_{s-1,p}^{k-1}$. In this
case $\delta(x_{p+1}^{k-1})$ cannot enclose a subset of holes from the primary
lobe of $\gamma_{s-1,p}^{k-1}$, since otherwise $\Gamma_{s-1,p+1}^{k-1}$ would
fail to satisfy (L3). It follows that $\delta(x_{p+1}^{k})$ is disjoint from
$\gamma_{s,p}^{k}$ and that it does not enclose any subset of holes from the
primary lobe of $\gamma_{s,p}^{k}$. Thus $\Gamma_{s,p+1}^{k}$ satisfies (L3).

(ii) $\delta(x_{p+1}^{k-1})$ intersects $\gamma_{s-1,p}^{k-1}$ and encloses at
least one hole below the primary lobe of $\gamma_{s-1,p}^{k-1}$. In this case
$\gamma_{s-1,p+1}^{k-1}$ would fail to be right-convex, which contradicts our
induction hypothesis. Hence this case cannot occur.

(iii) $\delta(x_{p+1}^{k-1})$ intersects $\gamma_{s-1,p}^{k-1}$ and encloses
at least one hole above the secondary lobe of $\gamma_{s-1,p}^{k-1}$. In this
case also $\gamma_{s-1,p+1}^{k-1}$ would fail to be right-convex, and hence
this case also cannot occur.

(iv) $\delta(x_{p+1}^{k-1})$ intersects $\gamma_{s-1,p}^{k-1}$ and encloses at
least one hole below the secondary lobe of $\gamma_{s-1,p}^{k-1}$. If
$\delta(x_{p+1}^{k-1})$ does not enclose all the holes contained in the
secondary lobe of $\gamma_{s-1,p}^{k-1}$, then either $\gamma_{s-1,p+1}^{k-1}$
would have more than two “lobes” or $\Psi_{r}^{k-1}(H_{s-1}^{k-1})$ would not
be the innermost hole of the primary lobe. Both of these cases contradict the
induction hypothesis hence they cannot occur. Thus the only possibility in
this case is that $\delta(x_{p+1}^{k-1})$ encloses all the holes contained in
the secondary lobe and at least one hole below it. This case is illustrated in
Figure 24(a). In this case $\delta(x_{p+1}^{k})$ will enclose all holes in the
secondary lobe of $\gamma_{s,p}^{k}$ and enclose at least one hole below it.
Hence $\Gamma_{s,p+1}^{k}$ will satisfy (L3) in this case also. This concludes
the analysis for the case $\delta(x_{p+1}^{k})$ not enclosing the hole
$\Psi_{p}^{k}(H_{s}^{k})$.

Now suppose that $\delta(x_{p+1}^{k})$ encloses the hole
$\Psi_{p}^{k}(H_{s}^{k})$. Then $\delta(x_{p+1}^{k-1})$ encloses the hole
$\Psi_{p}^{k-1}(H_{s-1}^{k-1})$. Suppose that $\gamma_{s,p}^{k}$ is right-
convex; again the left-convex case is similar. Then $\gamma_{s-1,p}^{k-1}$ is
also right-convex. By the induction hypothesis, the image of
$\gamma_{s-1,p}^{k-1}$ under the clockwise half-twist about
$\delta(x_{p+1}^{k-1})$ must be left-convex and the image of
$\Gamma_{s-1,p}^{k-1}$ must continue to satisfy (L3). In this case it can be
checked that if $\delta(x_{p+1}^{k-1})$ does not enclose
$\gamma_{s-1,p}^{k-1}$, then it must enclose all the holes of the secondary
lobe (and no holes above it) and any number of holes below
$\Psi_{p}^{k-1}(H_{s-1}^{k-1})$; this situation is illustrated in Figure
24(b). Since, by splitting vertically the hole $\Psi_{p}^{k-1}(H_{i+1}^{k-1})$
into two adjacent holes, we obtain $\gamma_{s,p}^{k}$ and
$\delta(x_{p+1}^{k})$ from $\gamma_{s-1,p}^{k-1}$ and $\delta(x_{p+1}^{k-1})$,
respectively, it follows that $\gamma_{s,p}^{k}$ and $\delta(x_{p+1}^{k})$
will have the same form as $\gamma_{s-1,p}^{k-1}$ and $\delta(x_{p+1}^{k-1})$,
that is, $\delta(x_{p+1}^{k})$ will enclose $\gamma_{s,p}^{k}$ or it will
enclose all the holes of the secondary lobe (and no holes above it) and any
number of holes below $\Psi_{p}^{k}(H_{s}^{k})$. It is now clear that, after
applying applying a clockwise half-twist about $\delta(x_{p+1}^{k})$ to
$\Gamma_{s,p}^{k}$, $\Gamma_{s,p+1}^{k}$ will continue to satisfy (L3).

We have thus checked that $\gamma_{s,r}^{k}$ and $\Gamma_{s,r}^{k}$ satisfy
(L1), (L2) and (L3) for $r\leq k-2$. We now check that they satisfy (L1), (L2)
and (L3) for $r=k-1$ also.

(a)(b)$D_{k}$$D_{k}$$\delta(x_{p+1}^{k-1})$$\delta(x_{p+1}^{k-1})$

Figure 24. Some possibilities for $\delta(x_{p+1}^{k-1})$.

As $\gamma_{s-1,k-2}^{k-1}$ will be in the left-most copy of $D_{k-1}$, by
Lemma 13, all the holes enclosed by $\gamma_{s-1,k-2}^{k-1}$ will be adjacent
and the hole $\Psi_{k-2}^{k-1}(H_{s-1}^{k-1})$ will be either at the top or
the bottom of these holes. Being one-sided convex (by the induction
hypothesis), the curve $\gamma_{s-1,k-2}^{k-1}$ must be convex and hence the
pair $(\gamma_{s-1,k-2}^{k-1},\Gamma_{s-1,k-2}^{k-1})$ must be as in Figure
25. Thus the curve $\gamma_{s,k-2}^{k}$ must also be convex as the holes it
encloses are obtained from the holes that $\gamma_{s-1,k-2}^{k-1}$ encloses by
splitting $\Psi_{k-2}^{k-1}(H_{i+1}^{k-1})$ into two adjacent holes. As the
holes enclosed by $\gamma_{s,k-1}^{k}$ must be adjacent with the hole
$\Psi_{k-1}^{k}(H_{s}^{k})$ at one end (again, by Lemma 13), it follows that
$\delta(x_{k-1}^{k})$ must be disjoint from $\gamma_{s,k-2}^{k}$. Hence the
curve $\gamma_{s,k-1}^{k}$ will be convex and statements (L1) and (L2) will
hold for $r=k-1$ also.

