# Constructions and Isotopies of High-dimensional Legendrian spheres

Agniva Roy  School of Mathematics
Georgia Institute of Technology
Atlanta
Georgia<EMAIL_ADDRESS>
###### Abstract.

We explore the construction of Legendrian spheres in contact manifolds of any
dimension. Two constructions involving open books work in any contact
manifold, while one introduced by Ekholm works only in $\mathbb{R}^{2n+1}$. We
show that these three constructions are isotopic (whenever defined). We also
recover and generalise a result of Courte and Ekholm, that shows Ekholm’s
doubling procedure produces the standard Legendrian unknot.

##  1\. Introduction

A contact manifold $(M,\xi)$ is a smooth manifold $M$ equipped with a nowhere
integrable hyperplane field $\xi$. The construction and investigtion of
contact manifolds has historically been aided by studying distinguished
submanifolds that interact suitably with the contact structure. In dimension
3, these submanifolds are either convex hypersurfaces, or Legendrian knots,
which are 1-dimensional submanifolds. These in conjunction have helped achieve
the classification of tight contact structures on several classes of
3-manifolds e.g. [Hon00a, Hon00b, Gir01, Gir00]. A general theme in most of
these results can be seen as follows: first understand Legendrian isotopy
classes of a family of Legendrian knots in $(S^{3},\xi_{st})$, then understand
the contact structures that appear by performing contact surgery on these
Legendrians.

In higher dimensions, the first hindrance to carrying out this plan is the
shortage of examples of Legendrian spheres. Some constructions of high
dimensional Legendrian spheres are explored in [EES05, Ekh16, DR11, BST15], in
$\mathbb{R}^{2n+1}$ with the standard contact structure. In this article, we
describe some general constructions of Legendrian spheres in
$(2n+1)$-dimensional contact manifolds from Lagrangian disks in pages of
supporting open books.

######  Construction 1.1.

A natural way to construct Legendrian submanifolds in a contact manifold is
via open book decompositions. Suppose $(M,\xi)$ is supported by the open book
$(B,\nu)$, where $\nu:M-B\to S^{1}$ is a fibration, and each page is
symplectomorphic to $(W,\omega)$. Consider a properly embedded Lagrangian
$n$-disk $L$ on the page. Then consider two pages $W,W^{\prime}$ and two
copies of the same Lagrangian, called $L,L^{\prime}$ on them. The disks can be
individually perturbed to give Legendrian disks in $M$. Then, by Lemma 3.1,
the open book can be perturbed so that Legendrian disks actually lie on the
page of the open book as Lagrangian disks. We will consider a thickened
binding $B$. We can further perturb $L$ and $L^{\prime}$ so they can be
smoothly joined to give the union, a Legendrian sphere $L\cup L^{\prime}$.
This now gives a closed Legendrian in $(M,\xi)$. We will call this
construction $S_{join}(L)$. Since $L^{\prime}$ is an isotopic copy of $L$, the
notation suppresses $L^{\prime}$. This will be described in more detail in
Section 4.

######  Construction 1.2.

Consider $(M,\xi)$ and $L$ similarly as above. Then, the open book can be
stabilised, by modifying the page by attaching a Weinstein $n$-handle along
$\partial L$, and then performing a positive Dehn twist along the resulting
Lagrangian $n$-sphere, obtained by taking the union of $L$ and the core of the
handle. This resulting manifold is contactomorphic to $(M,\xi)$. Also, the
Lagrangian sphere in the new page can be perturbed to a Legendrian sphere. We
will call this Legendrian $S_{stab}(L)$.

Our main result is that these constructions give Legendrian isotopic spheres
and are isotopic to the standard Legendrian unknot. The standard Legendrian
unknot is defined to be the Legendrian realisation of the $n$-dimensional
unknotted sphere in a Darboux neighbourhood of a contact manifold. Its front
projection can be inductively constructed by starting from the unknot with
maximal Thurston-Bennequin number in $(\mathbb{R}^{3},\xi_{st})$ and
successively spinning half of the front projection.

######  Theorem 1.3.

Consider a supporting open book decomposition of a contact manifold $(M,\xi)$.
Consider a Lagrangian disk $L$ in the page. Then $S_{join}(L)$ is Legendrian
isotopic to $S_{stab}(L)$, and they are both isotopic to the standard
Legendrian unknot.

A technical issue in the above proof is the following: $S_{join}$ is defined
with respect to a certain open book decomposition of $(M,\xi)$, whereas
$S_{stab}$ is defined with respect to the stabilisation of the open book
decomposition. While the open books support contactomorphic contact manifolds,
to prove Legendrian isotopy, one needs to see stabilisation of the open book
as an embedded operation. We will show the following in Section 3:

######  Theorem 1.4.

Assume $(M^{2n+1},\xi)$ admits the supporting open book $(B,\nu)$, whose
stabilisation along the Legendrian disk $L$, which is a Lagrangian in the
page, is $(B^{\prime},\nu^{\prime})$. Then $\nu$ and $\nu^{\prime}$ are
obtained from each other by surgering out a $(2n+1)$-disk neighbourhood of
$L$, and replacing it by another $(2n+1)$-disk.

######  Remark 1.5.

Although the theorems are stated for all dimensions, there is a caveat with
ambient dimension 3. As observed in Remark 4.3 of [vK17], the proof of Lemma
3.1 does not work similarly in this dimension. However, the theorems hold if
we start with Lagrangian arcs in pages that are also Legendrian in the
manifold, and that can always be arranged for some open book decomposition.

######  Construction 1.6.

This construction of Legendrian spheres in $(\mathbb{R}^{2n+1},\xi_{st})$ was
introduced by Ekholm in [Ekh16]. Start with a Lagrangian disk $L$ which is
cylindrical near its boundary, in the symplectisation of $\mathbb{R}^{2n-1}$.
Embed it in a hypersurface in $\mathbb{R}^{2n+1}$ transverse to the Reeb flow.
Then join the Legendrian lifts of this disk and a reflection of the disk in
the same hypersurface to obtain a Legendrian sphere $\Lambda(L,L)$. An example
of this construction is given in Figure 1.

\begin{overpic}[width=466.14127pt,tics=10]{Cobordism} \end{overpic} Figure 1.
Front projection of a Legendrian sphere $\Lambda(L,L)$ in
$(\mathbb{R}^{5},\xi_{s}t)$, where $L$ is the Lagrangian disk described by a
pinch move from the middle knot and going to two Legendrian unknots, which are
then capped off.

In [CE17], Courte-Ekholm show that $\Lambda(L,L)$ is isotopic to the standard
Legendrian unknot.

The contact manifold $(S^{2n+1},\xi_{st})$ is supported by the open book where
the pages are symplectomorphic to $(D^{2n},\omega_{st})$, and the monodromy is
the identity. The binding is contactomorphic to $(S^{2n-1},\xi_{st})$. Given a
Lagrangian disk $L$ in $(B^{2n},\omega_{st})$, one can construct all the three
Legendrians as mentioned above. We can show that Courte-Ekholm’s result is a
particular case of Theorem 1.3.

######  Corollary 1.7.

(Originally proven in [CE17]) Given a Lagrangian disk $L$ in
$(B^{2n},\omega_{st})$, $\Lambda(L,L)$ is isotopic to the standard Legendrian
unknot.

The paper is organised as follows: In Section 2, we introduce relevant
background and give references for further exposition. In Section 3, we prove
Theorem 1.4. Then in Section 4, we prove the first half of Theorem 1.3, namely
that the join and stabilisation constructions give isotopic spheres. Finally
in Section 5, we prove that the constructions give the standard Legendrian
unknot, completing the proof of Theorem 1.3 and also show how to recover
Courte-Ekholm’s result.

###  1.1. Acknowledgements

I would like to thank my advisor John Etnyre for the discussions that led to
asking the questions that this paper eventually answers, and also for his
patience and guidance in helping me come up with the proof strategies. I am
further indebted to his careful reading and comments on various drafts of this
paper, and for his assistance in organising the exposition. I thank Sylvain
Courte and Tobias Ekholm for answering my questions about their result over
email. I am also grateful to Hyunki Min and James Conway for insightful
conversations at the beginning of the project. This work was partially
supported by NSF grant DMS-1906414.

##  2\. Background

In this section we will give the necessary background to set up the outline
for the proofs of the main theorems.

###  2.1. Legendrian submanifolds, front projections, and Legendrian surgery

######  Definition 2.1.

Given a contact manifold $(M^{2n+1},\xi)$, an $n$-dimensional submanifold $L$
is called a Legendrian if $T_{x}(L)\subset\xi_{x}$ for every $x\in L$.

It is well-known that a Legendrian sphere in a contact manifold always has a
standard neighbourhood.

######  Lemma 2.2.

If $S$ is a Legendrian $n$-sphere in $(M^{2n+1},\xi)$, then in any open set
containing $S$ there is a neighborhood $N$ with boundary $\partial
N=S^{n}\times S^{n}$ contactomorphic to an $\epsilon$-neighbourhood
$N_{\epsilon}$ of the zero section $Z$ in the 1-jet space of $S^{n}$, denoted
$J^{1}(S^{n})$. We call $N$ a standard neighbourhood.

The model we will use to describe Legendrian surgery is understood as what is
happening on the boundary when a Weinstein handle is attached along the
Legendrian sphere. This is called the flat Weinstein model and is described in
Section 3 of [vK17] in more generality, for isotropic surgery along $S^{k}$
for $k\leq n$. Our description follows the exposition there.

Notation: To make the notation less cluttered when we talk about
$\mathbb{R}^{2n+2}$, we will write the coordinates
$(z_{1},w_{1},\cdots,z_{n+1},w_{n+1})$ as $(z,w)$. The symplectic form
$\omega_{0}=\sum_{i=1}^{n+1}dz_{i}\wedge dw_{i}$ will be referred to as
$dz\wedge dw$. Similar liberties will be taken with 1-jet space coordinates
where $(z,p_{1},q_{1},\cdots,p_{n},q_{n})$ will be truncated to $(z,p,q)$, and
the contact structure there is $\ker(dz+pdq)$. Products between vectors should
be thought of as dot products.

Consider the symplectic manifold $(\mathbb{R}^{2n+2},\omega_{0})$, where the
coordinates are $(n+1)$ pairs of $(z,w)$ coordinates, and $\omega_{0}=dz\wedge
dw$. The vector field $X=2z\partial_{z}-w\partial_{w}$ is Liouville. The set
$S_{-1}\coloneqq\\{(z,w)\mid|w|^{2}=1\\}$ is transverse to $X$ and inherits
the contact form $\alpha=2zdw+wdz$.

In $S_{-1}$, the sphere $\\{z=0,|w|^{2}=1\\}$ describes a Legendrian sphere.
Using $\psi_{W}:J^{1}(S^{n})\to S_{-1}$ given by $(z,q,p)\mapsto(zq+p,q)$, we
get a strict contactomorphism between $S_{-1}$ and the standard neighbourhood
described in Lemma 2.2. Thus $S_{-1}$ can be regarded as the standard
neighbourhood of a Legendrian sphere.

Now, Legendrian surgery along a Legendrian $S$ will involve removing a
neighbourhood of $S$ identified with $S_{-1}$ and gluing in another contact
hypersurface of $(\mathbb{R}^{2n+2},\omega_{0})$. The contact hypersurface
involved in that is called $S_{1}$ and we describe it here. Define functions
$f$ and $g$, described in Figure 2, to satisfy the following:

  * •

$f$ is increasing on $[1-\epsilon,\infty)$

  * •

$f(w)=1$ for $w\in[0,1-\epsilon),f(w)=w+\epsilon$ for $w>1-\frac{\epsilon}{2}$

  * •

$g$ is increasing on $(0,1+\epsilon)$

  * •

$g(z)=z$ for $z<1$, $g(w)=1+\epsilon$ for $w>1+\epsilon$

\begin{overpic}[width=466.14127pt,tics=10]{handle.jpeg}
\put(260.0,198.0){$1+\epsilon$} \put(272.0,188.0){$1$}
\put(70.0,198.0){$1+\epsilon$} \put(82.0,188.0){$1$}
\put(130.0,120.0){$1-\epsilon$} \put(150.0,120.0){$1$} \put(80.0,220.0){$f$}
\put(275.0,220.0){$g$} \put(344.0,120.0){$1$} \put(352.0,120.0){$1+\epsilon$}
\end{overpic} Figure 2. The functions $f$ and $g$ used to describe Legendrian
surgery.

Then, define the hypersurface $S_{1}\coloneqq\\{(z,w)\mid
f(w^{2})-g(z^{2})=0\\}$. As $X$ is transverse to $S_{1}$, it inherits a
contact structure. Then, Legendrian surgery along $S$ is removing $\nu(S)\cong
S_{-1}$ and gluing $S_{1}$ in its place. If $S\subset(M,\xi)$, and there is a
symplectic manifold $W$ obtained by attaching a Weinstein handle to part of
the symplectisation $(M\times[0,1],d(e^{t}\alpha))$, along $S$ in
$M\times\\{1\\}$, the Legendrian surgery along $S$ can be understood as the
upper boundary of $W$.

###  2.2. Weinstein handle attachment

We briefly review the notion of Weinstein handle attachments here, which we
will need to define stabilisation of open books. For more detailed exposition
the reader is encouraged to consult [CE12].

A Weinstein domain is the symplectic analogue of a smooth handlebody. For a
$2n$-dimensional domain, Weinstein $k$-handles can have index at most $n$, and
are attached along isotropic $(k-1)$-spheres in the convex boundary. Recall
that a submanifold $S$ of a contact manifold is called isotropic if
$T_{x}S\subset\xi_{x}$ for all $x\in S$.

######  Definition 2.3.

A Weinstein handle of index $k$ is $h^{k}=D^{k}\times D^{2n-k}$ with a
symplectic structure so that $\partial_{-}h^{k}=(\partial D^{k})\times
D^{2n-k}$ is concave, and $\partial_{+}h^{k}=D^{k}\times(\partial D^{2n-k})$
is convex. Moreover, $D^{k}\times\\{0\\}$ is isotropic and its intersection
with $\partial_{-}h^{k}$ is an isotropic $S^{k-1}$ in the contact structure
induced on $\partial_{-}h^{k}$. Thus, the attaching sphere of a Weinstein
$k$–handle is an isotropic $S^{k-1}$. Given an isotropic sphere $S^{k-1}$ in
the convex boundary of a symplectic manifold with a choice of trivialization
of its conformal symplectic normal bundle, one can attach a Weinstein k–handle
by identifying a neighborhood of the isotropic sphere with
$\partial_{-}h^{k}$.

