# Multisections with divides and Weinstein 4-manifolds

Gabriel Islambouli  and  Laura Starkston

LS was supported by NSF 1904074, NSF 2042345, and a Sloan research fellowship.

##  1\. Introduction

There are various methods to diagrammatically encode a $4$-dimensional
manifold, each of which is based on a decomposition theorem which breaks up
the manifold into simple pieces such that the diagram encodes the way these
pieces glue together (e.g. handle decompositions, Lefschetz
pencils/fibrations). The most recent such method is the development of
trisections by Gay and Kirby [GayKir16], generalized to multisections in
[IslNay20]. Symplectic structures have played a key role in 4-dimensional
topology, due to connections with gauge theory. Compatibility between
symplectic topology with handle decompositions arose from Weinstein’s
construction [Wei91] and in the $4$-dimensional case was diagrammatically
encoded by a Legendrian surgery diagram by Gompf [Gompf98]. Similarly
compatibility between symplectic manifolds and Lefschetz pencils and
fibrations was established [Don99, GompfStipsicz, LoiPie01], so a symplectic
manifold can be encoded by the fiber and base surfaces, pencil points, and
ordered vanishing cycles. A notion of compatibility between trisections and
symplectic manifolds was proposed in [LamMei18] and shown to exist in
[LamMeiSta20], but this compatibility did not yield a simple diagrammatic way
to encode a symplectic structure (rather it was motivated by attempts to
obtain genus bounds).

In this article we define a stronger compatibility between a multisection and
a symplectic structure, which can be diagrammatically encoded by collections
of curves on a surface. In addition to the diagrammatic data of the smooth
multisection, we keep track of another multi-curve representing the _dividing
set_ of convex surfaces in contact manifolds. Thus, we call our decomposition
of a symplectic manifold a _multisection with divides_. Our main result is
that every $4$-dimensional Weinstein domain admits a multisection with
divides.

######  Definition 1.1.

A multisection with divides of a symplectic filling $(W,\omega)$ with contact
boundary $(\partial W,\xi)$ is a decomposition $W=W_{1}\cup\cdots\cup W_{n}$,
such that

  * •

$W_{i}\cong\natural_{k_{i}}S^{1}\times D^{3}$.

  * •

$W_{i}\cap W_{i+1}=:H_{i+1}\cong\natural_{g}S^{1}\times D^{2}$ for
$i=1,\dots,n-1$.

  * •

$\Sigma:=W_{1}\cap\cdots\cap W_{n}=\partial H_{i}$ for all $i$

  * •

Each $(W_{i},\omega|_{W_{i}})$ is a symplectic filling of $(\partial
W_{i},\xi_{i})$.

  * •

$H_{i}\cup H_{i+1}$ is a contact Heegaard splitting of $(\partial
W_{i},\xi_{i})$

  * •

$H_{1}\cup H_{n+1}$ is a contact Heegaard splitting of $(\partial W,\xi)$

A _bisection with divides_ is a multisection with divides with $n=2$.

The advantage of using contact Heegaard splittings is that the handlebodies
each carry a standard positive _and_ a standard negative contact structure,
which are contactomorphic. This is one of the key ideas which we use to make
multisections compatible with symplectic and contact geometry.

######  Remark 1.2.

In the fourth bullet point of Definition 1.1, we ask for
$(W_{i},\omega|_{W_{i}})$ to be a symplectic filling. Note that in our setting
weak, strong, Liouville, and Weinstein fillability are all equivalent, because
there is a unique weak symplectic filling of $\\#_{k_{i}}S^{1}\times S^{2}$ up
to symplectic deformation, and this filling is actually Weinstein (thus strong
and Liouville) [NW].

Figure 1. Left: A schematic of a bisection with divides. Each $W_{i}$ is a
4-dimensional Weinstein 1-handlebody and each $H_{i}$ is a 3-dimensional
1-handlebody obtained as a neighbourhood of a Legendrian graph. Right: A
bisection diagram with divides of the unit cotangent bundle on $S^{2}$. The
red, blue, and green curves represent curves bounding compressing disks in the
respective handlebodies and the purple curves are the dividing set for the
surface.

An essential feature of these multisections with divides is that they can be
encoded as a sequence of cut systems together with a fixed dividing set on a
closed surface. An example of such a diagrammatic representation together with
a schematic of what this encodes can be seen in Figure 1.

We are able to encode symplectic geometric data diagrammatically because our
multisection with divides decompositions are geometrically restrictive by
asking each Heegaard splitting to be a _contact Heegaard splitting_ (see
section 2.1 for the definition). A typical multisection of a symplectic
manifold would be unlikely to satisfy this condition, even if it were a
“Weinstein multisection” as in [LamMei18, LamMeiSta20]. Therefore, it is
surprising that these geometrically restrictive multisection decompositions
actually exist quite generally. Our main theorem is the following.

######  Theorem.

Every compact 4-dimensional Weinstein domain admits a multisection with
divides.

We give two proofs of this theorem each with distinct advantages. Both proofs
also yield algorithmic methods to produce a diagram for the multisection with
divides. The first proof takes as input a Kirby-Weinstein diagram, and
produces a bisection with divides. The disadvantage of this algorithm is that
the core surface will generally have very high genus, which is typically
highly inefficient. On the other hand, the output only has two sectors,
instead of arbitrarily many.

The second proof takes as input a positive allowable Lefschetz fibration
(PALF) and produces a multisection with divides. In this case, the genus is
more controlled, being determined by the topology of the fiber of the PALF,
however there may be many sectors (potentially one for each Lefschetz
singularity). More specifically we prove the following.

######  Theorem.

Let $f:W^{4}\to D^{2}$ be a PALF whose regular fiber is a genus $g$ surface
with $b$ boundary components and $n$ singular fibers. Then $W^{4}$ admits a
genus $2g+b-1$ $n$-section with divides.

One can compare these results and Definition 1.1 to the definition of
Weinstein trisection for closed symplectic manifolds in [LamMei18,
LamMeiSta20]. The main difference is that those prior definitions do not
require any compatibility between the contact structure induced on the
boundary of each sector and the Heegaard splitting of the boundary induced by
the trisection. As a consequence, there is not easy diagrammatic data that
encodes the contact and symplectic topology in these prior definitions. (The
most likely candidate for such diagrammatic data is a weighted foliation for
each handlebody, but the data of a weighted foliation is not discrete.) By
contrast, in our more restrictive notion of multisection with divides, the
symplectic and contact geometry can be diagrammatically encoded by a single
dividing set on the core surface.

######  Remark 1.3.

Smooth multisections are compatible with both closed manifolds and manifolds
with boundary. In this article, we have given the definition of a multisection
for divides in the case that our symplectic manifold has contact boundary. The
definition naturally extends to closed manifolds. Although in this article we
establish the existence of our decompositions for Weinstein domains rather
than closed symplectic manifolds, this is a key step towards establishing
existence of the analogous multisections with divides for all closed
symplectic manifolds via results of Donaldson [Don96] and Giroux [Gir02,
Gir17].

The monodromy of a Lefschetz fibration is a product of right handed Dehn
twists. In general, there can be multiple ways to write the same mapping class
element as a product of right handed Dehn twists. Swapping out one of these
with another is called a _monodromy substitution_. A number of important
symplectic cut and paste operations like rational blow-down can be seen as a
monodromy substitution operation on a Lefschetz fibration [EndGur10] [EMV11].
By tracking the change induced by a monodromy substitution on a PALF through
our algorithm, we are able to realize these cut and paste operations on
multisections with divides. In Figure 15, we demonstrate this explicitly for
the monodromy substitution coming from the lantern relation, which induces a
$C_{2}$-rational blowdown.

We conclude the paper with a classification of genus-1 multisections with
divides. The smooth genus-1 multisections with boundary were previously
classified in [IKLM] to exist if and only if the manifold is a linear
plumbings of 2-spheres. Our requirement that the multisections be compatible
with genus-1 contact Heegaard splittings restricts this significantly more, as
in the following theorem.

######  Theorem.

Genus-1 $n$-sections with divides correspond to plumbings of $n-1$ disk
bundles over 2-spheres, each of Euler number $-2$.

The organization of this paper is as follows. Section 2 discusses the way we
will ask contact structures to be compatible with Heegaard splittings and
$3$-dimensional handlebodies. Section 3 gives our first proof of our main
theorem, showing how to turn a Kirby-Weinstein handlebody diagram into a
multisection with divides. Section 4 gives the second proof of our main
theorem, showing how to turn a PALF into a multisection with divides. Section
5 classifies genus-1 multisections with divides and shows how multisection
diagrams with divides can be stabilized to increase the genus of the surface.
Finally, in Section 6 we discuss some questions for future research.

##  2\. Contact geometry and Heegaard splittings

In this section, we explain the compatibility condition between contact
structures and Heegaard splittings which we will require on the boundary of
each sector. We also give a diagrammatic formulation of this compatibility. We
begin with some background on surfaces in contact manifolds.

A surface $\Sigma$ embedded in a contact 3-manifold $(Y,\xi)$ is said to be
convex if there exists a contact vector field $v$ for $(Y,\xi)$ such that $v$
is transverse to $\Sigma$. Convex surfaces are generic, meaning every smoothly
embedded surface has a $C^{\infty}$-small isotopy to a convex surface [Gir91,
Hon00]. Given a contact vector field $v$ transverse to $\Sigma$ we obtain a
multicurve called the dividing set, denoted $\Gamma_{\Sigma}$. This multicurve
is defined by $\Gamma_{\Sigma}=\\{x\in\Sigma|v_{x}\subset\xi\\}$ and the
isotopy class of this curve is an invariant of the embedding of $\Sigma$ up to
isotopy through convex surfaces.

