# Compatible Relative Open Books on Relative Contact Pairs
via Generalized Square Bridge Diagrams

İ. Özge Taşpınar  Dept. of Mathematics, Middle East Technical University,
Ankara, TURKEY<EMAIL_ADDRESS>and  M. Firat Arikan  Dept. of Mathematics,
Middle East Technical University, Ankara, TURKEY<EMAIL_ADDRESS>
###### Abstract.

Using square bridge position, Akbulut-Ozbagci and later Arikan gave algorithms
both of which construct an explicit compatible open book decomposition on a
closed contact $3$-manifold which results from a contact $(\pm 1)$-surgery on
a Legendrian link in the standard contact $3$-sphere. In this article, we
introduce the “generalized square bridge position” for a Legendrian link in
the standard contact $5$-sphere and partially generalize this result to the
dimension five via an algorithm which constructs relative open book
decompositions on relative contact pairs.

###### Key words and phrases:

Contact structure, open book, Legendrian, monodromy, square bridge, surgery

###### 2010 Mathematics Subject Classification:

57R65, 58A05, 58D27

##  1\. Introduction

Let $Y$ be an oriented $(2n+1)$-dimensional manifold. A (co-oriented positive)
contact structure on $Y$ is a maximally non-integrable hyperplane field
$\xi=\textrm{Ker}(\alpha)$ where $\alpha$ is a smooth $1$-form on $Y$ such
that $\alpha\wedge(d\alpha)^{n}>0$. The pair $(Y,\xi)$, or more simply
$Y_{\xi}$, is called a contact manifold and $\alpha$ is called a contact form
for $\xi$. An open book (decomposition) on $Y$ is a pair $(B,\pi)$ where $B$
(called the binding) is a codimension two submanifold with a trivial normal
bundle $B\times D^{2}$, and $\pi:Y\setminus B\rightarrow\mathbb{S}^{1}$ is a
smooth fibration map which agrees with the angular coordinate on the
$D^{2}-$factor near $B$. The closure of the fiber
$F=\overline{\pi^{-1}(\theta)}$ is called the page over
$\theta\in\mathbb{S}^{1}$. The monodromy of $(B,\pi)$ is the self-
diffeomorphism $h:F\to F$ defined by the return map of the flow of the vector
field $\partial/\partial\theta$. Up to contactomorphism (a diffeomorphism
respecting contact structures) an open book can be also given abstractly as a
pair $(F,h)$. An open book $(B,\pi)$ and $\xi$ on $Y$ are compatible if there
exists a contact form $\alpha$ for $\xi$ such that $\alpha$ is a contact form
on $B$, $d\alpha$ defines a symplectic structure on each page, and the
orientation on $B$ induced from $\alpha$ agrees with the one induced from
$d\alpha$ considering $B$ as the contact boundary of a symplectic page. For
more details see [13] and [8].

A $k$-dimensional ($k\leq n$) submanifold $\Sigma$ of $Y_{\xi}$ is isotropic
if $T_{p}\Sigma\subset\xi_{p}$ for all $p\in\Sigma$. An $n$-dimensional
isotropic submanifold is said to be Legendrian. Any Legendrian $\Sigma\subset
Y_{\xi}$ comes with a contact (or Thurston-Bennequin) framing given by a
vector field along $\Sigma$ which is transverse to $\xi$. For a Legendrian
knot $K$ in
$(\mathbb{R}^{3},\xi_{0}^{3}=\textrm{Ker}(\alpha_{0}^{3}=dz+x_{1}dy_{1}))$,
its contact framing can be computed as the integer

$tb(K)=writhe(K)-\frac{1}{2}\\#(\text{cusps of }K),$

called the Thurston-Bennequin number of $K$, where $writhe(K)$ is the sum of
the signs of the crossings in the front projection of $K$ onto the
$y_{1}z-$plane. In higher dimensional standard contact spaces, the Thurston-
Bennequin number of a Legendrian submanifold coincides with a topological
invariant and one can easily be computed (see [7]).

A contact surgery along a Legendrian $n$-knot $K\subset Y_{\xi}$ (i.e., the
image of an $n$-sphere under a Legendrian embedding) is an operation which
results in a new surgered contact manifold. The surgered contact manifold is
formed by first removing a tubular neighborhood $N$ of $K$ in $Y_{\xi}$, and
then gluing $S^{n}\times D^{n+1}$ back using a chosen diffeomorphism from
$S^{n}\times S^{n}$ onto $\partial N$ so that contact distributions of the
pieces smootly glued together. The contact surgery coeficient (which
determines the gluing diffeomorphism) is measured relative to the contact
framing of $K$. (See Section 4.2 for details). In dimension three, Ding and
Geiges proved the following result:

######  Theorem 1.1 ([6]).

Every closed contact $3$-manifold can be obtained by a contact $(\pm
1)$-surgery on a Legendrian link in the standard contact $3$-sphere
$\mathbb{S}^{3}_{st}=(\mathbb{S}^{3},\xi_{st}^{3})$.

This result implies that any closed contact $3$-manifold can be described by a
contact surgery diagram where contact surgery coefficients are $\pm 1$’s. Also
it was shown (see below) how contact $(\pm 1)$-surgery effects a compatible
open book when a surgery knot lies on a page:

######  Theorem 1.2 ([8]).

Let $(F,h)$ be an open book compatible with the contact manifold
$(M^{3},\eta)$. If a Legendrian knot $K$ lies on a page $F$, then the surgered
manifold $(M^{\pm 1},\eta^{\pm 1})$ obtained by the contact $(\pm 1)$-surgery
along $K$ has a compatible open book $(F,h\circ{\tau_{K}}^{\mp})$ where
$\tau_{K}^{\mp 1}$ denotes the negative/positive (or left-handed/right-handed)
Dehn twist along $K$, respectively.

Akbulut and Ozbagci gave an algorithm constructing an open book on
$\mathbb{S}^{3}_{st}$ so that a given Legendrian link lies on a page which is
the Seifert surface of a suitable $(p,q)$-torus link [1]. Arikan improved this
algorithm by using contact cell decompositions [2]. Combining these results
with Theorem 1.2, one can easily construct an open book on any closed contact
$3$-manifold obtained by a contact $(\pm 1)$-surgery along a Legendrian link
in $\mathbb{S}^{3}_{st}$.

In higher dimensions, some similar results along the same lines are as
follows: Koert constructed an open book on the standard contact $5$-sphere
$\mathbb{S}^{5}_{st}=(\mathbb{S}^{5},\xi_{st}^{5})$ such that its pages are
Briskorn varieties $V_{\epsilon}(p,q,2)$ and its binding is Seifert filling of
a Briskorn manifold $\Sigma(p,q,2)$ [14]. The right-handed (generalized) Dehn
twist $\tau:T^{*}S^{n}\rightarrow T^{*}S^{n}$ along the zero section of the
cotagent bundle of $n$-spheres and some of its applications were studied by
Seidel [20]. A definition of contact $(\pm 1)$-surgery on a Legendrian sphere
based on a generalized Seidel-Dehn twist was given by Avdek [3]. Lazarev
proved that if $(Y^{2n+1},\xi)$, $n\geqslant 2$ with an almost Weinstein
filling, then it can be obtained from $\mathbb{S}^{2n+1}_{st}$ by a sequence
of isotropic surgeries (of any index at most $n$) and a single coisotropic
surgery (of index $n$) [16]. Conway and Etnyre generalized Theorem 1.1 to any
odd dimension:

######  Theorem 1.3 ([5]).

Any oriented contact manifold $(Y^{2n+1},\xi)$ can be obtain from
$\mathbb{S}^{2n+1}_{st}$ by a sequence of isotropic and coisotropic surgeries
of all indicies.

In this paper, we focus on a certain class of closed contact $5$-manifolds
obtained by a contact $(\pm 1)$-surgeries along a special class of Legendrian
$2$-links (i.e., links of $2$-spheres) inside $\mathbb{S}^{5}_{st}$ as we
describe below. We introduce a new diagram for such a $2$-link which can be
considered as a generalization of square bridge diagram of a Legendrian
$1$-link in $\mathbb{R}^{3}$ defined by Lyon [17]. After presenting the
generalized square bridge diagram in Section 3, we will prove our main result
on the existence of open books on a certain class of closed contact
$5$-manifolds in Section 4. We will give examples in Section 5 to demonstrate
how our algorithm works. In order to state the main theorem, one needs to
introduce some relative notions first. We refer the reader to Section 2 for
more details.

Consider the standard contact structures
$\xi_{0}^{5}=\textrm{Ker}(\alpha_{0}^{5}=dz+x_{1}dy_{1}+x_{2}dy_{2})$ on
$\mathbb{R}^{5}$ with coordinates $(x_{1},y_{1},z,x_{2},y_{2})$, and
$\xi_{0}^{3}=\textrm{Ker}(\alpha_{0}^{3}=dz+x_{1}dy_{1})$ on $\mathbb{R}^{3}$
with coordinates $(x_{1},y_{1},z)$ as above. Note that
$(\mathbb{R}^{3},\xi_{0}^{3})$ (resp., $\mathbb{S}^{3}_{st}$) is a contact
submanifold of $(\mathbb{R}^{5},\xi_{0}^{5})$ (resp., $\mathbb{S}^{5}_{st}$).
Also recall that

$(\mathbb{R}^{3},\xi_{0}^{3})\cong\mathbb{S}^{3}_{st}\setminus\\{pt\\},\quad(\mathbb{R}^{5},\xi_{0}^{5})\cong\mathbb{S}^{5}_{st}\setminus\\{pt\\}.$

Therefore, any isotropic submanifold of $\mathbb{S}^{3}_{st}$ (resp.,
$\mathbb{S}^{5}_{st}$) is also an isotropic submanifold of
$(\mathbb{R}^{3},\xi_{0}^{3})$ (resp., $(\mathbb{R}^{5},\xi_{0}^{5})$), and
vice versa. This will enable us to work inside $(\mathbb{R}^{3},\xi_{0}^{3})$
and $(\mathbb{R}^{5},\xi_{0}^{5})$.

######  Definition 1.4.

Let
$\mathbb{L}=\bigsqcup_{i=1}^{r}L_{i}\subset(\mathbb{R}^{5},\xi_{0}^{5})\subset\mathbb{S}^{5}_{st}$
be a Legendrian $2$-link. $\mathbb{L}$ is called a relative link if its
equatorial link
$\mathbb{K}=\mathbb{L}\cap\mathbb{S}^{3}_{st}=\bigsqcup_{i=1}^{r}{K}_{i}$ is a
Legendrian $1$-link in
$(\mathbb{R}^{3},\xi_{0}^{2})\subset\mathbb{S}^{3}_{st}$. A relative
Legendrian $2$-link $\mathbb{L}$ is called admissible if the following two
conditions are satistified:

  * (i)

Each link component $L_{i}$ is Legendrian unknotted $2$-sphere in
$\mathbb{S}^{5}_{st}$.

  * (ii)

The Legendrian front of each link component
${K}_{i}={L}_{i}\cap\mathbb{S}^{3}$ onto the $y_{1}z$-plane has no self-
crossing.

A(n) (admissible) relative $2$-knot is a(n) (admissible) relative $2$-link
with a single component.

######  Definition 1.5 (See also Section 4.2).

Let $\mathbb{L}\subset\mathbb{S}^{5}_{st}$ be a(n) (admissible) relative
Legendrian $2$-link with its equatorial link
$\mathbb{K}\subset\mathbb{S}^{3}_{st}$. Suppose the (closed) contact
$5$-manifold $Y^{5}_{\xi}$ is obtained by only contact $(\pm 1)$-surgeries
along $\mathbb{L}$, and the corresponding contact $(\pm 1)$-surgeries along
$\mathbb{K}$ in $\mathbb{S}^{3}_{st}$ produce the (closed) contact submanifold
$M^{3}_{\eta}\subset Y^{5}_{\xi}$. Such a surgery will be called a(n)
(admissible) relative contact $(\pm 1)$-surgery along a pair
$(\mathbb{L},\mathbb{K})$. The pair $(Y^{5}_{\xi},M^{3}_{\eta})$ will be
called a(n) (admissible) relative contact pair.

We note that by forgetting all the contact geometric terms in the above
definitions, one can similarly introduce a relative (smooth) topological pair
$(Y^{5},M^{3})$ obtained by a relative topological surgery along a pair
$(\mathbb{L},\mathbb{K})\subset(\mathbb{S}^{5},\mathbb{S}^{3})$ where
$\mathbb{K}=\mathbb{L}\cap\mathbb{S}^{3}$ is the equatorial link.

A relative open book (decomposition) on a relative (smooth) topological pair
$(Y^{5},M^{3})$ is an open book $\mathcal{OB}_{Y}=(B,\pi)$ of $Y$ such that
$\mathcal{OB}_{M}=(B\cap M,\pi|_{M\setminus(B\cap M)})$ defines an open book
of $M$. For a relative contact pair $(Y^{5}_{\xi},M^{3}_{\eta})$, we say that
$(\xi,\eta)$ is compatible with the relative open book
$(\mathcal{OB}_{Y},\mathcal{OB}_{M})$ on $(Y^{5},M^{3})$ if the contact
structures are compatible with the corresponding open books. Now we are ready
to state our main theorems generalizing the results in [2]:

######  Theorem 1.6.

For any admissable Legendrian $2$-link $\mathbb{L}$ in $\mathbb{S}^{5}_{st}$
with its equatorial link $\mathbb{K}\subset\mathbb{S}^{3}_{st}$, there exists
a compatible relative open book
$(\mathcal{OB}_{\mathbb{S}^{5}},\mathcal{OB}_{\mathbb{S}^{3}})$ on the
(trivial) relative contact pair $(\mathbb{S}^{5}_{st},\mathbb{S}^{3}_{st})$
such that for any component $L$ of $\,\mathbb{L}$ and its equatorial knot
$K\subset\mathbb{K}$, we have

  1. (i)

$K$ lies on a Weinstein page $F_{\mathbb{S}^{3}}$ of the (compatible) open
book $\mathcal{OB}_{\mathbb{S}^{3}}$ of $\mathbb{S}^{3}_{st}$,

  2. (ii)

the page framing of $K$ is equal to its contact framing in
$\mathbb{S}^{3}_{st}$,

  3. (iii)

$L$ lies on a Weinsten page $F_{\mathbb{S}^{5}}$ of the (compatible) open book
$\mathcal{OB}_{\mathbb{S}^{5}}$ of $\mathbb{S}^{5}_{st}$,

  4. (iv)

the page framing of $L$ is equal to its contact framing in
$\mathbb{S}^{5}_{st}$, and

  5. (v)

$F_{\mathbb{S}^{3}}$ is symplectically and properly embedded in
$F_{\mathbb{S}^{5}}$, and the monodromies of
$\mathcal{OB}_{\mathbb{S}^{5}},\mathcal{OB}_{\mathbb{S}^{3}}$ are,
respectively, the products of right-handed Dehn twists

$h_{\mathbb{S}^{5}}=\tau_{\mathcal{D}_{1}}\circ\tau_{\mathcal{D}_{2}}\circ\cdots\circ\tau_{\mathcal{D}_{m}},\quad
h_{\mathbb{S}^{3}}=\tau_{\gamma_{1}}\circ\tau_{\gamma_{2}}\circ\cdots\circ\tau_{\gamma_{m}}$

(for some $m$) where each $\gamma_{k}$ is an isotropic equatorial circle of a
Legendrian sphere $\mathcal{D}_{k}$.

######  Theorem 1.7.

Let $(Y^{5}_{\xi},M^{3}_{\eta})$ be any admissible relative contact pair
obtained by an admissible relative contact $(\pm 1)$-surgery along an
admissible link $\mathbb{L}$ in $\mathbb{S}^{5}_{st}$. Then there exists a
compatible relative open book $(\mathcal{OB}_{Y},\mathcal{OB}_{M})$ on
$(Y^{5}_{\xi},M^{3}_{\eta})$ with Weinstein pages. Moreover, the pages and the
monodromies of the open books $\mathcal{OB}_{Y}$, $\mathcal{OB}_{M}$ are
precisely and systematically determined by the contact surgery along
$\mathbb{L}$.

##  2\. Stabilizations of Relative Open Books

It is known that any odd dimensional manifold has an open book decomposition.
Stabilization is a process which transforms a given open book of a manifold
into another open book for the same manifold. It changes both pages and the
monodromy. One can naturally generalize this process for the relative open
books on relative contact pairs as what we discuss below. To this end, let us
introduce relative open books in the most generality. As before we first
define them for (smooth) topological pairs and then make a connection with
contact geometry. Although any odd dimensions can be considered, we will
restrict our attention to the pairs of dimension three and five only.

######  Definition 2.1.

Let $M^{3}$ and $Y^{5}$ be closed (smooth) topological manifolds such that $M$
is a submanifold of $Y$. A relative open book (decomposition) on the pair
$(Y,M)$ is a quadruple $(B_{Y},B_{M},\pi_{Y},\pi_{M})$ such that

  * (i)

$(B_{M},\pi_{M})$ and $(B_{Y},\pi_{Y})$ are open book decompositions of $M$
and $Y$, resp., and

  * (ii)

$\pi_{M}=\pi_{Y}\circ i_{M}$ where $i_{M}:M\hookrightarrow Y$ is the embedding
map.

Note that the conditions imply that
$\overline{\pi^{-1}_{M}(t)}\subset\overline{\pi^{-1}_{Y}(t)}$ and
$\partial\overline{\pi^{-1}_{M}(t)}\subset\partial\overline{\pi^{-1}_{Y}(t)}$
for all $t\in S^{1}$. This means that $2$-dimensional pages of
$(B_{M},\pi_{M})$ are properly embedded in the corresponding $4$-dimensional
pages of $(B_{Y},\pi_{Y})$. Moreover, the monodromy of $(B_{Y},\pi_{Y})$
restricts to the monodromy of $(B_{M},\pi_{M})$. These suggest that relative
open books can be also defined in a purely abstract way as follows:

######  Definition 2.2.

Let $M^{3}$ and $Y^{5}$ be closed (smooth) topological manifolds such that $M$
is a submanifold of $Y$. An abstract relative open book (decomposition) on the
pair $(Y,M)$ is a quadruple $(F_{Y},F_{M},h_{Y},h_{M})$ such that

  * (i)

$(F_{M},h_{M})$ and $(F_{Y},h_{Y})$ are abstract open book decompositions of
$M$ and $Y$, resp.,

  * (ii)

the page $F_{M}$ is properly embedded in the page $F_{Y}$ (so, $\partial
F_{M}\subset\partial F_{Y}$),

  * (iii)

$h_{Y}|_{F_{M}}=h_{M}$.

