# Real tight contact structures on lens spaces and surface singularities

Si̇nem Onaran  Hacettepe Üni̇versi̇tesi̇, Department of Mathematics, TR-06800
Beytepe-Ankara, Turkey<EMAIL_ADDRESS>and  Feri̇t Öztürk
Boğazi̇çi̇ Üni̇versi̇tesi̇, Department of Mathematics, TR-34342 Bebek,
İstanbul, Turkey<EMAIL_ADDRESS>
###### Abstract.

We classify the real tight contact structures on solid tori up to equivariant
contact isotopy and apply the results to the classification of real tight
structures on $S^{3}$ and real lens spaces $L(p,\pm 1)$. We prove that there
is a unique real tight $S^{3}$ and $\mathbb{R}P^{3}$. We show there is at most
one real tight $L(p,\pm 1)$ with respect to one of its two possible real
structures. With respect to the other we give lower and upper bounds for the
count. To establish lower bounds we explicitly construct real tight manifolds
through equivariant contact surgery, real open book decompositions and
isolated real algebraic surface singularities. As a by-product we observe the
existence of an invariant torus in an $L(p,p-1)$ which cannot be made convex
equivariantly.

###### 2020 Mathematics Subject Classification:

Primary 57K33, 32S25; Secondary 57M50

##  1\. Introduction

A real structure on an oriented 3- or 4-manifold is an orientation preserving
involution. A real contact 3-manifold is a closed real 3-manifold where the
contact structure is anti-symmetric with respect to the real structure. They
naturally arise as the link manifolds of complex algebraic isolated surface
singularities given by zero loci of polynomials with real coefficients (in
this note we term those singularities as real algebraic surface
singularities); there the real structure is the complex conjugation and the
contact structure is the unique tight one determined by complex tangencies.
Without reference to the real structure, such a singularity is usually called
a Milnor filling of the given contact 3-manifold. So in the real setting we
may take the liberty to call the filling a real Milnor filling. More generally
real contact 3-manifolds appear as the boundary of real symplectic 4-manifolds
in the sense of [18].

It is known that the 3-manifolds that admit a real structure is scarce. “Most
manifolds have little symmetry”; more precisely there is a natural notion of
density in the set of all closed orientable 3-manifolds defined via the first
cohomology and this density vanishes for the subset of those manifolds that
admit a non-trivial involution [17]. Nevertheless once a 3-manifold admits a
real structure $c$ then it admits a $c$-real contact structure [16].

Having noted these, a possible path of investigation is real Milnor
fillability, i.e. the fillability of real tight contact 3-manifolds by
isolated real algebraic surface singularities, and more generally the
fillability by real symplectic 4-manifolds. Related to the latter, it has been
observed that there are real open books (hence real contact structures) on
real 3-manifolds which cannot be filled by a real Lefschetz fibration,
although it is filled by non-real Lefschetz fibrations [14].

We started this work with an interest in realizing real tight lens spaces as
the link manifolds of real algebraic surface singularities, in cases where
there is no contact topological obstruction. Recall that up to
contactomorphism there is at most one tight contact structure on a closed
oriented 3-manifold which is Milnor fillable [2]. Keeping that in mind, we
first attempt a classification of real tight structures over lens spaces up to
equivariant contact isotopy. In that regard the present work can be considered
as a continuation of [13] in which we introduced real contact 3-manifolds and
showed that there is a unique real tight 3-ball, which was accomplished
through a partial classification of real tight solid tori relative to the
invariant convex boundary.

Here in order to proceed towards the classification on lens spaces, we first
improve the classification on real tight solid tori. We also expect this
classification to be useful in studying the classification of real Legendrian
knots and hope to return this in a future work. On a solid torus there are
four real structures up to isotopy through real structures [7]. We denote them
and their restrictions on the boundary torus by $c_{j},j=1,2,3,4$ (see Section
2 for an explicit list). Here $c_{1}$ is the hyperelliptic involution. It
turns out that the $c_{2}$-, $c_{3}$\- and $c_{4}$-real tight solid tori with
convex boundary appear only as the standard real tight neighborhoods of
Legendrian and real knots (Theorem 6; see Section 2.1 for the discussion on
standard neighborhoods).

Using these results we deduce that there is a unique real tight structure on
$S^{3}$ (Theorem 15) and on $\mathbb{R}P^{3}$ (Theorem 16). These occur as the
link manifolds of a regular point and an $A_{1}$ singularity of a real
algebraic surface respectively (Section 4).