The dividing set captures all of the contact geometric information of $\Sigma$
in a neighbourhood $\Sigma\times[-\epsilon,\epsilon]$ obtained by flowing by
the contact vector field. More precisely we have the Giroux flexibility
theorem.

######  Theorem 2.1.

([Gir91]) Let $\Sigma$ be a closed orientable surface, and
$f_{0}:\Sigma\to(Y,\xi)$ and $g:\Sigma\to(Y^{\prime},\xi^{\prime})$ be convex
embeddings of $\Sigma$. Suppose $v$ is a contact vector field transverse to
$(Y,\xi)$. If the oriented multicurves $f_{0}^{-1}(\Gamma_{f_{0}(\Sigma)})$
and $g^{-1}(\Gamma_{g(\Sigma)})$ are isotopic, then there exists an isotopy
$f_{t}$ for $t\in[0,1]$ such that
$f_{1}^{*}(\xi|_{f_{1}(\Sigma)})=g^{*}(\xi|_{f_{1}(\Sigma)})$.

Given a handlebody, $H$, a spine for $H$ is a graph, $G$, such that $H$
retracts onto $G$. If $H$ additionally carries a contact structure, then a
spine, $G$, is said to be a Legendrian spine if each edge is a Legendrian arc
or knot. By combining Darboux’s theorem with the standard neighborhood for
Legendrians [Geiges, Theorem 2.5.8], we see that Legendrian spines have a
standard tight contact neighborhood, determined by the ribbon neighborhood of
the spine tangent to the contact planes.

###  2.1. Contact Heegaard splittings

######  Definition 2.2.

A _contact Heegaard splitting_ of a contact manifold $(Y,\xi)$ is a Heegaard
splitting $Y=H_{1}\cup_{\Sigma}H_{2}$ such that $H_{1}$ and $H_{2}$ are
contactomorphic to standard neighborhoods of Legendrian spines $L_{1}$ and
$L_{2}$.

We will call the handlebodies which are standard neighborhoods of Legendrian
graphs _standard contact handlebodies_. Note that a smooth handlebody of a
given genus typically has multiple different “standard” contact structures,
which are differentiated by the number of components of the dividing set on
the boundary. The different options come from the fact that there are
generally multiple different surfaces with boundary whose doubles are a genus
$g$ surface (determined by the number of boundary components).

This notion of contact Heegaard splittings originated with Giroux [Giroux] and
was also developed by Torisu [Tor00], and can be equivalently formulated as
follows.

######  Lemma 2.3 ([Giroux, Tor00]).

Let $H_{1}\cup_{\Sigma}H_{2}$ be a Heegaard splitting of $(Y,\xi)$ with
$\Sigma$ a convex surface. The following are equivalent

  1. (1)

$H_{1}$ and $H_{2}$ are standard neighborhoods of Legendrian graphs

  2. (2)

$H_{1}$ and $H_{2}$ are two halves of an open book decomposition supporting
$(Y,\xi)$

  3. (3)

For each handlebody $H_{i}$, there exists a set of compression disks cutting
$H_{i}$ into a ball, such that the boundary of each compression disk
intersects the dividing set on $\Sigma$ in exactly two points.

Though these equivalences are known, we review how to pass between them. If we
start with $H_{1}$ and $H_{2}$ as standard neighborhoods of Legendrian graphs,
we can see the page $F$ of the corresponding open book decomposition as a
contact framed ribbon of the Legendrian. The contact planes are tangent to $F$
along the Legendrian, so in a standard neighborhood, $d\alpha$ is a positive
area form when restricted to this page. Moreover, in the standard model, we
can identify $H_{i}=F\times I/\sim$ (where $(x,t)\sim(x,t^{\prime})$ for
$x\in\partial F$), such that $d\alpha$ is positive on each $F\times\\{t\\}$.
In this way, we see that the first characterization gives rise to the second
characterization. Conversely, given a supporting open book decomposition, we
can Legendrian realize a spine on two pages. Restricting the contact structure
defined on an abstract open book to half of the pages, we see that this is a
standard neighborhood of this Legendrian spine. The dividing set on the
boundary of a standard neighborhood of a Legendrian graph is precisely the
intersection of the contact framed ribbon with $\Sigma=\partial H_{i}$. The
meridian of each edge of the graph thus intersects the dividing set in exactly
two points. Similarly, if $H_{1}$ and $H_{2}$ are two halves of an open book
decomposition, the dividing set on $\Sigma=\partial H_{1}=\partial H_{2}$ is
the binding of the open book $\partial F$. A collection of compressing disks
for $F\times I/\sim$ is given by $a_{j}\times I/\sim$ where $\\{a_{j}\\}$ is a
collection of arcs on $F$ which cut $F$ into a disk. Each $a_{j}\times I/\sim$
intersects the binding in exactly two points (the endpoints of $a_{j}$ in
$\partial F$). If we have a Heegaard splitting with compression disks for each
handlebody which each intersect the dividing set in two points, we can cut
along these compressing disks to get a ball with the unique tight contact
structure. Reversing the cuts amounts to attaching standard contact
$1$-handles which gives a standard neighborhood of a Legendrian graph.

There is a fundamental challenge in obtaining compatibility between a
multisection and a symplectic structure. Each interior $3$-dimensional
handlebody in a multisection appears in the boundary of two $4$-dimensional
sectors, but the boundary orientations are opposite to each other. Viewing a
sector as a symplectic filling of its boundary induces a contact structure on
the boundary which is a _positive_ contact structure with respect to the
_boundary_ orientation. The sign of a contact structure (with respect to a
fixed orientation) is an inherent property of the contact planes which
measures the direction/handedness of the twisting of the contact planes. Note,
this is not the same as the co-orientation of the contact structure which
depends only on the contact form, not the contact structure. Therefore, we
cannot have identical contact structures on a fixed manifold realize both
positive and negative contact structures with respect to a fixed orientation.
In general this suggests that we would need two different contact structures
on each interior $H_{i}$ of a multisection. However, as we show in the
following lemma, there are both positive and negative _standard contact
handlebodies_ which are orientation reversing contactomorphic to each other.
Both the positive and negative contact structures are supported by the same
half open book.

######  Lemma 2.4.

Let $F$ be a surface with boundary. Let $\eta$ be a $1$-form on $F$ which
evaluates positively on the oriented boundary of $F$ such that $d\eta$ is an
area form on $F$. Consider $F\times I/\sim$ where $(x,t)\sim(x,t^{\prime})$
whenever $x\in\partial F$. Using $t$ as the coordinate parametrizing the $I$
direction, let

$\alpha^{\pm}=\pm dt+\eta.$

Then $\xi^{+}:=\ker(\alpha^{+})$ (respectively $\xi^{-}:=\ker(\alpha^{-})$) is
a positive (respectively negative) contact structure on $H$ supported by the
trivial half open book on $F\times I/\sim$ (with pages $F\times\\{t\\}$).

###### Proof.

$\xi^{\pm}$ is a $\pm$ contact structure because

$(\pm dt+\eta)\wedge d(\pm dt+\eta)=\pm dt\wedge d\eta$

is a $\pm$ volume form on $F\times I$. Note that $\pm dt+\eta$ is independent
of $t$ so the form is well-defined on the quotient $F\times I/\sim$.

To check that it is supported by the open book, we need to verify that
$\alpha^{\pm}$ evaluates positively on the binding and $d\alpha^{\pm}$ is a
positive area form on the pages. Indeed if $T$ is a vector positively tangent
to $\partial F$, $\alpha^{\pm}(T)=\eta(T)>0$, and $d\alpha^{\pm}=d\eta$ is a
positive area form on $F$ by assumption. ∎

######  Remark 2.5.

Note, it is always possible to find such a $1$-form $\eta$ on a surface $F$
with non-empty boundary. Also, observe the orientation reversing
diffeomorphism which sends $t\in I$ to $-t$ takes $\alpha^{+}$ to
$\alpha^{-}$.

###  2.2. Contact Heegaard diagrams

Motivated by the third characterization in Lemma 2.3, we define a diagrammatic
version of contact Heegaard splittings.

######  Definition 2.6.

A contact Heegaard diagram is a quadruple $(\Sigma,d,C_{1},C_{2})$ such that:

  * •

$\Sigma$ is a closed oriented surface.

  * •

$d$ is a multicurve which separates $\Sigma$ into two homeomorphic surfaces
with boundary.

  * •

$C_{i}$ $i\in\\{1,2\\}$ is a cut system for $\Sigma$ such that each curve
intersects $d$ twice.

Given a contact Heegaard diagram, we can reconstruct a contact manifold
together with a contact Heegaard splitting. In particular, if $F$ is one of
the halves of $\Sigma\setminus d$, then we endow each of the handlebodies the
standard contact structures coming from $F\times I$. This induces an open book
with page $F$ and binding $d$, which in turn induces a contact structure.
Conversely, every contact Heegaard splitting has a contact Heegaard diagram by
characterization (3) of Lemma 2.3.

This correspondence allows us to give a classification of genus-1 contact
Heegaard splittings of $S^{3}$ as we will see in Section 5.1. In particular,
by Euler characteristic considerations, genus-1 contact Heegaard splittings
correspond to open book decompositions of $S^{3}$ with an annular page. The
monodromy then consists or a left-handed or right handed Dehn twist about the
core of this annulus. The right handed Dehn twist gives the tight contact
structure on $S^{3}$ whereas the left handed Dehn twist gives an overtwisted
structure. These lead to the Heegaard diagrams shown in Figure 2.