In the presence of both geometric and topological structures, it is worth to
try to relate them in some way. Giroux showed that there is one such relation,
called compatibility as we have mentioned earlier, between contact structures
and open book decompositions, and proved that there is a one-to-one
correspondence between contact structures (up to isotopy) and their compatible
open books (up to positive stabilization) [11]. One can easily modify
compatibility notion for relative contact pairs and relative open book
decompositions as follows:

######  Definition 2.3.

Let $(Y^{5}_{\xi},M^{3}_{\eta})$ be a relative contact pair. A relative open
book decomposition $(B_{Y},B_{M},\pi_{Y},\pi_{M})$ on $(Y,M)$ is compatible
with the pair $(\xi,\eta)$ if

  * (i)

the open book $(B_{Y},\pi_{Y})$ is compatible with the contact structure $\xi$
on $Y$, and

  * (ii)

the open book $(B_{M},\pi_{M})$ is compatible with the contact structure
$\eta$ on $M$.

######  Example 2.4.

Consider the (trivial) relative contact pair
$(\mathbb{S}^{5}_{st},\mathbb{S}^{3}_{st})$ of standard contact spheres as
before. Let $\mathbb{S}^{2}$ be the standard Legendrian $2$-unknot in
$\mathbb{S}^{5}_{st}$ and $\mathbb{S}^{1}$ be its isotropic equatorial circle
which is the standard Legendrian $1$-unknot in $\mathbb{S}^{3}_{st}$ (see
Figure 1 for their front projections). Then it is a (customary) standard fact
that $(D^{*}\mathbb{S}^{2},\tau_{\mathbb{S}^{2}})$ and
$(D^{*}\mathbb{S}^{1},\tau_{\mathbb{S}^{1}})$ are compatible open books on
$\mathbb{S}^{5}_{st}$ and $\mathbb{S}^{3}_{st}$, respectively, where
$D^{*}\mathbb{S}^{n}$ denotes the unit disk cotangent bundle of the
$n$-sphere. It is also well known that $D^{*}\mathbb{S}^{1}$ is properly
embedded in $D^{*}\mathbb{S}^{2}$ and the right-handed Dehn twist
$\tau_{\mathbb{S}^{2}}:D^{*}\mathbb{S}^{2}\rightarrow D^{*}\mathbb{S}^{2}$
restricts to the right-handed Dehn twist
$\tau_{\mathbb{S}^{1}}:D^{*}\mathbb{S}^{1}\rightarrow D^{*}\mathbb{S}^{1}$.
Therefore,
$(D^{*}\mathbb{S}^{2},D^{*}\mathbb{S}^{1},\tau_{\mathbb{S}^{2}},\tau_{\mathbb{S}^{1}})$
is a compatible relative open book decomposition on the relative contact pair
$(\mathbb{S}^{5}_{st},\mathbb{S}^{3}_{st})$.

According to the Giroux correspondence, the positive stabilization does not
only fix the diffeomorphism type of the underlying smooth manifold, but it
also respects to the contact structure. More precisely, positively
stabilization of a compatible open book on a contact manifold is compatible
with a contact structure (on the same smooth manifold) which is isotopic to
the original one. This correspondence can be naturally generalized for
compatible relative open books on contact relative pairs. To this end, one
needs to define the notion of relative stabilization. For completeness, let us
first recall stabilization which arises as a special case of a more general
process, called plumbing:

Let $(Y^{2n+1}_{i},\xi_{i})$ be two closed contact $(2n+1)$-manifolds such
that each $\xi_{i}=\textrm{Ker}(\alpha_{i})$ is compatible with an open book
$(F_{i},h_{i})$ on $Y_{i}$. Suppose that $A_{i}$ is a properly embedded
Lagrangian $n$-disk in a symplectic page $(F_{i},d\alpha_{i})$ with Legendrian
boundary $\partial A_{i}\subset(\partial
F_{i},\textrm{Ker}(\alpha_{i}|_{\partial F_{i}}))$. By the Weinstein (i.e.,
Lagrangian) neighborhood theorem each $A_{i}$ has a standard neighborhood
$N_{i}$ in $F_{i}$ which is symplectomorphic to
$(T^{*}D^{n},d\lambda_{\textrm{can}})$ where
$\lambda_{\textrm{can}}=\textbf{p}\textbf{d}\textbf{q}$ is the canonical
$1$-form on $\mathbb{R}^{n}\times\mathbb{R}^{n}$ with coordinates
$(\textbf{q},\textbf{p})$. Then the _plumbing_ of the open books
$(F_{1},h_{1})$ and $(F_{2},h_{2})$ _along_ $A_{1}$ and $A_{2}$ is the open
book $(\mathcal{P}(F_{1},F_{2};A_{1},A_{2}),h)$ on the connected sum
$Y_{1}\\#Y_{2}$ with the pages obtained by gluing $F_{i}$’s together along
$N_{i}$’s by interchanging q-coordinates in $N_{1}$ with p-coordinates in
$N_{2}$, and vice versa. In order to describe $h$, extend each $h_{i}$ to
$\tilde{h}_{i}$ on the new page by requiring $\tilde{h}_{i}$ to be identity
map outside the domain of $h_{i}$. Then the monodrodmy $h$ is defined to be
$\tilde{h}_{1}\circ\tilde{h}_{2}$. For simplicity, “tilde” sign will be
dropped, and we will simply write $h=h_{1}\circ h_{2}$.

######  Definition 2.5 ([11]).

Suppose that $(F,h)$ is an open book compatible with the contact structure
$\xi=\textrm{Ker}(\alpha)$ on a closed $(2n+1)$-manifold $Y$. Let $A$ be a
properly embedded Lagrangian $n$-disk in a symplectic page $(F,d\alpha)$ such
that its boundary $\partial A$ is a Legendrian $(n-1)$-sphere in the contact
binding $(\partial F,\textrm{Ker}(\alpha|_{\,\partial F}))$. Then the positive
stabilization $\mathcal{S}^{+}[(F,h);A]$ of $(F,h)$ along $A$ is the open book
defined by

$\mathcal{S}^{+}[(F,h);A]\doteq(\mathcal{P}(F,D^{*}\mathbb{S}^{n};A,\textbf{D}),h\circ\tau_{\mathbb{S}^{n}})$

where $\textbf{D}\cong D^{n}$ is any fiber in the unit disk cotangent bundle
$D^{*}\mathbb{S}^{n}$ and $\tau_{\mathbb{S}^{n}}:D^{*}\mathbb{S}^{n}\to
D^{*}\mathbb{S}^{n}$ is the right-handed Dehn twist along the zero section
$\mathbb{S}^{n}$. Similarly, the negative stabilization
$\mathcal{S}^{-}[(F,h);A]$ of $(F,h)$ along $A$ is the open book defined by

$\mathcal{S}^{-}[(F,h);A]\doteq(\mathcal{P}(F,D^{*}\mathbb{S}^{n};A,\textbf{D}),h\circ\tau^{-1}_{\mathbb{S}^{n}})$

######  Remark 2.6.

When a compatible open book $(F,h)$ is stabilized, abstractly it means that
each symplectic page $(F^{2n},d\alpha)$ gains a Weinstein handle
$H=D^{n}\times D^{n}$ of index $n$. Attaching spheres of these handles are the
copies of Legendrian $(n-1)$-sphere $\partial A$ considered inside the
boundary $\partial F$ of each fiber. Also the notation used in the above
definition is quite misleading and is not appropriate for our purposes. When a
positive/negative stabilization is performed, abstractly it means that the
right-handed/left-handed Dehn twist that we adjoin to the monodromy is along
the Lagrangian $n$-sphere $L$ formed by gluing the Lagrangian $n$-disk
$(A,\partial A)\subset(F,\partial F)$ and the Lagrangian $n$-disk
$D^{2}\times\\{0\\}$ (the core disk of $H$) together along their common
boundary $\partial A\subset\partial F$, that is, $L=A\cup_{\partial
A}(D^{2}\times\\{0\\})$. In conclusion, we will use the following (abstract)
alternative (but equivalent) definition (notation) for positive/negative
stabilization of $(F,h)$ along $A$:

$\mathcal{S}^{\pm}[(F,h);A]\doteq(F\cup H,h\circ\tau^{\pm 1}_{L}).$

######  Theorem 2.7 ([11]).

In the setting of Definition 2.5, both $\mathcal{S}^{\pm}[(F,h);A]$ are open
books on the connected sum $Y\\#\mathbb{S}^{2n+1}(\cong Y)$. Indeed,
$\mathcal{S}^{+}[(F,h);A]$ is an open book on the contact connected sum
$Y_{\xi}\\#\mathbb{S}^{2n+1}_{st}(\cong Y_{\xi})$, in other words, it is a
compatible open book on $(Y,\xi)$.

Returning back to the dimension three and five, we now define relative
stabilization process:

######  Definition 2.8.

Suppose that $(F_{Y},F_{M},h_{Y},h_{M})$ is a relative open book compatible
with the pair of contact structures
$(\xi=\textrm{Ker}(\alpha),\eta=\textrm{Ker}(\alpha|_{M}))$ on a relative
contact pair $(Y^{5}_{\xi},M^{3}_{\eta})$. Let $A$ be a properly embedded
Lagrangian $2$-disk in a symplectic page $(F_{Y},d\alpha|_{F_{Y}})$ with
Legendrian boundary $\partial A$ (an unknot) in the contact binding $(\partial
F_{Y},\textrm{Ker}(\alpha|_{\,\partial F_{Y}}))$ such that $A\cap F_{M}$ is a
properly embedded Lagrangian $1$-disk in a symplectic page
$(F_{M},d\alpha|_{F_{M}})$ with Legendrian boundary $\partial(A\cap F_{M})$ (a
pair of points) in the contact binding $(\partial
F_{M},\textrm{Ker}(\alpha|_{\,\partial F_{M}}))$. Also consider the relative
Weinstein handle $H=(H^{4},H^{2})=(D^{2}\times D^{2},D^{1}\times D^{1})$ such
that the core and the cocore of $H^{2}$ is properly embedded in those of
$H^{4}$, that is, $D^{1}\times\\{0\\}\subset D^{2}\times\\{0\\}$,
$\\{0\\}\times D^{1}\subset\\{0\\}\times D^{2}$ and $\partial
D^{1}\times\\{0\\}\subset\partial D^{2}\times\\{0\\}$ and
$\\{0\\}\times\partial D^{1}\subset\\{0\\}\times\partial D^{2}$. Then the
positive relative stabilization
$\mathcal{S}^{+}[(F_{Y},F_{M},h_{Y},h_{M});A,A\cap F_{M}]$ of
$(F_{Y},F_{M},h_{Y},h_{M})$ along $(A,A\cap F_{M})$ is the open book defined
by

$\mathcal{S}^{+}[(F_{Y},F_{M},h_{Y},h_{M});A,A\cap F_{M}]\doteq(F_{Y}\cup
H^{4},F_{M}\cup H^{2},h_{Y}\circ\tau_{L},h_{M}\circ\tau_{K})$

where the framing of $H^{4}$ is one less than the contact framing of $\partial
A$ in $(\partial F_{Y},\textrm{Ker}(\alpha|_{\,\partial F_{Y}}))$, and
$K=A\cap F_{M}\cup_{\partial(A\cap F_{M})}(D^{1}\times\\{0\\})$ is the
Lagrangian $1$-sphere in the new symplectic page $F_{M}\cup H^{2}$ and
$L=A\cup_{\partial A}(D^{2}\times\\{0\\})$ is the Lagrangian $2$-sphere in the
new symplectic page $F_{Y}\cup H^{4}$. Similarly, the negative relative
stabilization $\mathcal{S}^{-}[(F_{Y},F_{M},h_{Y},h_{M});A,A\cap F_{M}]$ of
$(F_{Y},F_{M},h_{Y},h_{M})$ along $(A,A\cap F_{M})$ is the open book defined
by

$\mathcal{S}^{-}[(F_{Y},F_{M},h_{Y},h_{M});A,A\cap F_{M}]\doteq(F_{Y}\cup
H^{4},F_{M}\cup H^{2},h_{Y}\circ\tau^{-1}_{L},h_{M}\circ\tau^{-1}_{K})$

where this time the framing of $H^{4}$ is one more than the contact framing of
$\partial A$ in $(\partial F_{Y},\textrm{Ker}(\alpha|_{\,\partial F_{Y}}))$.

######  Remark 2.9.

From the definition it is understood that a relative stabilization consists of
two stabilizations one of which performed in dimension three and the other in
five, but there is only one stabilization indeed: we are only stabilizing the
open book $(F_{Y},h_{Y})$ along $A$, i.e.,

$\mathcal{S}^{\pm}[(F_{Y},F_{M},h_{Y},h_{M});A,A\cap
F_{M}]=\mathcal{S}^{\pm}[(F_{Y},h_{Y});A],$

and as a result of this $(F_{M},h_{M})$ is also simultaneously stabilized
along $A\cap F_{M}$.

The result of Giroux holds also in the relative setting. Namely,

######  Theorem 2.10.

In the setting of Definition 2.8, both
$\mathcal{S}^{\pm}[(F_{Y},F_{M},h_{Y},h_{M});A,A\cap F_{M}]$ are open books on
the connected sum $(Y\\#\mathbb{S}^{5},M\\#\mathbb{S}^{3})\cong(Y,M)$. Indeed,
$\mathcal{S}^{+}[(F_{Y},F_{M},h_{Y},h_{M});A,A\cap F_{M}]$ is an open book on
the contact connected sum
$(Y_{\xi}\\#\mathbb{S}^{5}_{st},M_{\eta}\\#\mathbb{S}^{3}_{st})\cong(Y_{\xi},M_{\eta})$,
in other words, it is a compatible relative open book on the relative contact
pair $(Y_{\xi},M_{\eta})$.

###### Proof.

Observe that both stabilizations
$\mathcal{S}^{\pm}[(F_{Y},F_{M},h_{Y},h_{M});A,A\cap F_{M}]$ correspond to the
stabilized open books $\mathcal{S}^{\pm}[(F_{Y},h_{Y});A]$ on
$Y\\#\mathbb{S}^{5}\cong Y$ and $\mathcal{S}^{\pm}[(F_{M},h_{M});A\cap F_{M}]$
on $M\\#\mathbb{S}^{3}\cong M$. Therefore, by Theorem 2.7, both are relative
open books on the relative connected sum
$(Y\\#\mathbb{S}^{5},M\\#\mathbb{S}^{3})\cong(Y,M)$, and also
$\mathcal{S}^{+}[(F_{Y},F_{M},h_{Y},h_{M});A,A\cap F_{M}]$ is an open book on
the contact connected sum
$(Y_{\xi}\\#\mathbb{S}^{5}_{st},M_{\eta}\\#\mathbb{S}^{3}_{st})\cong(Y_{\xi},M_{\eta})$.
Hence, Definition 2.3 applies. ∎

######  Example 2.11.

Consider the simplest compatible open books $(D^{4},id_{D^{4}})$ and
$(D^{2},id_{D^{2}})$ on the standard contact spheres $\mathbb{S}^{5}_{st}$ and
$\mathbb{S}^{3}_{st}$, respectively. Here we consider both disks with their
standard symplectic structures, and $D^{2}$ as a properly and symplectically
embedded in $D^{4}$. One easily checks the conditions in Definition 2.2 and
Definition 2.3, and so $(D^{4},D^{2},id_{D^{4}},id_{D^{2}})$ is a compatible
relative open book on the contact pair
$(\mathbb{S}^{5}_{st},\mathbb{S}^{3}_{st})$. Also it is a well-known
construction that $D^{*}\mathbb{S}^{2}$ and $D^{*}\mathbb{S}^{1}$ can be,
simultanaously, obtained from $D^{4}$ and $D^{2}$ by attachhing Weinstein
handles $H^{4}$ and $H^{2}$ along a Legendrian $1$-unknot $\partial A$ (the
boundary of a properly embedded Lagrangian disk $A$ in $D^{4}$) in
$\mathbb{S}^{3}_{st}(=\partial D^{4})$ and a Legendrian $0$-unknot
$\partial(L\cap D^{2})$ in $\mathbb{S}^{1}_{st}(=\partial D^{2})$. Assuming
the framing of $H^{4}$ is chosen correctly (one less than the contact framing
of $\partial A$ in $\mathbb{S}^{3}_{st}=\partial D^{4}$), we conclude, by
Definition 2.8, that
$(D^{*}\mathbb{S}^{2},D^{*}\mathbb{S}^{1},\tau_{\mathbb{S}^{2}},\tau_{\mathbb{S}^{1}})$
is a positive relative stabilization of $(D^{4},D^{2},id_{D^{4}},id_{D^{2}})$
along $(A,A\cap D^{2})$. That is,

$(D^{*}\mathbb{S}^{2},D^{*}\mathbb{S}^{1},\tau_{\mathbb{S}^{2}},\tau_{\mathbb{S}^{1}})=\mathcal{S}^{+}[(D^{4},D^{2},id_{D^{4}},id_{D^{2}});A,A\cap
D^{2}].$

On the other hand, if the framing of $H^{4}$ is chosen to be one more than the
contact framing of $\partial A$, then one concludes that

$(D^{*}\mathbb{S}^{2},D^{*}\mathbb{S}^{1},\tau^{-1}_{\mathbb{S}^{2}},\tau^{-1}_{\mathbb{S}^{1}})=\mathcal{S}^{-}[(D^{4},D^{2},id_{D^{4}},id_{D^{2}});A,A\cap
D^{2}].$

##  3\. Generalized Square Bridge Diagrams

Any Legendrian link $\mathbb{K}$ in
$(\mathbb{R}^{3},\xi_{0}^{3})\subset\mathbb{S}^{3}_{st}$ can be represented by
a square bridge diagram [17]. In this section, we will construct a generalized
square bridge diagram to represent any admissible Legendrian $2$-link
$\mathbb{L}$ in $(\mathbb{R}^{5},\xi_{0}^{5})\subset\mathbb{S}^{5}_{st}$ which
naturally contains a square bridge diagram of the equatorial link
$\mathbb{K}=\mathbb{L}\cap\mathbb{S}^{5}_{st}$ of $\mathbb{L}$.