On lens spaces $L(p,q)$ for $p>2$, $q=1,p-1$ up to equivariant isotopy there
are exactly two real structures with nonempty real part [8]. On a genus-1
Heegaard decomposition these act on the boundaries of the solid tori either as
$(c_{1},c_{1})$ or as $(c_{2},c_{3})$, called type $A$ and type $B$
respectively. Moreover each is equivariantly isotopic to a real Heegaard
decomposition of genus 1; i.e. the real Heegaard genus equals 1 (type $C$ or
$C^{\prime}$), See Section 5 for details. There we prove

######  Theorem 1.

(Theorem 17 in Section 5) There is a $B$-real tight structure on $L(p,q)$ if
and only if $q=1$ or $q=p-1$. Up to equivariant contact isotopy, there is a
unique $B$-real tight structure on $L(p,p-1)$ and on $L(p,1)$, $p$ even. They
are real Milnor fillable. There is at most one $B$-real tight structure on
$L(p,1)$, $p$ odd.

We explicitly construct the claimed structures via isolated real algebraic
surface singularities and equivariant contact diagrams (see [16] for the
foundations of equivariant contact diagrams).

As a curious by-product of this result, we observe that there is an invariant
Heegaard torus in an $L(p,p-1)$ which cannot be made convex equivariantly,
although of course it can always be made convex (Theorem 22). Equivalently the
unique $B$-real tight $L(p,p-1)$ does not admit a real contact Heegaard
decomposition of genus 1; i.e. its real contact Heegaard genus is greater than
1. The proof is through observing the mismatch between the Thurston-Bennequin
invariants of the real Legendrian knots in a $B$-real tight $L(p,q)$ and a
hypothetical $C$-real tight $L(p,q)$, which are known to be equivariantly
isotopic. (See Section 7 for the discussion on type $C,C^{\prime}$ and the
computation of the invariants.)

As for the classification in type $A$, one needs a classification for
$c_{1}$-real tight solid tori and thick tori. The latter is required in
peeling off basic $c_{1}$-slices from solid tori [9]. We prove that such a
peeling-off is impossible by showing that interestingly there is no
$c_{1}$-real tight basic slice (Proposition 12) although there are
$c_{1}$-real tight double slices (Proposition 13). Nevertheless we give a
partial classification for $c_{1}$-real tight solid tori and upper bounds for
the count (see Section 3.4).

Using these results we demonstrate upper and lower bounds for the count of
$A$-real tight lens spaces $L(p,q)$ for $q=1,p-1$ (Theorem 18). The lower
bounds are established by explicit construction through equivariant contact
diagrams, real open book decompositions (Section 6) and real algebraic
singularities (Section 8). The lower bound in each case agrees with the
definite count for the number of tight structures of Ko Honda [9]. We also
point out the cases of real tight lens spaces which are not real Milnor
fillable, the obstruction being contact topological. Thus we observe that
fillability by isolated real algebraic singularities is neither related with
the type of the real structure, nor with the real Heegaard genus.

##  2\. Preliminaries

The background for the 3-dimensional contact topology and the techniques we
use in the sequel can be found in several standard references such as [4],
[5], [6].

In order to fix the notation and refer properly, we recall the classification
of real structures on a solid torus up to isotopy through real structures [7].
We identify the oriented boundary a solid torus $S^{1}\times D^{2}$ with
$\mathbb{R}^{2}_{(x,y)}/\mathbb{Z}^{2}$ where $x$ and $y$ directions
correspond to the direction of the meridian $(1,0)$ and of longitude $(0,1)$
respectively. We fix the coordinates of $S^{1}\times D^{2}$ as $(y,(u,v))$
where $(u,v)$ is the rectangular coordinates over $D^{2}$. Then the four
possible real structures on the solid torus are given as follows:

(1) $c_{1}:(y,(u,v))\mapsto(-y,(-u,v))$ – hyperelliptic;
(2) $c_{2}:(y,(u,v))\mapsto(y,(-u,-v))$;
(3) $c_{3}:(y,(u,v))\mapsto(y+1/2,(u,v))$;
(4) $c_{4}=c_{2}\circ c_{3}:(y,(u,v))\mapsto(y+1/2,(-u,-v))$.

All real structures and real parts in the sequel will be sketched in red,
whenever color is applicable.