Figure 2. The two genus-1 convex Heegaard diagrams of $S^{3}$. The one on the
left is tight whereas the one on the right is overtwisted.

###  2.3. Dividing sets on standard neighborhoods from Legendrian front
projections

In Section 3, we will be looking at Heegaard splittings of $S^{3}$, or more
generally $\\#_{k_{1}}S^{1}\times S^{2}$ where one handlebody is a standard
neighborhood of a Legendrian graph $\widetilde{\Lambda}$ described via a front
projection, and the other handlebody is the complement. In order to verify
that the complement is a standard contact handlebody, it will be useful to
know exactly how to draw the dividing set on the boundary of the standard
neighborhood of an explicitly embedded Legendrian graph in terms of the front
projection.

We will mainly focus on trivalent Legendrian graphs. Higher valence vertices
can be split into trivalent vertices by growing additional Legendrian edges
via a Legendrian deformation which preserves the standard neighborhood and
thus, the dividing set on its boundary. Given a front projection representing
a Legendrian embedding in $\mathbb{R}^{3}$, we can draw the corresponding
dividing set on the boundary of a standard neighborhood. Recall that the
dividing set on the boundary of a standard contact handlebody is the boundary
of the page of the compatible open book decomposition. A page of the open book
is given by the contact framed ribbon of the Legendrian knot. Considering how
the contact framing wraps around at left and right cusps and at left and right
trivalent vertices, we obtain the local models for the dividing set as shown
in Figure 3. The first five models cover the generic front projections of a
trivalent Legendrian graph. The last model includes a Legendrian arc which
degenerately projects to a single point, whose two end points are trivalent
vertices. We can isotope this model to a generic front projection as in Figure
4, and thus derive its local model from the previous models. We include this
last “compound” model for convenience as we will use it extensively in
implementing our algorithm of Section 3.1.

Figure 3. Left: Local models for a trivalent Legendrian graph. Right: Local
models for the dividing set of the boundary of a regular neighbourhood of the
graph. To obtain the last model from the previous ones, see Figure 4.

##  3\. Kirby-Weinstein handlebody diagrams and multisections with divides

In this section we will show how to use a Kirby-Weinstein handlebody diagram
to produce a bisection with divides. A consequence of our proof is an
algorithm to obtain a multisection diagram with divides (defined in Section
3.2) from a Kirby-Weinstein diagram.

###  3.1. Existence of bisections with divides from Kirby-Weinstein handlebody
diagrams

######  Theorem 3.1.

Every compact $4$-dimensional Weinstein domain admits a bisection with
divides.

###### Proof.

By definition, a Weinstein $4$-manifold has a Weinstein handle structure. By
[Gompf98], this handle structure can be represented in a standard form by a
Legendrian front projection with $1$-handles (which we will call a Kirby-
Weinstein diagram). We will give an algorithm to convert a Kirby-Weinstein
diagram in Gompf standard form into a multisection diagram with divides. If we
ignore the symplectic and contact structure, the smooth part of this algorithm
essentially follows those in [GayKir16] and [MeiSchZup16] used for converting
a Kirby diagram into a trisection.

The union of the $0$\- and $1$-handles will be $W_{1}$. This is diffeomorphic
to $\natural_{k_{i}}S^{1}\times D^{3}$ and with the Weinstein structure of $W$
restricted to $W_{1}$, it is a Weinstein filling of its boundary $(\partial
W_{1},\xi_{1})$.

We will construct a contact Heegaard splitting $H_{1}\cup_{\Sigma}H_{2}$ of
$(\partial W_{1},\xi_{1})$ such that the Legendrian attaching spheres for the
$2$-handles of $W$ are a subset of the Legendrian core of $H_{2}$. Then we
will define $W_{2}$ to be a collar of $H_{2}$ together with the $2$-handles of
$W$. Because the attaching spheres of the $2$-handles are a subset of the
Legendrian core of $H_{2}$, we will see that $W_{2}$ will be diffeomorphic to
$\natural_{k_{2}}S^{1}\times D^{3}$. There is a naturally induced Heegaard
splitting of $\partial W_{2}$ given by $\overline{H}_{2}\cup H_{3}$ where
$H_{3}$ is obtained from $H_{2}$ by doing Legendrian surgery on the attaching
spheres of the $2$-handles. We will then show that this is also a contact
Heegaard splitting, and that $(W_{2},\omega|_{W_{2}})$ is a symplectic filling
of this contact manifold.

Let $\Lambda$ be the Legendrian attaching link for the $2$-handles of $W$ in
$\\#_{k_{1}}S^{1}\times S^{2}$. To construct the appropriate contact Heegaard
splitting $H_{1}\cup H_{2}$ of $(\partial W_{1},\xi_{1})$, we will add
Legendrian tunnels to $\Lambda$, yielding a Legendrian graph
$\widetilde{\Lambda}$ containing $\Lambda$. A standard contact neighborhood of
$\widetilde{\Lambda}$ will be $H_{2}$ and its complement will be $H_{1}$. The
first purpose of the tunnels is to ensure that the complement of the
neighborhood of $\widetilde{\Lambda}$ is a handlebody. Additional tunnels will
be added to ensure this handlebody has the standard contact structure.

The construction of $\widetilde{\Lambda}$ is as follows.

  1. (1)

Start with $\Lambda$ in Gompf standard form.

  2. (2)

If there is any $1$-handle of $W$ whose belt sphere is disjoint from
$\Lambda$, add a Legendrian circle which passes through that $1$-handle once.

  3. (3)

For each $1$-handle add Legendrian arcs to connect all the strands that pass
through that $1$-handle on the left and right as shown in Figure 6.

  4. (4)

At each crossing in the diagram, add a Legendrian arc which projects to the
crossing point connecting the over- and under-strands. See Figure 4 for
generic front projections for a Legendrian isotopic graph.

  5. (5)

Add Legendrian arcs to connect disconnected components until the graph is
connected.

  6. (6)

The resulting front projection divides up the plane into regions, such that
the boundary of each region is a Legendrian unknot. Add further Legendrian
arcs to cut up each region as in Figure 5 so that at the end, each region is
bounded by a Legendrian unknot with $tb=-1$. Namely, every region in the front
projection should have a unique “right cusp” and a unique “left cusp” where a
vertex is a right (resp. left) cusp of a region if the two edges on the
boundary of the region which meet in the vertex both approach the vertex from
the left (resp. right). Note that here we can treat each crossing as a single
vertex by shrinking the Legendrian arc connecting the two strands by a
Legendrian deformation.

Figure 4. Adding a tunnel at each crossing which projects to the crossing
point (top center) is Legendrian isotopic to the generic front projection
shown on the top right. The dividing set on the boundary of a neighborhood can
be determined using the standard models to obtain the lower right picture, and
then isotoped to obtain the lower center picture. Figure 5. We first flatten
each crossing in the knot by asdding a vertical Legendrian tunnel. To make
each region a Legendrian unknot with $tb=-1$ we can further add tunnels to
regions with additional cusps. Figure 6. Adding tunnels between parallel
strands passing through the same $1$-handle allows us to locate a curve on the
boundary of the neighborhood of the Legendrian graph $\widetilde{\Lambda}$
bounding a disk on the exterior coming from the surgery induced by the
$1$-handle. Choosing the arcs in precisely the manner shown ensures that we
can find such a curve intersecting the dividing set in exactly two points. The
lower part of the figure shows the relevant portion of the Heegaard surface
with the dividing set (in purple) and the compressing curve (in red).

Now if $H_{2}$ is a standard contact neighborhood of $\widetilde{\Lambda}$,
and $H_{1}$ is the complement, we want to show that $H_{1}\cup H_{2}$ is a
contact Heegaard splitting. If there are no $1$-handles in the Kirby-Weinstein
diagram, $H_{1}$ is a smooth handlebody because we put tunnels at each
crossing, so $H_{1}$ is diffeomorphic to a $3$-ball with one handle attached
for each bounded planar region in the diagram of $\widetilde{\Lambda}$. When
there are $1$-handles in the Kirby-Weinstein diagram, there is an additional
compressing disk for each $1$-handle as seen in Figure 7.

Figure 7. Additional compressing disk for each $1$-handle.

Using the models from Figure 3, we can draw the dividing set $d$ on the
boundary $\Sigma$ of $H_{2}$ in terms of the front projection. To see that
$H_{1}$ has the standard contact structure, we show that there is a set of
compressing curves for $H_{1}$ on $\Sigma$ such that each curve intersects the
dividing set in two points.

Let $\\{D_{j}^{\prime}\\}$ be the bounded regions of the complement of the
front projection of $\widetilde{\Lambda}$, and let $D_{j}=D_{j}^{\prime}\cap
H_{1}$. If there are no $1$-handles in the handle diagram for $W$, then
$\\{D_{j}\\}$ form a collection of compressing disks for $H_{1}$, which cut
$H_{1}$ into a ball. If there are $1$-handles, there is also a compressing
disk as in Figure 7 for each $1$-handle. After performing handleslides over
the regions that pass through the $1$-handle, we can realize this compressing
curve as in Figure 6 so that it intersects the dividing set in exactly two
points. To see that that boundary of each $D_{j}$ intersects the dividing set
in two points, we use the property from the last step of the construction of
$\widetilde{\Lambda}$, that each region is bounded by a Legendrian unknot with
$tb=-1$, meaning there is a unique left cusp (which may be a vertex) and a
unique right cusp (which may be a vertex). Any other vertices along the
boundary of the region have the two edges on the boundary of this region
entering the vertex from different sides. Examining all of the ways that our
local models in Figure 3 may appear as the boundary of a region, we see that
each cusp (either a standard cusp or a vertex cusp) contributes one
intersection point between the dividing set $\partial D_{j}$, and the
remaining edges and vertices in the boundary of the region do not contribute
any intersections between the dividing set and $\partial D_{j}$. Thus we see
that $H_{1}$ is a standard contact handlebody.