Let
$\Pi_{5}:\mathbb{R}^{5}\rightarrow\mathbb{R}^{3}:(x_{1},y_{1},z,x_{2},y_{2})\mapsto(y_{1},z,y_{2})$
and
$\Pi_{3}:\mathbb{R}^{3}\rightarrow\mathbb{R}^{2}:(x_{1},y_{1},z)\mapsto(y_{1},z)$
be the front projection maps. In order to picture Legendrian $2$-knots (resp.
Legendrian $1$-knots) in $(\mathbb{R}^{5},\xi_{0}^{5})$ (resp.
$(\mathbb{R}^{3},\xi_{0}^{3})$), we consider their fronts, i.e., their images
under the map $\Pi_{5}$ (resp. $\Pi_{3}$). Examples of such fronts are
depicted for Legendrian unknots in Figure 1.

Figure 1. On the left, the front of the admissible Legendrian $2$-unknot
$\mathbb{L}$, and, on the right, the front of its isotropic equator
$\mathbb{K}$ (Legendrian $1$-unknot in $(\mathbb{R}^{3},\xi_{0}^{3})$.

We first consider the square bridge diagram (in the $y_{1}z$-plane) of the
front of the equatorial $1$-link $\mathbb{K}=\bigsqcup_{i=1}^{r}K_{i}$ of a
given admissible Legendrian $2$-link $\mathbb{L}=\bigsqcup_{i=1}^{r}L_{i}$
such that it consists of the following two types of line segments:

$\displaystyle h_{a_{i}}$ $\displaystyle=$
$\displaystyle\\{(x_{1},y_{1},z,x_{2},y_{2})\in\mathbb{R}^{5}:\quad
x_{1}=x_{2}=y_{2}=0,\quad z=-y_{1}+a_{i}\\}$ $\displaystyle v_{b_{j}}$
$\displaystyle=$
$\displaystyle\\{(x_{1},y_{1},z,x_{2},y_{2})\in\mathbb{R}^{5}:\quad
x_{1}=x_{2}=y_{2}=0,\quad z=y_{1}+b_{j}\\}$

for some real numbers $a_{1}<a_{2}<\cdots<a_{p}$ and
$0<b_{1}<b_{2}<\cdots<b_{q}$, and it bounds a polygonal region in the
$y_{1}z-$plane as in Figure 2 (on the left). Secondly, divide each polygonal
region into finitely many square regions $R_{k}$ for $k=1,2,...,m$ as in
Figure 2 (on the right). Such a division can be achieved by applying an
appropriate Legendrian isotopy to $\mathbb{L}$ so that the gaps between
consequtive $a_{i}$’s and consequtive $b_{j}$’s are all equal to the same
fixed real number.

Figure 2. The square bridge diagram of the equatorial isotropic $1$-link
(demonstrated when $\mathbb{K}$ is the Hopf link) and its division into
squares (on the right).

Thirdly, we construct an octahedron for each square region $R_{k}$ so that
$\partial R_{k}$ is the equatorial square of the octahedron. This can be done
by composing (gluing) eight triangular plane segments of the following four
types:

$\displaystyle t_{a_{i}}^{1,1}\;$ $\displaystyle=$
$\displaystyle\\{(x_{1},y_{1},z,x_{2},y_{2})\in\mathbb{R}^{5}:\quad
x_{1}=x_{2}=0,\quad z=-y_{1}-y_{2}+a_{i}\\}$ $\displaystyle
t_{a_{i}}^{1,-1}\;$ $\displaystyle=$
$\displaystyle\\{(x_{1},y_{1},z,x_{2},y_{2})\in\mathbb{R}^{5}:\quad
x_{1}=x_{2}=0,\quad z=-y_{1}+y_{2}+a_{i}\\}$ $\displaystyle t_{b_{j}}^{-1,1}$
$\displaystyle=$
$\displaystyle\\{(x_{1},y_{1},z,x_{2},y_{2})\in\mathbb{R}^{5}:\quad
x_{1}=x_{2}=0,\quad z=y_{1}-y_{2}+b_{j}\\}$ $\displaystyle t_{b_{j}}^{-1,-1}$
$\displaystyle=$
$\displaystyle\\{(x_{1},y_{1},z,x_{2},y_{2})\in\mathbb{R}^{5}:\quad
x_{1}=x_{2}=0,\quad z=y_{1}+y_{2}+b_{j}\\}.$

A typical octahedron for a typical square region $R_{k}$ is depicted in Figure
3 which is indeed what we call as the generalized square bridge position of a
standard Legendrian $2$-unknot whose front has appeared in Figure 1 above.

Figure 3. An octahedron corresponding to $R_{k}$ (a generalized square bridge
position of a standard Legendrian $2$-unknot.

In Figure 3, the upper part of the octahedron contains the triangles
$t_{a_{1}}^{1,-1}$, $t_{a_{2}}^{1,1}$, $t_{b_{1}}^{-1,-1},t_{b_{2}}^{-1,1}$,
and the lower part contains the triangles $t_{a_{1}}^{1,1}$,
$t_{a_{2}}^{1,-1}$, $t_{b_{1}}^{-1,1},t_{b_{2}}^{-1,-1}$. Also the north and
south poles of the octahedron are $\displaystyle
N=(0,\frac{a_{1}-b_{1}}{2},\frac{a_{1}+b_{2}}{2},0,\frac{b_{2}-b_{1}}{2})$ and
$\displaystyle
S=(0,\frac{a_{1}-b_{1}}{2},\frac{a_{1}+b_{2}}{2},0,\frac{b_{1}-b_{2}}{2})$.

Finally, we define

######  Definition 3.1.

The generalized square bridge diagram, denoted by $\Delta(\mathbb{L})$, of
$\mathbb{L}$ is a cellular $2$-complex in the $(y_{1}z\,y_{2})$-space obtained
by considering all $R_{k}$’s (in the square bridge diagram of $\mathbb{K}$)
and taking the union of corresponding $m$ octahedrons (see Figure 4).

Figure 4. The generalized square bridge diagram $\Delta(\mathbb{L})$ of the
admissible Legendrian $2$-link $\mathbb{L}$ whose equatorial link $\mathbb{K}$
is the Hopf link.

Note that the square bridge diagram $\Gamma(\mathbb{K})$ of $\mathbb{K}$ is a
cellular $1$-subcomplex of $\Delta(\mathbb{L})$. Next, we will show that
starting from its generalized square bridge diagram
$\Delta(L)\subset\Delta(\mathbb{L})$, one can reconstruct each component $L$
of $\mathbb{L}$ in $(\mathbb{R}^{5},\xi_{0}^{5})\subset\mathbb{S}^{5}_{st}$.
More precisely,

######  Proposition 3.2.

For any component $L_{i}$ of $\mathbb{L}=\bigsqcup_{i=1}^{r}L_{i}$, starting
from $\Delta(L_{i})$, one can construct a piecewise smooth Legendrian
$2$-sphere $L^{\prime}_{i}$ in
$(\mathbb{R}^{5},\xi_{0}^{5})\subset\mathbb{S}^{5}_{st}$ by gluing appropriate
Legendrian and planar triangles and rectangles so that the smooth Legendrian
$2$-sphere obtained by rounding the corners of $L_{i}^{\prime}$ is Legendrian
isotopic to $L_{i}$. Moreover, the piecewise smooth equatorial $1$-link
$\mathbb{K}^{\prime}=\bigsqcup_{i=1}^{r}K^{\prime}_{i}$
($K^{\prime}_{i}=L^{\prime}_{i}\cap\mathbb{S}^{3}_{st}$) and $\mathbb{K}$ have
the same linking matrix.

###### Proof.

We first lift each line segment $h_{a_{i}}$ and $v_{b_{j}}$ in
$\Gamma(\mathbb{K})$ (divided into square regions) along the $x_{1}$-axis such
that the lifts are disjoint Legendrian line segments in
$(\mathbb{R}^{3},\xi_{0}^{3})\subset\mathbb{S}^{3}_{st}$ given by

$\displaystyle H_{a_{i}}$ $\displaystyle=$
$\displaystyle\\{(x_{1},y_{1},z,x_{2},y_{2})\in\mathbb{R}^{5}:\quad
x_{1}=1,x_{2}=y_{2}=0,\quad z=-y_{1}+a_{i}\\}$ $\displaystyle V_{b_{j}}$
$\displaystyle=$
$\displaystyle\\{(x_{1},y_{1},z,x_{2},y_{2})\in\mathbb{R}^{5}:\quad
x_{1}=-1,x_{2}=y_{2}=0,\quad z=y_{1}+b_{j}\\}.$

Then we connect each segment $H_{a_{i}}$ to certain segments $V_{b_{j}}$ via
Legendrian linear arcs $I_{k}^{l}$ which are parallel to the $x_{1}$-axis so
that all of the (piecewise smooth) Legendrian unknots $\gamma_{k}$ in
$\Gamma(\mathbb{K})$ are obtained in $(\mathbb{R}^{3},\xi_{0}^{3})$ as in [2].
The construction of $\gamma_{k}$ (for a typical square region $R_{k}$) is
illustrated in Figure 5. Note that by construction the Legendrian unknot
$\gamma_{k}$ is attached to $\gamma_{k+1}$ by either sidewise or by opening a
back gate. As a result, we have constructed the lift
$\widetilde{\Gamma}(\mathbb{K})$ of $\Gamma(\mathbb{K})$ as a Legendrian
cellular $1$-complex in
$(\mathbb{R}^{3},\xi_{0}^{3})\subset\mathbb{S}^{3}_{st}$ (which is also an
isotropic cellular $1$-complex in
$(\mathbb{R}^{5},\xi_{0}^{5})\subset\mathbb{S}^{5}_{st}$). Observe that by
construction there exists a piecewise smooth Legendrian $1$-link
$\mathbb{K}^{\prime}=\bigsqcup_{i=1}^{r}K^{\prime}_{i}\subset\widetilde{\Gamma}(\mathbb{K})$
which we can round its corners (in its $\epsilon$-neighborhood) so that the
resulting smooth Legendrian link is Legendrian isotopic to $\mathbb{K}$. In
particular, the linking matrices of $\mathbb{K}^{\prime}$ and $\mathbb{K}$ are
the same. Hence, the second claim in the statement will be satisfied once we
prove the first one (i.e., once we construct $L_{i}^{\prime}$) in such a way
that $K^{\prime}_{i}=L^{\prime}_{i}\cap\mathbb{S}^{3}_{st}$.

Figure 5. A typical square region $R_{k}$ and the Legendrian unknot
$\gamma_{k}$.

Next we will construct a Legendrian thickening
$\mathcal{L}(\widetilde{\Gamma}(\mathbb{K}))$ of
$\widetilde{\Gamma}(\mathbb{K})$ (the union of all $\gamma_{k}$’s) which will
contain the neighborhood of each (equatorial knot) $K_{i}^{\prime}$ inside
Legendrian $L_{i}^{\prime}$ that we will eventually construct. Although there
is no unique way of Legendrian thickening (need to choose a thickening
direction), we do this in a systematic way as follows: For each $k=1,...,m$,
we first construct the Legendrian thickening $M_{k}$ (which we call a middle
part) of $\gamma_{k}$ in the way explained below. Then we form
$\mathcal{L}(\widetilde{\Gamma}(\mathbb{K}))\doteq\bigsqcup_{k=1}^{m}M_{k}$
where the union is taken over the common parts of $M_{k}$’s (By the
construction below, if two $\gamma_{k}$’s have a common edge, the
corresponding $M_{k}$’s have a common rectangular part which is the Legendrian
thickening of the common edge.)

We construct each middle part $M_{k}$ so that it consists of 4 Legendrian
rectangular plates and 16 Legendrian triangular ones as shown in the Figure 6.
The whole $M_{k}$ lies in the hyperspace
$\\{y_{2}=0\\}\subset(\mathbb{R}^{5},\xi_{0}^{5})$, and $\gamma_{k}$ lies as
the core of $M_{k}$ where $x_{2}=y_{2}=0$.

Figure 6. A shematic picture of the middle part $M_{k}$ for a typical square
$R_{k}$ (on the left), and the isotropic unknot $\gamma_{k}$ (on the right).

For $H_{i}$ and $V_{j}$ parts of $\gamma_{k}$, we take their Legendrian
thickenings to be the $4$ Legendrian rectangles $S_{{H}_{i}}$ and
$S_{{V}_{j}}$ on which $x_{1}=\pm 1$ and $-1\leq x_{2}\leq 1$. On the other
hand, for ${I}_{k}$ parts of $\gamma_{k}$, their Legendrian thickenings
totally consist of $16$ Legendrian triangles $S_{{I}_{k}}$ on which both
$x_{1}$ and $x_{2}$ change between $-1$ and $1$ in such a way that each group
of $4$ triangles together join an $S_{{H}_{i}}$ to one of the two of its
neighboring $S_{{V}_{j}}$’s. In particular, $x_{1}=x_{2}=0$ at the points
$p_{1}$, $p_{2}$, $p_{3},p_{4}$ of $M_{k}$ which are the midpoints of the arcs
${I}_{k}$ in $\gamma_{k}$. (See Figure 7 for a shematic picture of this
construction performed in a four dimensional space
$\\{y_{2}=0\\}\subset(\mathbb{R}^{5},\xi_{0}^{5})$.)

Figure 7. A shematic picture of the construction of $M_{k}$. (The most left
and right vertical edges are identified.)

By construction, all $M_{k}$’s touch together along their common parts, and
they together form the Legendrian thickening
$\mathcal{L}(\widetilde{\Gamma}(\mathbb{K}))\doteq\bigsqcup_{k=1}^{m}M_{k}$.
We note that
$\mathbb{K}^{\prime}\subset\widetilde{\Gamma}(\mathbb{K})\subset\mathcal{L}(\widetilde{\Gamma}(\mathbb{K}))$.

Now we consider the triangles of the octahedrons in the generalized square
bridge diagram $\Delta(\mathbb{L})$. We lift each of the triangular plane
segments $t_{a_{i}}^{1,1}$, $t_{a_{i}}^{1,-1}$, $t_{b_{j}}^{-1,1}$,
$t_{b_{j}}^{-1,-1}$ (to an appropriate $x_{1}$ and $x_{2}$ levels) so that the
resulting lifts are disjoint Legendrian triangular plates lying on the
following $4$-types of planes:

$\begin{array}[]{rclll}T_{a_{i}}^{1,1}&=&\\{(x_{1},y_{1},z,x_{2},y_{2})\in\mathbb{R}^{5}:\quad
x_{1}=1,&x_{2}=1,&z=-y_{1}-y_{2}+a_{i}\\}\\\
T_{a_{i}}^{1,-1}&=&\\{(x_{1},y_{1},z,x_{2},y_{2})\in\mathbb{R}^{5}:\quad
x_{1}=1,&x_{2}=-1,&z=-y_{1}+y_{2}+a_{i}\\}\\\
T_{b_{j}}^{-1,1}&=&\\{(x_{1},y_{1},z,x_{2},y_{2})\in\mathbb{R}^{5}:\quad
x_{1}=-1,&x_{2}=1,&z=y_{1}-y_{2}+b_{j}\\}\\\
T_{b_{j}}^{-1,-1}&=&\\{(x_{1},y_{1},z,x_{2},y_{2})\in\mathbb{R}^{5}:\quad
x_{1}=-1,&x_{2}=-1,&z=y_{1}+y_{2}+b_{j}\\}.\end{array}$

The next lemma shows that the triangular planes listed above are all
Legendrian.

######  Lemma 3.3.

For any $c\in\mathbb{R}$, the planes

$T_{c}^{k,\ell}=\\{(x_{1},y_{1},z,x_{2},y_{2})\in\mathbb{R}^{5}:\quad
x_{1}=m,\quad x_{2}=n,\quad z=ky_{1}+\ell y_{2}+c\\}$

are Legendrian in $(\mathbb{R}^{5},\xi_{0}^{5})\subset\mathbb{S}^{5}_{st}$ if
and only if $k=-m$ and $\ell=-n$.

###### Proof.

Need to show any vector tangent to $T_{c}^{k,\ell}$ lies in
$\textrm{Ker}(\alpha_{0}^{5})$ where
$\alpha_{0}^{5}=dz+x_{1}dy_{1}+x_{2}dy_{2}$. The tangent space of
$T_{c}^{k,\ell}$ at any point $p\in T_{c}^{k,\ell}$ is spanned by the vectors
$\partial/\partial y_{1}+k\,\partial/\partial z$ and $\partial/\partial
y_{2}+\ell\,\partial/\partial z$. That is,

$T_{p}T_{c}^{k,\ell}=\left\langle\frac{\partial}{\partial
y_{1}}+k\frac{\partial}{\partial z},\;\frac{\partial}{\partial
y_{2}}+\ell\frac{\partial}{\partial z}\right\rangle.$

Then one can easily verify that

$\alpha_{0}^{5}\left(\frac{\partial}{\partial y_{1}}+k\frac{\partial}{\partial
z}\right)=k+m=0\Leftrightarrow
k=-m,\quad\alpha_{0}^{5}\left(\frac{\partial}{\partial
y_{1}}+\ell\frac{\partial}{\partial z}\right)=\ell+n=0\Leftrightarrow\ell=-n.$

Hence, the claim follows from linearity. ∎

Next, using the above triangles, for each square region $R_{k}$ we will obtain
two disjoint piecewise smooth disks $\mathcal{D}^{{}^{\prime}}_{k}$ and
$\mathcal{D}^{{}^{\prime\prime}}_{k}$ which will be the upper and lower part
of a piecewise smooth Legendrian $2$-unknot, called a Legendrian diamond in
$(\mathbb{R}^{5},\xi_{0}^{5})$ as shown in the Figure 8 and Figure 9.

Figure 8. The Legendrian piecewise smooth disk
$\mathcal{D}^{{}^{\prime}}_{k}$(on the left) and the Legendrian piecewise
smooth disk $\mathcal{D}^{{}^{\prime\prime}}_{k}$ (on the right).