For $a\in\mathbb{Z}-\\{0\\}$ we set

$\eta_{a}\doteq\ker(\cos(2\pi ay)du+\sin(2\pi ay)dv),$

and

$\eta_{0}\doteq\ker(du+\sin(2\pi ay)dv).$

Note that the contact solid torus $(S^{1}\times D^{2},\eta_{a})$ is $c_{1}$\-
and $c_{2}$-real tight; it is $c_{3}$-real tight for $a$ odd and $c_{4}$-real
tight for $a$ even. We set

$\zeta\doteq\ker(dy+udv).$

Note that $(S^{1}\times D^{2},\eta_{a})$ is $c_{1}$-real tight.

###  2.1. Real tight neighborhood theorems for knots

In [13], we have presented contact neighborhood theorems for real submanifolds
and invariant contact submanifolds up to equivariant contactomorphism. Also we
have given the classification of sufficiently small equivariant neighborhoods
of real knots in a real 3-manifold up to equivariant contact isotopy.

Here, we resolve shortly the remaining cases for the classification of real
tight neighborhoods of equivariant transverse and Legendrian knots up to
equivariant contactomorphism and up to isotopy through real tight structures.

######  Theorem 2.

(Real Contact Neighborhood Theorem for Equivariant Transverse Knots) An
equivariant transverse knot $K$ in a real contact 3-manifold $(M,\alpha,c)$
has a neighborhood equivariantly contactomorphic to a standard one, namely to
the real tight solid torus $(S^{1}\times D^{2},\zeta,c_{1})$.

Proof: By [13, Theorem 2.6], it is enough to classify the real conformal
symplectic 2-bundles over $S^{1}$ up to equivariant conformal bundle
isomorphism.

An involution on the unit circle $S^{1}$ in $\mathbb{C}$ is isotopic through
involutions either to the reflection with respect to the $x$-axis or to the
antipodal map. We denote respectively by $c_{1}$ and $c_{3}$ the corresponding
actions of the real structure $c$ on $K$. Note that $c_{3}$ is orientation
preserving on $K$ while $c_{1}$ is reversing.

We will argue that there is no real conformal symplectic 2-bundle over $S^{1}$
with its real structure inducing $c_{3}$ on the zero section. Let there be
such a bundle on $S^{1}$, with real structure still denoted by $c_{3}$ (with a
slight abuse of notation). The involution $c_{3}$ is orientation preserving on
the fibers, sending each fiber to the antipodal one. Recall that the
symplectic structure $\omega$ on the fibers is anti-symmetric, i.e.
$c_{3}^{*}\omega=-\omega$. Thus over the semi-circle, the 2-form $\omega$ is
isotoped through symplectic (area) forms to $-\omega$. This is impossible.

Now we want to classify the real conformal symplectic 2-bundles over $S^{1}$
with the real structure inducing $c_{1}$ on the zero section. Consider two
copies $N_{1}$ and $N_{2}$ of the canonically oriented trivial bundle
$[0,\pi]\times\mathbb{R}^{2}$ with the standard symplectic form on each fiber.
Then any real conformal symplectic 2-bundle $N$ can be built as a real mapping
torus as follows:

$N^{f,g}=\frac{N_{1}\amalg
N_{2}}{(\pi,(u,v))\sim(0,f(u,v)),(0,(u,v))\sim(\pi,g(u,v))}$

where $f,g\in\mbox{Sp}(2)$ and are equivariant with respect to the real
structure $c_{1}$ on $N$ given by

$c_{1}:N_{i}^{f,g}\rightarrow
N_{j}^{f,g},\;\;(y,(u,v))\mapsto(1-y,(u,-v)),\;\;\;i,j\in\\{1,2\\},i\neq j.$

Here $N_{i}^{f,g}$ ($i=1,2$) denote the projection of $N_{i}$ to $N^{f,g}$.
Then $f$ and $g$ can be either $+$id and $-$id. Note that with the choices of
orientation above, $N^{f,g}$ is canonically oriented and $c_{1}$ is
orientation preserving. Moreover the four possible bundles
$N^{++},N^{--},N^{+-},N^{-+},$ are equivariantly conformal isomorphic.
Explicitly $N^{++}$ and $N^{--}$ are equivariantly isomorphic via the
diffeomorphism

$\displaystyle N^{++}_{1}\rightarrow N^{--}_{1},(y,(u,v))\mapsto(y,u\sin
y+v\cos y,-u\cos y+v\sin y)$ $\displaystyle N^{++}_{2}\rightarrow
N^{--}_{2},(y,(u,v))\mapsto(y,u\sin y+v\cos y,-u\cos y+v\sin y).$