Next, we look at the second sector $W_{2}$. We need to show that
$W_{2}\cong\natural_{k_{2}}S^{1}\times D^{3}$, that there is a contact
structure $\xi$ on $\partial W_{2}$ with a contact Heegaard splitting
$\overline{H_{2}}\cup_{\Sigma}H_{3}$ such that $(W_{2},\omega|_{W_{2}})$ is a
symplectic filling of $(\partial W_{2},\xi)$. Recall that $W_{2}$ is obtained
from $H_{2}\times I$ by attaching $2$-handles along the Legendrian link
$\Lambda\subset\widetilde{\Lambda}$ with framing $ct-1$ where $ct$ is the
framing induced by the contact planes. Since $\Lambda$ is embedded in
$\widetilde{\Lambda}$ (the core of $H_{2}$) $W_{2}$ is smoothly diffeomorphic
to $\natural_{k_{2}}S^{1}\times D^{3}$. A natural Heegaard splitting of
$\partial W_{2}$ is given by $\overline{H_{2}}\cup_{\Sigma}H_{3}$ where
$H_{3}=(H_{2})_{ct-1}(\Lambda)$ ($ct-1$ surgery of $H_{2}$ along $\Lambda$).
There is a well-defined contact structure obtained by $ct\pm 1$ Legendrian
surgery, which agrees with the contact structure on $H_{2}$ near $\partial
H_{3}=\Sigma$ (since the surgery is performed on the interior). Therefore the
dividing set on $\partial H_{3}=\Sigma$ can be viewed as the same as the
dividing set on $\partial H_{2}=\Sigma$, but the compressing curves for
$H_{3}$ change based on the surgery. Namely, for each component $\Lambda_{k}$
of $\Lambda$, the compressing curve changes from a meridian of $\Lambda_{k}$
to a $ct-1$-framed copy of $\Lambda_{k}$ on $\Sigma$. Since the dividing set
is parallel to the $ct$ framing of $\Lambda_{k}$, the $ct-1$ framing of
$\Lambda_{k}$ intersects the dividing set exactly twice. Therefore, $H_{3}$ is
a standard contact handlebody with the contact structure induced by Legendrian
surgery on $H_{2}$.. Note that $\overline{H_{2}}$ is also a standard contact
handlebody, using the negative contact structure on $H_{2}$ from Lemma 2.4.
Putting these together, we get a contact Heegaard splitting of $\partial
W_{2}$.

It suffices to show that $(W_{2},\omega|_{W_{2}})$ is a Weinstein filling of
$\partial W_{2}$ where the contact structure on $\partial W_{2}$ is given by
the contact Heegaard splitting $\overline{H_{2}}\cup_{\Sigma}H_{3}$. For this,
notice that $H_{2}\times I$ is a $1$-handlebody, and if we restrict $\omega$
to this subset of $W_{2}$, up to shrinking $I$, the symplectic structure must
be a standard neighborhood of the isotropic spine of $H_{2}$. In other words,
$H_{2}\times I$ with the symplectic structure $\omega$ is symplectomorphic to
a Weinstein $1$-handlebody. Moreover, the induced unique tight contact
structure on $\partial(H_{2}\times I)=\\#_{k_{1}}S^{1}\times S^{2}$ is
supported by the open book $F\times S^{1}$ (viewing $H_{2}=F\times I$). This
open book with trivial monodromy gives rise exactly to the contact Heegaard
splitting $\overline{H_{2}}\cup H_{2}$ (where we collapse $I$ along $\partial
H_{2}$). Since $W_{2}$, (with the contact Heegaard splitting
$\overline{H_{2}}\cup_{\Sigma}H_{3}$) is obtained from the Weinstein
$1$-handlebody $H_{2}\times I$ (with the contact Heegaard splitting
$\overline{H_{2}}\cup_{\Sigma}H_{2}$) by attaching Weinstein $2$-handles along
Legendrian knots in $H_{2}$, we have that $W_{2}$ is a Weinstein filling of
the contact Heegaard splitting $\overline{H_{2}}\cup_{\Sigma}H_{3}$.

∎

###  3.2. Multisection diagrams with divides

A fundamental feature of multisections with divides is that they can be
completely encoded on a surface. In this section we define these diagrams and
show how the previous proof gives an algorithm for obtaining a bisection
diagram with divides from a Kirby-Weinstein handlebody diagram.

######  Definition 3.2.

A multisection diagram with divides is a closed orientable surface, together
with a set of dividing curves $d$ and cut systems $C_{1},C_{2},...,C_{n}$ such
that for all $i\in\\{1,2,...,n-1\\}$

  * •

Each curve in $C_{i}$ intersects $d$ in two points.

  * •

$(\Sigma,d,C_{i},C_{i+1})$ is a contact Heegaard splitting of the _tight_
contact structure on $\\#_{k_{i}}S^{1}\times S^{2}$ for some
$k_{i}\in\mathbb{N}$.

######  Remark 3.3.

The condition that each curve in each cut system intersects the dividing set
in two points is easily checked. However, it is potentially difficult to check
whether the union of two consecutive contact handlebodies forms the _tight_
contact structure on $\\#_{k_{i}}S^{1}\times S^{2}$.

The proof of Theorem 3.1 gives an algorithmic method to obtain a multisection
diagram with divides as follows.

Starting from a Kirby-Weinstein handlebody diagram, construct the Legendrian
graph $\widetilde{\Lambda}$ as in the proof. Use the models from Figure 3 to
draw the dividing set $d$ on $\Sigma$ as the boundary of the neighborhood of
$\widetilde{\Lambda}$ ($H_{2}$) in terms of the front projection.

We can describe cut systems $C_{i}$ $i\in\\{1,2,3\\}$ for the handlebodies
$H_{1}$, $H_{2}$, and $H_{3}$ as follows. The cut system $C_{1}$ is given by
taking the regions of the planar diagram together with a curve tracing through
the vertical tunnels added between the 2-handles passing through each
1-handle, as in Figure 6–note the curve in this figure intersects the dividing
set in two points. As the regions are diagrams of $tb=-1$ unknots, each of the
curves given by the boundary of a region intersects the dividing set twice.
This ensures that the pair $(\Sigma,C_{1})$ is a diagram of a standard contact
handlebody.

The cut system $C_{2}$ is given by taking a meridian of each tunnel together
with a meridian of each knot in the Kirby-Weinstein diagram. Using the local
model at the top of Figure 3 and Figure 8 we see that these curves intersect
the dividing set twice so that $(\Sigma,C_{2})$ is also a standard contact
handlebody. Because $(\Sigma,C_{1},C_{2},d)$ represents a Heegaard splitting
of the boundary of the $0$\- and $1$-handles of the Kirby-Weinstein handlebody
diagram, it is symplectically fillable and thus must support the tight contact
structure on a connected sum of copies of $S^{1}\times S^{2}$.

Figure 8. The meridian of the tunnel at a crossing projects as the blue curve
in the figure. Note this curve intersects the dividing set at two points.

To obtain $C_{3}$, we start by taking a Legendrian push off of each knot
component $K_{i}$ in the diagram. Each such component intersects its chosen
meridian in $C_{2}$ once and does not intersect the dividing set at all.
Adding a left handed twist about the meridian to each component $K_{i}$ gives
a curve $K_{i}^{\prime}$ whose surface framing is the contact framing $-1$ and
which intersects the dividing set twice. Then replacing each meridian in
$C_{2}$ with the corresponding $K_{i}^{\prime}$ gives a diagram
$(\Sigma,d,C_{2},C_{3})$ representing a contact Heegaard splitting of a
connected sum of copies of $S^{1}\times S^{2}$. Therefore
$(\Sigma,d,C_{1},C_{2},C_{3})$ is a bisection diagram with divides of the
given manifold.

This algorithm is carried out in Figures 9 and 10 for the result of attaching
a Weinstein handle to the max $tb$ right handed trefoil and in Figures 11 and
12 for the Weinstein domain $T^{*}\mathbb{R}\mathbb{P}^{2}$, which is a disk
bundle over $\mathbb{R}\mathbb{P}^{2}$.

Figure 9. Adding tunnels to the right handed Legendrian trefoil on the left
results in the graph on the right. Note that the exterior is unknotted and
that each bounded region in the diagram is a $tb=-1$ unknot.  Figure 10. Top:
A bisection diagram of the manifold obtained by attaching a Weinstein handle
to the max $tb$ trefoil. On this diagram, many green curves which are parallel
to blue curves are omitted for visual clarity. Bottom: The dividing set,
followed by each of the cut systems in order are drawn out individually.
Figure 11. Top: A Kirby-Weinstein diagram for $T^{*}\mathbb{R}P^{2}$. Bottom:
Following the proof of Theorem 3.1 we add Legendrian tunnels to the diagram
above so that the exterior of the tunnels drawn is a standard contact
handlebody.  Figure 12. Top: A bisection diagram of $T^{*}\mathbb{R}P^{2}$. On
this diagram, many green curves which are parallel to blue curves are omitted
for visual clarity. Bottom: The dividing set, followed by each of the cut
systems in order are drawn out individually.