To see that (L3) also holds for $r=k-1$, first suppose that $s>i+2$. Then
$\Psi_{k-2}^{k}(H_{s}^{k})$ will already be at the top or bottom of the holes
enclosed by $\gamma_{s,k-2}^{k}$. As the last marked point $x_{k-1}^{k}$ only
involves the wires having the geometric positions $\\{t\,|\,1\leq t\leq
i+2,t\neq i+1\\}$ on the right hand side, the last clockwise half-twist
$(\Delta_{k-1}^{k})^{-1}$ will be about a subdisk that does not enclose
$\Psi_{k-2}^{k}(H_{s}^{k})$. It easily follows that $\Gamma_{s,k-1}^{k}$ will
satisfy (L3).

Now suppose that $s=i+2$. Then the hole $\Psi_{k-2}^{k}(H_{s}^{k})$ will be
either one below the top hole or one above the bottom hole of
$\gamma_{s,k-2}^{k}$. The last clockwise half-twist will be about a subdisk
that encloses all the holes of $\gamma_{s,k-2}^{k}$ except the hole
$\Psi_{k-2}^{k}(H_{i+1}^{k})$, which will be at the top or the bottom. Again
it follow that $\Gamma_{s,k-1}^{k}$ will satisfy (L3). This completes the
proof of Case II.B.

$\gamma_{s-1,k-2}^{k-1}$$\gamma_{s-1,k-2}^{k-1}$$D_{k-1}$$D_{k-1}$

Figure 25. One arc of $\Gamma_{s-1,k-2}^{k-1}$ is shown in red.

Case II.C (Interior blowup, $s=i+1$): Suppose that $s=i+1$. By the proof of
Case II.B, we know that $\gamma_{i+2,r}^{k}$ and $\Gamma_{i+2,r}^{k}$ satisfy
(L1), (L2) and (L3) for $1\leq r\leq k-1$. Note that for $1\leq r\leq k-2$,
the holes $\Psi_{r}^{k}(H_{i+1}^{k})$ and $\Psi_{r}^{k}(H_{i+2}^{k})$ will be
adjacent, since the wires $w_{j}$ and $w_{k}$, having geometric positions
$i+1$ and $i+2$, respectively, on the right hand side of the diagram, will
remain consecutive up to the marked point $x_{k-1}$. Thus hole
$\Psi_{r}^{k}(H_{i+1}^{k})$ will be in the primary lobe of
$\gamma_{i+2,r}^{k}$ for $1\leq r\leq k-2$. It is now clear that
$\gamma_{i+1,r}^{k}$ is given by isotoping $\gamma_{i+2,r}^{k}$ over the hole
$\Psi_{r}^{k}(H_{i+2}^{k})$ from the side dictated by $\Gamma_{i+2,r}^{k}$ for
$1\leq r\leq k-2$; see Figure 26. It follows that statements (L1), (L2) and
(L3) hold for $\gamma_{i+1,r}^{k}$ and $\Gamma_{i+1,r}^{k}$ for $1\leq r\leq
k-2$.

$\gamma_{i+2,r}^{k}$$\gamma_{i+2,r}^{k}$$\gamma_{i+1,r}^{k}$$\gamma_{i+1,r}^{k}$

Figure 26. The case when $1\leq r\leq k-2$.

For $r=k-1$, we argue as follows: Since $\gamma_{i+1,k-2}^{k-1}$ will be
convex with the hole $\Psi_{k-2}^{k-1}(H_{i+1}^{k-1})$ at one end and the two
holes $\Psi_{r}^{k}(H_{i+1}^{k})$ and $\Psi_{r}^{k}(H_{i+2}^{k})$ will be
adjacent and both in the primary lobe of $\gamma_{i+2,k-2}^{k}$, the curve
$\gamma_{i+1,k-2}^{k}$ must have one of the two forms given in Figure 27. It
easily follows that $\gamma_{i+1,k-1}^{k}$ and $\Gamma_{i+1,k-1}^{k}$ satisfy
statements (L1), (L2) and (L3). This completes the proof of Case II.C. ∎

$\gamma_{i+2,k-2}^{k}$$\gamma_{i+2,k-2}^{k}$$D_{k}$$D_{k}$$\gamma_{i+1,k-2}^{k}$$\gamma_{i+1,k-2}^{k}$

Figure 27. The case when $r=k-1$.

####  5.2.2. The case of $\alpha$-curves:

We reformulate our claim in Proposition 18 (b) about $\alpha$-curves as
Proposition 26 below, where have we replaced
$\mathcal{W}_{\textbf{n}}(\textbf{m})$ by $\mathcal{W}_{\textbf{n}}$, since
the extension from $\mathcal{W}_{\textbf{n}}$ to
$\mathcal{W}_{\textbf{n}}(\textbf{m})$ is irrelevant. Recall that
$\overline{\alpha}$ denotes the mirror image of a given curve $\alpha$. In the
following, we will decorate each curve with a superscript to indicate the
number of holes in the disk in which they are embedded. For example, we will
use $\alpha^{k}_{t}$ to indicate that we are talking about the curve
$\alpha_{t}$ (see Section 3.2.1) in $D_{k}$. Similarly, we will decorate each
marked point in a wiring diagram with a superscript to indicate the number of
wires in the diagram.

######  Proposition 26.

Let $\mathcal{W}_{\textbf{n}}$ be a wiring diagram with $k\geq 2$ wires and
$k-1$ marked points $x^{k}_{k-1},\ldots,x^{k}_{1}$ (reading from left to
right) constructed as in Definition 17 with respect to some blowup sequence.
Then, for each $1\leq t\leq k-1$, the vanishing cycle $V(x^{k}_{t})$
associated to the marked point $x^{k}_{t}$ via the method of Plamenevskaya and
Starkston, is the mirror image $\overline{\alpha}^{k}_{t}$ of the curve
${\alpha}^{k}_{t}$ obtained by the stabilization algorithm described in
Section 3.2.1 with respect to the same blowup sequence.

###### Proof.

According to Definition 8, $V(x^{k}_{1})=\delta(x^{k}_{1})$ and
$V(x^{k}_{t})=\Delta^{k}_{1}\circ\cdots\circ\Delta^{k}_{t-1}(\delta(x^{k}_{t}))$
for $2\leq t\leq k-1$. Therefore, we need to verify that
$\delta(x^{k}_{1})={\alpha}^{k}_{1}$ (both curves are convex and
$\overline{\alpha}^{k}_{1}={\alpha}^{k}_{1}$) and
$\Delta^{k}_{1}\circ\cdots\circ\Delta^{k}_{t-1}(\delta(x^{k}_{t}))=\overline{\alpha}^{k}_{t},$
for $2\leq t\leq k-1$.