$N^{++}$ and $N^{+-}$ are equivariantly isomorphic via the diffeomorphism

$\displaystyle N^{++}_{1}\rightarrow
N^{+-}_{1},(y,(u,v))\mapsto(y,u\cos\frac{y}{2}+v\sin\frac{y}{2},-u\sin\frac{y}{2}+v\cos\frac{y}{2})$
$\displaystyle N^{++}_{2}\rightarrow
N^{+-}_{2},(y,(u,v))\mapsto(y,u\sin\frac{y}{2}-v\cos\frac{y}{2},u\cos\frac{y}{2}+v\sin\frac{y}{2}).$

(Similarly for the other cases.) $\Box$

As for an equivariant Legendrian knot $K$, as in [13, Corollary 2.5], we
observe that the proof of [13, Theorem 2.2] shows that the isotopy
classification of the real contact structures near $K$ is real topological.
Therefore $tw(K)$ with respect to a longitude and the action of the real
structure in a neighborhood of $K$ determine the standard tubular neighborhood
of $K$ up to equivariant contact isotopy. Thus we obtain the following
classification; the case (2) has already been presented as [13, Corollary 2.5]
except for $tw=0$.

######  Theorem 3.

(Real Contact Neighborhood Theorem for Equivariant Legendrian Knots) Let $K$
be an equivariant Legendrian knot in a real tight $(M,\xi,c)$. Let $U$ be a
sufficiently small real contact neighborhood of $K$ such that $\partial U$ is
an equivariant convex torus. Then $U$ is equivariantly contact isotopic
through real tight structures to one of the following:

  1. (1)

$c|_{K}=c_{1}$: $U\sim(S^{1}\times D^{2},\pm\eta_{tw(K)},c_{1})$. Here we
assume that the real arcs are oriented so that the real points on the boundary
are labeled. Then the sign $\pm$ is determined by the choice of sign
decoration on the regions of $\partial U-\Gamma$.

  2. (2)

$c|_{K}=id$: $K$ is real and $U\sim(S^{1}\times D^{2},\eta_{tw(K)},c_{2})$.

  3. (3)

$c|_{K}=c_{3}$: if $tw(K)$ is odd then $U\sim(S^{1}\times
D^{2},\eta_{tw(K)},c_{3})$; if $tw(K)$ is even then $U\sim(S^{1}\times
D^{2},\eta_{tw(K)},c_{4})$.

######  Remark 1.

Note that in case (1), the two contact structures $\pm\eta_{tw(K)}$ are not
equivariantly contact isotopic simply because the diffeomorphism $c_{1}$ that
sends one to the other is not equivariantly isotopic to the identity: the
labeled real points on the boundary are to be fixed. (A discrepancy between
real and non-real cases… These contact structures do not satisfy the Giroux’s
Flexibility in the real setting; e.g. the hypothesis of [13, Corollary 3.5] is
not satisfied.)

##  3\. Real tight solid tori

We gave a partial classification of $c_{3}$-real tight solid tori with
$\\#\Gamma=2$ in [13]. Here we give a complete classification for all
$c_{j}$-real tight solid tori $j=2,3,4$ with $\\#\Gamma=2$ and a partial
classification for $c_{1}$-real tight solid tori. First, let us set up the
required terminology.

Consider two points $a,b\in\mathbb{Q}$ located at the boundary of the
hyperbolic unit disk with the Farey tessellation. We consider the longest
counterclockwise steps in the Farey tessellation. Starting from $a$, if one
can reach $b$ after $d$ such steps, we say that the distance between $a$ and
$b$, denoted $dist(a,b)$, is $d$.

Below, we are going to split contact manifolds into building blocks. These are
either basic slices $T^{2}\times I$ as defined in [9] supplied with $c$-real
tight contact structures or $c$-invariant genuine double slices. Contact
topologically, a genuine double slice is nothing but two basic slices glued
together, which cannot be split into two $c$-invariant basic slices though. We
will show in the sequel that depending on $j$, $c_{j}$-invariant basic slices
or genuine double slices may or may not exist. We denote by $c$-$T(a,b)$ a
$c$-real tight, $c$-invariant thick torus with convex boundary with inner
slope $a$ and outer slope $b$. Note that $dist(a,b)$ on a basic slice and on a
double slice is $1$ and $2$ respectively.

On a real tight torus, let us denote by $\mu$ and $\lambda$ the meridian and
the longitude and by $\tau_{\mu}$ and $\tau_{\lambda}$ the Dehn twist along
each respectively. Observe that $\tau^{2}_{\mu}$ and $\tau_{\lambda}$ (resp.
$\tau_{\mu}$ and $\tau^{2}_{\lambda}$) can be realized in a $c_{3}$-invariant
way (resp. $c_{2}$-invariant). Both $\tau_{\mu}$ and $\tau_{\lambda}$ can be
made $c_{1}$-invariant.