A Weinstein handle of index $n$ is called a critical Weinstein handle, and is
attached along a Legendrian sphere. It will be useful for us to understand the
local model for attaching a critical Weinstein handle. Consider
$\mathbb{R}^{2n}$ with the symplectic structure $\sum_{i=1}^{n}dx_{i}\wedge
dy_{i}$. Now consider $H_{a,b}\coloneqq D_{a}\times D_{b}$, where $D_{a}$ is
the disk of radius $a$ in the $x_{i}$ subspace and $D_{b}$ the disk of radius
$b$ in the $y_{i}$ subspace. Then, $H_{a,b}$ is a model for the Weinstein
$n$-handle $h^{n}$. The expanding vector field
$v=\sum_{i=1}^{n}-y_{i}dy_{i}+2x_{i}dx_{i}$ induces contact structures on
$\partial_{-}h^{n}=(\partial D_{a})\times D_{b}$ and
$\partial_{+}h^{n}=D_{a}\times(\partial D_{b})$.

###  2.3. Contact open books

The background on contact open books is taken from the lecture notes by Van
Koert [vK17]. The reader is referred to the same for more details.

######  Definition 2.4.

An (abstract) contact open book $(\Sigma,\lambda,\phi)$, or
$Open(\Sigma,\phi)$ if we suppress the Liouville form from the notation,
consists of a compact exact symplectic manifold $(\Sigma,\lambda)$ and a
symplectomorphism $\phi:\Sigma\to\Sigma$ with compact support, i.e., it is
identity near $\partial\Sigma$.

######  Definition 2.5.

An (embedded) supporting open book for a contact manifold $(M,\xi)$ is a pair
$(\nu,B)$, where $B$ is a codimension-2 submanifold of $M$ with trivial normal
bundle, such that

  * •

$\nu:(M-B)\to S^{1}$ is a fiber bundle, such that $\nu$ gives the angular
coordinate of the $D^{2}$-factor of a neighbourhood $B\times D^{2}$ of $B$,
and

  * •

if $\alpha$ is a contact form for $\xi$, it induces a positive contact
structure on $B$ and $d\alpha$ induces a positive symplectic structure on each
fiber of $\nu$

The embedded open book constructed from Definition 2.4 is the manifold
$\Sigma\times[0,1]/\sim$, where the equivalence relation $\sim$ identifies all
points $(x,t)$ and $(x,t^{\prime})$ where $x\in\partial\Sigma$, and identifies
points $(x,0)$ with $(\phi(x),1)$. For our purpose, we will employ another
(but equivalent, up to contact isotopy) way of building a manifold from an
abstract open book, where we will have something called the thickened binding.
This construction will work as follows:

######  Definition 2.6.

A manifold constructed from the abstract open book $Open(\Sigma,\phi)$ with
thickened binding is the quotient of the disjoint union of the mapping torus
$\Sigma\times[0,1]/((x,0)\sim(\phi(x),1))$ and the thickened binding
$\partial\Sigma\times D^{2}$, under the identification $(x,t)\sim(x,1,t)$,
where $x\in\partial\Sigma$, and $\\{(x,r,\theta)\mid
r\in[0,1],t\in\mathbb{R}/\mathbb{Z}\\}$ are the coordinates on
$\partial\Sigma\times D^{2}$.

An open book with thickened binding can be given a compatible contact
structure, as shown in Section 2.2 of [vK17]. Every contact manifold has a
supporting open book decomposition, by work of Giroux. Further, the contact
structure supported by an open book is unique upto isotopy, as said by the
next theorem, due to Giroux.

######  Theorem 2.7 (Giroux).

If an open book $(\Sigma,\lambda,\phi)$ supports a contact structure
$(M,\xi_{1})$, and $\xi_{2}$ is another contact structure on $M$ supported by
an open book whose pages are symplectomorphic to $\Sigma$ and the monodromy is
isotopic through symplectomorphisms to $\phi$, then $\xi_{1}$ and $\xi_{2}$
are contactomorphic.

An abstract open book defines a supporting open book for the corresponding
contact manifold. This follows from work of Thurston-Winkelnkemper [TW75] and
Giroux. The reader can refer to [vK17] for a proof (originally by Giroux), and
more details. For open books, by a page we refer to $\Sigma$ for abstract open
books, and to the closure of a fiber of $\nu$ for embedded ones. In the
manifold built from the abstract open book, the equivalence class $[(x,t)]$
for $x\in\partial\Sigma$ is the binding. In the embedded case, $B$ is the
binding. In the thickened binding case, $\partial\Sigma\times D^{2}$ is the
binding. In the embedded case, as $M-B$ has the structure of a fibration over
$S^{1}$, it makes sense to talk about the monodromy of an open book. In the
abstract setting, $\phi$ is called the monodromy.

####  2.3.1. Relative Open Books

We will need the idea of open book decompositions of manifolds with boundary,
and how to glue such decompositions together. The difference from Definitions
2.4 and 2.5 in this case is that the page topology can change. This definition
is closely linked to the notions of partial open books [HKM09] and foliated
open books [LV20] in 3-dimensional contact geometry.

######  Definition 2.8.

An (embedded) supporting relative open book for a contact manifold
$(M,\partial M,\xi)$ is a pair $(\nu,B)$, where $B$ is a codimension-2
submanifold of $M$ with trivial normal bundle, such that

  * •

$\nu:(M-B)\to S^{1}$ is a circle-valued Morse function. Further, $\nu$ gives
the angular coordinate of the $D^{2}$-factor of a neighbourhood $B\times
D^{2}$ of $B$, and

  * •

if $\alpha$ is a contact form for $\xi$, it induces a positive contact
structure on $B$ and $d\alpha$ induces a positive symplectic structure on each
fiber of $\nu$

In our case, the change in page topology will only be effected by adding or
removing a Weinstein $n$-handle. To help the reader’s intuition, we will give
the notion of an abstract relative open book, along the lines of abstract
foliated open books [LV20] in 3 dimensions.

######  Definition 2.9.

An (abstract) supporting relative open book for a contact manifold is a tuple
$(\\{S_{i}\\}_{i=0}^{2k},h)$ where:

  * •

$S_{i}$’s are Liouville domains for odd (or even) $i$ such that $S_{i+1}$ is
obtained from $S_{i}$ by either attaching a standard $2n$-ball or removing
one, as a neighbourhood of a properly embedded Lagrangian $n$-disk, which
happens alternately.

  * •

The boundary of $S_{i}$ is a smoothing of $B\cup\alpha_{i}$, where before
smoothing, $B$ is a convex boundary component with an outward pointing
Liouville vector field, $\alpha_{i}$ is diffeomorphic to $(D^{n}\times
S^{n-1})$, and for alternate $i$ it is a convex or concave boundary component.
The $2n$-ball can be regarded as $D^{n}\times D^{n}$ with the smoothed
boundary having two convex parts, the attaching part $\partial_{-}$ which is
$D^{n}\times S^{n-1}$, and the upper part $S^{n-1}\times D^{n}$. At every
stage, to go from $S_{i}$ to $S_{i+1}$ the ball is either attached along a
concave $\alpha_{i}$, or the convex $\alpha_{i}$ is the upper part of the
boundary of the ball that gets removed.

  * •

$h$ is a symplectomorphism between $S_{2k}$ and $S_{0}$.

In this case, we construct the manifold similarly as before by gluing together
$S_{i}\times I$ pieces, then using $h$ to create a ”mapping torus”, and then
filling in the binding with $B\times D^{2}$. There is a natural correspondence
between the embedded and abstract descriptions. The schematic of the pages
changing is shown in Figure 3. We should point that we expect the full
generalisation of partial and foliated open books in high dimensions should
involve more ways of changing the page topology, and also the boundary can
possibly be enriched to be a convex hypersurface in the sense of Honda-Huang
[HH19]. However, for the purpose of this paper, and understanding
stabilisation, the simple notion described here suffices.

\begin{overpic}[width=466.14127pt,tics=10]{relobd.jpeg}
\put(220.0,205.0){cutting out a ball} \put(230.0,160.0){adding a ball}
\put(50.0,170.0){properly embedded} \put(50.0,160.0){Lagrangian disk}
\put(130.0,155.0){$L$} \put(130.0,60.0){$S_{2j}$} \put(330.0,60.0){$S_{2j+1}$}
\end{overpic} Figure 3. A schematic for how the page topology changes in a
relative open book. The darkened part of the boundary is the region $\alpha$.

######  Example 2.10.

We may obtain a relative open book decomposition of a contact ball $B^{2n+1}$,
obtained as the neighbourhood of a properly embedded Lagrangian disk in the
page of an open book, as follows. A schematic of this decomposition is Figure
7. We derive this in detail in Section 3.

Here $k$ = 4. The manifolds $S_{0}$, $S_{2}$, and $S_{4}$ are the disk
cotangent bundles of the $n$-disk $D^{n}$, while the manifolds $S_{1}$ and
$S_{3}$ are the disk cotangent bundles of the $n$-dimensional annulus
$S^{n-1}\times I$. The map $h$ is the identity. The Lagrangian disk in the
page shows up in this neighbourhood as the core $n$-disk of $S_{0}$, $S_{2}$,
and $S_{4}$.

Our main purpose to define these relative open books is to write down how to
glue two such decompositions of manifolds together to obtain a closed manifold
with an open book decomposition.

######  Lemma 2.11.

Consider two relative open book decompositions on $(M_{1},\partial M_{1})$
denoted $(\\{S_{i,1}\\}_{i=0}^{2k},h_{1})$, and on $M_{2},\partial M_{2}$
denoted $(\\{S_{i,2}\\}_{i=0}^{2k},h_{2})$. Suppose the page boundaries are
denoted $B_{1}\cup\alpha_{i,1}$ and $B_{2}\cup\alpha_{i,2}$. These induce
relative open book decompositions on the thickened boundary of the manifolds
$\partial M_{1}\times I$ and $\partial M_{2}\times I$. Suppose there is a
contactomorphism $\phi:\partial M_{1}\times I\to\partial M_{2}\times I$, which
restricts to each page boundary $\phi|_{(\alpha_{i,1}\times I)}$ as a local
symplectomorphism that identifies the concave boundary at $\alpha_{i,1}$ with
the convex boundary at $\alpha_{i,2}$. This extends the symplectic structures
on $S_{i,1}$ and $S_{i,2}$ to a Weinstein structure on $S_{i,1}\cup S_{i,2}$.
Then, $M_{1}\cup M_{2}$ is a contact manifold with supporting open book
decomposition where the pages are symplectomorphic to $S_{i,1}\cup S_{i,2}$,
and the monodromy is $h_{1}\cup h_{2}$.

###### Proof.

The fact that $M_{1}\cup_{\phi}M_{2}$ is a contact manifold follows because
the contact structures on each of them are identified by $\phi$ over the
gluing region. The relative open book fibrations, by the given conditions,
glue to give a fibration over the glued fibers. ∎

####  2.3.2. Generalised Dehn Twist

Suppose $(W,\omega)$ is a symplectic manifold with an embedded Lagrangian
sphere $L\subset W$. A neighbourhood $\nu_{W}(L)$ is symplectomorphic to a
neighbourhood of the zero section of the canonical symplectic structure on
$(T^{*}S^{n},d\lambda_{can})$, by the Weinstein neighbourhood theorem. The
cotangent bundle of the $n$-sphere $T^{*}S^{n}$ can be regarded as a
submanifold of $\mathbb{R}^{2n+2}$ as the set
$\\{(p,q)\in\mathbb{R}^{n+1}\times\mathbb{R}^{n+1}\mid q\cdot q=1,q\cdot
p=0\\}$. In these coordinates, $\lambda_{can}=pdq$. Define an auxiliary map
describing the normalised geodesic flow

$\sigma_{t}(q,p)=\begin{pmatrix}\cos t&|p|^{-1}\sin t\\\ -|p|\sin t&\cos
t\end{pmatrix}\begin{pmatrix}q\\\ p\end{pmatrix}$

Then define

$\tau(q,p)=\begin{cases}\sigma_{g_{1}(|p|)}&p\neq 0\\\ -Id&p=0\\\ \end{cases}$

$g_{1}$ is a smooth map as graphed in Figure 4. Since $\tau$ is identity
outside a neighbourhood of the Lagrangian $\\{p=0\\}$, it can be extended to
all of $(W,\omega)$ by the identity and defines a symplectomorphism. The map
$\tau$ is called the generalised Dehn twist about the Lagrangian sphere $L$.

\begin{overpic}[width=233.07063pt,tics=10]{dehntwist.jpeg}
\put(82.0,112.0){$\pi$} \put(82.0,122.0){$g_{1}$} \end{overpic} Figure 4. The
function $g_{1}$ parametrising a Dehn twist.

####  2.3.3. Stabilisation of open books

Given a contact open book $M=$ Open $(\Sigma^{2n},\phi)$, suppose $L$ is an
embedded Lagrangian $n$-disk in the page $\Sigma$ whose boundary $\partial L$
is a Legendrian sphere in the binding. Consider $\widetilde{\Sigma}$ to be the
manifold obtained by attaching a Weinstein $n$-handle to $\Sigma$ along
$\partial L$. Then, call $L_{S}$ the Lagrangian sphere in $\widetilde{\Sigma}$
defined by the union of $L$ and the core of the $n$-handle.

######  Definition 2.12.

The contact open book $\tilde{M}\coloneqq$ Open
$(\tilde{\Sigma},\phi\circ\tau_{L_{S}})$, where $\tau_{L_{S}}$ is the Dehn
twist along $L_{S}$, is called the stabilisation of Open$(\Sigma,\phi)$ along
$L$.

The following is a well-known statement due to Giroux. A proof can be found in
[vK17].

######  Proposition 2.13.

The stabilisation of a contact open book Open$(\Sigma,\phi)$ along a
Lagrangian disk $L$ bounding a Legendrian sphere in $\partial\Sigma$ is
contactomorphic to the contact manifold Open$(\Sigma,\phi)$.

In Section 4, we use the following folklore theorem (refer [vK17] for
details), that doing Legendrian surgery on a Legendrian sphere that lives on a
page is the same as changing the monodromy by a Dehn twist about that sphere.