For $k=2$, the wiring diagram has only two parallel wires and one marked point
$x^{2}_{1}$ at the bottom wire, corresponding to the initial blowup
$(0)\to(1,1)$. The statement holds for this case since
$\delta(x^{2}_{1})={\alpha}^{2}_{1}$. Now suppose that $k\geq 3$ and the
statement holds for any wiring diagram with $k-1$ wires and $k-2$ marked
points, constructed as in Definition 17 with respect to some blowup sequence.
We will prove that the statement holds for any wiring diagram with $k$ wires
and $k-1$ marked points obtained by inserting one more wire and a marked point
corresponding to the new blowup.

Case I (Exterior blowup): This is the easiest case. Suppose that the new wire
$w_{k}$ is inserted to the diagram with $k-1$ wires as a consequence of an
exterior blowup so that $w_{k}$ lies below all the wires and has no
“interaction” with the other wires. Recall that $w_{k}$ carries a free marked
point $x^{k}_{k-1}$ which is placed geometrically to the left of all the
previous marked points in the diagram.

Consider a fixed embedding of $D_{k-1}\subset D_{k}$, where $D_{k-1}$ includes
the top $k-1$ holes in $D_{k}$. In other words $D_{k}$ is obtained from
$D_{k-1}$ by inserting an extra hole, named $H^{k}_{k}$ by our conventions, at
the bottom. It is clear that for any $1\leq t\leq k-2$, the convex curve
$\delta(x^{k-1}_{t})\subset D_{k-1}$ can be identified with the convex curve
$\delta(x^{k}_{t})\subset D_{k}$ and hence $\Delta^{k}_{t}=\Delta^{k-1}_{t}$
for any $1\leq t\leq k-2$, under this embedding. Similarly, for any $1\leq
t\leq k-2$, ${\alpha}^{k}_{t}$ can be identified with ${\alpha}^{k-1}_{t}$, by
our stabilization algorithm in Section 3.2.1.

By induction, the property we want to verify holds for the wiring diagram with
$k-1$ wires before we insert $w_{k}$, i.e., we have
$\delta(x^{k-1}_{1})={\alpha}^{k-1}_{1}$ and
$\Delta^{k-1}_{1}\circ\cdots\circ\Delta^{k-1}_{t-1}(\delta(x^{k-1}_{t}))=\overline{\alpha}^{k-1}_{t},$
for $2\leq t\leq k-2$. Under the embedding above, these can be upgraded to
$\delta(x^{k}_{1})={\alpha}^{k}_{1}$ and
$\Delta^{k}_{1}\circ\cdots\circ\Delta^{k}_{t-1}(\delta(x^{k}_{t}))=\overline{\alpha}^{k}_{t},$
for $2\leq t\leq k-2$, by simply replacing the superscript $k-1$ with $k$. The
key point here is that the composition
$\Delta^{k}_{1}\circ\cdots\circ\Delta^{k}_{t-1}$ takes place in the fixed
embedded disk $D_{k-1}\subset D_{k}$.

To finish the proof of this case, we only have to verify the statement for
$t=k-1$. Note that $\delta(x^{k}_{k-1})=\alpha^{k}_{k-1}$, which by
definition, is the convex curve that encloses the last hole $H^{k}_{k}\subset
D_{k}$. We observe that

$\Delta^{k}_{1}\circ\cdots\circ\Delta^{k}_{k-2}(\delta(x^{k}_{k-1}))=\alpha^{k}_{k-1}$

so that the “last” vanishing cycle is
$\alpha^{k}_{k-1}=\overline{\alpha}^{k}_{k-1}$, which is consistent with our
stabilization algorithm in Section 3.2.1.

Case II (Interior blowup): Now, suppose that the new wire $w_{k}$ is
introduced into the diagram with $k-1$ wires as a consequence of an interior
blowup at the $i$th term so that $w_{k}$ is initially right below the
$(i+1)$st wire with respect to the geometric ordering of the wires on the
right-hand side of the diagram. Suppose this $(i+1)$st wire is $w_{j}$. Our
proof below splits into two subcases: the case where $1\leq t\leq k-2$, and
the case $t=k-1$.

Case II.A (Interior blowup, $1\leq t\leq k-2$): Now, imagine that we take a
step back in our blowup algorithm. In other words, we delete the last wire
$w_{k}$ (and the last marked point $x^{k}_{k-1}$ and the associated last
twisting) from the diagram. At the same time we remove the corresponding hole
from $D_{k}$ as follows. First we remove the $(i+2)$nd hole from $D_{k}$ to
obtain the rightmost copy of $D_{k-1}$. As we move from right to left in the
wiring diagram, every time we pass through a marked point, we have a new copy
of $D_{k-1}$ obtained by removing from $D_{k}$ the hole whose order is the
same as the local geometric order of the wire $w_{k}$. All these copies of
$D_{k-1}$ can of course be identified with the rightmost copy of $D_{k-1}$, in
which by induction, we have

$V(x^{k-1}_{t}):=\Delta^{k-1}_{1}\circ\cdots\circ\Delta^{k-1}_{t-1}(\delta(x^{k-1}_{t}))=\overline{\alpha}^{k-1}_{t}$

for $2\leq t\leq k-2$ and $\delta(x^{k-1}_{1})={\alpha}^{k-1}_{1}$. We claim
that we can upgrade the superscript $k-1$ to $k$ in the previous sentence by
splitting the $(i+1)$st hole in (the rightmost copy of) $D_{k-1}$, to create a
new hole right below it, and hence identifying the new disk as $D_{k}$ and
promoting the all the relevant curves from $D_{k-1}$ to $D_{k}$. Note that
this is precisely how ${\alpha}^{k}_{t}$ is obtained from ${\alpha}^{k-1}_{t}$
by the stabilization algorithm in Section 3.2.1. We just need to show that
$V(x^{k-1}_{t})$ can be promoted to $V(x^{k}_{t})$ in the same manner, for
each $1\leq t\leq k-2$. This is easy to see for $t=1$ since the convex curve
$\delta(x^{k-1}_{1})$ can be promoted to the convex curve $\delta(x^{k}_{1})$
by inserting a hole in $D_{k-1}$ either enclosed by $\delta(x^{k}_{1})$ or not
depending on whether $x^{k}_{1}$ (which is in fact the same marked point
$x^{k-1}_{1}$) belongs to $w_{j}$ or not.