The disk $\mathcal{D}^{{}^{\prime}}_{k}$ consists of $4$ disjoint Legendrian
triangular plates $T_{a_{1}}^{1,-1}$, $T_{a_{2}}^{1,1}$,
$T_{b_{1}}^{-1,-1}$,$T_{b_{2}}^{-1,1}$, $4$ Legendrian rectangular plates
(connecting them), and $4$ other Legendrian triangular plates at the top, and
likewise, the disk $\mathcal{D}^{{}^{\prime\prime}}_{k}$ consists of $4$
disjoint Legendrian triangular plates $T_{a_{1}}^{1,1}$, $T_{a_{2}}^{1,-1}$,
$T_{b_{1}}^{-1,1}$, $T_{b_{2}}^{-1,-1}$, $4$ Legendrian rectangular plates
(connecting them), and $4$ other Legendrian triangular plates at the botton
(Figure 8). For each disk, the rectangular plates connect certain two of the
disjoint triangular plates, and the four triangular plates connecting the four
Legendrian rectagular plates are the following:

$\displaystyle{\tiny P_{\pm 1}=\\{z=\pm y_{2}+\frac{a_{i}+b_{i}}{2},\quad
y_{1}=\frac{a_{i}-b_{i}}{2},\quad x_{2}=\mp 1,\quad-1\leq x_{1}\leq 1\\}}$
$\displaystyle P^{{}^{\prime}}_{\pm 1}=\\{y_{2}=\pm
y_{1}\pm\frac{b_{2}-a_{1}}{2},\quad z=\frac{a_{1}+b_{2}}{2}\quad-1\leq
x_{1}=\mp x_{2}\leq 1\\}.$

Also, for each disk, there are two plates are composed of $4$-triangular
plates on the top and bottom:

$\displaystyle P_{t}=\\{y_{1}=\frac{a_{1}-b_{1}}{2},\quad
z=\frac{a_{1}+b_{2}}{2},\quad y_{2}=\frac{b_{2}-b_{1}}{2},\quad-1\leq
x_{1},x_{2}\leq 1\\}$ $\displaystyle
P_{b}=\\{y_{1}=\frac{a_{1}-b_{1}}{2},\quad z=\frac{a_{1}+b_{2}}{2},\quad
y_{2}=\frac{b_{1}-b_{2}}{2},\quad-1\leq x_{1},x_{2}\leq 1\\}.$

Figure 9. The Legendrian diamond $\mathcal{D}_{k}$ (a piecewise smooth
Legendrian $2$-unknot).

From their constructions, it is straight forward to check that all the
triangular and rectangular plates in $\mathcal{D}^{\prime}_{k}$,
$\mathcal{D}^{\prime\prime}_{k}$ and $M_{k}$ meet each other along the common
parts of their boundaries appropriately, and they together form the Legendrian
diamond $\mathcal{D}_{k}\doteq\mathcal{D}^{{}^{\prime}}_{k}\cup
M_{k}\cup\mathcal{D}^{{}^{\prime\prime}}_{k}$ in
$(\mathbb{R}^{5},\xi_{0}^{5})\subset\mathbb{S}^{5}_{st}$. We note that the
equatorial link $\mathbb{K}=\bigsqcup_{i=1}^{r}K_{i}$ is reconstructed as the
piecewise smooth $1$-link
$\mathbb{K}^{\prime}=\bigsqcup_{i=1}^{r}K^{\prime}_{i}$ contained in the union
$\mathcal{L}(\widetilde{\Gamma}(\mathbb{K}))(=\bigsqcup_{k=1}^{m}M_{k})$ which
can be realized as the union of the neighborhoods of the isotropic equatorial
unknots inside the union $\bigsqcup_{k=1}^{m}\mathcal{D}_{k}$ of all
Legendrian diamonds.

To verify the first claim, let $L_{i}$ be any component of $\mathbb{L}$.
Consider the subcollection of all square regions
$\\{R_{i_{1}},R_{i_{2}},...,R_{i_{n}}\\}\subset\\{R_{1},R_{2},...,R_{m}\\}$
which are contained in the square bridge diagram
$\Gamma(K_{i})\subset\Gamma(\mathbb{K})$. Then the front $\Pi_{3}(K_{i})$ is
contained in the union $\bigsqcup_{j=1}^{n}R_{i_{j}}$, and so the
corresponding piecewise smooth isotropic unknot $K_{i}^{\prime}$ is contained
in the Legendrian thickening
$\mathcal{L}(\widetilde{\Gamma}(K_{i}))=\bigsqcup_{j=1}^{n}M_{i_{j}}\subset\bigsqcup_{j=1}^{n}\mathcal{D}_{i_{j}}$.
Let $\mathcal{L}(K_{i}^{\prime})\subset\mathcal{L}(\widetilde{\Gamma}(K_{i}))$
denote the Legendrian thickening of $K_{i}^{\prime}$ (i.e., the union of the
plates in $\bigsqcup_{j=1}^{n}M_{i_{j}}$ which retracts onto
$K_{i}^{\prime}$). The boundary $\partial(\mathcal{L}(K_{i}^{\prime}))$
consists of two (isotropic) push-offs of $K_{i}^{\prime}$ where
$\mathcal{L}(K_{i}^{\prime})$ meets with the corresponding rectangular and
triangular plates in the upper and lower disks
$\mathcal{D}^{{}^{\prime}}_{i_{j}}$, $\mathcal{D}^{{}^{\prime\prime}}_{i_{j}}$
of the Legendrian diamonds
$\mathcal{D}_{i_{1}},\mathcal{D}_{i_{2}},...,\mathcal{D}_{i_{n}}$. Finally, we
define $L_{i}^{\prime}$ as the union

$L_{i}^{\prime}\doteq\mathcal{L}(K_{i}^{\prime})\cup_{\partial(\mathcal{L}(K_{i}^{\prime}))}\bigsqcup_{j=1}^{n}(\overline{\mathcal{D}_{i_{j}}\setminus
M_{i_{j}}}).$

Clearly, $L_{i}^{\prime}$ is a piecewise smooth Legendrian $2$-unknot in
$\mathbb{S}^{5}_{st}$ with its equatorial $1$-unknot $K_{i}^{\prime}$ in
$\mathbb{S}^{3}_{st}$ as required. After rounding the corners (and sharp
edges) in $L_{i}^{\prime}$ (and so, in $K_{i}^{\prime}$) in a small
neighborhood of $L_{i}^{\prime}$, we obtain a smooth Legendrian admissible
$2$-unknot, say $L_{i}^{\prime\prime}$, with its equatorial $1$-unknot
$K_{i}^{\prime\prime}$ which is Legendrian isotopic to the equatorial
$1$-unknot $K_{i}$ of $L_{i}$. By extending this isotopy over the union of the
upper and the lower Legendrian disks of $L_{i}^{\prime\prime}$ (i.e., over
$L_{i}^{\prime\prime}\setminus K_{i}^{\prime\prime}$), we conclude that
$L_{i}^{\prime\prime}$ and $L_{i}$ are Legendrian isotopic in
$\mathbb{S}^{5}_{st}$. This finishes the proof of Proposition 3.2. ∎

######  Remark 3.4.

As observed in the proof of Proposition 3.2, the linking matrices of the links
$\mathbb{K}$, $\mathbb{K}^{\prime}$, and so
$\mathbb{K}^{\prime\prime}\doteq\bigsqcup_{i=1}^{r}K_{i}^{\prime\prime}$ are
all the same. Also if two line segments $h_{a_{i}}$, $v_{b_{j}}$ cross each
other in the square bridge diagram $\Gamma(\mathbb{K})$, the lift $H_{a_{i}}$
of $h_{a_{i}}$ crosses over the lift $V_{b_{j}}$ in the lifted diagram
$\widetilde{\Gamma}(\mathbb{K})$ as being located in a larger $x_{1}$-level.
Both lifts $H_{a_{i}}$, $V_{b_{j}}$ live on the core of the union
$\mathcal{L}(\widetilde{\Gamma}(\mathbb{K}))=\bigsqcup_{k=1}^{m}M_{k}$. By the
construction of each middle part $M_{k}$, the Legendrian thickening of
$H_{a_{i}}$ is still crossing over that of $V_{b_{j}}$ along the
$x_{1}$-direction because Legendrian thickenings (near such over-under
crossings) are performed along the $x_{2}$-direction only, and so they are
performed in the $x_{1}$-level where $H_{a_{i}}$, $V_{b_{j}}$ live. Moreover,
the components of $\mathbb{L}^{\prime}$ can be rounded (in their small
neighborhoods) in such a way that the resulting $L_{i}^{\prime\prime}$’s are
disjoint, and so we obtain a smooth Legendrian $2$-link
$\mathbb{L}^{\prime\prime}=\bigsqcup_{i=1}^{r}L_{i}^{\prime\prime}$ (with the
equatorial link $\mathbb{K}^{\prime\prime}$) which is Legendrian isotopic to
the original link $\mathbb{L}$. In particular, the linking matrices of
$\mathbb{L}$, $\mathbb{L}^{\prime\prime}$ are the same, and indeed, they
coinside with those of $\mathbb{K}$ and $\mathbb{K}^{\prime\prime}$. This is
one of the motivations of the autors in defining relative $2$-links and their
generalized square bridge diagrams.

##  4\. The Algorithm

###  4.1. Proof of Theorem 1.6

Start with the generalized square bridge diagram $\Delta(\mathbb{L})$ of a
given admissible Legendrian $2$-link $\mathbb{L}$ in
$(\mathbb{R}^{5},\xi_{0}^{5})$ which consists of the square bridge diagram
$\Gamma(\mathbb{K})$ of the isotropic equatorial link $\mathbb{K}$ which is
Legendrian in $(\mathbb{R}^{3},\xi_{0}^{3})$. As in the previous section, one
can construct a Legendrian diamond
$\mathcal{D}_{k}=\mathcal{D}^{{}^{\prime}}_{k}\cup
M_{k}\cup\mathcal{D}^{{}^{\prime\prime}}_{k}$ for each square region $R_{k}$
in $\Gamma(\mathbb{K})$. In what follows, we will construct a $4$-dimensional
ribbon $\mathcal{R}_{\mathcal{D}_{k}}$ for each such diamond. Then by
iteratively joining a ribbon $\mathcal{R}_{\mathcal{D}_{k}}$ to the union of
the ribbons constructed in the previous step, we will reach the union of all
ribbons so that at each step the union will be realized as a page of a
compatible open book on $\mathbb{S}^{5}_{st}$ such that it contains the
corresponding page of a simultanously evolving compatible open book on
$\mathbb{S}^{3}_{st}$. We will achieve this by making use of relative positive
stabilizations of compatible relative open book decompositions. In order to
construct each $\mathcal{R}_{\mathcal{D}_{k}}$, we will construct a suitable
ribbon for each Legendrian plates of $\mathcal{D}_{k}$ so that their union is
the required ribbon $\mathcal{R}_{\mathcal{D}_{k}}$.

For the middle part $M_{k}$ of the diamond $\mathcal{D}_{k}$, first consider
its $S_{{I}_{k}}$ part which consists of $16$ Legendrian triangles. We
construct $16$ ribbons $\mathcal{R}_{S_{{I}_{k}}}$ ($4$-dimensional compact
hypersurface in $(\mathbb{R}^{5},\xi_{0}^{5})$ with corners) by following the
$4$-dimensional contact planes along these $16$ triangular plates. More
precisely,

$\mathcal{R}_{S_{{I}_{k}}}\doteq\\{(p,v)\;:\;p\in
S_{{I}_{k}},\;\;v\in\xi_{0}^{5}|_{p},\;\;\|v\|\leq\epsilon\\}\subset\xi_{0}^{5}|_{S_{{I}_{k}}}.$

For each rectangular parts $S_{{H}_{i}}$, $S_{{V}_{j}}$ (there are 4 of them
in total) of $M_{k}$, we consider two types of ribbons in
$(\mathbb{R}^{5},\xi_{0}^{5})$ given by

$\displaystyle\mathcal{R}_{S_{{H}_{i}}}$ $\displaystyle\doteq$
$\displaystyle\\{z=-y_{1}+a_{i},\quad 1-\epsilon\leq x_{1}\leq
1+\epsilon,\quad-1\leq x_{2}\leq 1\\}$
$\displaystyle\mathcal{R}_{S_{{V}_{j}}}$ $\displaystyle\doteq$
$\displaystyle\\{z=y_{1}+b_{j},\quad-1-\epsilon\leq
x_{1}\leq-1+\epsilon,\quad-1\leq x_{2}\leq 1\\}$

which are topologically the $2$-dimensional thickenings of $S_{{H}_{i}}$,
$S_{{V}_{j}}$. One can check that the boundaries of these $20$ ribbons
$\mathcal{R}_{S_{{I}_{k}}},\mathcal{R}_{S_{{H}_{i}}},\mathcal{R}_{S_{{V}_{j}}}$
meet along their $3$-dimensional common parts which are the $2$-dimensional
thickenings of the common interval parts of the boundaries of
$S_{{I}_{k}},S_{{H}_{i}},S_{{V}_{j}}$. (Recall the picture of $M_{k}$ in
Figure 6 and 7.) Hence, we construct the ribbon $\mathcal{R}_{M_{k}}$ of
$M_{k}$ by gluing $20$ ribbons
$\mathcal{R}_{S_{{I}_{k}}},\mathcal{R}_{S_{{H}_{i}}},\mathcal{R}_{S_{{V}_{j}}}$
along the common parts of their boundaries, i.e.,

$\mathcal{R}_{M_{k}}=\mathcal{R}_{S_{{H}_{i}}}\cup\mathcal{R}_{S_{{I}_{K}}}\cup\mathcal{R}_{S_{{V}_{j}}}.$

By construction, $\mathcal{R}_{M_{k}}$ deformation retracts onto $M_{k}$, and
the contact planes are tangent to $\mathcal{R}_{M_{k}}$ only along its
Legendrian core $M_{k}$. Hence, $\mathcal{R}_{M_{k}}$ is the ribbon of $M_{k}$
in the contact geometric sense used by Giroux in [11]. Moreover, observe that
the ribbon $\mathcal{R}_{M_{k}}$ properly contains the $2$-dimensional ribbon
$\mathcal{R}_{\gamma_{k}}$ of the Legendrian unknot $\gamma_{k}$ (the
isotropic core of $M_{k}$) which is defined by taking the union
$\mathcal{R}_{I_{k}}\subset\mathcal{R}_{S_{{I}_{k}}}$ of the $4$ narrow bands
defined by following the $2$-dimensional contact planes along the $4$ arcs
$I_{k}$ in $(\mathbb{R}^{3},\xi_{0}^{3})$ and the following Legendrian planes:

$\displaystyle\mathcal{R}_{{H}_{a_{i}}}$ $\displaystyle\doteq$
$\displaystyle\\{z=-y_{1}+a_{i},\quad 1-\epsilon\leq x_{1}\leq
1+\epsilon,\quad x_{2}=y_{2}=0\\}\subset\mathcal{R}_{S_{{H}_{i}}}$
$\displaystyle\mathcal{R}_{{V}_{b_{j}}}$ $\displaystyle\doteq$
$\displaystyle\\{z=y_{1}+b_{j},\quad-1-\epsilon\leq x_{1}\leq-1+\epsilon,\quad
x_{2}=y_{2}=0\\}\subset\mathcal{R}_{S_{{V}_{j}}}.$

Next we construct the ribbons of the plates in the Legendrian disks
$\mathcal{D}^{{}^{\prime}}_{k}$, $\mathcal{D}^{{}^{\prime\prime}}_{k}$. For
the triangular plates
$T_{a_{i}}^{1,1},T_{a_{i}}^{1,-1},T_{b_{j}}^{-1,1},T_{b_{j}}^{-1,-1}$, we
define corresponding ribbons as

$\displaystyle\mathcal{R}_{T_{a_{i}}^{1,1}}$ $\displaystyle\doteq$
$\displaystyle\\{z=-y_{1}-y_{2}+a_{i},\quad 1-\epsilon\leq x_{1}\leq
1+\epsilon,\quad 1-\delta\leq x_{2}\leq 1+\delta\\}$
$\displaystyle\mathcal{R}_{T_{a_{i}}^{1,-1}}$ $\displaystyle\doteq$
$\displaystyle\\{z=-y_{1}+y_{2}+a_{i},\quad 1-\epsilon\leq x_{1}\leq
1+\epsilon,\quad-1-\delta\leq x_{2}\leq-1+\delta\\}$
$\displaystyle\mathcal{R}_{T_{b_{j}}^{-1,1}}$ $\displaystyle\doteq$
$\displaystyle\\{z=y_{1}-y_{2}+b_{j},\quad-1-\epsilon\leq
x_{1}\leq-1+\epsilon,\quad 1-\delta\leq x_{2}\leq 1+\delta\\}$
$\displaystyle\mathcal{R}_{T_{b_{j}}^{-1,-1}}$ $\displaystyle\doteq$
$\displaystyle\\{z=y_{1}+y_{2}+b_{j},\quad-1-\epsilon\leq
x_{1}\leq-1+\epsilon,\quad-1-\delta\leq x_{2}\leq-1+\delta\\}.$

for sufficiently small $\epsilon,\delta>0$. The ribbons
$\mathcal{R}_{T_{a_{i}}^{1,1}},\mathcal{R}_{T_{a_{i}}^{1,-1}},\mathcal{R}_{T_{b_{j}}^{-1,1}},\mathcal{R}_{T_{b_{j}}^{-1,-1}}$
are disjoint and they deformation retract onto
$T_{a_{i}}^{1,1},T_{a_{i}}^{1,-1},T_{b_{j}}^{-1,1},T_{b_{j}}^{-1,-1}$,
respectively.