###  3.1. Stripping-off $c_{3}$-real tight solid tori

We want to break a $c_{3}$-real tight solid torus with boundary slope $s$ into
equivariant slices. We proceed as in the proof of [13, Theorem 11]. After
applying a sequence of $c_{3}$-invariant Dehn twists $\tau^{2}_{\mu}$, we can
assume that $s\leq-1$ or $s\geq 1$. An equivariant Legendrian isotopic copy
$L$ of the core has a standard equivariant tight neighborhood with equivariant
and convex boundary. The boundary slope is $-\frac{1}{2k+1}$ (by Theorem 3 or
[13, Corollary 15]). We decompose the solid torus into two equivariant pieces:
$\nu(L)$ and $T(-\frac{1}{2k+1},s)$. The latter is minimally twisting by
Proposition 4.16 and the proof of Proposition 4.15 in [9]. We factor
$T(-\frac{1}{2k+1},s)$ into equivariant slices. If the final slice is a basic
slice of the form $T(-\frac{1}{2k+1},-\frac{1}{2k})$ or $T(-1,-2)$, we have
shown in [13] that this leads to an overtwisted structure.

Otherwise we have a final slice in one of the following forms:
(i) Either $T(-1,t)$ where $t\in\mathbb{Q}\cap[-3,-1)$ and $dist(-1,t)=1$ or
$2$ (single or double slice respectively);
(ii) Or $T(-\frac{1}{2k+1},-\frac{1}{2k-1})$.

Now via applying a chain of $c_{3}$-equivariant homeomorphisms, second case
above can be transformed to a thick torus $T(-1,p)$ with $p\in\mathbb{Z}^{+}$
and the double slices in the first case above can be transformed to such a
$T(-1,p)$ or a $T(-1,-\frac{p}{q})$ with $p=2k+1$, $q=k$,
$k\in\mathbb{Z}^{+}$. In particular there are two possible $c_{3}$-real tight
single slices: $T(-1,-2)$ or $T(-1,\infty)$, and the former was already proven
impossible in [13, Proposition 4.11].

###  3.2. Stripping-off $c_{2}$-real tight solid tori

With respect to the real structure $c_{2}$, $\tau_{\mu}$ is equivariant while
$\tau_{\lambda}$ is not. By means of a $c_{2}$-equivariant $\tau_{\mu}$ a
boundary slope can always be made less than or equal to $-1$.

Suppose $(S^{1}\times D^{2},\xi,c_{2})$ is a real tight solid torus with
convex boundary and with $\\#\Gamma=2$. Let $s=-p/q$
$(p,q\in\mathbb{Z},p>q\geq 0)$.

The core $L$ of the solid torus is real, hence Legendrian. Let
$k=\mbox{tw}(L)$. By [13, Corollary 2.5], a sufficiently small neighborhood
$U$ of $L$ is equivariantly contact isotopic to $U_{k}$. Excise $U$ to get an
equivariant real thick torus $(T^{2}\times I,\xi,c_{2})$ with convex boundary
components $T_{0}=T^{2}\times\\{0\\}$ and $T_{1}=T^{2}\times\\{1\\}$ with
$\\#\Gamma=2$ on each and with slopes $s_{0}=-1/k$ and $s_{1}=-p/q$
respectively. We denote such a thick torus by $T(-1/k,-p/q)$ as before.

We proceed very much similarly as in the above paragraph; we decompose
$T(-1/k,-p/q)$ into equivariant basic and double slices. Namely, suppose that
the equivariant characteristic foliation (Legendrian rulings) has slope 0. A
meridional annulus has Legendrian boundary with negative twisting with respect
to the annulus and thus can be perturbed relative to the boundary to obtain a
convex annulus $A$ ([9] or [10]). The inner boundary component of $A$ has tw
$-1$ while outer has $-p$. If $p>2$, there are at least two bypasses on $A$
for $T_{1}$. We can perform one bypass after the other on $T_{1}$ to obtain a
$c_{2}$-invariant convex torus $T^{\prime}_{1}$. Here we note that after the
first bypass, the shifted copy $\gamma$ of the meridian (the old meridional
arc union the inner part of the bypass halfdisk) is still a leaf of the
characteristic foliation. On the disk bounded by $\gamma$ there is still a
bypass which can be made symmetric with the first. The thick torus bounded by
$T_{1}$ and $T_{1}^{\prime}$ is a $c_{2}$-invariant double slice. Moreover,
the distance from $s_{1}$ to the slope $s_{1}^{\prime}$ on $T_{1}^{\prime}$ is
2.