For $2\leq t\leq k-2$, we argue as follows. In the aforementioned copies of
$D_{k-1}$, there is a hole whose order is the same as the local geometric
order of the wire $w_{j}$. Since, by the blowup algorithm, the wires $w_{j}$
and $w_{k}$ are geometrically consecutive throughout the diagram (except when
they intersect) the corresponding two holes will be adjacent in $D_{k}$,
interchanging their relative order at each intersection point of $w_{j}$ and
$w_{k}$ (which is indeed a marked point in the diagram). Moreover, for each
$1\leq t\leq k-2$, the convex curve $\delta(x^{k}_{t})$ assigned to
$x^{k}_{t}$ will either enclose both or neither of these holes, depending on
whether $x^{k}_{t}$ belongs to $w_{j}$ or not. Therefore, for each $2\leq
t\leq k-2$, any one of the curves

$\delta(x^{k}_{t}),\;\Delta^{k}_{t-1}(\delta(x^{k}_{t})),\;\ldots,\;\Delta^{k}_{1}\circ\cdots\circ\Delta^{k}_{t-1}(\delta(x^{k}_{t}))$

obtained iteratively by applying counterclockwise half-twists starting from
the convex curve $\delta(x^{k}_{t})$, will either enclose both or neither of
these two holes. This implies that we could just upgrade the curves

$\delta(x^{k-1}_{t}),\;\Delta^{k-1}_{t-1}(\delta(x^{k-1}_{t})),\;\ldots,\;\Delta^{k-1}_{1}\circ\cdots\circ\Delta^{k-1}_{t-1}(\delta(x^{k-1}_{t}))$

from $D_{k-1}$ to $D_{k}$ by inserting a new hole (corresponding to the local
geometric position of the new wire $w_{k}$) in each copy of $D_{k-1}$. Note
that the new hole corresponding to the new wire $w_{k}$ can be viewed as being
obtained by splitting the hole corresponding to $w_{j}$ at each step. We
observe that this new hole will appear right below the $(i+1)$st hole in the
rightmost copy of $D_{k-1}$ giving $D_{k}$, and these two holes in question
are enumerated as $H^{k}_{i+1}$ and $H^{k}_{i+2}$ in the rightmost copy of
$D_{k}$. This “splitting” of the $(i+1)$st hole is the crux of the matter in
our stabilization algorithm in Section 3.2.1.

The upshot of this discussion is that for $1\leq t\leq k-2$, the vanishing
cycle $V(x^{k}_{t})$ is the curve $\overline{\alpha}^{k}_{t}$, which is in
fact nothing but $\overline{\alpha}^{k-1}_{t}$ upgraded to $D_{k}$ from
$D_{k-1}$, in a manner consistent with our stabilization algorithm in Section
3.2.1. Finally, the case $t=k-1$ will be treated separately below.

Case II.B (Interior blowup, $t=k-1$): To finish the proof of Proposition 26,
we just have to verify that the “last” vanishing cycle $V(x^{k}_{k-1})$ is the
curve $\overline{\alpha}^{k}_{k-1}$. Equivalently, we need to verify that
$\Psi^{k}_{k-2}(\overline{\alpha}^{k}_{k-1})=\delta(x^{k}_{k-1})$, where
$\Psi^{k}_{k-2}=(\Delta_{k-2}^{k})^{-1}\circ\cdots\circ(\Delta_{1}^{k})^{-1}$,
by Definition 23. Here, instead of induction, we will rather use the statement
in Proposition 18 (a) for $\gamma_{i+1}$, which we already proved.

The standing assumption in Case II is that the new wire $w_{k}$ is introduced
into the diagram with $k-1$ wires as a consequence of an interior blowup at
the $i$th term so that $w_{k}$ is initially right below the $(i+1)$st wire
with respect to the geometric ordering of the wires on the right-hand side of
the diagram. Suppose that this $(i+1)$st wire, labelled $w_{j}$, has an odd
number of marked points. The case with an even number of points is very
similar and is left to the interested reader.

On the left in Figure 28, we depict the convex curves $\delta(y^{k}_{i+1})$
and $\delta(x^{k}_{k-1})$. This configuration can be deduced from the proof of
Lemma 13 for the case when $w_{j}$ has an odd number of marked points. For
$s=i+1$, Proposition 18 (a) implies that

$\delta(y^{k}_{i+1})=\Psi^{k}_{k-1}(\gamma^{k}_{i+1})=(\Delta^{k}_{k-1})^{-1}\Psi^{k}_{k-2}(\gamma^{k}_{i+1})$

and we set
$(\gamma^{k}_{i+1})^{\prime}=\Delta^{k}_{k-1}(\delta(y^{k}_{i+1}))=\Psi^{k}_{k-2}(\gamma^{k}_{i+1})$,
which is depicted on the right in Figure 28.

$\delta(y^{k}_{i+1})$$\Delta^{k}_{k-1}$$(\gamma^{k}_{i+1})^{\prime}$$\delta(x^{k}_{k-1})$$\beta^{k}_{\psi(i+2)}$$D_{k}$$D_{k}$

Figure 28. The curve
$(\gamma^{k}_{i+1})^{\prime}=\Delta^{k}_{k-1}(\delta(y^{k}_{i+1}))$.

Let $\beta^{k}_{i+1}$ be the convex curve in Figure 29 enclosing the holes
$H^{k}_{i+1}$ and $H^{k}_{i+2}$ and let $\Delta^{k}_{\beta_{i+1}}$ denote the
counterclockwise half-twist in the subdisk bounded by $\beta^{k}_{i+1}$. Note
that in $D_{k}$, we have
$\overline{\alpha}^{k}_{k-1}=\Delta^{k}_{\beta_{i+1}}(\gamma^{k}_{i+1})$ as
one can see from Figure 29.

$\overline{\alpha}^{k}_{k-1}$$\gamma^{k}_{i+1}$$\beta^{k}_{i+1}$$D_{k}$$H^{k}_{i+2}$

Figure 29. The curves $\gamma^{k}_{i+1}$, $\beta^{k}_{i+1}$, and
$\overline{\alpha}^{k}_{k-1}$ in $D_{k}$.

Hence we have

(1)
$\Psi^{k}_{k-2}(\overline{\alpha}^{k}_{k-1})=\Psi^{k}_{k-2}\circ\Delta^{k}_{\beta_{i+1}}(\gamma^{k}_{i+1})=\Delta^{k}_{\beta_{\psi(i+2)}}\circ\Psi^{k}_{k-2}(\gamma^{k}_{i+1})=\Delta^{k}_{\beta_{\psi(i+2)}}((\gamma^{k}_{i+1})^{\prime}),$

where $\beta_{\psi(i+2)}^{k}$ is defined as follows: The image of the curve
$\beta^{k}_{i+1}$ under $\Psi^{k}_{k-2}$ is the convex curve
$\beta^{k}_{\psi(i+2)}$ (depicted on the right in Figure 28) enclosing two
adjacent holes whose geometric orders are given as
$\\{\psi(i+2),\psi(i+2)+1\\}$. Here $\Delta^{k}_{\beta_{\psi(i+2)}}$ denotes
the counterclockwise half-twist in the subdisk bounded by
$\beta^{k}_{\psi(i+2)}$.