Now we consider $4$ other ribbon types $\mathcal{R}_{P_{\pm 1}}$,
$\mathcal{R}_{P^{{}^{\prime}}_{\pm 1}}$ whose cores are the Legendrian
rectangular plates $P_{\pm 1}$, $P^{{}^{\prime}}_{\pm 1}$ in
$\mathcal{D}^{{}^{\prime}}_{k}$ and $\mathcal{D}^{{}^{\prime\prime}}_{k}$.
Their tangent spaces at a point $q$ are given by

$\displaystyle T_{q}\mathcal{R}_{P_{\pm 1}}$ $\displaystyle=$
$\displaystyle\left\langle\frac{\partial}{\partial
y_{2}}\pm\frac{\partial}{\partial z},\quad\frac{\partial}{\partial
x_{1}},\quad\frac{\partial}{\partial x_{2}},\quad\frac{\partial}{\partial
z}-x_{1}\frac{\partial}{\partial y_{1}}\right\rangle$ $\displaystyle
T_{q}\mathcal{R}_{P^{{}^{\prime}}_{\pm 1}}$ $\displaystyle=$
$\displaystyle\left\langle\frac{\partial}{\partial
y_{1}}\pm\frac{\partial}{\partial y_{2}},\quad\frac{\partial}{\partial
x_{1}}-\frac{\partial}{\partial x_{2}},\quad\frac{\partial}{\partial
z}-x_{1}\frac{\partial}{\partial y_{1}},\quad\frac{\partial}{\partial
z}-x_{2}\frac{\partial}{\partial y_{2}}\right\rangle.$

As the final ribbon types we consider $\mathcal{R}_{P_{t}}$ and
$\mathcal{R}_{P_{b}}$ whose cores are the Legendrian top and bottom plates in
$\mathcal{D}^{{}^{\prime}}_{k}$ and $\mathcal{D}^{{}^{\prime\prime}}_{k}$. The
tangent spaces of them at a point $q$ are given by

$T_{q}\mathcal{R}_{P_{t}}=T_{q}\mathcal{R}_{P_{b}}=\left\langle\frac{\partial}{\partial
x_{1}},\quad\frac{\partial}{\partial x_{2}},\quad\frac{\partial}{\partial
z}-x_{1}\frac{\partial}{\partial y_{1}},\quad\frac{\partial}{\partial
z}-x_{2}\frac{\partial}{\partial y_{2}}\right\rangle.$

By taking appropriate $\epsilon$ and $\delta$, the ribbons
$\mathcal{R}_{T_{a_{i}}^{1,1}},\mathcal{R}_{T_{a_{i}}^{1,-1}},\mathcal{R}_{T_{b_{j}}^{-1,1}},\mathcal{R}_{T_{b_{j}}^{-1,-1}},\mathcal{R}_{P_{\pm
1}}$, $\mathcal{R}_{P^{{}^{\prime}}_{\pm
1}},\mathcal{R}_{P_{t}},\mathcal{R}_{P_{b}}$ meet along their $3$-dimensional
common boundary parts which are the $2$-dimensional thickenings of the common
interval parts of the boundaries of their cores
$T_{a_{i}}^{1,1},T_{a_{i}}^{1,-1},T_{b_{j}}^{-1,1},T_{b_{j}}^{-1,-1}$, $P_{\pm
1}$, $P^{{}^{\prime}}_{\pm 1},P_{t},P_{b}$. So, as before, gluing them along
their common boundary regions gives two disjoint ribbons (both are
topologically $D^{4}$) $\mathcal{R}_{\mathcal{D}^{{}^{\prime}}_{k}}$ and
$\mathcal{R}_{\mathcal{D}^{{}^{\prime\prime}}_{k}}$ whose cores are
$\mathcal{D}^{{}^{\prime}}_{k}$, $\mathcal{D}^{{}^{\prime\prime}}_{k}$,
respectively.

Finally, by taking the union of $\mathcal{R}_{M_{k}}$,
$\mathcal{R}_{\mathcal{D}^{{}^{\prime}}_{k}}$ and
$\mathcal{R}_{\mathcal{D}^{{}^{\prime\prime}}_{k}}$ (along the thickening of
the regions where
$M_{k},\mathcal{D}^{{}^{\prime}}_{k},\mathcal{D}^{{}^{\prime\prime}}_{k}$
meets), we construct the ribbon

$\mathcal{R}_{\mathcal{D}_{k}}=\mathcal{R}_{\mathcal{D}^{{}^{\prime}}_{k}}\cup\mathcal{R}_{M_{k}}\cup\mathcal{R}_{\mathcal{D}^{{}^{\prime\prime}}_{k}}$

whose core is the Legendrian diamond $\mathcal{D}_{k}$. By construction the
$4$-dimensional ribbon $\mathcal{R}_{\mathcal{D}_{k}}$ properly contains the
$2$-dimensional ribbon $R_{\gamma_{k}}$ which is diffeomorphic to
$D^{*}\mathbb{S}^{1}$. Indeed,

######  Lemma 4.1.

After rounding the corners, the ribbon $\mathcal{R}_{\mathcal{D}_{k}}$ is
diffeomorphic to $D^{*}\mathbb{S}^{2}$.

###### Proof.

First, we observe that the tangent bundle of the rectangular plates
$S_{{H}_{i}}$ and $S_{{V}_{j}}$ in the middle part $M_{k}$ are generated by
$\frac{\partial}{\partial y_{1}}\pm\frac{\partial}{\partial z}$ and
$\frac{\partial}{\partial x_{2}}$. The ribbons $\mathcal{R}_{S_{{H}_{i}}}$ and
$\mathcal{R}_{S_{{V}_{j}}}$ are, respectively, obtained by thickening of
$S_{{H}_{i}}$ and $S_{{V}_{j}}$ along the vectors $\frac{\partial}{\partial
x_{1}}$ and $\frac{\partial}{\partial y_{2}}-x_{2}\frac{\partial}{\partial
z}$. Also, we note that the tangent bundle of the triangular plates
$S_{{I}_{k}}$ in $M_{k}$ is generated by $\frac{\partial}{\partial x_{1}}$ and
$\frac{\partial}{\partial x_{2}}$, and the ribbon $\mathcal{R}_{S_{{I}_{k}}}$
is the thickening of ${I}_{k}$ along the two vectors $\frac{\partial}{\partial
y_{1}}-x_{1}\frac{\partial}{\partial z}$ and $\frac{\partial}{\partial
y_{2}}-x_{2}\frac{\partial}{\partial z}$.

Now, consider the isotropic unknot $\gamma_{k}$ in the middle part $M_{k}$.
The tangent bundle of ${H}_{i}$ and ${V}_{j}$ in $\gamma_{k}$ are generated by
$\frac{\partial}{\partial y_{1}}\pm\frac{\partial}{\partial z}$. The
$2$-dimesnsional ribbon $\mathcal{R}_{{H}_{i}}$ and $\mathcal{R}_{{V}_{j}}$
are obtained by thickening of ${H}_{i}$ and ${V}_{j}$ in the direction of
$\frac{\partial}{\partial x_{1}}$. Also, the tangent bundle of ${I}_{k}$ is
generated by $\frac{\partial}{\partial x_{1}}$. The $2$-dimesnsional ribbon
$\mathcal{R}_{{I}_{k}}$ is the thickening of ${I}_{k}$ along
$\frac{\partial}{\partial y_{1}}-x_{1}\frac{\partial}{\partial z}$. One can
observe that $\mathcal{R}_{\gamma_{k}}$ topologically positive Hopf band. So,
the $4$-dimensional ribbon $\mathcal{R}_{M_{k}}$ contains the $2$-dimensional
ribbon $\mathcal{R}_{\gamma_{k}}$ which is topologically
$D^{*}\mathbb{S}^{1}$, and the thickening directions of
$\mathcal{R}_{\gamma_{k}}$ are the cofiber direction of
$\mathcal{R}_{\gamma_{k}}$ and the manifold direction of
$\mathcal{R}_{M_{k}}$. Since the ribbon $\mathcal{R}_{\mathcal{D}_{k}}$ is
obtained by attaching two ribbons
$\mathcal{R}_{\mathcal{D}^{{}^{\prime}}_{k}}$ and
$\mathcal{R}_{\mathcal{D}^{{}^{\prime\prime}}_{k}}$, which are topologically
$D^{*}D^{2}$, to $\mathcal{R}_{M_{k}}$, $\mathcal{R}_{\mathcal{D}_{k}}$ is
topologically $D^{*}\mathbb{S}^{2}$. ∎

######  Remark 4.2.

In fact, we can say more here: By straigthforward computations one can check
that $d\alpha_{0}^{5}|_{\mathcal{D}_{k}}$ is non-degenerate on the tangent
space $T_{q}\mathcal{R}_{\mathcal{D}_{k}}$ for all
$q\in\mathcal{R}_{\mathcal{D}_{k}}$, and so the pair
$(\mathcal{R}_{\mathcal{D}_{k}},d\alpha_{0}^{5}|_{\mathcal{D}_{k}})$ is a
symplectic manifolds. Indeed, one can realize it as the unit disk cotangent
bundle of $\mathcal{D}_{k}\cong\mathbb{S}^{2}$. More precisely, by
construction we know that $\mathcal{R}_{\gamma_{k}}$ is a positive Hopf band
equipped with the symplectic structure
$d\alpha_{0}^{3}|_{\gamma_{k}}=d\alpha_{0}^{5}|_{\gamma_{k}}$, and so
$(\mathcal{R}_{\gamma_{k}},d\alpha_{0}^{3}|_{\mathcal{D}_{k}})$ is the unit
disk cotangent bundle of $\gamma_{k}\cong\mathbb{S}^{1}$ which is
symplectically and properly embedded in
$(\mathcal{R}_{\mathcal{D}_{k}},d\alpha_{0}^{5}|_{\mathcal{D}_{k}})$. Also by
construction the zero section $\gamma_{k}$ is properly embedded in
$\mathcal{D}_{k}$, and the contraction deforming
$\mathcal{R}_{\mathcal{D}_{k}}$ onto its core $\mathcal{D}_{k}$ is,
simultanously, deforming $\mathcal{R}_{\gamma_{k}}$ onto the core
$\gamma_{k}$. Hence,
$(\mathcal{R}_{\mathcal{D}_{k}},d\alpha_{0}^{5}|_{\mathcal{D}_{k}})$ must be
the unit disk cotangent bundle of the $2$-sphere $\mathcal{D}_{k}$.

By Lemma 4.1 and Remark 4.2, we conclude that, for each Legendrian diamond
$\mathcal{D}_{k}$, one can obtain a compatible open book
$(\mathcal{R}_{\mathcal{D}_{k}},\tau_{\mathcal{D}_{k}})\cong(D^{*}\mathbb{S}^{2},\tau_{\mathbb{S}^{2}})$
on $\mathbb{S}^{5}_{st}$ where we identify (after rounding corners)
$\mathcal{D}_{k}$ with the zero section $\mathbb{S}^{2}$ in
$D^{*}\mathbb{S}^{2}$, and $\tau_{\mathcal{D}_{k}}$ denotes the right-handed
Dehn twist along $\mathcal{D}_{k}$. Indeed, simultaneously, we also obtain a
compatible open book
$(\mathcal{R}_{\gamma_{k}},\tau_{\gamma_{k}})\cong(D^{*}\mathbb{S}^{1},\tau_{\mathbb{S}^{1}})$
on $\mathbb{S}^{3}_{st}$ where we identify (after rounding corners)
$\gamma_{k}$ with the zero section $\mathbb{S}^{1}$ in $D^{*}\mathbb{S}^{1}$,
and $\tau_{\gamma_{k}}$ denotes the right-handed Dehn twist along $\gamma_{k}$
(which is the equatorial circle of $\mathcal{D}_{k}$). Thus, for each
Legendrian diamond $\mathcal{D}_{k}$ we have, indeed, a compatible relative
open book on the (admissible) relative contact pair
$(\mathbb{S}^{5}_{st},\mathbb{S}^{3}_{st})$ given by

$(\mathcal{R}_{\mathcal{D}_{k}},\mathcal{R}_{\gamma_{k}},\tau_{\mathcal{D}_{k}},\tau_{\gamma_{k}})\cong(D^{*}\mathbb{S}^{2},D^{*}\mathbb{S}^{1},\tau_{\mathbb{S}^{2}},\tau_{\mathbb{S}^{1}}).$

Next, starting from the square bridge diagram $\Delta(\mathbb{L})$, we will
construct a compatible relative open book on
$(\mathbb{S}^{5}_{st},\mathbb{S}^{3}_{st})$:

######  Proposition 4.3.

For a given admissible Legendrian $2$-link $\mathbb{L}$ with its equatorial
$1$-link $\mathbb{K}$, suppose that there are $m$ square regions
$R_{1},R_{2},...,R_{m}$ in the square bridge diagram $\Delta(\mathbb{K})$, and
for each $k\in\\{1,2,...,m\\}$ let $\mathcal{D}_{k}$ be a smooth (rounded)
Legendrian $2$-unknot with its smooth (rounded) isotropic equatorial
$1$-unknot $\gamma_{k}$ constructed as above. Then there exists a compatible
relative open book
$(F_{\mathbb{S}^{5}},F_{\mathbb{S}^{3}},h_{\mathbb{S}^{5}},h_{\mathbb{S}^{3}})$
on the (admissible) relative contact pair
$(\mathbb{S}^{5}_{st},\mathbb{S}^{3}_{st})$ such that the pages
$F_{\mathbb{S}^{5}},F_{\mathbb{S}^{3}}$ are Weinstein, and the following are
satisfied:

  1. (i)

$F_{\mathbb{S}^{3}}$ is symplectically and properly embedded in
$F_{\mathbb{S}^{5}}$.

  2. (ii)

The monodromies of $(F_{\mathbb{S}^{5}},h_{\mathbb{S}^{5}})$,
$(F_{\mathbb{S}^{3}},h_{\mathbb{S}^{3}})$ are, respectively, given by

$h_{\mathbb{S}^{5}}=\tau_{\mathcal{D}_{1}}\circ\tau_{\mathcal{D}_{2}}\circ\cdots\circ\tau_{\mathcal{D}_{m}},\quad
h_{\mathbb{S}^{3}}=\tau_{\gamma_{1}}\circ\tau_{\gamma_{2}}\circ\cdots\circ\tau_{\gamma_{m}}.$

###### Proof.

We will introduce $R_{k}$’s to the square bridge diagram $\Delta(\mathbb{K})$
(and so the corresponding octahedrons to the generalized square bridge diagram
$\Delta(\mathbb{L})$) one by one (in the order of their indices), and each
step we will obtain a new compatible relative open book on
$(\mathbb{S}^{5}_{st},\mathbb{S}^{3}_{st})$ as the result of a positive
relative stabilization of the compatible relative open book constructed in the
previous step. As an initial step, if no squares is introduced yet, then we
consider the trivial compatible relative open book
$\mathcal{OB}_{0}\doteq(D^{4},D^{2},id_{D^{4}},id_{D^{2}})$ on
$(\mathbb{S}^{5}_{st},\mathbb{S}^{3}_{st})$ given in Example 2.4. When the
first square $R_{1}$ is introduced, from Example 2.11 we obtain

$\mathcal{OB}_{1}\doteq(\mathcal{R}_{\mathcal{D}_{1}},\mathcal{R}_{\gamma_{1}},\tau_{\mathcal{D}_{1}},\tau_{\gamma_{1}})=\mathcal{S}^{+}[(D^{4},D^{2},id_{D^{4}},id_{D^{2}});A_{1},A_{1}\cap
D^{2}]$

where $A_{1}$ is a properly embedded Lagrangian $2$-disk in $D^{4}$ with a
Legendrian boundary in $\partial D^{4}$ as described in the example. We note
that $\mathcal{R}_{\mathcal{D}_{1}}=D^{4}\cup\Sigma_{1}$ and
$\mathcal{R}_{\gamma_{1}}=D^{2}\cup\sigma_{1}$ where $\Sigma_{1}$ and
$\sigma_{1}$ are the corresponding Weinstein handles of the stabilizations
mentioned in Remark 2.6.

Now introducing $R_{2}$ yields

$\mathcal{OB}_{2}\doteq(\mathcal{R}_{\mathcal{D}_{1}}\cup\mathcal{R}_{\mathcal{D}_{2}},\mathcal{R}_{\gamma_{1}}\cup\mathcal{R}_{\gamma_{2}},\tau_{\mathcal{D}_{1}}\circ\tau_{\mathcal{D}_{2}},\tau_{\gamma_{1}}\circ\tau_{\gamma_{2}})\cong\mathcal{S}^{+}[(\mathcal{R}_{\mathcal{D}_{1}},\mathcal{R}_{\gamma_{1}},\tau_{\mathcal{D}_{1}},\tau_{\gamma_{1}});A_{2},A_{2}\cap\mathcal{R}_{\gamma_{1}}].$

Here
$\mathcal{R}_{\mathcal{D}_{1}}\cup\mathcal{R}_{\mathcal{D}_{2}}=\mathcal{R}_{\mathcal{D}_{1}}\cup\Sigma_{2}$
and
$\mathcal{R}_{\gamma_{1}}\cup\mathcal{R}_{\gamma_{2}}=\mathcal{R}_{\gamma_{1}}\cup\sigma_{2}$
where

$\Sigma_{2}=\mathcal{R}_{\mathcal{D}_{2}}\setminus\mathcal{R}_{M_{1}\cap
M_{2}},\quad\sigma_{2}=\mathcal{R}_{\gamma_{2}}\setminus\mathcal{R}_{\gamma_{1}\cap\gamma_{2}}$

are the corresponding Weinstein handles of the stabilizations so that the
properly embedded Lagrangian $2$-disk $A_{2}$ in the ribbon
$\mathcal{R}_{M_{1}\cap M_{2}}$ obtained by gluing (smoothy) the standard
Lagrangian cylinder $S^{1}\times[0,1]=\mathcal{R}_{M_{1}\cap
M_{2}}\cap\mathcal{D}_{2}$ to the Lagrangian disk $M_{1}\cap M_{2}$ so that
$S^{1}\times\\{0\\}=\partial(M_{1}\cap M_{2})$ and
$S^{1}\times\\{1\\}=\partial A_{2}$ is the attaching cirle of $\Sigma_{2}$
(and so $\sigma_{2}$ is attached along the pair of points
$\partial(A_{2}\cap\mathcal{R}_{M_{1}\cap M_{2}})=\partial
A_{2}\cap\partial\mathcal{R}_{M_{1}\cap M_{2}}$). This will attach the (new)
diamond $\mathcal{D}_{2}$ to the existent core $\mathcal{D}_{1}$, and they
together define the core $\mathcal{D}_{1}\cup\mathcal{D}_{2}$ of the new page
(see Figure 10).