This discussion shows that after a number of steps, we obtain a collection of
$c_{2}$-invariant double slices and possibly a single invariant basic slice.
Furthermore, the last slice is either $T(-\frac{1}{k},-\frac{1}{k-1})$ or
$T(-\frac{1}{k},-\frac{1}{k-2})$ or $T(-1,-\frac{p}{q})$, where the distance
from $-\frac{p}{q}$ to $-1$ is 1 or 2 and $-\infty\leq-\frac{p}{q}<-1$.
Whatever the last slice is, we can find a $c_{2}$-equivariant diffeomorphism
to change $s_{0}$ to $-1$ by $\tau_{\mu}$ and then modify $s_{1}$ by
$\tau^{2}_{\lambda}$ to obtain a real tight slice in one of the forms
$T(-1,-2)$ or $T(-1,-\infty)$ or $T(-1,-\frac{p}{q})$ or $T(-1,p)$ where
$p\in\mathbb{Z}^{+}$ and $-3\leq-\frac{p}{q}<-2$, $dist(-1,-\frac{p}{q})=2$.
In particular, $T(-\frac{1}{k},-\frac{1}{k-1})$ is equivariantly
contactomorphic to $T(-1,-\infty)$ and $T(-\frac{1}{k},-\frac{1}{k-2})$ to
$T(-1,+1)$.

We claim that a $c_{2}$-real, tight, minimally twisting $T(-1,-2)$ does not
exist. In fact, consider a convex equivariant meridional annulus. The inner
boundary component of the annulus has $tw=-1$ while outer has $-2$. The
dividing set on the annulus must be $c_{2}$-symmetric and, because of the
minimally twisting property, it must have at least one component connecting
the inner boundary to the outer. However such a dividing set is impossible.

###  3.3. Classification of $c_{2}$-, $c_{3}$\- and $c_{4}$-real tight solid
tori

First we will note the obvious correspondence between $c_{2}$-real tight thick
tori and $c_{3}$-real tight thick tori.

Let $T(-1,b)$ be a $c_{3}$-real tight thick torus. The map
$\phi:T(a,b)\rightarrow T^{2}\times I$ given by the matrix
$\begin{pmatrix}0&1\\\ -1&0\end{pmatrix}$ supplies the thick torus with a
$c_{2}$-real tight structure. With that structure, it is a $c_{2}$-real
$T(1,-1/b)$. Via the $c_{2}$-equivariant diffeomorphisms
$\begin{pmatrix}1&-2\\\ 0&1\end{pmatrix}$, we can turn that thick torus into a
$c_{2}$-real $T(-1,-1/(b+2))$. Meanwhile via the $c_{2}$-equivariant
diffeomorphisms $\begin{pmatrix}1&0\\\ -2&1\end{pmatrix}$, we get a
$c_{2}$-real $T(-1,-(2b+1)/b)$. Note that the above discussion is still valid
if we replace the roles of $c_{3}$ and $c_{2}$. Thus we have proven

######  Lemma 4.

  1. (1)

There is a $c_{3}$-real (respectively $c_{2}$-real) tight thick torus
$T(-1,k)$, $(k\in\mathbb{Z}^{+})$ if and only if there is a $c_{2}$-real
(respectively $c_{3}$-real) tight thick torus $T(-1,-(2k+1)/k)$.

  2. (2)

There is a $c_{3}$-real (respectively $c_{2}$-real) tight thick torus
$T(-1,k)$, $(k\in\mathbb{Z}^{-},k<2)$ if and only if there is a $c_{2}$-real
(respectively $c_{3}$-real) tight thick torus $T(-1,-1/(k+2))$.

Hence we obtain

######  Theorem 5.

There is no $c_{2}$\- and $c_{3}$-real tight double slice.

Proof: There are two possible sorts of $c_{3}$-real tight double slices. For
$k\in\mathbb{Z}^{+}$, a $c_{3}$-$T(-1,k)$ to exist, there must exist a
$c_{2}-T(-1,-1/(k+2))$ by Lemma 4.(2). Stripping off this latter thick torus
into basic slices, at the end of the first step we obtain a tight $T(-1,0)$
which does not exist. Therefore there is no
$c_{3}$-$T(-1,k),(k\in\mathbb{Z}^{+})$.