Note that the order of the purple coloured hole depicted in the copy of
$D_{k}$ carrying $(\gamma^{k}_{i+1})^{\prime}$ in Figure 28 is $\psi(i+2)$,
and the blue hole right below it has order $\psi(i+2)+1$. In fact, $\psi(i+2)$
is the local geometric order of the wire $w_{k}$, and $\psi(i+2)+1$ is the
local geometric order of the wire $w_{j}$, before the last twisting. Recall
that these two wires are always geometrically consecutive and they swap their
order each time they intersect.

In the second equality in (1) above, we used the fact that

$\Psi^{k}_{k-2}\circ\Delta^{k}_{\beta_{i+1}}\circ(\Psi^{k}_{k-2})^{-1}=\Delta^{k}_{\beta_{\psi(i+2)}}$

which can be easily verified.

Finally, applying $\Delta^{k}_{\beta_{\psi(i+2)}}$ (on the subdisk bounded by
$\beta^{k}_{\psi(i+2)}$ enclosing only the blue and the purple holes) to
$(\gamma^{k}_{i+1})^{\prime}$ in Figure 28 we get the dashed curve
$\delta(x^{k}_{k-1})$. Therefore, coupled with (1), we conclude that
$\Psi^{k}_{k-2}(\overline{\alpha}^{k}_{k-1})=\delta(x^{k}_{k-1})$. ∎

###  5.3. Proofs of the corollaries and some examples

###### Proof of Corollary 2.

According to Theorem 1, for each Stein filling $W_{p,q}(\textbf{n})$ of the
contact $3$-manifold $(L(p,q),\xi_{can})$, there is an (unbraided) wiring
diagram $\mathcal{W}_{\textbf{n}}({\textbf{m}})$, constructed as in Definition
17, based on the blowup sequence $(0)\to(1,1)\to\cdots\to\textbf{n}$ and
$\textbf{m}=\textbf{b}-\textbf{n}$, whose associated planar Lefschetz
fibration $f\colon W\to D^{2}$ obtained by the method of Plamenevskaya and
Starkston [10, Section 5.2] is equivalent to the one constructed by the
authors [2]. Moreover, using [10, Proposition 5.5], the wiring diagram
$\mathcal{W}_{\textbf{n}}({\textbf{m}})$, viewed as a collection of
intersecting curves in ${\mathbb{R}}\times{\mathbb{C}}$, can be extended to an
explicit collection of symplectic graphical disks
$\Gamma_{1},\ldots,\Gamma_{n}$ in ${\mathbb{C}}\times{\mathbb{C}}$, with
marked points $p_{1},\ldots,p_{m}\in\bigcup_{i}\Gamma_{i}$ including all the
intersection points of these disks. Note that we can assume that each
intersection of these disks is positive and transverse, and non-intersection
points are allowed as free marked points. Furthermore, by [10, Proposition
5.6], the Stein filling $W_{p,q}(\textbf{n})$ is supported by the restriction
of the Lefchetz fibration

$\pi_{x}\circ\Pi\colon{\mathbb{C}}^{2}\\#m\hbox{$\overline{{\mathbb{C}}P^{2}}$}\setminus(\widetilde{\Gamma}_{1}\cup\cdots\cup\widetilde{\Gamma}_{n})\to{\mathbb{C}}$

to a Milnor ball (to get compact fibers), where
$\pi_{x}\colon{\mathbb{C}}^{2}\to{\mathbb{C}}$ denotes the projection onto the
first coordinate, $\widetilde{\Gamma}_{1},\ldots,\widetilde{\Gamma}_{n}$ are
the proper transforms of $\Gamma_{1},\ldots,\Gamma_{n}$, and $\Pi$ is the
blowup map. Finally, the Lefschetz fibrations $f$ and $\pi_{x}\circ\Pi$ are
equivalent by the discussion in [10, Section 5.4]. Note that the last
statement in Corollary 2 follows immediately from [10, Proposition 5.8]. ∎

###### Proof of Corollary 3.

Let $W_{p,q}(\textbf{n})$ be the Stein filling of $(L(p,q),\xi_{can})$, and
let $\mathcal{W}_{\textbf{n}}({\textbf{m}})$ be the (unbraided) wiring diagram
constructed as in Definition 17 based on the blowup sequence
$(0)\to(1,1)\to\cdots\to\textbf{n}$ and $\textbf{m}=\textbf{b}-\textbf{n}$.
Let $\Gamma=\Gamma_{1}\cup\cdots\cup\Gamma_{n}$ be the collection of
symplectic graphical disks, with marked points $p_{1},\ldots,p_{m}\in\Gamma$
as in Corollary 2. Note that the incidence matrix
$\mathcal{I}_{\textbf{n}}({\textbf{m}}):=\mathcal{I}(\Gamma,\\{p_{j}\\})$ can
be read off from the wiring diagram $\mathcal{W}_{\textbf{n}}({\textbf{m}})$,
where each wire $w_{i}$ is identified with the disk $\Gamma_{i}$ and the $x$\-
and $y$-type marked points are identified with $p_{1},\ldots,p_{m}$.

To compute the incidence matrix $\mathcal{I}_{\textbf{n}}({\textbf{m}})$, we
enumerate the wires from top to bottom with respect to their geometric order
on the right-hand side of the diagram
$\mathcal{W}_{\textbf{n}}({\textbf{m}})$, and we enumerate the $x$\- and
$y$-type marked points using their natural geometric order from right to left.
The matrix $\mathcal{I}_{\textbf{n}}({\textbf{m}})$ can be viewed as an
extension of the incidence matrix $\mathcal{I}_{\textbf{n}}$ corresponding to
the wiring diagram $\mathcal{W}_{\textbf{n}}$, which carries only the $x$-type
marked points. In the following, first we inductively construct
$\mathcal{I}_{\textbf{n}}$ depending on the blowup sequence
$(0)\to(1,1)\to\cdots\to\textbf{n}$, starting from the matrix

$\blockarray{cc}&x_{1}\\\ \block{c(c)}w_{1}0\\\ w_{2}1\\\ $

corresponding to the wiring diagram with two wires $w_{1},w_{2}$ and one
marked point $x_{1}$, obtained by the initial blowup $(0)\to(1,1)$. Suppose
that we have constructed the $r\times(r-1)$ incidence matrix for a wiring
diagram with $r$ wires and $r-1$ marked points $x_{1},\ldots,x_{r-1}$. Assume
that we insert the next wire $w_{r+1}$ into the diagram according to an
exterior blowup. Then we “blowup the incidence matrix” at hand by adding a
last row and a last column which consist of all zeros except a $1$ at the
lower right corner, to obtain the $(r+1)\times r$ incidence matrix.