Note that the cut systems $C_{2}$ and $C_{3}$ which are output from our
algorithm have a very particular form. More specifically, each component
$\gamma$ of $C_{3}$ either agrees with or is dual to a component $\beta$ of
$C_{2}$. In the latter case, there exists a curve $V$ in $\Sigma$ which is
disjoint from the dividing set, dual to the component $\beta$ with respect to
$C_{2}$, such that $\gamma=\tau^{-1}_{\beta}(V)=\tau_{V}(\beta)$ where
$\tau_{A}$ is a right-handed Dehn twist about $A$ with respect to the
orientation on $\Sigma$ induced as the boundary of $H_{2}$. Let’s call two cut
systems related in this way _standard Weinstein cobordant_. By isotoping each
$V$ into the Legendrian core of the handlebody $H_{2}$, we find a Legendrian
link in $H_{2}$ such that the $ct-1$ framing for this link is represented by
the corresponding $\gamma$ curves in $C_{3}$, ($V$ represents the contact
framing, and $\beta$ is a meridian of the surgery torus so
$\tau^{-1}_{\beta}(V)$ represents the contact framing $-1$). Thus we see that
if two cut systems with a dividing set are standard Weinstein cobordant, the
corresponding sector can be endowed with the structure of a Weinstein
cobordism from $H_{i}$ to $H_{i+1}$ obtained by attaching Weinstein
$2$-handles to $H_{i}\times I$. We can always endow the first sector with the
structure of a Weinstein $1$-handlebody (since by definition
$(\Sigma,d,C_{1},C_{2})$ is a contact Heegaard splitting of
$\\#_{k_{1}}S^{1}\times S^{2}$ with the tight (fillable) contact structure).
Therefore, we have an (overly strong) condition that ensures a multisection
diagram with divides corresponds to a Weinstein $4$-manifold.

######  Proposition 3.4.

Let $(\Sigma,C_{1},\dots,C_{n},d)$ be a multisection diagram with divides such
that $(C_{i},C_{i+1},d)$ are standard Weinstein cobordant for $1<i<n$. Then
$(\Sigma,C_{1},\dots,C_{n},d)$ corresponds to a Weinstein $4$-manifold.

In Section 4, we will see another condition that ensures a multisection
diagram with divides corresponds to a Weinstein $4$-manifold. It is an
interesting question to ask whether there is a general characterization of all
multisection diagrams with divides which correspond to Weinstein
$4$-manifolds. In general, we only expect a multisection diagram with divides
to correspond to a symplectic $4$-manifold, which may not admit any global
Weinstein structure.

##  4\. PALFs, monodromy substitution and multisections with divides

###  4.1. PALFS and monodromy factorizations

Fibration structures on symplectic manifolds have a long history of study,
dating to Donaldson’s work in [Don99] where it was shown that every closed
symplectic 4-manifold admits a Lefschetz pencil. Conversely, Gompf proved that
every 4-manifold with a Lefschetz pencil admits a symplectic structure
[GompfStipsicz]. The corresponding objects in the Weinstein category are
positive allowable Lefschetz fibrations.

######  Definition 4.1.

A _Lefschetz fibration_ on $X$ is a map $\pi:X\to B$ to a surface $B$ such
that near each critical point of $\pi$, there are local orientation preserving
coordinates such that $\pi$ is modeled on $(z_{1},z_{2})\mapsto z_{1}z_{2}$. A
positive allowable Lefschetz fibration (PALF) is a Lefschetz fibration whose
base $B$ is $D^{2}$ and whose regular fiber is a compact surface with boundary
such that every vanishing cycle is homologically essential (allowable).

Following Loi and Piergallini [LoiPie01], every Weinstein domain admits a
PALF. Conversely, every PALF supports a Weinstein structure [GompfStipsicz].

In this section, we will show how to use the PALF structure to obtain a
decomposition of a Weinstein domain as a multisection with divides. As in
Figure 13, we cut the disk $D^{2}$ into closed subdisks, $D_{1},\dots,D_{n}$,
such that

  * •

Each $D_{i}$ contains a unique critical value of $\pi$,

  * •

$D_{i}\cap D_{i+1}$ is diffeomorphic to an interval for $i=1,\cdots,n-1$,

  * •

$\partial D^{2}\subset X_{1}\cup X_{n}$, and

  * •

$D_{1}\cap\cdots\cap D_{n}=\\{(0,1),(0,-1)\\}$.

Then let $X_{i}=\pi^{-1}(D_{i})$, $H_{1}=\pi^{-1}(\partial D^{2}\cap D_{1})$,
$H_{i}=\pi^{-1}(D_{i-1}\cap D_{i})$ for $i=2,\dots,n$,
$H_{n+1}=\pi^{-1}(\partial D^{2}\cap D_{n})$. Then $H_{i}=X_{i-1}\cap X_{i}$
and $X_{1}\cap\cdots\cap X_{n}=\partial H_{i}$ (after quotienting by the
$D^{2}$ factor along points in $\partial F\times D^{2}$ which is a Weinstein
homotopic domain).

######  Theorem 4.2.

The decomposition $(X_{1},\dots,X_{n})$ is a multisection with divides for
$X$.

###### Proof.

Each $H_{i}$ is a $3$-dimensional handlebody since it is diffeomorphic to
$F\times I$. Thus to check this is a multisection with divides it suffices to
check that (1) each $X_{i}$ is symplectomorphic to a $4$-dimensional Weinstein
$1$-handlebody and (2) that $H_{i}\cup H_{i+1}$ is a contact Heegaard
splitting of $\partial X_{i}$.

First we look at each $X_{i}$. We will use $F:=\pi^{-1}((0,1))$ as the regular
fiber. The vanishing cycles are curves $(c_{1},\dots,c_{n})$ in $F$ which
collapse to the critical point under parallel transport from $(0,1)$ to the
critical value. The model for Lefschetz singularities shows that $X_{i}$ is
diffeomorphic to the manifold obtained from $H_{i}\times I$ by attaching a
$2$-handle along $c_{i}$ with framing given by one less than the page framing.
$H_{i}\times I$ is certainly a $1$-handlebody, so the result will still be a
$4$-dimensional $1$-handlebody if this $2$-handle cancels with one of the
$1$-handles of $H_{i}\times I$. Thus to see that $X_{i}$ is diffeomorphic to a
$1$-handlebody it suffices to check that there is a meridional disk in $H_{i}$
which intersects $c_{i}$ exactly once.

As $\pi$ is a PALF, $c_{i}\subset F$ is homologically essential. Thus, by
Poincare-Lefschetz duality, there exists an arc $a_{i}\subset F$ such that
$|a_{i}\cap c_{i}|=1$. Parallel transport defines diffeomorphisms
$\Psi_{i}:F\times I\to H_{i}$ which identify $F\times\\{0\\}$ with $F\subset
H_{i}$. Then $\Psi_{i}(a\times I)$ is a meridional disk of $H_{i}$ which
intersects $c_{i}\subset F=\Psi_{i}(F\times\\{0\\})$ at a single point. Thus
$X_{i}$ is a $4$-dimensional $1$-handlebody.

Since $X_{i}=\pi^{-1}(D_{i})$ and $D_{i}$ is a disk, $X_{i}$ has the structure
of a Weinstein manifold induced by the (restricted) PALF. This agrees with the
symplectic structure on $X$ since both are compatible with the Lefschetz
fibration.

To see that $H_{i}\cup H_{i+1}$ gives a contact Heegaard splitting of
$\partial X_{i}$ with the contact structure induced by the Weinstein structure
on $X_{i}$, we use the open book construction of contact Heegaard splittings.
Restricting $\pi$ to $\partial X_{i}$ gives an open book decomposition of
$\partial X_{i}$ which supports the contact structure induced by the Weinstein
structure on $X_{i}$ since the Weinstein structure comes from the Lefschetz
fibration structure. $H_{i}$ and $H_{i+1}$ are precisely the two halves of the
open book which give a contact Heegaard splitting. ∎

######  Remark 4.3.

Using the fact that all Weinstein manifolds are supported by Lefschetz
fibrations, Theorem 4.2 gives a similar result to Theorem 3.1. The main
difference is that Theorem 3.1 yields a _bisection_ , whereas Theorem 4.2 will
usually have many more than two sectors.

###  4.2. Multisection diagrams with divides from a PALF

A PALF can be encoded combinatorially through the fiber surface $F$ and the
ordered set of vanishing cycles $(c_{1},\dots,c_{n})$. In this section we show
how to use the combinatorial data of a PALF to obtain the combinatorial data
of the multisection diagram with divides corresponding to the decomposition
from Theorem 4.2.

The monodromy about a Lefschetz critical value with vanishing cycle $c$ is a
right handed Dehn twist about $c$, which we denote by $\tau(c)$. Thus a PALF
can equivalently be encoded by an ordered sequence of right handed Dehn twists
about the vanishing cycles called a monodromy factorization.

The core surface of the multisection $\Sigma$ is diffeomorphic to the union of
two copies of $F$ glued together along their boundary. The dividing set on
$\Sigma$ is given by the boundary of $F$. More precisely, if $\Psi_{i}:F\times
I\to H_{i}$ is the diffeomorphism defined by parallel transport,
$\Sigma=\Psi_{i}(F\times\\{0\\}\cup
F\times\\{1\\})=F\cup\Psi_{i}(F\times\\{1\\})$ (note we are suppressing the
quotient of the $I$ direction at points in $\partial F$).