We remark that taking the union of the Lagrangian core disk (say $C_{2}$) of
$\Sigma_{2}$ and $A_{2}=(M_{1}\cap
M_{2})\cup_{S^{1}\times\\{0\\}}S^{1}\times[0,1]$ gives the Lagrangian
$2$-sphere

$\mathcal{D}_{2}=(M_{1}\cap
M_{2})\cup_{S^{1}\times\\{0\\}}S^{1}\times[0,1]\cup_{S^{1}\times\\{1\\}}C_{2}$

in the new symplectic (indeed, Weinstein) page
$\mathcal{R}_{\mathcal{D}_{1}}\cup\mathcal{R}_{\mathcal{D}_{2}}\subset\mathbb{S}^{5}_{st}$
along which a new right-handed Dehn twist is adjoined to the previous
monodromy. In particular,

$\mathcal{D}_{2}\cap\mathbb{S}^{3}_{st}=[(M_{1}\cap
M_{2})\cup_{S^{1}\times\\{0\\}}S^{1}\times[0,1]\cup_{S^{1}\times\\{1\\}}C_{2}]\cap\mathbb{S}^{3}_{st}=\gamma_{2}$

is the Lagrangian $1$-sphere in the new symplectic (Weinstein) page
$\mathcal{R}_{\gamma_{1}}\cup\mathcal{R}_{\gamma_{2}}\subset\mathbb{S}^{3}_{st}$
along which a new right-handed Dehn twist is adjoined to the previous
monodromy.

Figure 10. An attaching a new diamond $\mathcal{D}_{k+1}$ to an existent
$\mathcal{D}_{k}$.

Iteratively we continue as follows: Assuming $R_{k}$ has been introduced, and
so the corresponding open book
$\mathcal{OB}_{k}\doteq(\bigsqcup_{i=1}^{k}\mathcal{R}_{\mathcal{D}_{i}},\bigsqcup_{i=1}^{k}\mathcal{R}_{\gamma_{i}},\tau_{\mathcal{D}_{1}}\circ\cdots\circ\tau_{\mathcal{D}_{k}},\tau_{\gamma_{1}}\circ\cdots\circ\tau_{\gamma_{k}})$
has been constructed, introducing $R_{k+1}$ yields the compatible relative
open book

$\displaystyle\mathcal{OB}_{k+1}$ $\displaystyle\doteq$
$\displaystyle((\bigsqcup_{i=1}^{k}\mathcal{R}_{\mathcal{D}_{i}})\cup\mathcal{R}_{\mathcal{D}_{k+1}},(\bigsqcup_{i=1}^{k}\mathcal{R}_{\gamma_{i}})\cup\mathcal{R}_{\gamma_{k+1}},(\tau_{\mathcal{D}_{1}}\circ\cdots\circ\tau_{\mathcal{D}_{k}})\circ\tau_{\mathcal{D}_{k+1}},(\tau_{\gamma_{1}}\circ\cdots\circ\tau_{\gamma_{k}})\circ\tau_{\gamma_{k+1}})$
$\displaystyle=$
$\displaystyle\mathcal{S}^{+}[(\bigsqcup_{i=1}^{k}\mathcal{R}_{\mathcal{D}_{i}},\bigsqcup_{i=1}^{k}\mathcal{R}_{\gamma_{i}},\tau_{\mathcal{D}_{1}}\circ\cdots\circ\tau_{\mathcal{D}_{k}},\tau_{\gamma_{1}}\circ\cdots\circ\tau_{\gamma_{k}});A_{k+1},A_{k+1}\cap\bigsqcup_{i=1}^{k}\mathcal{R}_{\gamma_{i}}]$

As before we have

$(\bigsqcup_{i=1}^{k}\mathcal{R}_{\mathcal{D}_{i}})\cup\mathcal{R}_{\mathcal{D}_{k+1}}=(\bigsqcup_{i=1}^{k}\mathcal{R}_{\mathcal{D}_{i}})\cup\Sigma_{k+1},\quad(\bigsqcup_{i=1}^{k}\mathcal{R}_{\gamma_{i}})\cup\mathcal{R}_{\gamma_{k+1}}=(\bigsqcup_{i=1}^{k}\mathcal{R}_{\gamma_{i}})\cup\sigma_{k+1}$

where

$\Sigma_{k+1}=\mathcal{R}_{\mathcal{D}_{k+1}}\setminus\mathcal{R}_{(\bigsqcup_{i=1}^{k}M_{i})\cap
M_{k+1}},\quad\sigma_{k+1}=\mathcal{R}_{\gamma_{k+1}}\setminus\mathcal{R}_{(\bigsqcup_{i=1}^{k}\gamma_{i})\cap\gamma_{k+1}}$

are the corresponding Weinstein handles of the stabilizations so that the
properly embedded Lagrangian $2$-disk $A_{k+1}$ in the ribbon
$\mathcal{R}_{(\bigsqcup_{i=1}^{k}M_{i})\cap M_{k+1}}$ obtained by gluing
(smoothy) the standard Lagrangian cylinder
$S^{1}\times[0,1]=\mathcal{R}_{(\bigsqcup_{i=1}^{k}M_{i})\cap
M_{k+1}}\cap\mathcal{D}_{k+1}$ to the Lagrangian disk
$(\bigsqcup_{i=1}^{k}M_{i})\cap M_{k+1}$ so that
$S^{1}\times\\{0\\}=\partial((\bigsqcup_{i=1}^{k}M_{i})\cap M_{k+1})$ and
$S^{1}\times\\{1\\}=\partial A_{k+1}$ is the attaching cirle of
$\Sigma_{k+1}$, and so $\sigma_{k+1}$ is attached along the pair of points

$\partial(A_{k+1}\cap(\bigsqcup_{i=1}^{k}M_{i})\cap M_{k+1})=\partial
A_{2}\cap\partial\mathcal{R}_{(\bigsqcup_{i=1}^{k}M_{i})\cap M_{k+1}}.$

This will attach the (new) diamond $\mathcal{D}_{k+1}$ to the existent core
$\bigsqcup_{i=1}^{k}\mathcal{D}_{i}$, and they together define the core
$(\bigsqcup_{i=1}^{k}\mathcal{D}_{i})\cup\mathcal{D}_{k+1}$ of the new page
(see Figure 11 where we take $m=4$).

Figure 11. Attaching process along the middle parts $M_{i}$’s of
$\mathcal{D}_{i}$’s.

Similar to $k=2$ case, taking the union of the Lagrangian core disk $C_{k+1}$
of $\Sigma_{k+1}$ and $A_{k+1}=((\bigsqcup_{i=1}^{k}M_{i})\cap
M_{k+1})\cup_{S^{1}\times\\{0\\}}(S^{1}\times[0,1])$ gives the Lagrangian
$2$-sphere

$\mathcal{D}_{k+1}=((\bigsqcup_{i=1}^{k}M_{i})\cap
M_{k+1})\cup_{S^{1}\times\\{0\\}}(S^{1}\times[0,1])\cup_{S^{1}\times\\{1\\}}C_{k+1}$

in the new symplectic (Weinstein) page
$(\bigsqcup_{i=1}^{k}\mathcal{R}_{\mathcal{D}_{i}})\cup\mathcal{R}_{\mathcal{D}_{k+1}}\subset\mathbb{S}^{5}_{st}$
along which a new right-handed Dehn twist is adjoined to the previous
monodromy. In particular,

$\mathcal{D}_{k+1}\cap\mathbb{S}^{3}_{st}=[((\bigsqcup_{i=1}^{k}M_{i})\cap
M_{k+1})\cup_{S^{1}\times\\{0\\}}(S^{1}\times[0,1])\cup_{S^{1}\times\\{1\\}}C_{k+1}]\cap\mathbb{S}^{3}_{st}=\gamma_{k+1}$

is the Lagrangian $1$-sphere in the new symplectic (Weinstein) page
$(\bigsqcup_{i=1}^{k}\mathcal{R}_{\gamma_{i}})\cup\mathcal{R}_{\gamma_{k+1}}\subset\mathbb{S}^{3}_{st}$
along which a new right-handed Dehn twist is adjoined to the previous
monodromy.

Hence, by taking $k=m$, i.e., after introducing all $R_{k}$’s, we obtain a
relative open book

$\displaystyle(F_{\mathbb{S}^{5}},F_{\mathbb{S}^{3}},h_{\mathbb{S}^{5}},h_{\mathbb{S}^{3}})$
$\displaystyle\doteq$ $\displaystyle\mathcal{OB}_{m}$ $\displaystyle=$
$\displaystyle\mathcal{S}^{+}[\cdots[\mathcal{S}^{+}[(D^{4},D^{2},id_{D^{4}},id_{D^{2}});A_{1},A_{1}\cap
D^{2}];\cdots];A_{m},A_{m}\cap\bigsqcup_{i=1}^{m-1}\mathcal{R}_{\gamma_{i}}]$

which is compatible with the admissible relative contact pair
$(\mathbb{S}^{5}_{st},\mathbb{S}^{3}_{st})$ by Theorem 2.10. Moreover, by
construction, the pages $F_{\mathbb{S}^{3}},F_{\mathbb{S}^{5}}$ are Weinstein
(as being constructed from Weinstein handles), and the conditions (i), (ii)
are satisfied as required. ∎

Returning back to the proof of Theorem 1.6, starting from the generalized
square bridge diagram $\Delta(\mathbb{L})$, consisting of $m$ octahedrons, of
the given link, we know, by Proposition 4.3, that there exists a compatible
relative open book
$(F_{\mathbb{S}^{5}},F_{\mathbb{S}^{3}},h_{\mathbb{S}^{5}},h_{\mathbb{S}^{3}})$
on $(\mathbb{S}^{5}_{st},\mathbb{S}^{3}_{st})$ with Weinstein pages and
satisfying (v) in the statement. Moreover, for any component $L$ of
$\,\mathbb{L}$ and its equatorial knot $K\subset\mathbb{K}$, we know, by
Proposition 3.2, that we can reconstruct $L$ as a piecewise smooth Legendrian
$2$-sphere $L^{\prime}\subset\mathbb{S}^{5}_{st}$ and by rounding the corners
of $L^{\prime}$, one can obtain a smooth Legendrian $2$-sphere which is
Legendrian isotopic to $L$. Indeed, by construction, $L^{\prime}$ (and so its
smoothening) is contained in the core $\bigsqcup_{i=1}^{m}\mathcal{D}_{i}$ of
the page
$F_{\mathbb{S}^{5}}=\bigsqcup_{i=1}^{m}\mathcal{R}_{\mathcal{D}_{i}}$. In
particular, the piecewise smooth Legendrian equatorial $1$-sphere
$K^{\prime}\subset\mathbb{S}^{3}_{st}$ (and its smoothening) is contained in
the core $\bigsqcup_{i=1}^{m}\gamma_{i}$ of the page
$F_{\mathbb{S}^{3}}=\bigsqcup_{i=1}^{m}\mathcal{R}_{\gamma_{i}}$. We depict
this reconstruction for the standard Legendrian $2$-unknot in Figure 12.

Figure 12. On the left, the front projection of the (admissible) standard
Legendrian $2$-unknot $L$, and on the right, the corresponding piecewise
smooth Legendrian $2$-unknot $L^{\prime}$ (reconstruction of $L$).

So far, we have proved the statements (i) and (iii) (considering $L$ as the
rounded $L^{\prime}$). Since the reconstruction of $K$ (rounded $K^{\prime}$)
in the page $F_{\mathbb{S}^{3}}$ is based on the square bridge diagram
$\Delta(K)$, the statement (ii) follows directly from the same lines in [2].
In order to prove (iv), we observe that the arguments used in proving (ii) can
be generalized to high dimensions in a natural way. More precisely, since $L$
is contained in the core $\bigsqcup_{i=1}^{m}\mathcal{D}_{i}$, its ribbon
$\mathcal{R}_{L}$ is contained in the page $F_{\mathbb{S}^{5}}$ (the ribbon of
$\bigsqcup_{i=1}^{m}\mathcal{D}_{i}$). Note that reconstruction process does
not change the Legendrian isotopy type of $L$, so the contact (Thurston-
Bennequin) framing of $L$ is still the same as what it was at the beginning.
Moreover, consider tubular neighborhood $N(L)$ of $L$ in the page
$F_{\mathbb{S}^{5}}$. Being a codimension zero submanifold of the ribbon
$F_{\mathbb{S}^{5}}$, $N(L)$ is the ribbon of $L$. By definition, the contact
framing of $L$ is the one coming from the ribbon $N(L)$. Therefore, the
contact framing and the $N(L)$-framing (page framing) of $L$ are the same.
Since $N(L)\subset F_{\mathbb{S}^{5}}$, the framing which $L$ gets from the
page $F_{\mathbb{S}^{5}}$ is the same as the contact framing of $L$. This
completes the proof of Theorem 1.6.

∎

###  4.2. Admissible Relative Contact Surgery and Proof of Theorem 1.7

In this section, we will describe a(n) (admissible) relative contact
$r-$surgery along a(n) (admissible) relative Legendrian $2$-knot $L$ in
$(\mathbb{S}^{5}_{st},\mathbb{S}^{3}_{st})$ which gives us a new relative
contact pair, and then we will consider a special type of it which consists of
a contact (Legendrian) $(\pm 1)$-surgery along $L$ in $\mathbb{S}^{5}_{st}$
and a simultaneously performed contact (Legendrian) $(\pm 1)$-surgery along
the equatorial $1$-knot $K$ of $L$ in $\mathbb{S}^{3}_{st}$, and prove Theorem
1.7.

A contact surgery along a Legendrian $1$-knot in a contact $3$-manifold is a
topolological surgery where the surgery coefficient measured with respect to
contact framing: Let $K$ be an Legendrian $1$-knot in a contact manifold
$(M^{3},\eta)$ with a trivial neigborhood
$N(K)\cong\mathbb{D}^{2}\times\mathbb{S}^{1}$ with convex boundary. The
boundary $\partial N(K)$ has two homology generators represented by a meridian
$\mu$ and a longitude $\lambda$ where $\lambda$ is a parallel copy of $K$
obtained by pushing $K$ using its contact framing. Choose a trivilization
$\phi:\mathbb{D}^{2}\times\mathbb{S}^{1}\rightarrow N(K)$ such that
$\phi_{*}(\partial\mathbb{D}^{2}\times\\{pt\\})=\mu$ and
$\phi_{*}(\\{pt\\}\times\mathbb{S}^{1})=\lambda$ where
$\partial\mathbb{D}^{2}\times\\{pt\\}$ and $\\{pt\\}\times\mathbb{S}^{1}$ are
the generators of $H_{1}(\mathbb{S}^{1}\times\mathbb{S}^{1})$. A contact
$r=(\frac{p}{q})-$surgery along $K$ gives a new (not necessarily unique)
contact manifold
$\displaystyle(\mathbb{D}^{2}\times\mathbb{S}^{1})\cup_{\phi_{r}}(M^{3}\setminus
N(K))$ where
$\phi_{r}:\partial(\mathbb{D}^{2}\times\mathbb{S}^{1})\rightarrow\partial(M^{3}\setminus
N(K))$ is a gluing map such that
$(\phi_{r})_{*}(\partial\mathbb{D}^{2}\times\\{pt\\})=p\mu+q\lambda$. In the
case of $r=\pm 1$, a contact surgery produces a unique (up to isotopy) contact
structure on the surgered (resulting) $3$-manifold. For details see [12] and
[19].

In higher dimensions, one could also consider contact surgeries along
isotropic and coisotropic spheres (see [5]). However, a contact surgery along
a Legendrian $n$-knot is defined in a similar way: It is a topological surgery
where the surgery coefficient is determined relative to contact framing. Let
$\mathcal{S}$ be an Legendrian $n$-knot in a contact manifold $(Y^{2n+1},\xi)$
with a trivial neigborhood
$N(\mathcal{S})\cong\mathbb{D}^{n+1}\times\mathbb{S}^{n}$ with convex
boundary. The boundary $\partial N(\mathcal{S})$ has two homology generators
represented by a meridian $m$ and a longitude $\ell$ which is a parallel copy
of $\mathcal{S}$ obtained by pushing $\mathcal{S}$ using its contact framing.
Choose a trivilization $\Phi:\mathbb{D}^{n+1}\times\mathbb{S}^{n}\rightarrow
N(\mathcal{S})$ such that $\Phi_{*}(\partial\mathbb{D}^{n+1}\times\\{pt\\})=m$
and $\Phi_{*}(\\{pt\\}\times\mathbb{S}^{n})=\ell$ where
$\partial\mathbb{D}^{n+1}\times\\{pt\\}$ and $\\{pt\\}\times\mathbb{S}^{n}$
are the generators of $H_{n}(\mathbb{S}^{n}\times\mathbb{S}^{n})$. A contact
$r=(\frac{p}{q})-$surgery along $\mathcal{S}$ gives a new (not necessarily
unique) contact manifold
$\displaystyle(\mathbb{D}^{n+1}\times\mathbb{S}^{n})\cup_{\Phi_{r}}(Y^{2n+1}\setminus
N(\mathcal{S}))$ where
$\Phi_{r}:\partial(\mathbb{D}^{n+1}\times\mathbb{S}^{n})\rightarrow\partial(Y^{5}\setminus
N(\mathcal{S}))$ is a gluing map such that
$(\Phi_{r})_{*}(\partial\mathbb{D}^{n+1}\times\\{pt\\})=p\mu+q\lambda$. As
before, in the case of $r=\pm 1$, a contact surgery produces a unique (up to
isotopy) contact structure on the surgered (resulting) $(2n+1)$-manifold.