Now assume that the last wire $w_{r+1}$ is inserted into the diagram according
to an interior blowup at the $i$th term. Then we blowup the incidence matrix
at hand by inserting a new row below the $(i+1)$st row and a last column so
that the new row is copied from the row above and the last column is of the
form

$[\underbrace{1,\ldots,1}_{i},0,1,\underbrace{0,\ldots,0}_{\geq 0}]^{T}$

to obtain the $(r+1)\times r$ incidence matrix.

The matrix $\mathcal{I}_{\textbf{n}}({\textbf{m}})$ can be obtained from
$\mathcal{I}_{\textbf{n}}$ in a standard way based only on the $k$-tuple m. To
extend $\mathcal{I}_{\textbf{n}}$ to $\mathcal{I}_{\textbf{n}}({\textbf{m}})$,
for each $1\leq s\leq k$, we just insert a column labelled with $y_{s}$, so
that the first $s$ entries from the top of the column labelled with $y_{s}$ is
$1$, and the rest are $0$. If $y_{s}$ has multiplicity $m_{s}$, then we repeat
$m_{s}$-times the column labelled with $y_{s}$. ∎

Here we illustrate the proof of Corollary 3 for the wiring diagram in Figure
14. The incidence matrix $\mathcal{I}_{\textbf{n}}$ can be obtained
algorithmically from the blowup sequence
$(1,1)\to(1,2,1)\to(2,1,3,1)\to(2,1,4,1,2)=\textbf{n}$ used to construct
$\mathcal{W}_{\textbf{n}}$ as follows.

$\blockarray{cc}&x_{1}\\\ \block{c(c)}w_{1}0\\\ w_{2}1\\\
\;\xrightarrow[\text{blowup}]{\text{exterior}}\blockarray{ccc}&x_{1}x_{2}\\\
\block{c(cc)}w_{1}0{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bf
0}\\\
w_{2}1{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bf
0}\\\
w_{3}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bf
0}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bf 1}\\\
\;\xrightarrow[\text{blowup}]{\text{interior}}\blockarray{cccc}&x_{1}x_{2}x_{3}\\\
\block{c(ccc)}w_{1}00{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bf
1}\\\
w_{2}10{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bf
0}\\\
w_{4}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bf
1}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bf
0}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bf 1}\\\
w_{3}01{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bf
0}\\\
\;\xrightarrow[\text{blowup}]{\text{interior}}\blockarray{ccccc}&x_{1}x_{2}x_{3}x_{4}\\\
\block{c(cccc)}w_{1}001{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bf
1}\\\
w_{2}100{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bf
1}\\\
w_{4}101{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bf
1}\\\
w_{3}010{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bf
0}\\\
w_{5}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bf
0}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bf
1}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bf
0}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bf 1}\\\
$

The first arrow above corresponds to an exterior blowup, where we insert the
last row and the last column which has a $1$ in the corner and $0$ everywhere
else. The second arrow corresponds to an interior blowup at the first term,
and we insert the third row and the last column so that the first two entries
in the third row are copied form the row above and the entries in the last
column are $1,0,1,0$ from the top. The last arrow corresponds to an interior
blowup at the third term, and we insert the fifth row and the last column so
that the first two entries in the third row are copied form the row above and
the entries in the last column are $1,1,1,0,1$ from the top.

Let $\textbf{m}=(0,1,1,1,1)$. To extend the $5\times 4$ incidence matrix
$\mathcal{I}_{\textbf{n}}$ for $\mathcal{W}_{\textbf{n}}$ to the $5\times 8$
incidence matrix $\mathcal{I}_{\textbf{n}}({\textbf{m}})$ for
$\mathcal{W}_{\textbf{n}}(\textbf{m})$, we insert the columns labelled with
$y_{2},y_{3},y_{4},y_{5}$ where the first $s$ entries from the top of the
column labelled with $y_{s}$ is $1$, and the rest are $0$.

$\blockarray{ccccccccc}&x_{1}x_{2}x_{3}x_{4}y_{2}y_{3}y_{4}y_{5}\\\
\block{c(cccc|cccc)}w_{1}00111111\\\ w_{2}10011111\\\ w_{4}10110111\\\
w_{3}01000011\\\ w_{5}01010001\\\ =\mathcal{I}_{\textbf{n}}({\textbf{m}})$

###### Proof of Corollary 4.

A matrix $M$ with $r\geq 2$ rows

$v_{i}=(v_{ij}),\quad v_{ij}\in\\{0,1\\}$

is called a CQS matrix (see [5, Definition 6.8]) if $\langle
v_{i},v_{j}\rangle=\langle v_{i},v_{i}\rangle-1$ holds for all $1\leq i<j\leq
r$, where $\langle\cdot,\cdot\rangle$ denotes the standard inner product, and
CQS stands for cyclic quotient singularity.

Let $X$ be a cyclic quotient singularity whose singularity link is
$(L(p,q),\xi_{can})$. According to [5, Theorem 6.18], there is a bijection
between the smoothing components of $X$ and incidence matrices of the picture
deformations of the decorated curve $(\mathcal{C},l)$ with smooth branches
representing $X$. Moreover, the incidence matrices are in one-to-one
correspondence with CQS matrices, up to permutations of the columns.

Let $\mathcal{I}^{R}_{\textbf{n}}(\textbf{m})$ be the matrix obtained from
$\mathcal{I}_{\textbf{n}}(\textbf{m})$ by reversing the order of the rows. As
in [5, Lemma 6.11], one can check that
$\mathcal{I}^{R}_{\textbf{n}}(\textbf{m})$ is a CQS matrix and that there is a
bijection between CQS matrices and the set of matrices of the form
$\mathcal{I}^{R}_{\textbf{n}}(\textbf{m})$. The reason that we had to reverse
the order of the rows of the matrix $\mathcal{I}_{\textbf{n}}(\textbf{m})$ is
simply because in the present paper, to construct
$\mathcal{I}_{\textbf{n}}(\textbf{m})$, we enumerated the wires in
$\mathcal{W}_{\textbf{n}}(\textbf{m})$ from top to bottom, as opposed to
bottom to top, with respect to their geometric order on the right-hand side of
the diagram.

Finally, we give an explicit one-to-one correspondence between the Stein
fillings of the contact singularity link $(L(p,q),\xi_{can})$ and the Milnor
fibers of the associated cyclic quotient singularity. Let
$W_{p,q}(\textbf{n})$ be the Stein filling given by Lisca obtained by the
blowup sequence $(0)\to(1,1)\to\cdots\to\textbf{n}$. Using the same blowup
sequence and $\textbf{m}=\textbf{b}-\textbf{n}$, we can construct a CQS matrix
$\mathcal{I}^{R}_{\textbf{n}}(\textbf{m})$ as in the proof of Corollary 3,
which corresponds to a picture deformation, and hence gives a Milnor fiber as
in [5]. The Stein filling $W_{p,q}(\textbf{n})$ is diffeomorphic to the Milnor
fiber, because while the wiring diagram
$\mathcal{W}_{\textbf{n}}({\textbf{m}})$ determines $W_{p,q}(\textbf{n})$, the
picture deformation, which is in the same combinatorial equivalence class as
the configuration of symplectic graphical disks arising from
$\mathcal{W}_{\textbf{n}}({\textbf{m}})$, determines the Milnor fiber.