To understand a multisection diagram with divides, we want to fix an
identification of $\Sigma$, and then draw cut systems for each handlebody
$H_{i}$. We will use $\Psi_{1}$ to identify $\Sigma=\partial H_{1}$ as
$F_{0}\cup F_{1}$. Then the restriction of $\Psi_{i}$ to $F_{0}\cup F_{1}$
gives a diffeomorphism from $\Sigma$ to $\partial H_{i}$.

Let $\\{a_{1},\dots,a_{k}\\}$ be a complete arc system for $F$ i.e. a
collection of properly embedded arcs which cut $F$ into a disk. Then
$\\{\Psi_{i}(a_{1}\times I),\dots,\Psi_{i}(a_{k}\times I)\\}$ gives a cut
system of disks for $H_{i}$. We want to see the boundaries of these disks on
our fixed identification of $\Sigma$. Namely, we want to describe the curves
$\Psi_{i}^{-1}(a_{j}\times\\{0\\}\cup a_{j}\times\\{1\\})$ in $F_{0}\cup
F_{1}$. Each $\Psi_{i}$ is the identity on $F_{0}$, and defines parallel
transport along the arc from $(0,1)$ to $(0,-1)$ over which $H_{i}$ lies.
Since the $\Psi_{i}$ define parallel transport, and we are using $\Psi_{1}$ to
identify $\Sigma$ with $F_{0}\cup F_{1}$, we see that
$\Psi_{i}(a_{j}\times\\{1\\})$ in $\Sigma$ is the image of $a_{j}$ under the
monodromy around the curve which goes from $(0,-1)$ to $(0,1)$ along the
$H_{i}$ curve and then goes from $(0,1)$ to $(0,-1)$ along the $H_{1}$ curve.
This monodromy is $\tau(c_{1}),...,\tau(c_{i-1})$. Thus, the cut system for
$H_{i}$ is obtained from the cut system for $H_{i-1}$ by applying the right
handed Dehn twist $\tau(c_{i-1})$ where $c_{i-1}\subset F_{1}$. Any choice of
complete arc system defines a cut system for $H_{1}$ by gluing together the
same arcs on $F_{0}$ and $F_{1}$.

To summarize, given a PALF with fiber $F$ and ordered vanishing cycles
$\\{c_{1},\dots,c_{n}\\}$, the corresponding multisection with divides is
given by

  * •

$\Sigma=F_{0}\cup_{\partial}F_{1}$ where $F_{0}$ and $F_{1}$ are copies of $F$
($F_{0}$ is oppositely oriented).

  * •

The dividing set $d$ is $\partial F=F_{0}\cap F_{1}$.

  * •

The cut system for $H_{1}$ is $\\{a_{1}\cup a_{1},\dots,a_{g}\cup a_{g}\\}$
where $\\{a_{1},\dots,a_{g}\\}$ is a complete arc system for $F$.

  * •

For $i>1$, the cut system for $H_{i}$ is obtained from the cut system for
$H_{i-1}$ by applying a right handed Dehn twist about $c_{i-1}\subset F_{1}$.

Note that each cut curve intersects the dividing set in exactly two points
(the end points of the arc $a_{j}$). (This is clearly true for $H_{1}$, and it
remains true for each $H_{i}$ because the Dehn twists are applied in the
interior of $F_{1}$ which is disjoint from the dividing set.)

Figure 13. Decomposing a PALF into pieces containing exactly one Lefschetz
singularity yields a multisection. The multisection surface is two fiber
surfaces glued along their binding and each handlebody is a product region
between these two surfaces.

######  Remark 4.4.

Note that the output of a PALF yields a slightly more general condition which
ensures that a multisection with divides corresponds to a Weinstein manifold.
In this case, every consecutive pair of cut systems differs by a right handed
Dehn twist about a curve which lies entirely in the $\Sigma_{+}$ side of the
dividing set. Let’s call such $(C_{i},C_{i+1},d)$ _generalized standard
Weinstein cobordant_ (gsWc). (This generalizes the notion of standard
Weinstein cobordant from Proposition 3.4.) If consecutive cut systems $C_{i}$
and $C_{i+1}$ are gsWc where the curve defining the Dehn twist relating them
is dual to one of the components of $C_{i}$, they define a smooth multisection
by Lemma 4.5. Furthermore, whenever we have gsWc cut systems representing
handlebodies $H_{i}$ and $H_{i+1}$ forming the Heegaard splitting on the
boundary of a sector $W_{i}$, we can again interpret $W_{i}$ as a Weinstein
cobordism from $H_{i}$ to $H_{i+1}$. This is because we can use the Lefschetz
fibration interpretation of $W_{i}$ and then view the Lefschetz critical point
as an attachment of a Weinstein $2$-handle to $H_{i}\times I$. Since
$H_{i}\times I$ can either be interpreted as a filling or a Weinstein
cobordism, the sector $W_{i}$ also supports a Weinstein structure making it a
filling of its boundary, and a Weinstein structure making it a cobordism from
$H_{i}$ to $H_{i+1}$. This shows that Proposition 3.4 holds under the more
general assumption that consecutive pairs of cut systems are gsWc, and the
curve defining the Dehn twist for a pair of cut systems is dual to one of the
curves from the first cut system.

The previous construction suggests that multisection can also be encoded by a
monodromy factorization, and here, we show that this is indeed the case.
Through some care, one can determine a particular monodromy factorization from
a multisection, however for the present paper it will be sufficient to have a
family of possible factorizations.

Fixing a surface $\Sigma$, a system of cut curves defines a handlebody with
boundary $\Sigma$. Given a handlebody, we can choose any system of cut curves
which bound disks in the handlebody to encode it. If we specify a
diffeomorphism $\Phi$ of $\Sigma$, we can apply that diffeomorphism to a cut
curve system, to produce a new cut curve system. If the original cut curve
system defines a handlebody $H$, we let $H_{\Phi}$ denote the handlebody
defined by the image of a cut-system for $H$ under the map $\Phi$. Note this
is independent of the choice of cut-system for $H$, because two different cut
systems $\\{\alpha_{i}\\}$ and $\\{\alpha_{i}^{\prime}\\}$ for $H$ will be
related by some sequence of handle slides, and thus $\\{\Phi(\alpha_{i})\\}$
will be related to $\\{\Phi(\alpha_{i}^{\prime})\\}$ by a sequence of handle
slides as well. We begin with a lemma which yields a sufficient condition for
$H$ and $H_{\Phi}$ to form a Heegaard splitting of $\\#_{g-1}S^{1}\times
S^{2}$ when $\Phi$ is a Dehn twist.

######  Lemma 4.5.

Let $\Sigma$ be a closed genus $g$ surface, $c$ be a simple closed curve on
$\Sigma$, and $\tau_{c}$ be a (right or left handed) Dehn twist about $c$. Let
$H$ be a handlebody with boundary $\Sigma$. Suppose there exists a properly
embedded disk $D\subset H$ whose boundary, $\partial D=E_{1}$, is non-
separating on $\Sigma$ such that $|E_{1}\cap c|=1$. Then
$H\cup_{\Sigma}H_{\tau_{c}}$ is a Heegaard splitting of $\\#_{g-1}S^{1}\times
S^{2}$.

###### Proof.

We will produce a Heegaard diagram of $H\cup_{\Sigma}H_{\tau_{c}}$ which
consists of $g-1$ parallel curves and two curves intersecting once, which
proves the lemma. Extend $E_{1}$ to a cut system, $E$, for $H$. Now each point
of $c\cap(E\backslash\\{E_{1}\\})$ can be eliminated by the following process:
start at the point $c\cap E_{1}$ and follow $c$ until we meet the first point
of $c\cap(E\backslash\\{E_{1}\\})$ call the resulting arc $a$ and the curve in
$E$ at this intersection $E_{i}$. Sliding $E_{i}$ over $E_{1}$ along $a$
eliminates this intersection and does not introduce any new intersections
between $E$ and $c$. Repeating this process, using narrower and narrower
sliding bands, we may eliminate all intersections of $c$ and $E$ except for
the point $c\cap E_{1}$.

We may then obtain a Heegaard diagram for $H\cup_{\Sigma}H_{\tau_{c}}$ by
taking the Dehn twist of the cut system resulting from these slides on $E$
about $c$. As $c$ is disjoint from all of the curves other than $E_{1}$ there
are $g-1$ curves which are unchanged by the twisting and are therefore
parallel. On the other hand $E_{1}$ and $\tau_{c}(E_{1})$ intersect once,
providing the desired Heegaard diagram. ∎

Using the above lemma, we obtain a sufficient criteria for Dehn twists on cut
systems to yield two sequential handlebodies in a multisection. Conversely,
the following proposition shows that the sequential handlebodies in all
multisections can be obtained in the fashion.

######  Proposition 4.6.

Let $\mathfrak{M}=X_{1}\cup X_{2}\cdots\cup X_{n}$ be a genus-$g$ multisection
with multisection surface $\Sigma$, and with
$X_{i}\cong\natural_{k_{i}}S^{1}\times D^{3}$. Let
$H^{1},H^{2},\dots,H^{n},H^{n+1}$ be the 3-dimensional handlebodies lying at
the boundaries of the $X_{i}$. Then there exist curves
$c_{1}^{1},c_{2}^{1},\dots,c_{g-k_{1}}^{1},c_{1}^{2},c_{2}^{2},\dots,c_{g-k_{2}}^{2},\dots,c_{1}^{n},c_{2}^{n},\dots,c_{g-k_{n}}^{n}$
such that
$H^{i}_{\tau(c_{1}^{i})\circ\tau(c_{2}^{i})\dots\cdot\circ\tau(c_{g-k_{i}}^{i})}=H^{i+1}$.