Similarly a $c_{3}$-$T(-1,-(2k+1)/k),(k\in\mathbb{Z}^{+})$ exists if and only
if a $c_{2}-T(-1,k)$ exists (Lemma 4.(1)) if and only if a
$c_{3}$-$T(-1,-1/k)$ exists (Lemma 4.(2)). Then the proof follows as above.
$\Box$

The discussion above helps us prove that $c_{2}$-, $c_{3}$\- and $c_{4}$-real
tight solid tori are scarce in the following sense:

######  Theorem 6.

The classification of $c_{2}$-, $c_{3}$\- and $c_{4}$-real tight solid tori
with convex boundary with $\\#\Gamma=2$:

The standard neighbourhoods of Legendrian knots with convex boundary on which
the real structure is $c_{3}$ (respectively $c_{4}$) are the only $c_{3}$-real
(respectively $c_{4}$-real) tight solid tori up to equivariant contact
isotopy.

Similarly the standard neighbourhoods of real knots with convex boundary are
the only $c_{2}$-real tight solid tori up to equivariant contact isotopy.

Proof: Given a $c_{2}$-, $c_{3}$ or $c_{4}$\- real tight solid torus, we take
a close, Legendrian copy of the core, excise its standard equivariant
neighbourhood and start peeling-off equivariant slices from the remaining
thick torus. The discussion in Sections 3.1 and 3.2 has already eliminated all
the possibilities except a $c_{2}$-real single slice $T(-1,-\infty)$, the
existence of which is equivalent to the existence of a $c_{3}$-real single
slice $T(-1,-2)$ (Lemma 4.(2)), which does not exist by [13, Proposition 4.11]

Similarly instead of peeling-off equivariant slices in case of $c_{4}$, we
turn the initial $c_{4}$-real tight thick torus into a $c_{3}$-real tight
thick torus via a diffeomorphism which acts as $\tau_{\mu}$ on each
$T^{2}\times\\{*\\}$. Since a $c_{3}$-real tight basic or double slice does
not exist (Theorem 5), the proof for the case $c_{4}$ follows. $\Box$

In particular, we have

######  Corollary 7.

A $c_{3}$-real tight solid torus can only have slopes $1/k$, $k$ an odd
integer (Theorem 3 and [13, Corollary 4.4]). A $c_{4}$-real tight solid torus
can only have slopes $1/k$, $k$ an even integer (Theorem 3). A $c_{2}$-real
tight solid torus can only have slopes $1/k$, $k\in\mathbb{Z}$ [13, Corollary
2.5].

We also note

######  Corollary 8.

Up to equivariant isotopy through real tight structures relative to the
boundary there is a unique $c_{i}$-real tight contact structure, $i=1,2,3,4$,
on $S^{1}\times D^{2}$ with convex boundary with $s=-1$ and $\\#\Gamma=2$.

###### Proof.

The case of $i=3$ is shown in [13]. Indeed similar idea applies to the other
cases. ∎

######  Remark 2.

There is a subtility in this uniqueness to be emphasized. For $j=2,3$ or 4,
let $T$ be a $c_{j}$-invariant convex torus in standard form in a contact
3-manifold. There are exactly two possible sign configurations on the regions
of $T$ determined by the linear dividing set on $T$. A priori these need not
be $c_{j}$-equivariantly contact isotopic, although they are in non-real
setting. However one can explicitly see that there is indeed a
$c_{j}$-equivariant contact isotopy between the two sign configurations. In
this way the hypothesis of [13, Corollary 3.5] is satisfied so that we may
employ Giroux’s Flexibility in the real setting and get the freedom to choose
signs in one way or the other. Compare with Remark 1.

###  3.4. $c_{1}$-real tight solid tori

First we refine [12, Theorem 1.2]. Let $\gamma$ be a Legendrian arc in a tight
3-ball. We denote the number of half twists along $\gamma$ by
$t(\gamma)=\lfloor 2tw(\gamma)\rfloor/2\in\mathbb{Z}+\frac{1}{2}$. Here
$tw(\gamma)$ is with respect to the standard framing and relative to the end
points.

######  Lemma 9.

Let $B$ be a $c_{st}$-real tight 3-ball with real points $r_{1},r_{2}$ on
$\partial B$ and the real arc $r_{1}r_{2}$. Then up to equivariant contact
isotopy fixing $\xi$ at $r_{1}$ and $r_{2}$, there is a unique such
$c_{st}$-real tight 3-ball. Each component of the dividing curve on the
oriented annulus $\partial B-(\nu(r_{1})\cup\nu(r_{2}))$ has $t(r_{1}r_{2})+1$
half twists.