Conversely, for any Milnor fiber, which is obtained from a picture deformation
whose incidence matrix is a CQS matrix, one can read off the pair
$(\textbf{n},\textbf{m})$ as in [5, Proposition 6.12], and therefore construct
the wiring diagram $\mathcal{W}_{\textbf{n}}({\textbf{m}})$, so that the
configuration of symplectic graphical disks arising from
$\mathcal{W}_{\textbf{n}}({\textbf{m}})$, is in the same combinatorial
equivalence class as the picture deformation. ∎

###### Proof of Corollary 5.

It is well-known that for any contact singularity link $(L(p,q),\xi_{can})$,
the Milnor fiber of the Artin smoothing component of the corresponding cyclic
quotient singularity gives the Stein filling $W_{(p,q)}((1,2,\ldots,2,1))$,
which is Stein deformation equivalent to the one obtained by deforming the
symplectic structure on the minimal resolution of the singularity; see [3].

We describe a decorated germ $(\mathcal{C},l)$ associated to the pair of
coprime integers $p>q\geq 1$ such that $(\mathcal{C},l)$ determines the cyclic
quotient singularity with link $L(p,q)$ via the construction of de Jong and
van Straten [5]. We follow the description given in [8]. Suppose that
$\frac{p}{q}=[a_{1},\ldots,a_{m}]$. Let $G$ be the decorated linear graph
having $m$ vertices $v_{1},\ldots,v_{m}$ with the vertex $v_{i}$ weighted by
the integer $-a_{i}$. Then $G$ is the dual resolution graph of a cyclic
quotient singularity with link $L(p,q)$. Let $G^{\prime}$ be the simple graph
obtained from $G$ by attaching $a_{1}-1$ new vertices to $v_{1}$ and $a_{i}-2$
new vertices to each vertex $v_{i}$ for $i>1$. Assign weight $-1$ to each new
vertex. Finally, let $G^{\prime\prime}$ denote the graph obtained from
$G^{\prime\prime}$ by endowing an arrowhead to each new vertex. Then
$G^{\prime\prime}$ is an embedded resolution graph of a germ of a plane curve
$\mathcal{C}=\bigcup_{i}C_{i}$ with _smooth_ irreducible components
corresponding to the arrowheads in $G^{\prime\prime}$. The order of
intersection of $C_{i}$ and $C_{j}$ corresponding to two distinct arrows is
the number of vertices on the intersection of the geodesics between the arrows
and $v_{m}$. Also $l=(l_{i})$, where the weight $l_{i}$ is the number of
vertices on the geodesic from the arrowhead corresponding to $C_{i}$ to
$v_{m}$.

Now every continued fraction expansion $[c_{1},\ldots,c_{m}]$, with $c_{i}\geq
2$ for all $i$, can be obtained from $[2]$ by repeated applications of the
following two operations:

  1. (i)

$[a_{1},\ldots,a_{r}]\mapsto[a_{1},\ldots,a_{r}+1]$,

  2. (ii)

$[a_{1},\ldots,a_{r}]\mapsto[a_{1},\ldots,a_{r},2]$.

Let $[b_{1},\ldots,b_{s}]$ denote the dual string of $[a_{1},\ldots,a_{r}]$,
i.e. if $\frac{p}{q}=[a_{1},\ldots,a_{r}]$, then
$\frac{p}{p-q}=[b_{1},\ldots,b_{s}]$. Then the dual string changes for each of
these two operations in the following way:

  1. (i)

$[b_{1},\ldots,b_{s}]\mapsto[b_{1},\ldots,b_{s},2]$,

  2. (ii)

$[b_{1},\ldots,b_{s}]\mapsto[b_{1},\ldots,b_{s}+1]$.

This follow immediately from a consideration of Riemenschneider’s point
diagrams. We check that if the statement of the corollary holds for a pair
$(p,q)$ with $\frac{p}{q}=[a_{1},\ldots,a_{r}]$, then it continues to hold if
we replace $(p,q)$ by $(p^{\prime},q^{\prime})$, where
$\frac{p^{\prime}}{q^{\prime}}=[a_{1},\ldots,a_{r}+1]$ or
$\frac{p^{\prime}}{q^{\prime}}=[a_{1},\ldots,a_{r},2]$. As the statement holds
trivially if $\frac{p}{q}=[2]$, by induction, it will follow that the
statement holds for every pair of coprime integers $p>q\geq 1$.

Suppose first that $\frac{p^{\prime}}{q^{\prime}}=[a_{1},\ldots,a_{r}+1]$.
Then we have
$\frac{p^{\prime}}{p^{\prime}-q^{\prime}}=[b_{1},\ldots,b_{s},2]$, where
$[b_{1},\ldots,b_{s}]=\frac{p}{p-q}$. Notice that in the wiring diagram for
the planar Lefschetz fibration

$W_{(p,q)}((1,2,\ldots,2,1))\to D^{2}$

constructed according to Theorem 1, there is a final marked point $y_{s}$
through which all the wires pass. It is easy to see that that the wiring
diagram for the planar Lefschetz fibration
$W_{(p^{\prime},q^{\prime})}((1,2,\ldots,2,1))\to D^{2}$ is given by taking
the wiring diagram for the planar Lefschetz fibration
$W_{(p,q)}((1,2,\ldots,2,1))\to D^{2}$ and inserting a new wire $w_{s+1}$ at
the bottom of the diagram on the right hand side, which runs under the wiring
diagram up to the point that it passes through the marked point $y_{s}$
(turning it into a $y_{s+1}$-type marked point) and then runs above the
diagram. In addition, a single new marked point is placed on the new wire
$w_{s+1}$.