###### Proof.

For each handlebody, $H^{i}$, we will show how to produce the curves
$c_{i}^{1}...c_{i}^{g-k_{i}}$. Recall that the handlebodies $H_{i}$ and
$H_{i+1}$ meet at the multisection surface $\Sigma$ to form a a Heegaard
splitting of $\\#^{k_{i}}S^{1}\times S^{2}$. Consider a Heegaard diagram of
$H_{i}\cup H_{i+1}$. By Waldhausen’s theorem [Wald68], after a sequence of
handle slides there is a cut system of curves $a_{1},$ … $,a_{g}$ for $H_{i}$
and $b_{1}...b_{g}$ for $H_{i+1}$ such that $a_{i}=b_{i}$ for $0\leq i\leq
k_{i}$ and $|a_{n}\cap b_{m}|=\delta_{n,m}$ for for $k_{i+1}\leq n,m\leq g$.

For $k_{i}\leq j\leq g$ we let $c_{i}^{j}=\tau_{b_{j}}(a_{j})$. Then,
$\tau_{c_{i}^{j}}(a_{j})=b_{j}$. Moreover, since $c_{j}^{i}$ does not
intersect any of the other $a_{k}$ for $k\neq j$,
$\tau_{c_{j}^{i}}(a_{k})=a_{k}$ for $k\neq j$. Then the product
$\Pi_{j=1}^{k_{i}}\tau_{c_{j}^{i}}(a_{k})$ takes a cut system for $H_{i}$ to a
cut system for $H_{i+1}$. ∎

We call the product $\Pi_{i=1}^{n}\Pi_{j=1}^{g-k_{i}}\tau(c_{j}^{i})$ a
monodromy factorization for $\mathfrak{M}$. By following through our
construction in Theorem 4.2, we can track the monodromy of a PALF onto the
monodromy of a multisection which immediately yields the following.

######  Corollary 4.7.

Suppose that $(X^{4},\omega)$ is a Weinstein manifold which admits a PALF with
fiber surface $F$ and monodromy factorization $P=\Pi_{i=1}^{n}\tau(c_{i})$.
Then $(X^{4},\omega)$ admits an n-section with divides with multisection
surface $\Sigma=F\cup-F$, dividing set $\partial F$, and monodromy
factorization $P^{\prime}$ obtained by applying the Dehn twists of $P$ to
$F\subset\Sigma$.

###  4.3. Monodromy Substitution

In this section we will demonstrate how a monodromy substitution affects a
multisection with divides. We begin with the analogous construction for PALFs.

######  Definition 4.8.

Let $f:M^{4}\to D^{2}$ be a PALF with fiber surface $\Sigma$ and monodromy
factorization $\Pi_{i=1}^{n}\tau(c_{i})$. Suppose that for some $k,l,m,n$ and
curves $c_{m}^{\prime}...c_{n}^{\prime}$ we have that, as mapping classes,
$\Pi_{i=k}^{l}\tau(c_{k})=\Pi_{j=m}^{n}\tau(c_{j}^{\prime})$. Then we may
obtain a new Lefschetz fibration with monodromy factorization given by

$\Pi_{i=1}^{k}\tau(c_{i})\Pi_{j=m}^{n}\tau(c_{j}^{\prime})\Pi_{i=l+1}^{n}\tau(c_{i}).$

We say that the new Lefschetz fibration is obtained by a monodromy
substitution on $f$.

Monodromy substitution has been used extensively to produce new symplectic
manifolds from existing ones. In particular, in [EndGur10], the authors show
that the lantern relation, pictured in Figure 14, can be used to perform a
rational blowdown on the configuration $C_{2}$ (see [GomSti99] Section 8.5 for
an exposition on these operations). This was later generalized in [EMV11] to
realize an infinite family of rational blowdowns as monodromy substitutions
using daisy relations. In general, any monodromy substitution can be thought
of as some symplectic cut-and-paste operation.

Figure 14. The lantern relation in the mapping class group in a 4-holed sphere
states that right handed Dehn twists about the red curves gives the same
mapping class as the right handed Dehn twists about the blue curves.

There is an analogous process of monodromy substitution on a multisection.

######  Definition 4.9.

Let $\mathfrak{M}$ be a multisection (not necessarily with boundary) starting
at the handlebody $H$ with monodromy factorization given by
$\Pi_{i=1}^{n}\tau(c_{i})$. Let $H^{k}$ be the handlebody
$H_{\tau(c_{1})\cdot\tau(c_{2})\cdot\dots\cdot\tau(c_{k})}$ and suppose that
$c_{k+1}^{\prime},c_{k+2}^{\prime},\dots,c_{j^{\prime}}^{\prime}$ is a
sequence of curves such that for all $l\in\\{k+1,k+2,\dots,j^{\prime}\\}$ we
have that $c_{l}^{\prime}$ is dual to some disk in
$H^{k}_{\tau(c_{k+1}^{\prime})\cdot\dots\cdot\tau(c_{l-1}^{\prime})}$ (this
will guaranteed that the assumptions of Lemma 4.5 hold). Suppose further that
$H^{k}_{\tau(c_{k+1})\cdot\tau(c_{k+2})\cdot\dots\cdot\tau(c_{j})}=H^{k}_{\tau(c_{k+1}^{\prime})\cdot\tau(c_{k+2}^{\prime})\cdot\dots\tau(c_{j^{\prime}}^{\prime})}$.
Then we may obtain a new multisection $\mathfrak{M}_{S}$ starting at the
handlebody $H$ and specified by the monodromy factorization
$\Pi_{i=1}^{k}\tau(c_{i})\Pi_{i=k+1}^{j^{\prime}}\tau(c_{i}^{\prime})\Pi_{i=j}^{n}\tau(c_{i})$.
We call $\mathfrak{M}_{s}$ a monodromy substitution of $\mathfrak{M}$.

It follows immediately from Corollary 4.7 that we can find monodromy
substitutions by doubling a PALF and a monodromy substitution of that PALF.
Carrying this out for the lantern relation gives us a monodromy substitution
on a multisection with divides yielding the operation outlined in Figure 15.

Figure 15. The three handlebodies on the top row yield a bisection whose
monodromy is the double of the right handed Dehn twists about the red curves
in Figure 14. Replacing the middle handlebody by the one below it yields a
multisection whose monodromy is the double of the right handed Dehn twists
about the blue curves in Figure 14. The overall change in a bisection
containing these handlebodies is a $C_{2}$ rational blowdown.

##  5\. Genus-1 multisections and stabilizations

###  5.1. Classification of genus 1 multisections with divides

In this subsection we will provide a characterization of genus-1 multisections
with divides. For examples of the diagrams for the unique 2- and 3-sections
with divides, see Figure 16. Smooth genus-1 multisections are well
characterized by their diagrams, which consist of sequences
$(\alpha_{1},...\alpha_{n})$ with $|\alpha_{i}\cap\alpha_{i+1}|=1.$ In [IKLM],
the authors show that smooth genus-1 $n$-sections with boundary correspond to
linear plumbings of $(n-1)$ disk bundles over the sphere. Moreover, given the
oriented sequence of cut curves $(\alpha_{1},\alpha_{2},...\alpha_{n})$
defining a genus-1 multisection diagram, the Euler number of the $i^{th}$ disk
bundle is given by the algebraic intersection
$\langle\alpha_{i-2},\alpha_{i}\rangle$.

######  Proposition 5.1.

There is a unique genus-1 n-section with divides for each $n\geq 2$. These
correspond to the linear plumbing of $(n-1)$ disk bundles of Euler number $-2$
over the sphere ($T^{*}S^{2}$’s).

###### Proof.

First we observe that the linear plumbing of $(n-1)$ disk bundles of Euler
number $-2$ supports a PALF structure whose fiber is an annulus with $n$
vanishing cycles all parallel to the core circle of the annulus. By the
algorithm in Theorem 4.2, these Weinstein domains have genus $1$
multisections.

Now we show that these are the only genus $1$ multisections with divides.

For a contact Heegaard splitting of $S^{3}$, the dividing set consists of two
parallel curves. Fixing coordinates $(a,b)$ for $H_{1}(T^{2})$, we may assume,
after an orientation preserving homeomorphism that $\alpha_{1}=(0,1)$ and
$\alpha_{2}=(1,0)$. The dividing set will be two parallel curves of slope $d$.
Note that $\alpha_{i}$ intersects the dividing set twice if and only if
$|\alpha_{i}\cap d|=1$. Therefore $d=(1,\pm 1)$. Since $d=(1,1)$ corresponds
to a contact Heegaard splitting for an overtwisted contact structure on
$S^{3}$, we must have $d=(1,-1)$.

We first treat the case $n=2$, and then proceed inductively. In this case, we
seek to find the possible slopes for $\alpha_{3}$. As $|\alpha_{2}\cap d|=1$,
all curves which intersect $\alpha_{2}$ once are given by Dehn twists of
$\alpha_{2}$ about $d$. In addition the requirement that $|\alpha_{3}\cap
d|=1$ means that $\alpha_{3}$ is a single Dehn twist of $\alpha_{2}$ about
$d$. If this Dehn twist is left handed, then the quadruple
$(\Sigma,d,\alpha_{2},\alpha_{3})$ gives a diagram for the overtwisted
$S^{3}$, so the Dehn twist must be right handed. Therefore $\alpha_{3}=(1,-2)$
and by the classification of smooth genus 1 multisections
$(\Sigma,\alpha_{1},\alpha_{2},\alpha_{3},d)$ gives a bisection with divides
of the disk bundle of Euler number $-2$ over the sphere.