######  Theorem 2.14.

Let Open$(\Sigma,\phi)$ be a contact open book with a Legendrian sphere
$L_{S}$, which is also a Lagrangian sphere in $\Sigma$. Denote the contact
manifold obtained from Open$(\Sigma,\phi)$ by Legendrian surgery along $L_{S}$
by $\widetilde{\text{Open}(\Sigma,\phi)}_{L_{S}}$. Then, the contact manifolds

$\text{Open}(\Sigma,\phi\circ\tau_{L_{S}})\simeq\widetilde{\text{Open}(\Sigma,\phi)}_{L_{S}}$

are contactomorphic.

######  Example 2.15.

This is the higher dimensional analogue of Example 6.4 in [LV20]. The standard
contact $(2n+1)$-sphere is supported by the open book
$Open(D(T^{*}S^{n}),\tau_{S})$. The page $D(T^{*}S^{n})$ can be parametrised
as $\\{(p,q)\in\mathbb{R}^{n+1}\times\mathbb{R}^{n+1}\mid|q|^{2}=1,p\cdot
q=0,|p|^{2}\leq 1\\}$, and $\tau_{S}$ represents a positive Dehn twist about
the sphere $S=\\{p_{i}=0\\}$ in the page. This open book decomposition can be
obtained by gluing together two relative open book decompositions on $(2n+1)$
balls. The first one is the relative decomposition in Example 2.10, but shift
the indices on the pages by 1 (we can do that since the monodromy is
identity). So the page $S_{i,1}$ is really the page $S_{i-1}$ in Example 2.10.
Here the Lagrangian is $S$. The second one is the complement of the
neighbourhood of $S_{0}$. It can be denoted
$(\\{S_{i,2}\\}_{i=0}^{4},\tau_{S_{0}})$, where $S_{1,2}$ and $S_{3,2}$ are
the disk cotangent bundles of $n$-disks, and $S_{0,2}$, $S_{2,2}$, and
$S_{4,2}$ are disk cotangent bundles of the $n$-sphere. The schematic of the
gluing is given in Figure 5. This example is understood in more detail in
Section 3.

\begin{overpic}[width=466.14127pt,tics=10]{ballglue.jpeg}
\put(150.0,190.0){$S_{2j+1,2}$} \put(100.0,190.0){$\alpha_{2j+1,2}$}
\put(250.0,200.0){$S_{2j+1,1}$} \put(150.0,50.0){$S_{2j,2}$}
\put(110.0,50.0){$\alpha_{2j,2}$} \put(250.0,90.0){$S_{2j,1}$}
\put(360.0,110.0){$D(T^{*}S^{n})$} \end{overpic} Figure 5. The schematic for
gluing two relative open books to obtain $S^{2n+1}$ as in Example 2.15. The
concave and convex parts of the boundaries, that are relevant for gluing the
open books, are labeled. The unlabeled boundaries constitute the binding and
are convex.

##  3\. Understanding stabilisation as a surgery

In this section, we will prove Theorem 1.4. We will need a lemma, proved in
[vK17] that will allow us to assume that if $L$ is a Lagrangian disk on the
page of an open book, and $L^{\prime}$ is the associated Legendrian, the open
book pages can be perturbed, without changing the page topology or the
monodromy, so that $L^{\prime}$ is a Lagrangian on a page as well as a
Legendrian in the manifold.

######  Lemma 3.1 (Lemma 4.2 in [vK17]).

Suppose $(M,\xi)$ is a contact manifold of dimension greater than 3 with a
supporting open book where the pages are $\Sigma$ and the monodromy is $\psi$,
i.e. $M=Open(\Sigma,\psi)$. If $L$ is a Lagrangian sphere in the page, then we
can isotope the contact structure on $M$ and find a supporting open book with
symplectomorphic page and isotopic monodromy such $L$ becomes Legendrian in
$M$.

We will now discuss the proof of Theorem 1.4.

###  3.1. Outline of the idea:

We will describe stabilisation of an open book supporting $(M,\xi)$ by
removing and regluing a $(2n+1)$-ball in the manifold. The balls will have two
different relative open book decompositions. Extending the induced relative
open book fibration map of the complement of the first ball, over the new ball
will show that after regluing, we get an open book decomposition of the
manifold that has the same abstract open book that describes the stabilised
manifold. We will glue the relative open books using Lemma 2.11. By
Proposition 2.13, we will conclude that the supported contact structure on $M$
by this open book is contactomorphic to $\xi$.

Now we discuss how we obtain the $(2n+1)$-ball to surger on, as described in
the above paragraph. The manifold $M$ has the open book given by $\nu:M-B\to
S^{1}$, where $\nu^{-1}(pt)\simeq\Sigma$. $L$ is a Lagrangian disk in a page
which is also a Legendrian in $M$. We can also assume that $B$ here is a
tubular neighbourhood of the binding.

Also, consider the abstract open book $Open(D(T^{*}(S^{n})),\tau_{S_{0}})$
which is contactomorphic to $(S^{2n+1},\xi_{st})$. Here $D(T^{*}(S^{n}))$
refers to the disk cotangent bundle of the sphere, which can be parametrised
as $\\{(p,q)\in\mathbb{R}^{n+1}\times\mathbb{R}^{n+1}\mid|q|^{2}=1,p\cdot
q=0,|p|^{2}\leq 1\\}$, and $\tau_{S_{0}}$ represents a positive Dehn twist
about the sphere $S_{0}=\\{p_{i}=0\\}$ in the page. The symplectic form on the
page is given by $\omega=dp\wedge dq$. The Lagrangian disk
$S\coloneqq\\{(p,q)\mid q=(0,\dots,0,1)\\}$ is a Legendrian in the manifold.
Here also, we can assume that the binding is thickened.

By the Weinstein neighbourhood theorems, given two Legendrian disks in
different contact manifolds, they have contactomorphic neighbourhoods. We will
take a $(2n+1)$-ball neighbourhood of $S$ and identify that with a
neighourhood of $L$. We will call the neighbourhood of $S$ as $D_{S}$, and
that of $L$ as $D_{L}$. Assume the pages in the open books are indexed by
$S^{1}=\mathbb{R}/\mathbb{Z}$. This neighbourhood $D$ will be built by taking
thickened neighbourhoods of $S$ (or neighbourhoods of $L$) in pages $(0,0.5)$,
and neighbourhoods of $\partial S$ in pages $(0.5,1)$. This gives $D$ the
structure of a relative open book, and its intersection with the bindings of
the ambient manifolds creates the binding. Hence, it makes sense to talk about
pages of $D$, and other than two singular ones, the pages of $D$ will have two
kinds of topology. They will either be $T^{*}D^{n}=D^{n}\times D^{n}$, or
$T^{*}(S^{n-1}\times I)=D^{n+1}\times S^{n-1}$. Thus $D_{L}=N_{L}\cup
N_{\partial L}\cup B_{L}$, where $N_{L}$ is the union of the neighbourhoods of
$L$, $N_{\partial L}$ is the union of the neighbourhoods of $\partial L$, and
$B_{L}$ is the portion of the binding that comes within $D_{L}$. Similarly,
$D_{S}=N_{S}\cup N_{\partial S}\cup B_{S}$. This is the same relative open
book decomposition that was mentioned in Example 2.10.

\begin{overpic}[width=466.14127pt,tics=10]{Pagemodified}
\put(80.0,150.0){$S_{2j+1,2}$} \put(200.0,150.0){$S_{2j+1,3}$}
\put(80.0,50.0){$S_{2j,2}$} \put(200.0,50.0){$S_{2j,3}$} \end{overpic} Figure
6. This is a schematic of how the pages are being modified under the
stabilisation operation. On the left are pages of $M$, while the annuli
represent $D(T^{*}S^{n})$, their core being $S_{0}$. The blue arc in the
annuli is $S$. In the page of $M$, the blue arc represents $L$. The shaded
portion in the top row left represents a page in $N_{L}$, while that on the
top row middle is a page in $N_{\partial S}$. For the gluing in the top row,
the complement of the shaded region in the small rectangular portion of the
annulus is first identified with the neighbourhood of $L$, then the remaining
region glues on as a critical Weinstein handle. The gluing in the second row
is similar, without the identification step.

Figure 7. The smooth ball $D_{L}$ that gets removed. The foliation is from its
intersection with the pages, and the points where the arcs meet are the
binding. On the right, is a schematic of the neighbourhoods whose intersection
with $D$ is shown in the left figure. The top three are the neighbourhoods of
$L$, the leftmost corresponding to two blue arcs near the equator, the one to
its right two blue arcs nearer the poles, and the rightmost being one of the
neighbourhoods contributing the black arcs. Similarly, the lower three
represent, from left to right, the neighbourhoods of $\partial L$ contributing
green arcs near the equator, ones that go higher (or lower), and the black
arcs.

Here we describe what we meant by _twisting the pages_. From the pages of $M$
where _neighbourhood of $L$_ were removed, we glue in the complements of
$\partial S$ neighbourhood, and the remaining pages, we glue in the complement
of _neighbourhood of $S$_. Which is to say, $\partial N_{L}$ glues to
$\partial N_{\partial S}$, $\partial N_{\partial L}$ glues to $\partial
N_{S}$.

This gluing is done so that for the _complement of $\partial S$
neighbourhood_, which is essentially still $D(T^{*}S^{n})$, a portion is
identified with $D(T^{*}D^{n})$, such that it seems as if $L$ is identified
with the core of this piece, and the remaining $D(T^{*}D^{n})\sim D^{n}\times
D^{n}$ is glued on like attaching a Weinstein $n$-handle along the isotropic
sphere $\partial L$. In the sense of Section 2.3.1, the initial open book
$(B,\nu)$ on $M$ is split as the union of two relative open books on
$M\setminus D_{L}$ and $D_{L}$. The abstract relative open book description of
$D_{L}$ is the same as in Example 2.10, denoted $(\\{S_{i,1}\\}_{i=1}^{4},id)$
while the abstract relative open book description of $M\setminus D_{L}$ is
$(\\{S_{i,2}\\}_{i=1}^{4},\phi_{\nu})$, where $S_{2j+1,2}$ is the complement
of $L$, while $S_{2j,2}$ is the complement of $\partial L$. Similarly, the
open book on $(S^{2n+1},\xi_{st})$ is split into $D_{S}$ and
$S^{2n+1}\setminus D_{S}$, as in Example 2.15. Denote the relative open book
on $S^{2n+1}\setminus D_{S}$ as $(\\{S_{i,3}\\}_{i=1}^{4},\phi_{\nu})$, where
$S_{2j,3}$ is the complement of $S$, while $S_{2j+1,3}$ is the complement of
$\partial S$. Then, the relative open books on $M\setminus D_{L}$ and
$S^{2n+1}\setminus D_{S}$ are glued, matching the indices. The shift in
indices from $D_{L}$ to $D_{S}$ is what we refer to as twisting. A schematic
of what is going on is shown in Fig 6.

The rest of the section is devoted to working out the technical details of
this gluing, and proving that this operation does indeed yield the stabilised
open book as defined in Definition 2.12. For $n=1$, this can be done using
convex surfaces and foliated open books, and is mentioned in [LV20] by Licata-
Vertesi. To help the reader’s intuition, we describe that argument first.

###### Proof of Thm 1.4 in the case $n=1$ (Example 6.6, [LV20]):.

The Legendrian $L$ is a properly embedded arc living on the page of an open
book supporting $(M,\xi)$, while $S$ is a Legendrian arc in $(S^{3},\xi_{st})$
is $S$, which is the Legendrian corresponding to a fiber in the
$D(T^{*}S^{1})$ page of the open book supporting $(S^{3},\xi_{st})$. This is
exactly the annulus open book for $(S^{3},\xi_{st})$, and $S$ is the co-core
arc.

As discussed above, a neighbourhood of $L$ in $(M,\xi)$ is $D_{L}$ is
contactomorphic to $D_{S}$ which is a neighbourhood of $S$. The balls $D_{S}$
and $D_{L}$ can be chosen so that their boundaries are convex, and the leaves
of the characteristic foliation correspond to their intersection with the
pages. This is pictured in Figure 7. The blue represent the intersection of
$\partial D_{S}$ with pages that have a neighbourhood of $S$ inside $D$, i.e.
pages in $N_{S}$, while the green represent the intersection with pages that
have a neighbourhood of $\partial S$ inside $D$, i.e., the pages in
$N_{\partial S}$. The foliation on $\partial D_{L}=\partial(M-D_{L})$ looks
the same. Now glue $M-D_{L}$ to $S^{3}-D_{S}$ by a rotation that maps the
green leaves to the blue leaves. As the leaves of the characteristic foliation
are matched up, the contact structure extends over the gluing and the new open
book on $M$ is modified exactly as described above. ∎

###  3.2. Coordinates on the pages, and neighbourhoods of $L$ and $\partial
L$.

The pages $D(T^{*}D^{n})$ intially are parametrised with coordinates from
$\mathbb{R}^{2n+2}$, as
$\\{(p_{1},p_{2},\cdots,p_{n+1},q_{1},q_{2},\cdots,q_{n+1})\mid\sum
p_{i}^{2}\leq 1,\sum q_{i}^{2}=1,\sum p_{i}q_{i}=0\\}$. The canonical
primitive of the symplectic form is $\lambda=\sum p_{i}dq_{i}$. The disk $S$,
with which the Lagrangian $L$ in the page in $M$ gets identified, is
parametrised as: $S=\\{q=(0,\cdots,0,1)\\}.$

Consider
$\psi:\mathbb{R}\times(T^{*}S^{n}\cap\\{q_{n+1}>0\\})\to\mathbb{R}^{2n+1}$,
given by

$\psi(z,p_{1},p_{2},\cdots,p_{n+1},q_{1},q_{2},\cdots,q_{n+1})=(z,x_{1},\cdots,x_{n},y_{1},\cdots,y_{n})\mbox{
where }x_{i}=q_{n+1}p_{i},y_{i}=\frac{q_{i}}{q_{n+1}}$

This allows us to put coordinates $(x_{1},\cdots,x_{n},y_{1},\cdots,y_{n})$ on
a neighbourhood of $S$ on the page, such that
$D(T^{*}S^{n})\cap\\{q_{n+1}>0\\}$ is identified with

$\\{(x,y)\in\mathbb{R}^{2n}\mid\left(\sum x_{i}^{2}+\left(\sum
x_{i}y_{i}\right)^{2}\right)\left(\sum y_{i}^{2}+1\right)\leq 1\\}$

and $S$ becomes $\\{y_{i}=0\\}$. The following calculation verifies this is a
symplectomorphism:

$\displaystyle\psi^{*}(\sum_{i=1}^{n}x_{i}dy_{i})$
$\displaystyle=\sum_{i=1}^{n}(q_{n+1}p_{i})d(\frac{q_{i}}{q_{n+1}})$
$\displaystyle=\sum_{i=1}^{n}(q_{n+1}p_{i})\frac{q_{n+1}dq_{i}-q_{i}dq_{n+1}}{q_{n+1}^{2}}$
$\displaystyle=\sum_{i=1}^{n}p_{i}dq_{i}-\frac{dq_{n+1}}{q_{n+1}}\sum_{i=1}^{n}p_{i}q_{i}$
$\displaystyle=\sum_{i=1}^{n}p_{i}dq_{i}+p_{n+1}dq_{n+1}$
$\displaystyle=\sum_{i=1}^{n+1}p_{i}dq_{i}$

The above computation uses that $\sum_{i=1}^{n+1}p_{i}q_{i}=0$ implies
$p_{n+1}=-\frac{1}{q_{n+1}}\sum_{i=1}^{n}p_{i}q_{i}$.