Returning back to the dimension three and five, we now introduce a relative
$r-$surgery along a relative $2$-knot. Although it can be defined in a purely
topological way, we give the definition in a contact setting:

######  Definition 4.4.

Let $L$ be a relative Legendrian $2$-knot in $\mathbb{S}^{5}_{st}$. By
Legendrian neighborhood theorem, $L$ has a neighborhood
$(N(L),\xi^{5}_{st}|_{N(L)})$ contactomorphic to a neighborhood of the zero
section in the $1$-jet space
$(T^{*}L\times\mathbb{R},\textrm{Ker}(dz-\sum_{i=1}^{2}p_{i}dq_{i}))$. The
contact $r-$surgery along $L$ gives a new (not unique) contact $5$-manifold
$(Y^{5},\xi)$ obtained by removing $N(L)$ from $\mathbb{S}^{5}_{st}$, and
gluing $\mathbb{D}^{3}\times\mathbb{S}^{2}$ back via a diffeomorphism inducing
a map $\Phi_{r}$ on the boundary (as above). Since $L$ is a relative knot, it
has an equatorial circle $K$ which is Legendrian in $\mathbb{S}^{3}_{st}$ and
its trivial neighborhood $(N(K),\xi_{st}^{3}|_{N(K)})$ is contactomorphic to a
tubular neighborhood of the zero section in
$\displaystyle(T^{*}K\times\mathbb{R},\textrm{Ker}(dz-p_{1}dq_{1}))$ which is
contactly embedded in
$(T^{*}L\times\mathbb{R},\textrm{Ker}(dz-\sum_{i=1}^{2}p_{i}dq_{i}))$. Since
$K$ is the isotropic equator of $L$, we have $T^{*}K\times\mathbb{R}\subset
T^{*}L\times\mathbb{R}$, and indeed, the neighborhood $N(K)\cong K\times
D^{2}$ is a contact submanifold of $N(L)\cong L\times D^{3}$ where
$\\{a\\}\times D^{2}\subset\\{a\\}\times D^{3}$ for any point $a\in K$. So,
when we remove $N(L)$ from $\mathbb{S}^{5}_{st}$, we simultaneously remove
$N(K)$ from $\mathbb{S}^{3}_{st}$. Therefore, by gluing
$\mathbb{D}^{3}\times\mathbb{S}^{2}$, and so
$\mathbb{D}^{2}\times\mathbb{S}^{1}(\subset\mathbb{D}^{3}\times\mathbb{S}^{2})$,
back via diffeomorphisms
$\Phi_{r}:\partial(\mathbb{D}^{3}\times\mathbb{S}^{2})\rightarrow\partial(\mathbb{S}^{5}_{st}\setminus
N(L))$ and its restriction
$\Phi_{r}|_{\partial(\mathbb{D}^{2}\times\mathbb{S}^{1})}:\partial(\mathbb{D}^{2}\times\mathbb{S}^{1})\rightarrow\partial(\mathbb{S}^{3}_{st}\setminus
N(K))$, we obtain the surgered manifolds

$(Y^{5},\xi):=(\mathbb{S}^{5}_{st}\setminus{N}(L))\cup(\mathbb{D}^{3}\times\mathbb{S}^{2}),\quad
M^{3}:=(\mathbb{S}^{3}_{st}\setminus
N(K))\cup(\mathbb{D}^{2}\times\mathbb{S}^{1}).$

Note that we obtain a relative (smooth) topological pair $(Y,M)$. Indeed, one
can check (see Lemma 4.5) that the $2$-plane distribution $\eta:=\xi\cap TM$
defines a contact structure on $M$ (so $(M,\eta)$ is a contact submanifold of
$(Y,\xi)$), and $(M,\eta)$ is the result of a contact $r-$surgery performed
simultaneously along $K$ in $\mathbb{S}^{3}_{st}$. In particular, we have
obtained a relative contact pair $(Y_{\xi},M_{\eta})$ which is said to be
obtained by a relative contact $r$-surgery along a relative Legendrian
$2$-knot $L$ in $\mathbb{S}^{5}_{st}$.

######  Lemma 4.5.

In the setting of Definition 4.4, the $2$-plane distribution $\eta=\xi\cap TM$
is a contact structure on $M$. Moreover, $(M,\eta)$ is the result of a contact
$r-$surgery along a Legendrian $1$-knot $K$ (the equator of $L$) in
$\mathbb{S}^{3}_{st}$.

###### Proof.

First we note that the contact embedding of $N(K)\cong K\times D^{2}$ into
$N(L)\cong L\times D^{3}$ is also a proper embedding, i.e.,

$\partial N(K)\cong K\times S^{1}\subset L\times S^{2}\cong\partial
N(L),\quad\textrm{with}\quad\\{a\\}\times S^{1}\subset\\{a\\}\times
S^{2},\quad\forall a\in K,$

and the characteristic foliation on the convex boundary $\partial
N(K)\subset\mathbb{S}^{3}_{st}$ is the restriction of the one on $\partial
N(L)\subset\mathbb{S}^{5}_{st}$. On the other hand, the contact structure on
$\mathbb{D}^{3}\times\mathbb{S}^{2}$ (the one chosen to perform the contact
$r$-surgery along $L$ resulting $(Y,\xi)$) produces the characteristic
foliation on the convex boundary $\mathbb{S}^{2}\times\mathbb{S}^{2}$. This
foliation restricts to the one on the convex boundary
$\mathbb{S}^{1}\times\mathbb{S}^{1}$ of the properly and contactly embedded
submanifold $\mathbb{D}^{2}\times\mathbb{S}^{1}$ of
$\mathbb{D}^{3}\times\mathbb{S}^{2}$. Moreover, when
$\mathbb{D}^{3}\times\mathbb{S}^{2}$ is glued to
$\mathbb{S}^{5}_{st}\setminus{N}(L)$ via
$\Phi_{r}:\mathbb{S}^{2}\times\mathbb{S}^{2}\to\partial(\mathbb{S}^{5}_{st}\setminus{N}(L))$,
the characteristic foliations on
$\mathbb{S}^{2}\times\mathbb{S}^{2},\partial(\mathbb{S}^{5}_{st}\setminus{N}(L))$
are identified with each other so that a contact structure $\xi$ is obtained
on the surgered manifold $Y$. In particular, the restriction diffeomorphism
$\Phi_{r}|_{\mathbb{S}^{1}\times\mathbb{S}^{1}}:\mathbb{S}^{1}\times\mathbb{S}^{1}\to\partial(\mathbb{S}^{3}_{st}\setminus{N}(K))$
glues the contact submanifolds
$\mathbb{D}^{2}\times\mathbb{S}^{1}\subset\mathbb{D}^{3}\times\mathbb{S}^{2}$
and
$\mathbb{S}^{3}_{st}\setminus{N}(K)\subset\mathbb{S}^{5}_{st}\setminus{N}(L)$
together so that the characteristic sub-foliations on
$\mathbb{S}^{1}\times\mathbb{S}^{1}$ and
$\partial(\mathbb{S}^{3}_{st}\setminus{N}(K))$ are identified with each other.
This means that when a contact surgery performed along $L$, a contact
structure $\eta=\xi\cap TM$ is, simultaneously, constructed on the surgered
manifold $M$ as claimed.

Now, the homology group $H_{2}(\partial N(L))$ has two generators represented
by a meridian $m$ and a longitude $\ell$ which is a parallel copy of $L$ on
$\partial N(L)$ obtained by pushing $L$ using its contact framing. Choose a
trivilization

$\Phi:\mathbb{D}^{3}\times\mathbb{S}^{2}\rightarrow N(L)$

such that $\Phi_{*}(\partial\mathbb{D}^{3}\times\\{pt\\})=m$ and
$\Phi_{*}(\\{pt\\}\times\mathbb{S}^{2})=\ell$ where
$\partial\mathbb{D}^{3}\times\\{pt\\}$ and $\\{pt\\}\times\mathbb{S}^{2}$ are
the generators of $H_{2}(\mathbb{S}^{2}\times\mathbb{S}^{2})$. If $p$ and $q$
are coprime integers such that $r=\frac{p}{q}$, then the result of the contact
$r-$surgery to $\mathbb{S}^{5}_{st}$ is the manifold

$(Y,\xi):=(\mathbb{S}^{5}_{st}\setminus
N(L))\cup_{\Phi_{r}}(\mathbb{D}^{3}\times\mathbb{S}^{2})$

where $(\Phi_{r})_{*}(\partial\mathbb{D}^{3}\times\\{pt\\})=pm+q\ell$. As
explained above simultaneously a surgery is being performed along the
equatorial Legendrian circle $K$ in $\mathbb{S}^{3}_{st}$: Replacing $N(L)$
with $\mathbb{D}^{3}\times\mathbb{S}^{2}$ corresponds to replacing $N(K)$ with
$\mathbb{D}^{2}\times\mathbb{S}^{1}$. For the homology group $H_{1}(\partial
N(K))\cong\mathbb{Z}\oplus\mathbb{Z}$, consider the following two generators:
Let $\mu\subset m$ be the meridian $\mu=m\cap\partial N(K)$, and
$\lambda\subset\ell$ be the longitude $\lambda=\ell\cap\partial N(K)$. Note
that $\mu$ is an equatorial circle of $m$ as $N(K)$ is properly embedded in
$N(L)$, also that $\lambda$ is an equatorial circle of $\ell$ as $N(K)$ is
contactly embedded in $N(L)$, so in particular, it is a parallel copy of $K$
produced by the contact framing of $K$ in $\mathbb{S}^{3}_{st}$. Now the
trivilization $\Phi$ restricts to a trivilization

$\phi\doteq\Phi|_{\mathbb{D}^{2}\times\mathbb{S}^{1}}=\mathbb{D}^{2}\times\mathbb{S}^{1}\rightarrow
N(K)$

such that $\phi_{*}(\partial\mathbb{D}^{2}\times\\{pt\\})=\mu$ and
$\phi_{*}(\\{pt\\}\times\mathbb{S}^{1})=\lambda$ where
$\partial\mathbb{D}^{3}\times\\{pt\\}$ and $\\{pt\\}\times\mathbb{S}^{2}$ are
the generators of $H_{1}(\mathbb{S}^{1}\times\mathbb{S}^{1})$. Finally,
observe that the gluing via the restriction map
$\Phi_{r}|_{\partial(\mathbb{D}^{2}\times\mathbb{S}^{1})}$ (resulting the
contact submanifold $(M,\eta)$) is equivalent to gluing via the map

$\phi_{r}:\partial(\mathbb{D}^{2}\times\mathbb{S}^{1})\rightarrow\partial(\mathbb{S}^{3}_{st}\setminus
N(K))\quad\textrm{with}\quad\phi_{*}(\partial\mathbb{D}^{2}\times\\{pt\\})=p\mu+q\lambda.$

(Since the $2$-sphere $\Phi_{r}(\partial\mathbb{D}^{3}\times\\{pt\\})$ travels
$p$-times around the meridian $m$ and $q$-times around the longitude $\ell$,
so does its equator $\Phi_{r}(\partial\mathbb{D}^{2}\times\\{pt\\})$, i.e., it
travels $p$-times around the meridian $\mu$ and $q$-times around the longitude
$\lambda$.) Hence, the simultaneous surgery along $K$ constructing the contact
submanifold $(M,\eta=\xi\cap TM)$ is also a contact $r-$surgery. ∎

###### Proof of Theorem 1.7.

Suppose that $(Y^{5}_{\xi},M^{3}_{\eta})$ is a given admissible relative
contact pair obtained by an admissible relative contact $(\pm 1)$-surgery
along an admissible Legendrian relative $2$-link
$\mathbb{L}=\bigsqcup_{i=1}^{r}L_{i}$ in $\mathbb{S}^{5}_{st}$ whose
equatorial isotropic $1$-link is
$\mathbb{K}=\mathbb{L}\cap\mathbb{S}^{3}_{st}=\bigsqcup_{i=1}^{r}{K}_{i}$. We
know, by Theorem 1.6, that there exists a compatible relative open book
$(F_{\mathbb{S}^{5}},F_{\mathbb{S}^{3}},h_{\mathbb{S}^{5}},h_{\mathbb{S}^{3}})$
on the relative contact pair $(\mathbb{S}^{5},\mathbb{S}^{3})$ with the
Weinstein pages $F_{\mathbb{S}^{5}},F_{\mathbb{S}^{3}}$ and the monodromies

$h_{\mathbb{S}^{5}}=\tau_{\mathcal{D}_{1}}\circ\tau_{\mathcal{D}_{2}}\circ\cdots\circ\tau_{\mathcal{D}_{m}},\quad
h_{\mathbb{S}^{3}}=\tau_{\gamma_{1}}\circ\tau_{\gamma_{2}}\circ\cdots\circ\tau_{\gamma_{m}}$

(as constructed in the proof of the theorem) such that each reconstructed and
smoothened $K_{i}$ lies on $F_{\mathbb{S}^{3}}$ and the page framing of
$K_{i}$ is equal to its contact framing in $\mathbb{S}^{3}_{st}$, and
similarly, each reconstructed and smoothened $L_{i}$ lies on
$F_{\mathbb{S}^{5}}$ and the page framing of $L_{i}$ is equal to its contact
framing in $\mathbb{S}^{5}_{st}$. From know on, all reconstructed links and
their reconstructed components are also assumed to be smoothened, and by abuse
of notation, we’ll denote them by the same letters.

By construction, all components of the reconstructed $2$-link $\mathbb{L}$
sits on a single page, say $F^{0}_{\mathbb{S}^{5}}$, as embedded Legendrian
$2$-unknots (of $\mathbb{S}^{5}_{st}$) in such a way that all components of
the reconstructed equatorial $1$-link $\mathbb{K}$ lie on a page, say
$F^{0}_{\mathbb{S}^{3}}$ (a Weinstein submanifold of $F^{0}_{\mathbb{S}^{5}}$)
as embedded Legendrian $1$-unknots (of $\mathbb{S}^{3}_{st}$). Being
Legendrian on a symplectic page of a compatible open book, all $L_{i}$’s
(resp. $K_{i}$’s) are Lagrangian in $F^{0}_{\mathbb{S}^{5}}$ (resp. in
$F^{0}_{\mathbb{S}^{3}}$). Recall that, by construction, the components of the
reconstructed $\mathbb{K}$ are still mutually disjoint in
$F^{0}_{\mathbb{S}^{3}}$ and the linking matrix of the reconstructed
$\mathbb{K}$ is the same as the linking matrices of the original $\mathbb{K}$
and $\mathbb{L}$. However, the components of reconstructed $\mathbb{L}$ have
some overlaping regions. In order to get rid of such regions, we will realize
each reconstructed component $L_{i}$ on a different page, call
$F^{i}_{\mathbb{S}^{5}}$, of the open book
$(F_{\mathbb{S}^{5}},h_{\mathbb{S}^{5}})$ in such a way that each
corresponding equatorial component $K_{i}$ will be contained on a (different)
page $F^{i}_{\mathbb{S}^{3}}$ of the the open book
$(F_{\mathbb{S}^{3}},h_{\mathbb{S}^{3}})$ which is a Weinstein subdomain of
$F^{i}_{\mathbb{S}^{5}}$.

First, note that since all reconstructed components of $\mathbb{L}$ are
contained in the Lagrangian core of the page $F^{0}_{\mathbb{S}^{5}}$, they do
not intersect the binding, so they miss a page of
$(F_{\mathbb{S}^{5}},h_{\mathbb{S}^{5}})$, and hence we may assume that they
are contained in a closed neighborhood
$N_{\mathbb{S}^{5}}(F^{0}_{\mathbb{S}^{5}})$ of $F^{0}_{\mathbb{S}^{5}}$ in
$\mathbb{S}^{5}$. This neighborhood can be contactly identified with
$(F\times[-1,1],\textrm{Ker}(\alpha_{0}^{5}+dt))$ where $t$ is the coordinate
on $[-1,1]$, $F=F^{0}_{\mathbb{S}^{5}}$ is equipped with the symplectic form
$d\alpha_{0}^{5}$, and we identify $F^{0}_{\mathbb{S}^{5}}$ with the page
$F\times\\{0\\}$. Thus, we have a contactomorphism (contact identification)

$\Theta:(N_{\mathbb{S}^{5}}(F^{0}_{\mathbb{S}^{5}}),\textrm{Ker}(\alpha_{0}^{5}|_{N_{\mathbb{S}^{5}}(F^{0}_{\mathbb{S}^{5}})})\longrightarrow(F\times[-1,1],\textrm{Ker}(\alpha_{0}^{5}+dt)).$

The contactomorphism $\Theta$ identifies the pages (fibers) in
$N_{\mathbb{S}^{5}}(F^{0}_{\mathbb{S}^{5}})$ with those in $F\times[-1,1]$,
and the Reeb vector field of $N_{\mathbb{S}^{5}}(F^{0}_{\mathbb{S}^{5}})$
corresponds to the Reeb vector field $\partial/\partial t$ of $F\times[-1,1]$.

Similarly, all reconstructed components of $\mathbb{K}$ are contained in the
Lagrangian core of the page $F^{0}_{\mathbb{S}^{3}}$, and they miss a page of
$(F_{\mathbb{S}^{3}},h_{\mathbb{S}^{3}})$, and so they are contained in a
closed neighborhood $N_{\mathbb{S}^{3}}(F^{0}_{\mathbb{S}^{3}})$ of
$F^{0}_{\mathbb{S}^{3}}$ in $\mathbb{S}^{3}$ which can be contactly identified
with $(G\times[-1,1],\textrm{Ker}(\alpha_{0}^{3}+dt))$ where
$G=F^{0}_{\mathbb{S}^{3}}$ is equipped with the symplectic form
$d\alpha_{0}^{3}$, and we identify $F^{0}_{\mathbb{S}^{3}}$ with the page
$G\times\\{0\\}$. In fact, $G$ is a properly embedded Weinstein subdomain of
$F$, and the restriction of $\Theta$ gives a contactomorphism

$\Theta|_{N_{\mathbb{S}^{3}}(F^{0}_{\mathbb{S}^{3}})}:(N_{\mathbb{S}^{3}}(F^{0}_{\mathbb{S}^{3}}),\textrm{Ker}(\alpha_{0}^{3}|_{N_{\mathbb{S}^{3}}(F^{0}_{\mathbb{S}^{3}})})\longrightarrow(G\times[-1,1],\textrm{Ker}(\alpha_{0}^{3}+dt)).$

Likewise the restricted $\Theta$ identifies the pages (fibers) in
$N_{\mathbb{S}^{3}}(F^{0}_{\mathbb{S}^{3}})$ with those in $G\times[-1,1]$,
and the Reeb vector field of $N_{\mathbb{S}^{3}}(F^{0}_{\mathbb{S}^{3}})$
corresponds to the Reeb vector field $\partial/\partial t$ of $G\times[-1,1]$.
(Note that $(G\times[-1,1],\textrm{Ker}(\alpha_{0}^{3}+dt))$ is contactly and
properly embedded in $(F\times[-1,1],\textrm{Ker}(\alpha_{0}^{5}+dt))$ and
their Reeb vector fields coincide, namely $\partial/\partial t$.)