In general suppose that $(\alpha_{1}...\alpha_{n-1})$ is a sequence of curves
defining a $(n-1)$-section with divides. Then, as in the base case,
$\alpha_{n-1}$ is a right handed Dehn twist of $\alpha_{n-2}$ about $d$ and
$\alpha_{n}$ is a right handed Dehn twist of $\alpha_{n-1}$ about $d$.
Therefore, $\langle\alpha_{n-2},\alpha_{n}\rangle=-2$ so that we have indeed
plumbed an additional $-2$-sphere. ∎

Figure 16. Left: The unique genus-1 bisection with divides corresponding to
the disk bundle over the sphere of Euler number $-2$. Right: Adding this curve
to the decomposition yields the unique genus-1 3-section with divides.

###  5.2. Stabilization

Here, we will introduce an operation on multisections with divides which takes
a genus-$g$ $n$-section and produces a genus-$(g+1)$ $(n+1)$-section. An
explicit example of this process applied to the genus-1 bisection of
$T^{*}S^{2}$ can be seen in Figure 18.

This stabilization operation can be seen from both the perspective of a handle
decomposition, as in Section 3.1 or from the perspective of a PALF. We will
focus on the second perspective, and we recall the definition of a
stabilization of a PALF.

######  Definition 5.2.

Let $f:M^{4}\to D^{2}$ be a PALF with fiber surface $\Sigma$ monodromy
factorization $\Pi_{i=1}^{n}\tau(c_{i}).$ Then a stabilization of $f$ is a
PALF with fiber surface $\Sigma^{\prime}$ and monodromy factorization
$\Pi_{i=1}^{n+1}\tau(c_{i})$ where $\Sigma^{\prime}$ is obtained by attaching
a 2-dimensional 1-handle to $\Sigma$ and $c_{n+1}$ is a curve on
$\Sigma^{\prime}$ intersecting the belt sphere of the attached 1-handle
geometrically once.

Figure 17. The annulus used to perform a stabilization of a multisection with
divides. The existing cut systems $C_{1}...C_{n}$ each receive a new curve
$c_{i}^{g+1}$ to form cut systems $C_{i}^{\prime}$ for $i\in\\{1,...,n\\}$.
The arc $A$ glues with an arc in the existing multisection which is disjoint
from the dividing set to yield a curve $C$. Performing a Dehn twist about $C$
to each curve in the cut system $C_{n}^{\prime}$ yields a new handlebody
$C_{n+1}^{\prime}$ so that the sequence of cut systems
$(C_{1}^{\prime},...,C_{n+1}^{\prime})$ is a new multisection diagram.

By doubling the 1-handle used in a stabilization of a PALF, we obtain the
diagram in Figure 17. When glued to a multisection with divides appropriately
we obtain a new multisection with divides, whose construction is outlined in
the following definition.

######  Definition 5.3.

Let $\mathfrak{M}$ be a multisection with divides with a diagram given by
$(\Sigma,d,C_{1},...,C_{n})$. Suppose $C_{i}$ is made up of the curves
$(c_{i}^{1},...,c_{i}^{g})$. Let $p_{1}$ and $p_{2}$ be two points on $d$ and
$A^{\prime}$ be an arc between $p_{1}$ and $p_{2}$ which is contained in
$\Sigma^{+}$. Let $\Sigma^{\prime}$ be the surface obtained by removing
neighbourhoods of $p_{1}$ and $p_{2}$ and gluing in the surface shown in
Figure 17 to the resulting boundary where the gluing is performed so that the
arc $A$ meets the arc $A^{\prime}$ to form a curve $c$. By slight abuse of
notation, we will keep the notation $c_{i}^{k}$ for the simple closed curve on
$\Sigma^{\prime}$ after performing surgery. We then obtain a new multisection
$\mathfrak{M}_{S}$ with diagram given by
$(\Sigma^{\prime},d,C_{1}^{\prime},...,C_{n}^{\prime},C_{n+1}^{\prime})$
where, for i $\in\\{1,...,n\\}$,
$C_{i}^{\prime}=(c_{i}^{1},...,c_{i}^{g},c_{i}^{g+1})$ and
$C_{n+1}^{\prime}=(\tau_{c}(c_{n}^{1}),...,\tau_{c}(c_{n}^{g}),\tau_{c}(c_{n}^{g+1}))$.

We can easily see that $\mathfrak{M}_{S}$ is still a multisection diagram with
divides. The condition that each curve in each cut system intersects the
dividing set in two points is ensured by looking at the model in Figure 17 and
the fact that the curve we Dehn twist about is disjoint from the dividing set.
That $\mathfrak{M}_{S}$ still represents a multisection smoothly follows from
Lemma 4.5. That $(C_{n}^{\prime},C_{n+1}^{\prime},d)$ represents a contact
Heegaard splitting of the _tight_ contact structure on $\\#_{g-1}S^{1}\times
S^{2}$ follows from the fact that we can obtain this contact Heegaard
splitting as the boundary of a PALF sector as in the proof of Theorem 4.2.

Observe that the manifold represented by
$(\Sigma^{\prime},d,C_{1}^{\prime},\dots,C_{n}^{\prime})$ is related to the
manifold represented by $(\Sigma,d,C_{1},\dots,C_{n})$ by attaching a single
$1$-handle. Thus if $(\Sigma,d,C_{1},\dots,C_{n})$ represents a Weinstein
domain, then $(\Sigma^{\prime},d,C_{1}^{\prime},\dots,C_{n}^{\prime})$ also
represents a Weinstein domain obtained by attaching a single Weinstein
$1$-handle. (Note that there is no constraint on the attaching data for a
$1$-handle attachment to be Weinstein.) By Remark 4.4, the new sector $W_{n}$
amounts to attaching a Weinstein cobordism to
$W^{\prime\prime}=W_{1}^{\prime}\cup\dots\cup W_{n-1}^{\prime}$. Therefore,
the stabilized diagram
$(\Sigma^{\prime},d,C_{1}^{\prime},\dots,C_{n}^{\prime},C_{n+1}^{\prime})$
also represents a Weinstein domain. Furthermore, the Weinstein cobordism from
the sector $W_{n}$ attaches a Weinstein $2$-handle which cancels with the
added $1$-handle (the attaching sphere of the $2$-handle intersects the belt
sphere of the $1$-handle in one point). Therefore, in total, we have added a
trivial Weinstein cobordism (one which is Weinstein homotopic to a trivial
cobordism). Namely, if before stabilization, the multisection diagram with
divides represented a Weinstein domain, then after stabilization, the
multisection diagram with divides represents the same Weinstein domain up to
Weinstein homotopy. We summarize in the following proposition.

######  Proposition 5.4.

Let $\mathfrak{M}$ be a multisection with divides which admits a multisection
diagram with divides given by $(\Sigma,d,C_{1},...,C_{n})$. Let
$\mathfrak{M}^{\prime}$ be a stabilization of $\mathfrak{M}$ with a
multisection diagram for $\mathfrak{M}^{\prime}$ given by
$(\Sigma^{\prime},d,C_{1}^{\prime},...,C_{n+1}^{\prime})$. Then,
$\mathfrak{M}^{\prime}$ is a multisection diagram with divides, and the
Weinstein manifolds encoded by $\mathfrak{M}$ and $\mathfrak{M}^{\prime}$ are
Weinstein homotopic.

Figure 18. This figure gives the process for stabilizing the genus-1
multisection diagram for $T^{*}S^{2}$. Top: The genus-1 multisection diagram
for $T^{*}S^{2}$ together with two points on the dividing set and an arc $A$
whose interior is disjoint from the dividing set. Bottom left: Gluing the
stabilizing annulus in Figure 17 yields the first three cut systems for the
stabilization, together with a curve $C$. Bottom Right: Dehn twisting the
green cut system about the curve $C$ yields the final cut system for the
stabilized multisection.

##  6\. Questions

Donaldson [Don96] and Giroux [Gir02, Gir17] proved that every symplectic
manifold admits a symplectic divisor such that the complement of a standard
neighborhood of the divisor is a Weinstein domain. In search of a diagrammatic
theory for closed symplectic manifolds, one strategy would be to find a
suitable structure on the neighborhood of the divisor,and glue as in
[IslNay20] to a multisection with divides for the Weinstein complement. This
leads us to the following questions.

######  Question 6.1.

Can we construct analogous multisections with divides for symplectic
$4$-manifolds with _concave_ boundary? In particular, concave neighborhoods of
symplectic divisors. Do we need different diagrammatic information to encode
concave boundary? How do we specify in a multisection diagram how to
symplectically glue convex pieces to concave pieces?

The results of Section 4 primarily consisted of using PALFs to obtain
information about multisections with divides, but in favorable conditions (see
Remark 4.4), this construction can be reversed to obtain a PALF from a
multisection with divides. It is an open question as to whether two PALFs
corresponding to the same Weinstein 4-manifold are related by stabilization,
Hurwitz equivalence, and an overall conjugation. We have translated the
stabilization move in Section 5.2, and using a similar approach, the other two
moves can readily be translated into moves on multisections with divides.
Here, techniques used in the stable equivalence of trisections in [GayKir16]
could prove fruitful in addressing the following question.