The boundary of $D(T^{*}(S^{n}))$, which is given by $|p|^{2}=1$, now becomes

$\\{(x,y)\mid b(x,y)=1\\}\mbox{ where }b(x,y)=\left(\sum x_{i}^{2}+\left(\sum
x_{i}y_{i}\right)^{2}\right)\left(\sum y_{i}^{2}+1\right)$

Figure 8 illustrates how this looks for $n=1$ under the coordinate
transformation by $\psi.$ The red is the binding.

\begin{overpic}[width=233.07063pt,tics=10]{Functions}
\put(140.0,20.0){$b(x,y)=1$, binding} \put(30.0,150.0){$b(x,y)=1$, binding}
\end{overpic} Figure 8. Coordinates on a piece of $D(T^{*}(S^{n}))$, which is
inside the red

The blue in Figure 8 is the binding ”rotated”, i.e.,
$\\{(x,y)\mid\\{b(y,x)=1\\}$. The intersection of the blue and red is
transverse. The yellow is the set of points ”inside the page” that satisfy
$\\{b(x,y)=b(y,x)\\}$. Consider the sets $\\{b(x,y)-b(y,x)=t\\}$, as $t$ runs
from 0 to $\frac{1}{2}$. Also, consider a smooth function $\rho(x,y)$ which is
0 at $\\{b(x,y)=b(y,x)=1\\}$, and 1 on a large sphere inside the page away
from the binding. Now, consider the sets

$B_{t}\coloneqq\\{b(y,x)=\rho(x,y)(b(x,y)+t)+(1-\rho(x,y))\\}$

For $t$ nonzero, they are indicated by the green in the schematic figure.
Define a 1-parameter family of neighbourhoods of $L$ by

$\nu_{t}(L)\coloneqq\\{(x,y)\mid b(x,y)\leq
1,b(y,x)\leq\rho(x,y)(b(x,y)+t)+(1-\rho(x,y))b(x,y)\\}$

Similarly, define a 1-parameter family of neighbourhoods of $\partial L$ by

$\nu_{t}(\partial L)\coloneqq\\{(x,y)\mid b(x,y)\leq
1,b(x,y)\geq\rho(x,y)(b(y,x)+t)+(1-\rho(x,y))b(y,x)\\}$

Here, $t$ belongs to the interval $(0,0.5)$. An indicative picture for the 3
dimensional case is Figure 8.

######  Remark 3.2.

The neighbourhoods $\nu_{t}(L)$ and $\nu_{t}(\partial L)$ are precisely the
”pages” that build up $N_{L}$ and $N_{\partial L}$, respectively, from 3.1
above.

###  3.3. Coordinates on $D$.

Now, we are in a position to exactly describe the parametrised smooth
neighbourhood of the Lagrangian $L$ that will be removed for stabilisation.
Note that the above coordinates parametrise a neighbourhood of $L$, a
Lagrangian disk, sitting on a page in an open book decomposition of $M$.

In $M$, suppose the open book is given by $\nu:(M-B)\to S^{1}$, while suppose
the open book on $S^{2n+1}$ is given by $\nu_{1}:(S^{2n+1}-B^{\prime})\to
S^{1}$. Let us first focus on $D$ as a neigbourhood of $S$ on a page in
$S^{2n+1}$, and obtain a parametrisation of it. For the Legendrian $L$ which
is Lagrangian in the page, a neighbourhood in $M$, call it $D$, can be given
standard coordinates from $I\times\mathbb{R}^{2n}$ modulo some identifications
and a standard binding piece, (we assume that the monodromy effects are
localised in a place away from this neighbourhood). More precisely, if we
assume that the page coordinate or $S^{1}$ factor of the open book on
$S^{2n+1}$ is given by $t\in\mathbb{R}/\mathbb{Z}$, then the monodromy affects
the pages in the $t\in[0.7,0.8]$ interval, and even in those places, it leaves
substantial (as required by the next paragraph) neighbourhoods of $\partial L$
unaffected, in the sense that they can be parametrised directly as standard
pieces of $I\times\mathbb{R}^{2n}$. What we want is to ensure that the pieces
$D_{L}$ or $D_{S}$ can be described without the monodromy of the open book
affecting them.

Suppose in $M$, the pages of the open book decomposition are parametrised in
the $S^{1}$ direction by $t\in\mathbb{R}/\mathbb{Z}$. Consider an interval’s
worth of $L$-neighbourhoods parametrised by $t$, given by:

$N_{L}\coloneqq\\{(x,y,t)\mid(x,y)\in\nu_{\beta(t)}(L),t\in[0,0.5]\\},$

and an interval’s worth of $(\partial L)$-neighbourhoods parametrised by $t$,
given by:

$N_{\partial L}\coloneqq\\{(x,y,t)\mid(x,y)\in\nu_{\beta(t)}(\partial
L),t\in[0.5,1]\\}$

Here $\beta(t)=(0.5)\sin(2\pi t)$. Also, we can assume that in this local
scenario, the binding $B$ is thickened, which allows us to choose the standard
primitive of the symplectic form, $\sum x_{i}dy_{i}$ on the page, giving the
contact form $dt+\sum x_{i}dy_{i}$ on $N_{L}\cup N_{\partial L}$ and match it
up with an appropriate choice of contact form on the binding, for example as
done in Section 2.2, page 4, of [vK17], to get an explicit description of the
contact form restricted to $N_{L}\cup N_{\partial L}\cup B_{L}\subset M$. The
thickened binding inside $D_{L}$ can be parametrised as a quotient set
$B=\\{(x,y,r,t)\mid(x,y)\in\mathbb{R}^{2n},b(x,y)=1,r\in[0,1],t\in\mathbb{R}/\mathbb{Z},(x,y,0,t)\sim(x,y,0,t^{\prime})\\}$,
and in $D_{L}$, the points in $N_{L}$ (or $N_{\partial L}$) and $\partial B$
overlap as: $\\{(x,y,t)\in N_{L}\mid b(x,y)=1\\}$ are matched with
$\\{(x,y,1,t)\in B\\}$ (similarly for $N_{\partial L}$).

We claim that $D_{L}=N_{L}\cup N_{\partial L}\cup B_{L}$ is a smooth
$(2n+1)$-ball. For this, it suffices to check that the boundary $2n$-sphere
$\partial D_{L}$ is smooth. The boundary $\partial D_{L}$, in these
coordinates, can be described as: $\partial N_{L}\cup\partial N_{\partial
L}\cup\partial B_{L}$, modulo equivalences that go into the interior of
$B_{L}$. Note that $\nu_{\beta(0)}(L)=\nu_{\beta(0)}(\partial L)$, and
$\nu_{\beta(0.5)}(L)=\nu_{\beta(0.5)}(\partial L)$. $\partial D_{L}$ can be
described as:

$\displaystyle\bigcup_{t\in[0,0.5]}\\{(x,y,t)\mid
b(y,x)=\rho(x,y)(b(x,y)+\beta(t))+(1-\rho(x,y))\\}$ $\displaystyle\cup$
$\displaystyle\bigcup_{t\in[0.5,1]}\\{(x,y,t)\mid
b(x,y)=\rho(x,y)(b(y,x)+\beta(t-0.5))+(1-\rho(x,y))\\}\cup\\{(x,y,r,t)\in
B\mid b(y,x)=1\\}$

Let us focus on the region where $\rho=1$ and check it is smooth. In that
region $\partial D_{L}$ can be described as:

$\displaystyle\bigcup_{t\in[0,0.5]}\\{(x,y,t)\mid b(y,x)=b(x,y)+\beta(t)\\}$
$\displaystyle\cup$ $\displaystyle\bigcup_{t\in[0.5,1]}\\{(x,y,t)\in\mid
b(x,y)=b(y,x)+\beta(t-0.5)\\}\cup\\{(x,y,r,t)\in B\mid b(y,x)=1\\}$

modulo the identifications mentioned above, and $||(x,y)||$ everywhere
sufficiently small. Other than the binding, the set can be described as a
quotient of $\bigcup_{t\in[0,1]}\\{(x,y,t)\mid b(y,x)-b(x,y)=(0.5)\sin(2\pi
t)\\}$. Thus this is smooth. The smoothness of the whole can be easily checked
as it is the quotient under smooth equivalences of smooth pieces. The same
holds for $D_{S}$.

###  3.4. The surgery step.

The above establishes coordinates on the boundary of $D_{L}$, and they work
for both $D_{L}$ and $D_{S}$. What we will do now is remove $D_{L}$ from $M$
and $D_{S}$ from $S^{2n+1}$, then glue the complements after shifting indices.

We will surger $D_{L}$ out and glue in $S^{2n+1}\setminus D_{S}$ by a map $F$
which will give a contactomorphism between a neighbourhoods of
$\partial(M\setminus D_{L})$ and $(S^{2n+1}\setminus\partial D_{S})$. The
details follow.

Notation: Since both $D_{L}$ and $D_{S}$ are being given the same coordinates,
we will drop the subscript from $D$ and construct $F$ from $\partial D\times
I_{1}$ to $\partial D\times I_{2}$, for two intervals $I_{1}$ and $I_{2}$.

Coordinates on $\partial D\times I$. As mentioned above, $\partial D$ can be
described as a quotient of:

$\displaystyle\bigcup_{t\in\mathbb{R}/\mathbb{Z}}$ $\displaystyle\\{(x,y,t)\in
N_{L}\cup N_{\partial L}\mid b(y,x)=b(x,y)+\rho(x,y)\beta(t))\\}$
$\displaystyle\bigcup$ $\displaystyle\\{(x,y,r,t)\in B\mid
b(y,x)=1,r\in[0,1]\\}$

where the points $(x,y,t)$, when $b(x,y)=b(y,x)=1$, are identified with
$(x,y,1,t)$, and $(x,y,0,t)\sim(x,y,0,t^{\prime})$.

On a collar neighbourhood of the boundary in $S^{2n+1}-D$, call it $\partial
D\times[0,1]$, where $s$ represents the $I$ direction, a contact form is given
by $dt+e^{s}\sum x_{i}dy_{i}$ on the page part, i.e., $(\partial
D\times[0,1])\cap(N_{L}\cup N_{\partial L})$, which is extended to the
interior of the binding part, i.e., $(\partial D\times[0,1])\cap B$, as
$e^{s}h_{1}(r)\sum x_{i}dy_{i}+h_{2}(r)dt$, where $h_{1}$ and $h_{2}$ are
described in the following figure. The $s=1$ end represents the boundary of
the manifold.

\begin{overpic}[width=233.07063pt,tics=10]{sympl_handle_vk.jpeg}
\put(30.0,100.0){$h_{1}$} \put(135.0,100.0){$h_{2}$}
\put(92.0,55.0){$\frac{1}{2}$} \put(200.0,55.0){$\frac{1}{2}$} \end{overpic}
Figure 9. The functions $h_{1}$ and $h_{2}$ for the contact form near the
binding.

###  3.5. Gluing the complements of $D$.

Extend the collar neighbourhood of $\partial(M-D)$ to look like $\partial
D\times[-2,0]$, with the coordinate $s$ representing the interval direction,
and so, on $[-2,-1]$, the contact form on the page part is given by
$e^{s}dt+\lambda_{s}$, and on the binding part by
$e^{s}h_{1}(r)\lambda_{s}+h_{2}(r)dt$. Here $\lambda_{s}\coloneqq(-1-s)(\sum
x_{i}dy_{i})+(s+2)(\sum(-y_{i})dx_{i})$, and the $s=0$ end represents the
boundary of the manifold. On $s\in[-1,0]$, the contact structure is given by
$e^{s}dt+\lambda_{-1}$ on the page part, and by
$e^{s}h_{1}(r)\lambda_{-1}+h_{2}(r)dt$ on the binding. The $s=0$ end
represents the boundary of the manifold.

Now, define the contactomorphism between a neighbourhood of
$\partial(S^{2n+1}-D)$ and $\partial(M-D)$ as follows:

$\displaystyle F:\partial D\times[0,1]$ $\displaystyle\to\partial
D\times[-1,0]$ $\displaystyle(x,y,t,s)$ $\displaystyle\mapsto(-y,x,t+0.5,-s)$

Using this, we will show in the following section that we can apply Lemma 2.11
to glue $S^{2n+1}-D$ to $M-D$, to get back $M$. We will see that this gives an
open book decomposition of $M$.

###  3.6. An open book decomposition of $M$.

We would like to show that the above operation induces an honest open book
decomposition of $M$ that supports the contact structure on $M$. The fact that
the operation smoothly does not change $M$ is because we removed a ball and
glued it back in by a map on the boundary (rotation) which is smoothly
isotopic to the identity. We need to ensure that the conditions for Lemma 2.11
are met.

The contact manifold $M$ originally has an open book fibration given by
$\nu:(M-B)\to S^{1}$. In the neighbourhood of $L$ we called $D$, the map $\nu$
was given on $D\setminus B$ by $(x,y,t)\mapsto t$. The standard
$(2n+1)$-sphere $S^{2n+1}$ originally has an open book fibration given by
$\nu_{1}:S^{2n+1}-B^{\prime}\to S^{1}$, which in the neighbourhood
$(D-B^{\prime})$ was again given by $(x,y,t)\mapsto t$.