Let $(\mathcal{C},l)$ denote the decorated curve associated to the pair of
integers $(p,q)$ constructed as above. Then $\mathcal{C}$ will consist of $s$
curvettas by the induction hypothesis. Let $(\mathcal{C}^{\prime},l^{\prime})$
denote the decorated curve associated to the pair $(p^{\prime},q^{\prime})$.
Then it is easy to see that $\mathcal{C}^{\prime}$ can be obtained from
$\mathcal{C}$ by adding an extra curvetta $C_{s+1}$ which has intersection
multiplicity $1$ with each curvetta $C_{i},i\leq s$. In addition, $l^{\prime}$
is given by setting $l^{\prime}_{i}=l_{i}$ for $i\leq s$ and $l_{s+1}=2$.
Since the Scott deformation of $(\mathcal{C},l)$ is combinatorially equivalent
to the symplectic disk arrangement associated to the wiring diagram for
$W_{(p,q)}((1,2,\ldots,2,1))$ (by the induction hypothesis), it is easy to see
that the Scott deformation of $(\mathcal{C}^{\prime},l^{\prime})$ is
combinatorially equivalent to the symplectic disk arrangement associated to
the wiring diagram for $W_{(p^{\prime},q^{\prime})}((1,2,\ldots,2,1))$.

Now suppose that $\frac{p^{\prime}}{q^{\prime}}=[a_{1},\ldots,a_{r},2]$. Then
we have $\frac{p^{\prime}}{p^{\prime}-q^{\prime}}=[b_{1},\ldots,b_{s}+1]$. In
this case, the wiring diagram associated to
$W_{(p^{\prime},q^{\prime})}((1,2,\ldots,2,1))$ can be obtained from the
wiring diagram associated to $W_{(p,q)}((1,2,\ldots,2,1))$ by adding one new
simultaneous intersection point $y_{s}$ of all the wires. Also the decorated
curve $(\mathcal{C}^{\prime},l^{\prime})$ associated to the pair
$(p^{\prime},q^{\prime})$ can be obtained from the decorated curve
$(\mathcal{C},l)$ by increasing by $1$ the order of intersection between each
pair of distinct curvettas $C_{i}$ and $C_{j}$ and by increasing each $l_{i}$
by $1$. Again it is easy to check that the Scott deformation of
$(\mathcal{C}^{\prime},l^{\prime})$ is combinatorially equivalent to the
symplectic disk arrangement associated to the wiring diagram for
$W_{(p^{\prime},q^{\prime})}((1,2,\ldots,2,1))$. ∎

In the following we give an example to illustrate the proof of Corollary 5.
Note that the Stein filling $W_{(p,q)}((1,2,\ldots,2,1))$ is obtained via the
sequence of exterior blowups

$(0)\to(1,1)\to(1,2,1)\to\cdots\to(1,2,\ldots,2,1),$

and the corresponding unbraided wiring diagram can be drawn easily. The wiring
diagram corresponding to $W_{(56,17)}((1,2,2,2,1))$, for instance, is depicted
in Figure 30.

$y_{3}$$y_{3}$$y_{3}$$y_{5}$$y_{5}$$x_{3}$$x_{2}$$x_{1}$$x_{4}$$y_{1}$$w_{1}$$w_{2}$$w_{3}$$w_{4}$$w_{5}$

Figure 30. Wiring diagram for the Stein filling $W_{(56,17)}((1,2,2,2,1))$—the
symplectic resolution.

$-2$$-2$$-4$$-4$$-2$$-2$$-2$$-2$$-1$$-2$$-2$$-1$$-2$$-1$$\boxed{2}$$\boxed{2}$$\boxed{2}$$\boxed{3}$$\boxed{3}$$\boxed{3}$$\boxed{4}$$\boxed{4}$$\boxed{4}$$\boxed{6}$$\boxed{6}$$\boxed{6}$$\boxed{3}$$\boxed{3}$$-1$$-1$$-1$$-1$$-1$352$-4$$-2$$-2$$-4$$-2$$-4$$-2$$-2$$-4$$-2$dual
resolution graphScottdeformation$-1$$-1$$-1$$-1$$-1$

Figure 31. From dual resolution graph of a cyclic quotient singularity to the
Scott deformation of the decorated plane curve representing the singularity.

Next, we show that the arrangement of symplectic disks arising from the wiring
diagram in Figure 30 is combinatorially equivalent to the Scott deformation of
$(\mathcal{C},l)$. First we note that $\frac{56}{17}=[4,2,2,4,2]$ and
construct the curve $(\mathcal{C},l)$, as depicted in Figure 31.

Note that $(\mathcal{C},l)$ consists of five curvettas with indicated weights
and tangencies using the notation in [10], where boxed numbers indicate the
weights, circled numbers indicate the tangencies, and the other numbers are
the self-intersection numbers of the rational curves.

We also depicted the Scott deformation of $(\mathcal{C},l)$ at the bottom in
Figure 31. It follows that, after enumerating the smooth branches of the Scott
deformation of $(\mathcal{C},l)$ from top to bottom as they appear on the
right-hand side, the incidence matrix arising from the Scott deformation
coincides with the incidence matrix arising from the wiring diagram in Figure
30. Note that each branch of the Scott deformation $(\mathcal{C},l)$ includes
a free marked point. The $5\times 10$ incidence matrix
$\mathcal{I}_{Artin}:=\mathcal{I}_{\textbf{n}}(\textbf{m})$, where
$\textbf{n}=(1,2,2,2,1)$ and $\textbf{m}=(1,0,3,0,2)$, is given as follows.

$\blockarray{cccccccccccc}&x_{1}x_{2}x_{3}x_{4}y_{1}y_{3}y_{3}y_{3}y_{5}y_{5}\\\
\block{c(cccc|ccccccc)}w_{1}0000111111\\\ w_{2}1000011111\\\
w_{3}0100011111\\\ w_{4}0010000011\\\ w_{5}0001000011\\\ =\mathcal{I}_{Artin}$

Finally, one can easily obtain the disjoint vanishing cycles in $D_{5}$ of the
planar Lefschetz fibration $W_{(56,17)}((1,2,2,2,1))\to D^{2}$, which we
depicted in Figure 32. As observed in [10], this Lefschetz fibration agrees
with the Lefschetz fibration constructed by Gay and Mark [6], using the
plumbing description $W_{(56,17)}((1,2,2,2,1))$ given by the dual resolution
graph.

Figure 32. The vanishing cycles of the planar Lefschetz fibration
$W_{(56,17)}((1,2,2,2,1))\to D^{2}$.

Note that the total monodromy of the planar Lefschetz fibration
$W_{(56,17)}((1,2,2,2,1))\to D^{2}$, the product of Dehn twists along the
disjoint vanishing cycles depicted in Figure 32, is the monodromy of the open
book compatible with $(L(56,17),\xi_{can})$. This monodromy has another
positive factorization

$D(\gamma_{5})D(\gamma_{4})D(\gamma_{3})D(\gamma_{2})D(\overline{\alpha}_{4})D(\overline{\alpha}_{3})D(\overline{\alpha}_{2})D(\overline{\alpha}_{1}),$

which is the total monodromy of the planar Lefschetz fibration
$W_{(56,17)}((2,1,4,1,2))\to D^{2}$ we discussed in Section 5.1.

## References

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