Fix a metric $d$ on $F$ (and so on $G$) and let $d_{0}$ be the standard metric
on $[-1,1]$. Consider the pull-back metric (distance function)
$\textrm{dist}\doteq\Theta^{*}(d\times d_{0})$ on
$N_{\mathbb{S}^{5}}(F^{0}_{\mathbb{S}^{5}})$ where is the identification map.
Let $\epsilon>0$ be a sufficiently small number such that
$\epsilon<\textrm{dist}(K_{i},K_{j})$ for all $i\neq j$, and choose a
partition
$\\{-\epsilon=\epsilon_{1}\lneqq\epsilon_{2}\lneqq\cdots\lneqq\epsilon_{r}=\epsilon\\}$
of the interval $[-\epsilon,\epsilon]$.

Next, for each $i=1,2,...r$, consider the image $\Theta(L_{i})$ which is
Lagrangian in $(F\times\\{0\\},d\alpha_{0}^{5})$ and Legendrian in
$(F\times[-1,1],\textrm{Ker}(\alpha_{0}^{5}+dt))$. Using the flow of the Reeb
vector field $\partial/\partial t$ (which is transverse to the pages) we push
the $2$-unknot $\Theta(L_{i})$ untill we obtain a new smooth $2$-unknot
$U_{i}$ which is completely contained in the page $F\times\\{\epsilon_{i}\\}$.
Note that since $\partial/\partial t$ is a contact vector field,
$\Theta(L_{i})$ and $U_{i}$ are Legendrian isotopic in
$(F\times[-1,1],\textrm{Ker}(\alpha_{0}^{5}+dt))$. The restriction of this
isotopy to the equator $\Theta(K_{i})$ defines a Legendrian isotopy in
$(G\times[-1,1],\textrm{Ker}(\alpha_{0}^{3}+dt))$ from $\Theta(K_{i})$ to a
new smooth Legendrian $1$-unknot $V_{i}\subset
G\times\\{\epsilon_{i}\\}\subset F\times\\{\epsilon_{i}\\}$ which is, indeed,
the isotropic equator of $U_{i}$. Observe that $U_{i}$’s (and so $V_{i}$’s)
are mutually disjoint as they lie on different pages. Also by the choice of
$\epsilon$, the new smooth $2$-link $\mathbb{U}=\bigsqcup_{i=1}^{r}U_{i}$ and
its smooth equatorial $1$-link $\mathbb{V}=\bigsqcup_{i=1}^{r}V_{i}$ have a
common linking matrix which is, indeed, equal to the common linking matrix of
the originally given link $\mathbb{L}$ and its equatorial link $\mathbb{K}$.

Now consider the pull-back link
$\mathbb{L}^{\prime}\doteq\Theta^{-1}(\mathbb{U})\subset
N_{\mathbb{S}^{5}}(F^{0}_{\mathbb{S}^{5}})\subset\mathbb{S}^{5}_{st}$ and its
equatorial link $\mathbb{K}^{\prime}\doteq\Theta^{-1}(\mathbb{V})\subset
N_{\mathbb{S}^{3}}(F^{0}_{\mathbb{S}^{3}})\subset\mathbb{S}^{3}_{st}$. Since
$\Theta$ is a contactomorphism, the following are immediate:

  * (1)

Each component $L^{\prime}_{i}\doteq\Theta^{-1}(U_{i})$ of
$\mathbb{L}^{\prime}$ is Lagrangian on
$F^{i}_{\mathbb{S}^{5}}\doteq\Theta^{-1}(F\times\\{\i\\})$, and Legendrian in
$\mathbb{S}^{5}_{st}$, and Legendrian isotopic to original $L_{i}$.

  * (2)

Each component $K^{\prime}_{i}\doteq\Theta^{-1}(V_{i})$ of
$\mathbb{K}^{\prime\prime}$ is Lagrangian on
$F^{i}_{\mathbb{S}^{3}}\doteq\Theta^{-1}(G\times\\{i\\})$, and Legendrian in
$\mathbb{S}^{3}_{st}$, and Legendrian isotopic to original $K_{i}$.

  * (3)

The $2$-link $\mathbb{L}^{\prime}=\bigsqcup_{i=1}^{r}L^{\prime}_{i}$ and its
equatorial $1$-link $\mathbb{K}^{\prime}=\bigsqcup_{i=1}^{r}K^{\prime}_{i}$
have a common linking matrix which coincides with the common linking matrix of
$\mathbb{L}=\bigsqcup_{i=1}^{r}L_{i}$ and
$\mathbb{K}=\bigsqcup_{i=1}^{r}K_{i}$.

As a result, we conclude that the relative $2$-links $\mathbb{L}$,
$\mathbb{L}^{\prime}$ and their equatorial $1$-links $\mathbb{K}$,
$\mathbb{K}^{\prime}$ are Legendrian isotopic in $\mathbb{S}^{5}_{st}$ and
$\mathbb{S}^{3}_{st}$, respectively. Therefore, the relative contact $\pm
1$-surgery along $\mathbb{L}^{\prime}$ also results in the same (up to
isotopy) relative contact pair $(Y^{5}_{\xi},M^{3}_{\eta})$. On the other
hand, each component of $\mathbb{L}^{\prime}$ (resp. $\mathbb{K}^{\prime}$)
sits as a Lagrangian $2$-unknot (resp. Lagrangian $1$-unknot) on the pages of
the compatible open book $(F_{\mathbb{S}^{5}},h_{\mathbb{S}^{5}})$ on
$\mathbb{S}^{5}_{st}$ (resp. $(F_{\mathbb{S}^{3}},h_{\mathbb{S}^{3}})$ on
$\mathbb{S}^{3}_{st}$). Therefore, by “surgery on a page” theorems (Theorem
1.2 for dimension three, and see Theorem 4.6 in [15] for dimension five), one
immediately gets a compatible relative open book on the relative contact pair
$(Y^{5}_{\xi},M^{3}_{\eta})$ given by

$(F_{\mathbb{S}^{5}},F_{\mathbb{S}^{3}},h_{\mathbb{S}^{5}}\circ\tau_{\mathbb{L}^{\prime}},h_{\mathbb{S}^{3}}\circ\tau_{\mathbb{K}^{\prime}})$

where $\tau_{\mathbb{L}^{\prime}}$ (resp. $\tau_{\mathbb{K}^{\prime}}$)
denotes the product (composition) of the Dehn twists $\tau^{\mp
1}_{L^{\prime}_{i}}$ (resp. $\tau^{\mp 1}_{K^{\prime}_{i}}$) about the
components $\mathbb{L}^{\prime}$ (resp. $\mathbb{K}^{\prime}$) along which the
contact $(\pm 1)$-surgeries, resulting $Y^{5}_{\xi}$ (resp. $M^{3}_{\eta}$),
are performed. ∎

##  5\. Examples

######  Example 5.1.

Consider the admissible Legendrian $2$-link
$\mathbb{L}=\bigsqcup_{i=1}^{2}L_{i}$ whose equatorial link $\mathbb{K}$ is
the Hopf link such that its square bridge diagram $\Gamma(\mathbb{K})$ is the
one given in Figure 2. The generalized square bridge diagram
$\Delta(\mathbb{L})$ of $\mathbb{L}$ is given Figure 4. Using these diagrams,
for each $k$, we construct the Legendrian unknot $\gamma_{k}$ in
$(\mathbb{R},\xi_{0}^{3})\subset\mathbb{S}^{3}_{st}$ and the middle part
(Legendrian thickening) $M_{k}$ of $\gamma_{k}$ as in Figure 13.

Figure 13. (a) The middle part of the admissible Legendrian $2$-link
$\mathbb{L}$, (b) the equatorial link $\mathbb{K}$ of $\mathbb{L}$ (the Hopf
link) contained in the core of the $2$-dimensional ribbon.

Then one can obtain the piecewise smooth (Legendrian) reconstruction
$\mathbb{L}^{\prime}$ (with some overlapping regions) of $\mathbb{L}$ by first
building a Legendrian diamond $\mathcal{D}_{k}$ for each square region $R_{k}$
in $\Gamma(\mathbb{K})$ and then patching their appropriate parts together as
exaplained in Proposition 3.2 (see Figure 14). By construction,
$\mathbb{L}^{\prime}$ is contained in the union of
$\mathcal{R}_{\mathcal{D}_{k}}$’s ($4$-dimensional ribbons), and its equator
$\mathbb{K}^{\prime}$ is contained in the union of
$\mathcal{R}_{\gamma_{k}}$’s ($2$-dimensional ribbons).

Figure 14. The piecewise smooth Legendrian $2$-link $\mathbb{L}^{\prime}$
(with overlapping regions) and its equatorial link $\mathbb{K}^{\prime}$ (a
piecewise smooth Hopf link).

When each octahedron is introduced to the generalized square bridge diagram
$\Delta(\mathbb{L})$ of $\mathbb{L}$ (step by step) for each square region
$R_{k}$ in $\Gamma(\mathbb{K})$, a new compatible relative open book on
$(\mathbb{S}^{5}_{st},\mathbb{S}^{3}_{st})$ is obtained as the result of a
positive relative stabilization of the compatible relative open book in the
previous step.

More precisely, the trivial compatible relative open book
$\mathcal{OB}_{0}\doteq(D^{4},D^{2},id_{D^{4}},id_{D^{2}})$ on
$(\mathbb{S}^{5}_{st},\mathbb{S}^{3}_{st})$ is considered as an initial step.
When we reach the last step $k=7$ , we have the following compatible relative
open book on $(\mathbb{S}^{5}_{st},\mathbb{S}^{3}_{st})$:

$\displaystyle(F_{\mathbb{S}^{5}},F_{\mathbb{S}^{3}},h_{\mathbb{S}^{5}},h_{\mathbb{S}^{3}})$
$\displaystyle\doteq$ $\displaystyle\mathcal{OB}_{7}$ $\displaystyle=$
$\displaystyle\mathcal{S}^{+}[\cdots[\mathcal{S}^{+}[(D^{4},D^{2},id_{D^{4}},id_{D^{2}});A_{1},A_{1}\cap
D^{2}];\cdots];A_{7},A_{7}\cap\bigsqcup_{i=1}^{6}\mathcal{R}_{\gamma_{i}}]$

where $F_{\mathbb{S}^{5}}=\bigsqcup_{k=1}^{7}\mathcal{R}_{\mathcal{D}_{k}}$
and $F_{\mathbb{S}^{3}}=\bigsqcup_{k=1}^{7}\mathcal{R}_{\gamma_{k}}$ are the
corresponding Weinstein pages, the monodromies
$h_{\mathbb{S}^{5}}=\tau_{\mathcal{D}_{1}}\circ\cdots\circ\tau_{\mathcal{D}_{7}}$
and $h_{\mathbb{S}^{3}}=\tau_{\gamma_{1}}\circ\cdots\circ\tau_{\gamma_{7}}$
are the product of right-handed Dehn twists along $\mathcal{D}_{k}$ and
$\gamma_{k}$, respectively, and $A_{k}$ is the properly embedded Lagrangian
$2$-disk in the ribbon $\mathcal{R}_{(\bigsqcup_{i=1}^{k-1}M_{i})\cap M_{k}}$
as described in Proposition 4.3 for $k=1,\cdots,7$. Note that rounded
(smoothened) $\mathbb{L}^{\prime}$ and $\mathbb{K}^{\prime}$ are smoothly
embedded in the cores $\bigsqcup_{k=1}^{7}\mathcal{D}_{k}$ and
$\bigsqcup_{k=1}^{7}\gamma_{k}$ of
$F_{\mathbb{S}^{5}}=\bigsqcup_{k=1}^{7}\mathcal{R}_{\mathcal{D}_{k}}$ and
$F_{\mathbb{S}^{3}}=\bigsqcup_{k=1}^{7}\mathcal{R}_{\gamma_{k}}$,
respectively.

Let $(Y_{\xi}^{5},M_{\eta}^{3})$ be any admissible relative contact pair
obtained by an admissible relative contact $(\pm 1)$-surgery along the given
admissible link $\mathbb{L}$ in $\mathbb{S}_{st}^{5}$. By Theorem 1.7, there
exists a compatible relative open book

$(\mathcal{OB}_{Y},\mathcal{OB}_{M})\doteq(F_{\mathbb{S}^{5}},F_{\mathbb{S}^{3}},h_{\mathbb{S}^{5}}\circ\tau_{\mathbb{L}^{\prime}},h_{\mathbb{S}^{3}}\circ\tau_{\mathbb{K}^{\prime}})$

on $(Y_{\xi}^{5},M_{\eta}^{3})$ with Weinstein pages where
$\tau_{\mathbb{L}^{\prime}}$ and $\tau_{\mathbb{K}^{\prime}}$ denote the
product of the Dehn twists $\tau_{\mathbb{L}_{i}^{\prime}}^{\mp 1}$ and
$\tau_{\mathbb{K}_{i}^{\prime}}^{\mp 1}$ as described in the previous section
when the contact $(\pm 1)$-surgeries are applied.

######  Example 5.2.

Let $L$ be the admissible standard loose Legendrian $2$-unknot in
$(\mathbb{R}^{5},\xi_{0}^{5})\subset\mathbb{S}^{5}_{\xi_{st}}$ which is
obtained by taking $\mathbb{S}^{1}$-symmetry of the Legendrian $1$-unknot $K$
in $(\mathbb{R}^{3},\xi_{0}^{3})\subset\mathbb{S}^{3}_{\xi_{st}}$ with
$tb(K)=-3$ as depicted in Figure 15.

Figure 15. (a) The front of the admissible standard loose Legendrian
$2$-unknot $L$, (b) the front of its isotropic equator $K$.

The generalized square bridge diagram $\Delta(L)$ is given in Figure 16. By
using this diagram, one can obtain the piecewise smooth representative
$L^{\prime}$ (as depicted in Figure 17) by first constructing a Legendrian
diamond $\mathcal{D}_{k}$ for each square region $R_{k}$ in $\Gamma(K)$, and
then patching their appropriate parts together as exaplained in Proposition
3.2.

Figure 16. The generalized square bridge diagram of the admissible standard
loose Legendrian $2$-unknot $L$. Figure 17. The piecewise smooth standard
loose Legendrian $2$-unknot $L^{\prime}$.

As before, we consider the trivial compatible relative open book
$\mathcal{OB}_{0}\doteq(D^{4},D^{2},id_{D^{4}},id_{D^{2}})$ on
$(\mathbb{S}^{5}_{st},\mathbb{S}^{3}_{st})$ when no squares is introduced.
Then for each new octahedron, a new compatible relative open book on
$(\mathbb{S}^{5}_{st},\mathbb{S}^{3}_{st})$ is obtained by positively
stabilizing the compatible open book in the previous step. At the last step
$k=4$, the following compatible open book

$\displaystyle(F_{\mathbb{S}^{5}},F_{\mathbb{S}^{3}},h_{\mathbb{S}^{5}},h_{\mathbb{S}^{3}})$
$\displaystyle\doteq$ $\displaystyle\mathcal{OB}_{4}$ $\displaystyle=$
$\displaystyle\mathcal{S}^{+}[\cdots[\mathcal{S}^{+}[(D^{4},D^{2},id_{D^{4}},id_{D^{2}});A_{1},A_{1}\cap
D^{2}];\cdots];A_{4},A_{4}\cap\bigsqcup_{i=1}^{3}\mathcal{R}_{\gamma_{i}}]$

on $(\mathbb{S}^{5}_{st},\mathbb{S}^{3}_{st})$ is obtained where
$F_{\mathbb{S}^{5}}=\bigsqcup_{k=1}^{4}\mathcal{R}_{\mathcal{D}_{k}}$ and
$F_{\mathbb{S}^{3}}=\bigsqcup_{k=1}^{4}\mathcal{R}_{\gamma_{k}}$ are the
corresponding Weinstein pages,
$h_{\mathbb{S}^{5}}=\tau_{\mathcal{D}_{1}}\circ\cdots\circ\tau_{\mathcal{D}_{4}}$
and $h_{\mathbb{S}^{3}}=\tau_{\gamma_{1}}\circ\cdots\circ\tau_{\gamma_{4}}$
are the product of right-handed Dehn twists along $\mathcal{D}_{k}$ and
$\gamma_{k}$, respectively, and $A_{k}$ is the properly embedded Lagrangian
$2$-disk in the ribbon $\mathcal{R}_{(\bigsqcup_{i=1}^{k-1}M_{i})\cap M_{k}}$
as described in Proposition 4.3 for $k=1,2,3,4$. The smoothening of
$L^{\prime}$ (resp. $K^{\prime}$) are contained in the cores of
$F_{\mathbb{S}^{5}}$ (resp. $F_{\mathbb{S}^{3}}$).

Now, as an example, consider the admissible relative contact pair
$(Y_{\xi}^{5},M_{\eta}^{3})$ obtained by applying an admissible relative
contact $(+1)$-surgery along the given admissible link $L$ in
$\mathbb{S}_{st}^{5}$. By Theorem 1.7, there exists a compatible relative open
book

$(\mathcal{OB}_{Y},\mathcal{OB}_{M})\doteq(F_{\mathbb{S}^{5}},F_{\mathbb{S}^{3}},h_{\mathbb{S}^{5}}\circ\tau_{L^{\prime}}^{-1},h_{\mathbb{S}^{3}}\circ\tau_{K^{\prime}}^{-1})$

on $(Y_{\xi}^{5},M_{\eta}^{3})$ . Note that $Y_{\xi}^{5}$ is overtwisted by
Casal, Murphy and Presas [4] since it is obtained by contact $(+1)$-surgery
along a loose Legendrian $2$-knot $L^{\prime}$. In particular, it contains the
overtwisted contact submanifold $M_{\eta}=(L(2,1)=\mathbb{RP}^{3},\eta)$ which
is obtained by contact $(+1)$-surgery along the equatorial $1$-knot
$K^{\prime}$.

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