Call the fibers of $\nu$ to be all $2n$-dimensional Liouville domains
symplectomorphic to $X$. Then, $L$ is a Lagrangian $n$-disk in $X$, and
$\partial L$ is a Legendrian $(n-1)$-sphere in $\partial X$. To index the
different pages, call $\nu^{-1}(t)=X_{t}$. For $t\in(0,0.5)$, $X_{t}-D$ is a
smooth manifold from which a neighbourhood of $L$ has been removed. For
$t\in(0.5,1)$, $X_{t}-D$ is a smooth manifold from which a neighbourhood of
the $(n-1)$-sphere $\partial L$ has been removed. A neighbourhood of $L$ that
contains $X_{t}\cap D$ is parametrised as $\\{(x,y)\in\mathbb{R}^{2n}\mid
b(x,y)\leq 1\\}$, where locally the symplectic form is $dx\wedge dy$ and $L$
is given by $\\{y=0\\}$.

The fibers of $\nu_{1}$ are disk cotangent bundles of the $n-$sphere, call
them $Y$ and index them as $Y_{t}$ as above. Recall that they can be described
as subsets of $\mathbb{R}^{2n+2}$ in the following way:
$Y=\\{(p,q)\in\mathbb{R}^{n+1}\times\mathbb{R}^{n+1}\mid|q|^{2}=1,p\cdot
q=0,|p|^{2}\leq 1\\}$ with the symplectic form $dp\wedge dq$. As in Section
3.2, we can parametrise a part of $Y$, i.e., where $q_{n+1}>0$, as subsets of
$\mathbb{R}^{2n}$ with the symplectic form $dx\wedge dy$, as
$S=Y|_{q_{n+1}>0}=\\{(x,y)\in\mathbb{R}^{2n}\mid b(x,y)\leq 1\\}$. This region
contains the Lagrangian disk $S=\\{q=(0,\dots,0,1)\\}$ which becomes
$\\{y=0\\}$ in $S$. For $t\in(0,0.5)$, $Y_{t}\setminus
D_{S}=Y_{t}\setminus\nu_{\beta(t)}(S)$ has a neighbourhood of $S$ removed,
thus is the unit cotangent bundle over the disk. For $t\in(0.5,1)$,
$Y_{t}\setminus D_{S}=Y_{t}\setminus\nu_{\beta(t)}(\partial L)$ is obtained
from $Y_{t}$ by removing a neighbourhood of the $(n-1)$-knot $\partial S$ from
$\partial Y_{t}$, and the core $S^{n}$ stays. Thus it is still symplectic
deformation equivalent to $Y_{t}$.

After gluing in $S^{2n+1}\setminus D_{S}$, we want to define an open book
fibration on the manifold $(M\setminus D_{L})\cup(S^{2n+1}\setminus D_{S})$.
On the two pieces, we had the restrictions of the fibrations $\nu$ and
$\nu_{1}$, defined on the complements of the respective bindings $B$ and
$B^{\prime}$.

Define the open book fibration on $((M\setminus D_{L})\setminus
B)\cup((S^{2n+1}\setminus D_{S})\setminus B^{\prime})$, denoted $\nu_{new}$,
by $\nu$ on $((M\setminus D_{L})\setminus B)$ and $(\nu_{1}-0.5)$ on
$((S^{2n+1}\setminus D_{S})\setminus B^{\prime})$. The map extends smoothly.
In the following we verify that it is a fibration. i.e., $\nu_{new}^{-1}(t)$
are symplectomorphic for all $t$. For that, we need to check that
$X_{t}\cup_{F}Y_{t+0.5}$ are symplectomorphic for all $t$.

Consider $U_{1}\subset X$ to be the region described by our coordinates as
$\\{b(x,y)\leq 1\\}\cap\\{b(y,x)\leq 1\\}$. In $Y$ identify a similar
neighbourhood of $S$ and call it $U_{1,S}$. Both $U_{1}$ and $U_{1,S}$ are
diffeomorphic to $D^{n}\times D^{n}$, with symplectic form $\omega=\sum
dx_{i}\wedge dy_{i}$, with Liouville forms $\sum x_{i}dy_{i}$ and
$\sum(-y_{i}dx_{i})$ respectively, and respective Liouville vector fields
$\sum x_{i}\partial_{x_{i}}$ and $\sum y_{i}\partial_{y_{i}}$. The Liouville
vector field points out of the boundary of $U_{1}$.

Now, $Y-U_{1,S}$ can be regarded as a Weinstein $n$-handle, with the same
Liouville vector field, with its attaching region being $Y\cap\\{b(y,x)=1\\}$.

Near $\partial(Y-U_{1,S})$ we have the coordinates
$(x_{1},\dots,x_{n},y_{1},\dots,y_{n})\in\mathbb{R}^{2n}$, the attaching
sphere being given by $\\{x_{i}=0,\sum y_{i}^{2}=1\\}$, and the attaching
boundary of the handle is given by $\\{x_{i}=0,b(y,x)=1\\}$. Consider now
$G:\partial(Y-U_{1,S})\to\partial(X\cap U_{1})$ given by $G(x,y)=(-y,x)$.
Using $G$, one can attach the Weinstein $n$-handle $Y-U_{1,S}$ to $X$ along
$\partial L$. Call this manifold $X_{L}$.

Consider the following: $Y_{t+0.5}-D$ attaching to $X_{t}-D$. Suppose the
gluing of $Y_{t+0.5}-D$ to $X_{t}-D$ happens in two steps. First the gluing
under the restriction of $F$ of $(Y_{t+0.5}-D)\cap U_{1,S}$ to $X_{t}-D$, and
then the remainder of $Y_{t+0.5}-D$ is glued on. After the first step, for
every $t$, we get back $X$. Then the second step is gluing $Y-U_{1,S}$ to it
via $G$. Thus, for every $t$, we get $X$ with a Weinstein handle attached
along $\partial L$.

By construction, the contact structure on $M$ post gluing is supported by this
open book. The monodromy of the new open book decomposition, by construction,
changes by a positive Dehn twist along the sphere $L\cup C$ in the page, where
$C$ is the core of the $n$-handle attached to the page. As this open book has
the same pages and monodromy as the stabilisation of the open book $\nu$
supporting $(M,\xi)$, the contact structure on $M$ is still contactomorphic to
$\xi$, by Theorem 2.7.

This finishes the proof of Theorem 1.4.

##  4\. The isotopy between constructions Join and Stab

In this section, we will first rigorously define the constructions $S_{join}$
and $S_{stab}$ in open books supporting $(M,\xi)$. Then, we shall establish an
isotopy between them, after first ensuring that there is a manifold
contactomorphic to $(M,\xi)$ where both the constructions make sense.

At various points in this section, we will do some perturbations by using
Legendrians defined by piecewise functions. For that we need to carefully
understand how to read the coordinates of such Legendrians in $J^{1}(S^{n})$,
which we will address in the following lemma.

######  Lemma 4.1.

Consider $J^{1}(S^{n})$ parametrised as $\\{(z,q,p)\mid q^{2}=1,p\cdot
q=0\\}$. Given a function $f\in C^{\infty}_{\mathbb{R}}(S^{n})$, $j^{1}(f)$
defines a Legendrian in $J^{1}(S^{n})$, whose coordinates are given by
$(z,q,p)$ such that $z=f(q)$ and $p_{i}=-\frac{\partial f}{\partial
q_{i}}+(df\cdot q)q_{i}$, where $df$ is the vector given by
$(df)_{i}=\frac{\partial f}{\partial q_{i}}$.

###### Proof.

The idea here is simply that when a function $f$ is defined on the coordinates
$(q_{1},\dots,q_{n+1})\in\mathbb{R}^{n+1}$, the vector
$df|_{\mathbb{R}^{n+1}}\coloneqq(\frac{\partial f}{\partial
q_{1}},\dots,\frac{\partial f}{\partial q_{n+1}})$ belongs to
$T^{*}(\mathbb{R}^{n+1})$. Thus for $f\in C^{\infty}_{\mathbb{R}}(S^{n})$ but
defined on the coordinates $(q_{1},\dots q_{n+1})$, $df\in T^{*}(S^{n})$ is
given by the projection of $df|_{\mathbb{R}^{n+1}}$ from
$T^{*}(\mathbb{R}^{n+1})$ to $T^{*}(S^{n})$. ∎

###  4.1. Making sense of join Legendrian.

The join Legendrian $L_{1}\cup L_{2}$ is described in $M$ (pre-stabilisation)
in the following way. Suppose $L_{1}$ is a Lagrangian $n$-disk on the page,
and is also a Legendrian in the manifold. By Lemma 3.1, this can be arranged.
Consider two copies $L_{1}$ and $L_{2}$ in two pages. They can be glued
through the binding. The details follow, and the schematic is given in Figure
10.

\begin{overpic}[width=233.07063pt,tics=10]{sjoin}
\put(105.0,100.0){$L_{1,\epsilon}$} \put(100.0,150.0){$S_{stab}$ in page
containing $L_{1}$} \put(110.0,50.0){$L_{2,\epsilon}$}
\put(170.0,35.0){$q_{n+1}$} \put(198.0,95.0){$f(q_{n+1})$}
\put(40.0,90.0){$S_{join}$} \put(-5.0,174.0){region where boundary of}
\put(-5.0,167.0){thickened binding meets} \put(-5.0,160.0){the boundary of a
page} \put(-5.0,153.0){(pre-stabilisation)} \put(75.0,165.0){$\rightarrow$}
\end{overpic} Figure 10. The above schematic will describe both pre- and post-
stabilisation scenarios. Before stabilisation, the left of the vertical
represents the thickened binding near $J^{1}(L)\subset M$. The horizontal
lines represent $(z=c)$ slices, and the $q_{n+1}$ coordinate is plotted
horizontally. After stabilisation, the page extends to the left, and the
picture represents $J^{1}(S_{Stab}(L))$.

The open book on $M$ before stabilisation, by definition, has identity
monodromy near the boundaries of the pages. In particular, if we consider all
the pages and look near $\partial L_{1}$, we can parametrise it as
$J^{1}(D^{n})$, where $\\{z=c\\}$ represents the pages and the Legendrians
$\\{(z,q,0)\mid z\text{ constant}\\}$ represent copies of $L$ near the
boundary in different pages. Once two copies of $L$ enter the binding, they
can be joined up using the 1-jet of functions defined on $q_{n+1}$ as depicted
in Figure 10. Here we are assuming that the contact form on the binding is
adjusted so that it still looks like $J^{1}(D^{n})$ near the boundaries of the
pages. This amounts to a careful choice of the functions $h_{1}$ and $h_{2}$
in Figure 9. This joined up sphere is $S_{join}(L)$. The portion to the left
of the vertical dotted line in Figure 10 is the binding, while to the right
are the pages. We will parametrise the $n$-disk as
$\\{q\in\mathbb{R}^{n+1}\mid q^{2}=1,q_{n+1}\geq 0\\}$.

Now, the way stabilisation is described in Section 3, the pages are modified
in a small neighourhood of $\partial L$. Thus, we can interpret in these
coordinates as saying that as long as $\sum p_{i}^{2}$ is large enough, the
part of the page is unchanged after stabilisation. In more precise terms, we
make the assumption that $\\{(z,q,p)\in T^{*}S^{n}\mid
q\leq\sqrt{1+\epsilon}\\}$ is the region that is affected. Also, we can assume
that this $\epsilon$ is the same for the Legendrian surgery that happens
during stabilisation, as described in Section 2.1.

This is the choice of $\epsilon$ that will be used for the remainder of this
section.

Thus, we can modify $S_{join}(L)$ by choosing 1-jets of appropriate functions.
At the final stage of this modification, we get $S_{join}(L)$ obtained by
gluing two copies of $L$ ”through the pages”. Since by Lemma 4.1 the $p_{i}$
coordinates depend on this slope, by choosing a high enough slope for the
function we can ensure that the $p_{i}$ coordinates are large enough, and
hence this perturbation of $S_{join}(L)$ lives completely outside the region
which is cut out and replaced by stabilisation, and thus is well-defined after
stabilisation as well. Also, to make it easier for us to define the isotopy
later, we can ensure that $S_{join}(L)$ does not intersect the interior of the
region which is identified with $S_{-1}$ and replaced with $S_{1}$.

We can choose a Legendrian using a function that depends only on $q_{n+1}$. By
Lemma 4.1, for a Legendrian $j^{1}(H(q_{n+1}))\subset J^{1}(S^{n})$,
$p_{i}=\frac{dH}{dq_{n+1}}q_{n+1}q_{i}$ for $i=1,\dots,n$, and
$p_{n+1}=\frac{dH}{dq_{n+1}}(-1+q^{2}_{n+1})$. Thus,

$z^{2}+\sum_{i=1}^{n+1}p_{i}^{2}=H^{2}+\left(\frac{dH}{dq_{n+1}}\right)^{2}q_{n+1}^{2}(1-q_{n+1}^{2})+\left(\frac{dH}{dq_{n+1}}\right)^{2}(1-q_{n+1}^{2})^{2}=H^{2}+\left(\frac{dH}{dq_{n+1}}\right)^{2}(q_{n+1}^{2}+1)$

By choosing $H$ defined on $\\{q_{n+1}\geq\epsilon\\}$ such that
$H^{2}+(\frac{dH}{dq_{n+1}})^{2}(q_{n+1}^{2}+1)>(1+\epsilon)$, and such that
$H(\epsilon)=H^{\prime}(\epsilon)=0$, we can join it with the disk
$\\{q_{n+1}\geq\epsilon\\}$ in the zero section to get a perturbed $S_{join}$.
Such an $H$ would need to satisfy

$\displaystyle\left(\frac{dH}{dq_{n+1}}\right)^{2}\geq\frac{\sqrt{1+\epsilon}(1-t)}{1+q_{n+1}^{2}}\mbox{
whenever $H=t\sqrt{1+\epsilon}$}$

Such an $H$ can be found, e.g. as shown in Figure 10. Consider a real valued
function $H(x)$ on [$0,1$] that starts at 0 with value $-\sqrt{1+\epsilon}$
and slope 0, then increases fast enough so that its slope beats the above
inequality, and then reduces to 0 when the value of the function reaches
$\sqrt{1+\epsilon}$ at $x=\frac{1}{2}$.

This perturbed $S_{join}$ can be seen to be the union of two disks, all
described in Figure 10, namely:

  * •

$L_{1,\epsilon}$, the disk corresponding to $j^{1}(H)$, i.e., the top portion
of Figure 10

  * •

$L_{2,\epsilon}\coloneqq\\{(c_{2},q,0)\mid q_{n+1}\geq\epsilon\\}$ in page
that contains $L_{2}$

The isotopy from $S_{join}$ as originally defined to this can be done in $M$
before stabilisation.

###  4.2. The stabilisation Legendrian

Our goal is to show that in the stabilised manifold, $S_{join}(L)$ is isotopic
through smooth Legendrians to $S_{stab}(L)$. Recall that when $M$ is
stabilised, each page is modified by adding a Weinstein $n$-handle along
$\partial L$, and the core of this handle glues to $L$ to give a Legendrian
sphere, which we call $S_{stab}(L)$.

First note that locally, a neighbourhood of $S_{stab}(L)$ can be identified
with $J^{1}(S^{n})$ with coordinates $\\{(z,q,p)\mid q^{2}=1\\}$, where
$S_{stab}(L)$ is given by $(0,q,0)$. By the description of stabilisation in
Section 3, we can assume that the disk $L$ is exactly the part of
$S_{stab}(L)$ corresponding to $q_{n+1}\geq 0$, and that the page outside a
small neighboourhood of the sphere $\\{q_{n+1}=0\\}$ is exactly the page of
the open book before stabilisation.

In Figure 10, $S_{stab}(L)$ built from the same page that contained $L_{1}$
(above), is seen as the union of the following two disks:

  * •

$L_{\epsilon}\coloneqq\\{(c_{1},q,0)\mid q_{n+1}\geq-\epsilon\\}$

  * •

$C_{\epsilon}\coloneqq\\{(c_{2},q,0)\mid q_{n+1}\leq-\epsilon\\}$

###  4.3. Proof of isotopy.

In the stabilised manifold, we now have the perturbed $S_{join}(L)$ and the
$S_{stab}(L)$, as described in Sections 4.1 and 4.2 respectively. The
perturbed $S_{join}(L)$ can be seen as the union of two disks as described in
Section 4.1, while the perturbed $S_{stab}(L)$ can be seen as the union of two
disks as described in Section 4.2. The proof of the isotopy will involve
moving $C_{\epsilon}$ to $L_{2,\epsilon}$ and $L_{\epsilon}$ to
$L_{1,\epsilon}$, while taking care that their overlaps move consistently. The
following subsection is a reading guide for the rest of the proof.

###  4.4. Proof outline

  1. (1)

We will be using different local models at different parts of the argument. As
shown in Figure 11, a neighbourhood of $S_{stab}(L)$ can be thought of as
$I\times T^{*}(S^{n})$ with a piece cut out and replaced. The orange region is
like a portal that represents the ”surgery torus” that gets glued in. Recall
from Section 2, that Legendrian surgery is done by identifying a neighbourhood
of the Legendrian sphere with $S_{-1}$, then replacing $S_{-1,c}$ by
$S_{1,c}$. We will use a perturbed region called $S_{1}^{st}$ to construct the
first half of the isotopy, then move the pieces to $S_{1,c}$. What will be
important is that $S_{-1}\cap S_{1}=S_{-1}\cap S^{st}_{1}$.

Using the contactomorphisms $\psi_{W}:J^{1}(S^{n})\to S_{-1}$, and
$\psi:S^{st}_{1}\to J^{1}(S^{n})$, we will get local coordinates from
$J^{1}(S^{n})$ on the two pieces
$(J^{1}(S^{n})\setminus\psi_{W}^{-1}(S_{-1,c}))$, and $\psi(S^{st}_{1})$. We
carefully constructed $S_{join}(L)=L_{1,\epsilon}\cup L_{2,\epsilon}$ so that
$L_{1,\epsilon}$ lives on
$\partial(J^{1}(S^{n})\setminus\psi_{W}^{-1}(S_{-1,c}))$. The map
$\psi\circ\psi_{W}$ and its inverse makes sense on $\psi^{-1}(S_{-1}\cap
S_{1}$.

  2. (2)

In Section 4.5.1 we will construct a family of Legendrian disks that live in
$S_{1}^{st}$, with boundary on $S_{-1}\cap S_{1}^{st}$, that define an isotopy
between $\psi_{W}(C_{\epsilon})$ and $\psi_{W}(L_{2,\epsilon})$. We first
construct the disks in $\psi(S_{1}^{st})$, and then bring them to $S_{1}^{st}$
using $\psi^{-1}$. A nice observation is that in $\psi(S_{1}^{st})$, these two
are disks that appear in the front projection as shown in Figure 12, and the
isotopy is essentially twisting their front projections once to a point and
then back to the opposite orientation. In some sense, these are a pair of
Reidemeister 1-moves.

  3. (3)

In Section 4.5.2 we show that the above disks can be used to get a family of
Legendrian disks in $S_{1,c}$, with boundary on $S_{-1}\cap S_{1}$, that
define an isotopy between $\psi_{W}(C_{\epsilon})$ and
$\psi_{W}(L_{2,\epsilon})$. One can imagine that at this point, in Figure 11,
we have a 1-parameter family of disks indexed by $t\in[-1,1]$, such that for
$t\in(-1,1)$, they live inside the orange portal, but we can see their
boundaries at the boundary of the orange region. At $\\{t=1\\}$ we get
$C_{\epsilon}$, while at $\\{t=-1\\}$ we get $L_{2,\epsilon}$.

  4. (4)

The last step, done in Section 4.5.3, is to define a family of disks that will
give an isotopy between the other halves of $S_{stab}(L)$ and $S_{join}(L)$,
i.e., $L_{\epsilon}$ and $L_{1,epsilon}$. What we want is a family of disks,
that in Figure 11, will have their boundaries match up with the boundaries of
the disks living inside the orange, and their interiors will be disjoint. The
way we do that is ensure we can define disks with the boundary condition of
smoothly matching up with the boundaries of the disks found in Section 4.5.2,
such that their interiors live in
$(J^{1}(S^{n})\setminus\psi_{W}^{-1}(S_{-1,c}))$.

###  4.5. Modelling the Legendrian surgery:

We will assume that in the stabilised manifold the Legendrian surgery
corresponding to the Dehn twist about $S_{stab}(L)$ happens away from the
other monodromy of the open book. Stabilisation can be thought of as a two
step process happening on the top convex boundary of the symplectisation
$(I\times M)$, where the first step is attaching an index $(n)$ Weinstein
handle along the isotropic sphere $\partial L$ living in the binding, and the
next step is attaching an $(n+1)$-handle along $L\cup C$. After the first
step, on the boundary, we get a manifold $M^{\prime}$ presented by an open
book $\nu_{int}$ (as in the open book in the intermediate step) whose pages
have been modified by adding an $n$-handle to each, while the monodromy has
been extended by identity over this handle. Then, the next step modifies the
monodromy by adding a Dehn twist. We can call $L\cup C$ inside $M^{\prime}$ as
$S_{stab}^{\prime}(L)$. After the surgery, we get to $M$ and the Legendrian
sphere $S^{\prime}_{stab}(L)$ becomes $S_{stab}(L)$.

Recall that the identification of $S_{-1}$ with the standard neighbourhood of
a Legendrian sphere, $\mathbb{R}\times T^{*}S^{n}$, in Section 2.1, is done
via the following map:

$\displaystyle\psi_{W}:\mathbb{R}\times T^{*}S^{n}$ $\displaystyle\to S_{-1}$
$\displaystyle(z,q,p)$ $\displaystyle\to(zq+p,q)\mbox{ which implies}$
$\displaystyle\psi_{W}^{-1}(z_{1},w_{1})$ $\displaystyle=(z_{1}\cdot
w_{1},w_{1},z_{1}-(z_{1}\cdot w_{1})w_{1})$

Locally, we can think of a neighbourhood
$[-\sqrt{1+\epsilon},\sqrt{1+\epsilon}]\times T^{*}S^{n}$ of
$S_{stab}^{\prime}(L)\subset M^{\prime}$, across pages indexed by
$t\in[0.4,0.6]$ which is removed and reglued to achieve the above, i.e., under
the contactomorphism between $\mathbb{R}\times T^{*}(S^{n})$ and $S_{-1}$,
$[-\sqrt{1+\epsilon},\sqrt{1+\epsilon}]\times T^{*}S^{n}$ is the region that
corresponds to $\\{S_{-1}\mid z^{2}\leq 1+\epsilon\\}$, and this corresponds
to the neighbourhoods of $L$ in pages in $[0.4,0.6]$. As the same indexing of
pages outside $t\in(0.4,0.6)$ carries over to $M$, the following sentence
makes sense. We will consider $S_{stab}(L)$ living in $t=0.4$ and
$S_{join}(L)$ formed by joining copies of $L$ in $t=0.4$ and $t=0.6$.

Now, we will perturb both $S_{join}(L)$ and $S_{stab}(L)$ as described in 4.1
and 4.2 respectively, and see them both as unions of two pieces. We will
construct the isotopy by constructing isotopies between the two pieces and
matching them up along their overlaps. The main idea will be to construct
families of functions whose 1-jets will build the isotopies piecewise and
ensure they glue together.

####  4.5.1. Isotopy of $C_{\epsilon}$ to $L_{2,\epsilon}$

Here we first isotope half of $S_{join}(L)$ to half of $S_{stab}(L)$. These
halves are seen described in Figure 10. $C_{\epsilon}$ is the part of
$S_{stab}(L)$ that lives to the left of the picture. It is part of the core of
the Weinstein handle that was attached to the page during the stabilisation
procedure for the open book.

$\displaystyle L_{2,\epsilon}$
$\displaystyle=\\{(z,q,p)\in(-\sqrt{1+\epsilon},\sqrt{1+\epsilon})\times
T^{*}S^{n}\mid z=-\sqrt{1+\epsilon},q_{n+1}\geq\epsilon,p\equiv 0\\}$
$\displaystyle C_{\epsilon}$
$\displaystyle=\\{(z,q,p)\in[-\sqrt{1+\epsilon},\sqrt{1+\epsilon}]\times
T^{*}S^{n}\mid z=\sqrt{1+\epsilon},q_{n+1}\leq-\epsilon,p\equiv 0\\}$

The idea will be to construct an isotopy between $L_{2,\epsilon}$ and
$C_{\epsilon}$ inside the piece that is glued in during the Legendrian
surgery. A schematic is shown in Fig 11.

\begin{overpic}[width=233.07063pt,tics=10]{isotopy_schematic}
\put(80.0,150.0){$C_{\epsilon}$} \put(150.0,150.0){$L_{\epsilon}$}
\put(160.0,19.0){$L_{2,\epsilon}$} \put(40.0,40.0){$\Big{\uparrow}$z}
\end{overpic} Figure 11. This is a schematic representing the region where the
Legendrian surgery happens. This is a $J^{1}(S^{n})$ neighbourhood of
$S_{stab}(L)$, and is a 3-D representation of Figure 10. The orange is a cross
section at $\\{q=c\\}$ of $\psi_{W}^{-1}(S_{-1,c})$, which is removed during
Legendrian surgery. So only the closure of the complement of the orange and
the solid torus it sweeps out is actually part of the manifold $M$. The cores
of the annuli represent isotopic copies of the Legendrian $S_{stab}(L)$. The
purple lines and their union with a part of the blue arc represents
$L_{1,\epsilon}$. The purple is chosen to lie in the closure of the complement
of the orange region, which represents the fact that this copy of
$S_{join}(L)$ was constructed to lie out of the interior of the Legendrian
surgery region.

We will follow the convention and notation from 2.1 for Legendrian surgery.
First, we consider $S_{stab}^{\prime}(L)$ lying on a page of $M^{\prime}$ with
the open book $\nu_{int}$. A neighbourhood of $S_{stab}^{\prime}(L)$ is
contactomorphic to $S_{-1}=\\{(z,w)\in\mathbb{R}^{2n+2}\mid|w|^{2}=1\\}$ with
the contact form $2zdw+wdz$, induced by the symplectic dilation
$X=2z\partial_{z}-w\partial_{w}$. This is a contact hypersurface of
$(\mathbb{R}^{2n+2},\omega_{st})$. The surgery will be replacing $S_{-1}$ by
$S_{1}$, another contact hypersurface, given by
$S_{1}=\\{(z,w)\in\mathbb{R}^{2n+2}\mid f(|w|^{2})-g(|z|^{2})=0\\}$, with a
similarly induced contact form by $X$. The intersection of $S_{1}$ with
$S_{-1}$ is $\\{(z,w)\in\mathbb{R}^{2n+2}\mid|z|^{2}\geq
1+\epsilon,|w|^{2}=1\\}$. The surgery replaces $\\{S_{-1}\mid|z|^{2}\leq
1+\epsilon\\}$ with $\\{S_{1}\mid|z|^{2}\leq 1+\epsilon\\}$. We will call
these $S_{-1,c}$ and $S_{1,c}$ respectively.

The disks we are interested in are described as follows in $S_{-1}$:

$\psi_{W}(L_{2,\epsilon})=\\{(-\sqrt{1+\epsilon})q,q)\mid
q_{n+1}\geq\epsilon\\}$

$\psi_{W}(C_{\epsilon})=\\{(\sqrt{1+\epsilon})q,q)\mid
q_{n+1}\leq-\epsilon\\}$

Clearly, they can be seen as living in $S_{-1}\cap
S_{1}=\\{(z,w)\in\mathbb{R}^{2(n+1)}\mid|z|^{2}=(1+\epsilon),|w|^{2}=1\\}$.

Finding isotopy inside $S_{1}$: Define
$S_{1}^{st}\coloneqq\\{(z,w)\mid|z|^{2}=1+\epsilon\\}$. Clearly, $S_{1}^{st}$
is also transverse to $X$ and hence inherits a contact structure. Now,
consider the following contactomorphism

$\psi:S^{st}_{1}\to\mathbb{R}\times T^{*}(S^{n})$

given by

(1)
$\psi(z,w)=\bigg{(}\frac{2z.w}{\sqrt{1+\epsilon}},-\frac{z}{\sqrt{1+\epsilon}},w-\frac{(z.w)z}{1+\epsilon}\bigg{)}$

Using this, we can see the disks we are interested in inside
$J^{1}(S^{n})=\mathbb{R}\times T^{*}(S^{n})$.

$\psi\circ\psi_{W}(L_{2,\epsilon})=\\{(-2,q,0)\mid q_{n+1}\geq\epsilon\\}$

$\psi\circ\psi_{W}(C_{\epsilon})=\\{(2,-q,0)\mid q_{n+1}\leq-\epsilon\\}$

######  Remark 4.2.

At this point, one could scale the $z$ and $q$ coordinates to get the above
two isotopic as sets. However, for our purpose, we need the isotopy to respect
the parametrisation and the induced orientation, i.e., under the isotopy, we
want the point $(-2,q,0)\in\psi\circ\psi_{W}(L_{2,\epsilon})$ to flow to
$(2,-q,0)\in\psi\circ\psi_{W}(C_{\epsilon})$. So some more work needs to be
done. The rest of the isotopy is essentially two Reidemeister twists in
$J^{1}(S^{n})$. A schematic is shown in Figure 12.

\begin{overpic}[width=233.07063pt,tics=10]{Reidemeister}
\put(20.0,120.0){$D_{1}=\psi\circ\psi_{W}(C_{\epsilon})$}
\put(80.0,67.0){$D_{0}$}
\put(20.0,13.0){$D_{-1}=\psi\circ\psi_{W}(L_{2,\epsilon})$}
\put(160.0,105.0){$D_{1}=\psi\circ\psi_{W}(C_{\epsilon})$}
\put(190.0,65.0){$D_{0}$}
\put(160.0,30.0){$D_{-1}=\psi\circ\psi_{W}(L_{2,\epsilon})$} \end{overpic}
Figure 12. This is a schematic of what happens in $\psi(S_{1}^{std})$. The
vertical direction is the $z$ direction, and the figure represents what
happens in the isotopy, and how it is really a Reidemeister 1-move. The right
hand side represents the front projection, by projecting out the $p$
coordinates.

Consider the unit $n$-disk $D_{model}\coloneqq\\{(x_{1},\dots,x_{n})\mid\sum
x_{i}^{2}\leq 1\\}$.

For notational convenience, we will define the following

$\epsilon_{t}\coloneqq(\epsilon-1)|t|+1$

Also, for $t\in(0,1]$, consider a sequence of smooth functions $F_{t}$ defined
on the 1-parameter family of disks $\\{(2t,q,0)\mid
q_{n+1}\geq\epsilon_{t}\\}$ in $[-2,2]\times T^{*}S^{n}$, that satisfy the
piecewise properties

$\displaystyle F_{t}$ $\displaystyle\equiv 2t-\delta$ when
$q_{n+1}>\epsilon_{t}+\delta$
$\displaystyle=\frac{\sqrt{1-t^{2}}}{\sqrt{1-\epsilon_{t}^{2}}K_{\epsilon}(t)}\sum_{i=1}^{n}\frac{q_{i}^{2}}{2}+R_{t}$
near $q_{n+1}=\epsilon_{t}$

$F_{1}$ is the constant function 2. $R_{t}$ is there to ensure that at the
boundary, the value of $F_{t}$ is exactly $2t$. $K_{\epsilon}(t)$ will be
defined to satisfy a boundary condition for the disks. $\delta$ is small and
arbitrary.

Now, define a 1-parameter family of maps $D_{t}:D_{model}\to[-2,2]\times
T^{*}S^{n}$ by

$q_{i}(D_{t}(x_{1},\dots,x_{n}))\coloneqq x_{i}\sqrt{1-\epsilon_{t}^{2}}$

for $i=1,\dots,n$ and

$q_{n+1}(D_{t}(x_{1},\dots,x_{n}))\coloneqq\sqrt{1-\sum_{i=1}^{n}q_{i}^{2}}$

,

$z(D_{t}(x_{1},\dots,x_{n}))\coloneqq F_{t}(q)$

$\displaystyle p_{i}(D_{t}(x_{1},\dots,x_{n}))$ $\displaystyle=-\frac{\partial
F_{t}}{\partial q_{i}}+(dF_{t}.q)q_{i}\quad\mbox{when
$q_{n+1}\geq\epsilon_{t}$}$ $\displaystyle\implies$ when
$q_{n+1}=\epsilon_{t}$ $\displaystyle p_{i}$
$\displaystyle=-\frac{\sqrt{1-t^{2}}}{\sqrt{1-\epsilon_{t}^{2}}K_{\epsilon}(t)}q_{i}+\frac{\sqrt{1-t^{2}}}{\sqrt{1-\epsilon_{t}^{2}}K_{\epsilon}(t)}(1-\epsilon_{t}^{2})q_{i}$
$\displaystyle=-\frac{\sqrt{1-t^{2}}}{K_{\epsilon}(t)}\epsilon_{t}^{2}x_{i}$
for $i=1,\dots,n$, and $\displaystyle p_{n+1}$
$\displaystyle=\frac{\sqrt{1-t^{2}}}{\sqrt{1-\epsilon_{t}^{2}}K_{\epsilon}(t)}(1-\epsilon_{t}^{2})\epsilon_{t}$

From 1, we can see that

(2) $\psi^{-1}(z,q,p)=\bigg{(}(-\sqrt{1+\epsilon})q,p-\frac{zq}{2}\bigg{)}$

So $\psi(S_{1}\cap S_{-1})$ is exactly the set
$\frac{z^{2}}{4}+\sum_{i=1}^{n+1}p_{i}^{2}=1$. Setting this condition for the
disks at the extremes, i.e. when $q_{n+1}(D_{t})=\epsilon_{t}$, we will derive
the value of $K_{\epsilon}(t)$, as follows. (Note that at these points
$\sum_{i=1}^{n}x_{i}^{2}=1$):

$\displaystyle\frac{z^{2}}{4}+\sum_{i=1}^{n+1}p_{i}^{2}=1\implies\sum_{i=1}^{n+1}p_{i}^{2}=1-t^{2}$
plugging in the values of $p_{i}$’s, we get
$\displaystyle\sum_{i=1}^{n}(-\frac{\sqrt{1-t^{2}}}{\sqrt{1-\epsilon_{t}^{2}}K_{\epsilon}(t)}q_{i}+\frac{\sqrt{1-t^{2}}}{\sqrt{1-\epsilon_{t}^{2}}K_{\epsilon}(t)}(1-\epsilon_{t}^{2})q_{i})^{2}+(\frac{\sqrt{1-t^{2}}}{\sqrt{1-\epsilon_{t}^{2}}K_{\epsilon}(t)}(1-\epsilon_{t}^{2})\epsilon_{t})^{2}=1-t^{2}$
$\displaystyle\implies$
$\displaystyle\frac{1-t^{2}}{(1-\epsilon_{t}^{2})K_{\epsilon}(t)^{2}}(\epsilon_{t}^{4}\sum_{i=1}^{n}q_{i}^{2}+(1-\epsilon_{t}^{2})^{2}\epsilon_{t}^{2})=1-t^{2}$
$\displaystyle\implies$
$\displaystyle\frac{1}{K_{\epsilon}(t)^{2}}(\epsilon_{t}^{4}\sum_{i=1}^{n}x_{i}^{2}+(1-\epsilon_{t}^{2})\epsilon_{t}^{2})=1\implies
K_{\epsilon}(t)^{2}=\epsilon_{t}^{2}$

By Lemma 4.1, the $D_{t}$’s define a 1-parameter family of smooth Legendrian
disks in $[-2,2]\times T^{*}S^{n}$. Further, if we define
$D_{0}:D_{model}\to[-2,2]\times T^{*}S^{n}$ by

$z(D_{0})=q_{i}(D_{0})=0\text{ for }i=1,\dots,n,\\\
q_{n+1}(D_{0})=1,p_{i}(D_{0})=-x_{i}\text{ for }i=1,\dots,n,p_{n+1}(D_{0})=0$

then we have a smooth family of disks $D_{t}$ for $t\in[0,1]$. $D_{1}$ is
exactly the disk $\psi\circ\psi_{W}(C_{\epsilon})$.

The above thus defines an isotopy from $\psi\circ\psi_{W}(C_{\epsilon})$ to
$D_{0}$, which one can see in Figure 12 as being an isotopy from the top disk
which is part of the core of $T^{*}S^{n}$, to the disk ”transverse” to the
core. One can think of this as one Reidemeister-1 move, as suggested in Figure
12. Now extend by an isotopy from $D_{0}$ to $D_{-1}$ where $D_{-1}$ is
exactly $\psi\circ\psi_{W}(L_{2,\epsilon})$, defined as follows:

For $t\in(0,-1]$, define the functions as follows:

$\displaystyle F_{t}$ $\displaystyle=2t+\delta$ when
$q_{n+1}>\epsilon_{t}+\delta$
$\displaystyle=-\frac{\sqrt{1-t^{2}}}{\sqrt{1-\epsilon_{t}^{2}}K_{\epsilon}(t)}\sum_{i=1}^{n}\frac{q_{i}^{2}}{2}+R_{t}$
when $q_{n+1}=\epsilon_{t}$

Now, analogous to above, for $t\in(0,-1])$ define a 1-parameter family of maps
$D_{t}:D_{model}\to[-2,2]\times T^{*}S^{n}$ by

$q_{i}(D_{t}(x_{1},\dots,x_{n}))\coloneqq-x_{i}\sqrt{1-\epsilon_{t}^{2}}$

for $i=1,\dots,n$ and

$q_{n+1}(D_{t}(x_{1},\dots,x_{n}))\coloneqq\sqrt{1-\sum_{i=1}^{n}q_{i}^{2}}$

,

$z(D_{t}(x_{1},\dots,x_{n}))\coloneqq F_{t}(q)$

$\displaystyle p_{i}(D_{t}(x_{1},\dots,x_{n}))$ $\displaystyle=-\frac{\partial
F_{t}}{\partial q_{i}}+(dF_{t}.q)q_{i}$ when $q_{n+1}\geq\epsilon_{t}$

This will define an isotopy from $D_{0}$ to
$\psi\circ\psi_{W}(L_{2,\epsilon})$, and putting everything together, we have
defined a smooth 1-parameter family of Legendrians that give an isotopy
between $\psi\circ\psi_{W}(C_{\epsilon})$ and
$\psi\circ\psi_{W}(L_{2,\epsilon})$, with their boundaries on $\psi(S_{1}\cap
S_{-1})$. This family can be pulled back via $\psi^{-1}$ to an isotopy in
$S_{1}^{std}$ between $\psi_{W}(C_{\epsilon})$ and $\psi_{W}(L_{2,\epsilon})$,
with their boundaries on $S_{1}\cap S_{-1}$. We will denote this family of
disks in $S_{1}^{st}$ by $D_{t}^{st}$, where
$D_{1}^{st}=\psi_{W}(C_{\epsilon})$ and
$D_{-1}^{st}=\psi_{W}(L_{2,\epsilon})$.

####  4.5.2. Contact isotopy from $S_{1}^{std}$ to $S_{1}$

Consider a sequence of subsets of $\mathbb{R}^{2n+2}$ defined by
$S_{1,t}=\\{(z,w)\in\mathbb{R}^{2n+2}\mid
f_{t}(|w|^{2})-g_{t}(|z|^{2})=0,|z|^{2}\leq 1+\epsilon\\}$, where $f_{1}=f$
and $g_{1}=g$ are the functions pictured in Fig 9, $f_{0}(x)\equiv
1+\epsilon,g_{0}(x)=x$, and $f_{t}$ and $g_{t}$ are endpoint preserving smooth
homotopies between $f_{0}$ and $f_{1}$, and $g_{0}$ and $g_{1}$, respectively,
in the region $\\{|z|^{2}\leq 1+\epsilon\\}$. Thus,
$S_{1,0}=\\{(z,w)\in\mathbb{R}^{2n+2}\mid|z|^{2}=1+\epsilon\\}=S_{1}^{st}$,
and $S_{1,1}=S_{1}$. As $f_{t}$ and $g_{t}$ have the same values when their
arguments take the extremal values 1 and $1+\epsilon$ respectively, so all the
$S_{1,t}$ have the same boundary at $S_{1}\cap S_{-1}$. All the $S_{1,t}$ are
transverse to the symplectic dilation $X$, thus are a smooth family of contact
embeddings of $I\times D(T^{*}S^{n})$ in $\mathbb{R}^{2n+2}$, that agree on
the boundary $S^{n}\times S^{n}$. Pulling the contact form on $S_{1,t}$ back
via the embedding to $S^{n}\times D^{n+1}$, we get a 1-parameter family of
contact forms that agree on the boundary, call them $\alpha_{t}$. Now, Moser’s
method constructs on $S^{n}\times D^{n+1}$ a vector field that is 0 where the
forms agree, and flowing along this vector field gives a contactomorphism
between $(S^{n}\times D^{n+1},ker(\alpha_{0}))$ and $(S^{n}\times
D^{n+1},ker(\alpha_{1}))$, since the support of the vector field is compact.
This gives a contact isotopy taking $S_{1}^{st}$ to $S_{1}$, and we can use
that to get a family of disks that live in $S_{1}$.

This means that the isotopy in $S_{1}^{st}$ can be identified with an isotopy
in $S_{1}$. Thus, we have a family of disks $D^{1}_{t}$ in $S_{1}$ such that
$\partial D^{1}_{t}=\partial D_{t}^{st}$ for all $t$. Moreover,
$D^{1}_{1}=D_{1}^{st}=\psi_{W}(C_{\epsilon})$ and
$D^{1}_{-1}=D_{-1}^{st}=\psi_{W}(L_{2,\epsilon})$.

Our next step will be to build a family of disks to isotope $L_{\epsilon}$ to
$L_{2,\epsilon}$, but ensuring that the boundaries of these disks match up
with the boundaries of the $\psi_{W}^{-1}(D^{1}_{t})$ we have found till now.

####  4.5.3. Isotopy of $L_{\epsilon}$ to $L_{1,\epsilon}$

In this subsection we will construct an isotopy between the remaining halves
of $S_{join}(L)$ and $S_{stab}(L)$. We will have to be extra careful, as the
family of Legendrian disks we will construct will have to join up with the
family of Legendrian disks found above in subsections 4.5.1 and 4.5.2, to give
a family of spheres between $S_{join}(L)$ and $S_{stab}(L)$.

We will first identify the boundaries of the disks $D_{t}$ for $t\in[-1,1]$.
By the description above, we can see that for $t\in[0,1]$,

$\displaystyle\partial(D_{t})$
$\displaystyle=\\{(z_{t},q_{t},p_{t})\\}\quad\mbox{ where}$ $\displaystyle
z_{t}$ $\displaystyle=2t$ $\displaystyle q_{i,t}$
$\displaystyle=q_{i}(S_{t}(x_{1},\dots,x_{n}))=x_{i}\sqrt{1-\epsilon_{t}^{2}},\mbox{
for $i=1,\dots,n$}$ $\displaystyle q_{n+1,t}$ $\displaystyle=\epsilon_{t}$
$\displaystyle p_{i,t}$
$\displaystyle=p_{i}(S_{t}(x_{1},\dots,x_{n}))=-\epsilon_{t}(\sqrt{1-t^{2}})x_{i}\mbox{
where $\sum x_{i}^{2}=1$, for $i=1,\dots,n$,}$ $\displaystyle p_{n+1,t}$
$\displaystyle=\sqrt{(1-t^{2})(1-\epsilon_{t}^{2})}$

while for $t\in[-1,0]$

$\displaystyle\partial(D_{t})$
$\displaystyle=\\{(z_{t},q_{t},p_{t})\\}\quad\mbox{ where}$ $\displaystyle
z_{t}$ $\displaystyle=2t$ $\displaystyle q_{i,t}$
$\displaystyle=q_{i}(S_{t}(x_{1},\dots,x_{n}))=-x_{i}\sqrt{1-\epsilon_{t}^{2}},\mbox{
for $i=1,\dots,n$}$ $\displaystyle q_{n+1,t}$ $\displaystyle=\epsilon_{t}$
$\displaystyle p_{i,t}$
$\displaystyle=p_{i}(S_{t}(x_{1},\dots,x_{n}))=-\epsilon_{t}(\sqrt{1-t^{2}})x_{i}\mbox{
where $\sum x_{i}^{2}=1$, for $i=1,\dots,n$,}$ $\displaystyle p_{n+1,t}$
$\displaystyle=-\sqrt{(1-t^{2})(1-\epsilon_{t}^{2})}$

We will want to build a family $(D^{n})_{t}$ for $t\in[-1,1]$ living in
$S_{-1}$, such that $(\partial D^{n})_{t}=\partial D_{t}$ as described above,
$(D^{n})_{1}=\psi_{W}(L_{\epsilon})$, $(D^{n})_{-1}=\psi_{W}(L_{2,\epsilon})$.
We will first see these boundaries in $\psi_{W}^{-1}(S_{-1})$, and for that we
will see the images of the above via
$\psi_{W}^{-1}\circ\psi^{-1}:\mathbb{R}\times
T^{*}S^{n}\mid_{\frac{z^{2}}{4}+p^{2}=1}\to\mathbb{R}\times T^{*}S^{n}$, which
is given by:

$\psi_{W}^{-1}\circ\psi^{-1}(z,q,p)=\bigg{(}\frac{z\sqrt{1+\epsilon}}{2},p-\frac{zq}{2},-q\sqrt{1+\epsilon}-\frac{z\sqrt{1+\epsilon}}{2}(p-\frac{zq}{2})\bigg{)}$

The coordinates of $\psi_{W}^{-1}(\partial D_{t})$ are as follows:

for $t\in[0,1]$,

$\displaystyle\psi_{W}^{-1}\circ\psi^{-1}(\partial D_{t})$
$\displaystyle=\\{(z_{t},q_{t},p_{t})\\}\quad\mbox{ where}$ $\displaystyle
z_{t}$ $\displaystyle=t\sqrt{1+\epsilon}$ $\displaystyle q_{i,t}$
$\displaystyle=-\epsilon_{t}(\sqrt{1-t^{2}})x_{i}-t(\sqrt{1-\epsilon_{t}^{2}})x_{i}$
$\displaystyle q_{n+1,t}$
$\displaystyle=\sqrt{(1-t^{2})(1-\epsilon_{t}^{2})}-t\epsilon_{t}$
$\displaystyle p_{i,t}$
$\displaystyle=(\sqrt{1+\epsilon})x_{i}\bigg{(}-\sqrt{1-\epsilon_{t}^{2}}-t(-\epsilon_{t}(\sqrt{1-t^{2}})-t(\sqrt{1-\epsilon_{t}^{2}}))\bigg{)}$
$\displaystyle p_{n+1,t}$
$\displaystyle=(\sqrt{1+\epsilon})\bigg{(}-\epsilon_{t}-t(\sqrt{(1-t^{2})(1-\epsilon_{t}^{2})}-t\epsilon_{t})\bigg{)}$

for $t\in[-1,0]$,

$\displaystyle\psi_{W}^{-1}\circ\psi^{-1}(\partial D_{t})$
$\displaystyle=\\{(z_{t},q_{t},p_{t})\\}\quad\mbox{ where}$ $\displaystyle
z_{t}$ $\displaystyle=t\sqrt{1+\epsilon}$ $\displaystyle q_{i,t}$
$\displaystyle=-\epsilon_{t}(\sqrt{1-t^{2}})x_{i}+t(\sqrt{1-\epsilon_{t}^{2}})x_{i}$
$\displaystyle q_{n+1,t}$
$\displaystyle=-\sqrt{(1-t^{2})(1-\epsilon_{t}^{2})}-t\epsilon_{t}$
$\displaystyle p_{i,t}$
$\displaystyle=(\sqrt{1+\epsilon})x_{i}\bigg{(}\sqrt{1-\epsilon_{t}^{2}}-t(-\epsilon_{t}(\sqrt{1-t^{2}})+t(\sqrt{1-\epsilon_{t}^{2}}))\bigg{)}$
$\displaystyle p_{n+1,t}$
$\displaystyle=(\sqrt{1+\epsilon})\bigg{(}-\epsilon_{t}-t(-\sqrt{(1-t^{2})(1-\epsilon_{t}^{2})}-t\epsilon_{t})\bigg{)}$

By choosing functions appropriately we can define an isotopy from
$L_{\epsilon}$ to $L_{1,\epsilon}$, by constructing a family of disks whose
boundaries match up with the above. First, note that we can extend the disks
by a function that depends only on $q_{n+1}$. Now, note that $L_{\epsilon}$
lives on $\psi_{W}^{-1}(\\{S_{1}\mid|z|^{2}=(1+\epsilon)\\})$, while
$L_{1,\epsilon}$ meets $\psi_{W}^{-1}(\\{S_{1}\mid|z|^{2}\leq(1+\epsilon)\\})$
only at $\partial L_{1,\epsilon}$. We will choose the family of disks going
between them such that their interiors are disjoint from the interior of
$\psi_{W}^{-1}(S_{1})$. This will automatically ensure that when these disks
are joined with $D^{1}_{t}$, we get a family of embedded Legendrians.

The boundary conditions above will ensure that the boundaries live on
$\psi_{W}^{-1}(S_{1})$, so in order that the interior of the disks do not, we
need $\\{z^{2}+p^{2}\geq 1+\epsilon\\}$ for $(z,q,p)$ coordinates on the
interiors of the disks.

By Lemma 4.1 and the discussion in 4.1, the required family of Legendrians can
be described by $H_{t}(q_{n+1})$ defined on $\\{q_{n+1}\geq q_{n+1,t}\\}$
which satisfy $H_{t}(q_{n+1,t})=t\sqrt{1+\epsilon}$,
$H^{\prime}_{t}(q_{n+1,t})=\frac{\sqrt{1-t^{2}}}{\epsilon_{t}\sqrt{1-t^{2}}+|t|\sqrt{1-\epsilon_{t}^{2}}}$,
$H_{t}^{2}+(\frac{dH_{t}}{dq_{n+1}})^{2}(q_{n+1}^{2}+1)\geq(1+\epsilon)$.
Also, $H_{-1}=H$ from 4.1, while $H_{1}\equiv 0$, so as to give
$L_{2,\epsilon}$ and $L_{\epsilon}$, respectively.

These initial points are plotted in Figure 13 for $\epsilon=0.1$, while the
initial slopes are plotted in Figure 14.

Figure 13.  Plot of $q_{n+1,t}$ vs $t$ when $\epsilon=0.1$.
$q_{n+1,t}=\pm\sqrt{(1-t^{2})(1-\epsilon_{t}^{2})}-t\epsilon_{t}$ Figure 14.
Plot of $H^{\prime}_{t}$ vs $t$ when $\epsilon=0.1$.
$H^{\prime}_{t}(q_{n+1,t})=\frac{\sqrt{1-t^{2}}}{\epsilon_{t}\sqrt{1-t^{2}}+|t|\sqrt{1-\epsilon_{t}^{2}}}$.

The choice of generating functions $H_{t}$ that will now work, i.e.
$j^{1}(H_{t})$ will describe an isotopy between $L_{\epsilon}$ and
$L_{2,\epsilon}$, can be described by the following method.

Similar to the function $H$ that was chosen at the end of 4.1, consider a
smoothly varying family $H_{t}(X)$ of functions defined on [$q_{n+1,t},1$]
that start with the specified slopes, that increase fast enough, beating the
required inequality, and then reduce to 0 when the function value reaches
$\sqrt{1+\epsilon}$.

A schematic of the graphs of $H_{t}$ can be seen in Figure 15.

\begin{overpic}[width=233.07063pt,tics=10]{isotopy2}
\put(120.0,70.0){$H_{-1}$} \put(90.0,145.0){$H_{1}$} \end{overpic} Figure 15.
Here we can see the family $H_{t}$ that will be used to describe the family of
disks isotoping $L_{\epsilon}$ to $L_{1,\epsilon}$, drawn near their boundary
so the picture does not get cluttered. Plotted are the initial points and
slopes, following how $q_{n+1,t}$ and $p_{n+1,t}$ change as per the plotted
graphs. Now, to describe the full disks, we can pick any family of homotopic
functions that start at $H_{1}$, which is the constant 0 function at the top,
then plot functions such that their slopes are large enough to ensure
$z^{2}+p^{2}\geq 1$, and then $H_{-1}$ is the purple graph plotted.

##  5\. The constructions give the standard unknot

In this section we show that $S_{join}(L)$ is isotopic to the standard
Legendrian unknot, thus completing the proof of Theorem 1.3. Then we connect
our constructions and results to Courte-Ekholm’s work in [CE17] and prove
Corollary 1.7. We will show that via a sequence of contactomorphisms and
isotopies, $S_{join}(L)$ in $(S^{2n+1},\xi_{st})$ can be identified with
$\Lambda(L,L)$.

###  5.1. Isotoping $S_{join}$ to the unknot

###### Proof of Theorem 1.3..

In Section 4, we established that $S_{join}(L)$ and $S_{stab}(L)$ are
isotopic. All that remains now is to show that $S_{join}$ is the unknot. Using
generating functions as described in Figure 10, $S_{join}(L)$ can be isotoped
to $L_{2,\epsilon}$ joined with a pushoff arbitrarily close to it. In the
figure, that amounts to bringing the top strand labelled $L_{1,\epsilon}$
arbitrarily close to the lower one. Thus $S_{join}(L)$ is described in a
standard neigbourhood of the Legendrian disk $L$, and hence the isotopy class
of the sphere depends on the isotopy class of the disk. By Gromov’s
h-principle for Legendrian immersions, any two Legendrian disks are isotopic.
Thus we can consider the same construction in a Darboux neighbourhood for the
standard Legendrian disk, for which this construction gives the standard
Legendrian unknot. This proves that $S_{join}(L)$ is isotopic to the unknot. ∎

###  5.2. Identifying with $\Lambda(L,L)$ in the $(S^{2n+1},\xi_{st})$ case

We will quickly review Ekholm’s [Ekh16] construction and Courte-Ekholm’s proof
strategy [CE17]. To start with, one considers a codimension 1 space
$W_{\rho}=\\{z=(\frac{\rho(x_{n})}{\rho^{\prime}(x_{n})})y_{n}\\}\cap\\{0<x_{n}<1\\}$
in $(\mathbb{R}^{2n+1},\xi_{st})$ which is transverse to the Reeb flow. The
function $\rho$ can be considered a smoothing of the function $(1-|x|)$. A
Lagrangian disk $L$ with a cylindrical end can be embedded in $W_{\rho}$ with
its cylindrical end approaching $x_{n}=0$. Reflecting the $x_{n},y_{n}$
coordinates, another copy of $L$, $L^{-}$, can be similarly embedded with its
cylindrical end approaching that of $L$. Taking the Legendrian lift of these
and joining along the ends gives the Legendrian sphere $\Lambda(L,L)$.
Deforming the hypersurface $W_{\rho}$ to $\\{z=0\\}$, while staying transverse
to $\partial_{z}$, recovers the construction of $\Lambda(L,L)$ as originally
described in [Ekh16].

To show that this is the unknot, they describe the construction in
$(\mathbb{R}^{2n+1},ker(dz-\sum_{i=1}^{n-1}y_{i}dx_{i}+r_{n}^{2}d\theta_{n})$.
Then, they modify the contact structure so that two halves of the sphere can
be brought close to each other by flowing along $\partial_{\theta_{n}}$,
sketching out a pre-Lagrangian $(n+1)$-disk foliated by Legendrian disks in
the process.

###### Proof of Corollary 1.7.

By a contactomorphism (refer Example 2.1.10 in [Gei08]), we can identify a
hemisphere of $(S^{2n+1},\xi_{st})$ with $(\mathbb{R}^{2n+1},ker(dz+\sum
r_{i}^{2}d\theta_{i}))$. Under this, the modified $S_{join}(L)$, as in the
proof above, which is the Legendrian lift of $L$ in a page, joined with a
pushoff, is identified with the Legendrian lift of a disk in
$\\{z=0\\}\cap\\{0<x_{n}<1\\}$, joined with a pushoff. (Note that in the proof
above we have modified the open book pages so that the Lagrangian in the page
is a Legendrian, but for the contactomorphism we want to respect the open book
structure given by $\theta_{n}$, and hence have to use Legendrian lifts.)

Then, by a further sequence of contactomorphisms, we can get to
$(\mathbb{R}^{2n+1},ker(dz-\sum_{i=1}^{n-1}y_{i}dx_{i}+r_{n}^{2}d\theta_{n})$,
where the Legendrian sphere is still given by the Legendrian lift of a disk in
$\\{z=0\\}$ joined with its pushoff. Now deforming the hypersurface
$\\{z=0\\}\cap\\{0<x_{n}<1\\}$ to $W_{\rho}$, and tracing back through Courte-
Ekholm’s proof, it is clear that this sphere is in fact isotopic to
$\Lambda(L,L)$. Since it is the image under a contactomorphism of the unknot,
$\Lambda(L,L)$ is in fact the unknot. This completes the proof of Corollary
1.7. ∎

######  Remark 5.1.

Courte and Ekholm also use the h-principle for their proof. Hence, the proof
here does not use a radically different argument. The point is to see their
result as a part of a more general picture.

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