The proof is an appropriate modification of that of [12, Theorem 1.2]. It
follows simply from the existence of a standard equivariant contact
neighborhood $N$ of the real arc $r_{1}r_{2}$ with a fixed twisting ([12,
Corollary 2.5]) (with the extra condition that the contact structure is fixed
in a neighborhood of $r_{1}$ and $r_{2}$) and the nonexistence of a
$c_{3}$-real tight solid torus $B-N$ with boundary slope $l\in\mathbb{Z}$,
except $l=\pm 1$ in which case there is a unique such solid torus. ([12,
Theorem 4.1]).

These said, we label the real points on the fixed boundary of a $c_{1}$-solid
torus as $r_{1},\ldots,r_{4}$, with $r_{1}r_{2}$ and $r_{3}r_{4}$ being the
real arcs.

######  Proposition 10.

There are exactly two $c_{1}$-real tight solid tori with slope $1/k$,
$k\in\mathbb{Z}$, up to equivariant isotopy. These are distinguished by the
sign decoration on the boundary or equally on a horizontal disk. For both
tori, the real arcs $r_{1}r_{2}$ and $r_{3}r_{4}$ have $t=-1$.

Proof: Since $\tau_{\mu}$ can be made $c_{1}$-equivariant, it suffices to
prove the claim for slope $-1$. Let $S$ be such a solid torus with linear,
equivariant dividing set $\Gamma$ of two connected components and with the
slope of the Legendrian ruling made 0. Note that $\Gamma$ contains the real
points and recall that there are two possible sign decorations on the regions
$\partial S-\Gamma$. We take an equivariant pair of disjoint, meridional disks
$D$ and $D^{\prime}$ in $S$ with Legendrian boundary and perturb them convex
equivariantly relative to the boundary so that they reside away from the real
points. The sign decoration on $D$ and $D^{\prime}$ (once oriented) are
determined by that on $\partial S$ (and vice versa). A sufficiently small open
neighborhood $\nu(D)$ of $D$ is isotopic to the standard tight 3-ball.
Excising $\nu(D)$ and its disjoint copy $\nu(D^{\prime})$ from $S$, we get two
$c_{st}$-equivariant 3-balls $B_{1}$ and $B_{2}$ with connected dividing sets
and a prescribed sign decoration on their boundary.

Up to equivariant contact isotopy fixing the contact structure on $\partial
B_{1}$ there is a unique $c_{st}$-real tight contact structure on $B_{1}$. It
follows from Lemma 9 that $t(r_{1}r_{2})+1=0$ so that $t(r_{1}r_{2})=-1$ (with
respect to the horizontal framing). The same discussion applies for $B_{2}$.
Hence there are at most two $c_{1}$-real tight structures, distinguished by
the sign decoration on $D$. These are in fact $\pm\eta_{-1}$ that appear in
Theorem 3.(1). Note that $c_{2}^{*}\eta_{-1}=-\eta_{-1}$. $\Box$

######  Proposition 11.

Let $S$ be a $c_{1}$-real tight solid torus with real arcs $r_{1}r_{2}$ and
$r_{3}r_{4}$ and with convex boundary on which the equivariant dividing set
has exactly two components and slope $-m\in\mathbb{Z}^{-}$. The number of such
$c_{1}$-real tight solid tori up to equivariant isotopy relative to the fixed
boundary is at most the $(m-1)^{st}$ Catalan number $C_{m-1}$. These are
distinguished by the configuration of the dividing set on a horizontal disk.
For all, the real arcs $r_{1}r_{2}$ and $r_{3}r_{4}$ have $t=-1$

Proof: We follow the notation in the previous paragraphs. The first paragraph
in the previous proof applies verbatim here. Having said that, we investigate
the possible dividing set configurations on $D$, which consists of properly
embedded $m$ arcs, connecting $2m$ fixed points on $\partial D$. The dividing
set configuration must be counted up to rotation. So one boundary parallel arc
connecting neighboring points can be fixed. Then the count is the number of
expressions containing $m$ pairs of parentheses which are correctly matched;
this number is nothing but $C_{m-1}$. A sample is as in Figure 1. Again up to
equivariant contact isotopy fixing that boundary there is a unique
$c_{st}$-real tight contact structure on $B_{1}$. Whatever the dividing set
configuration on $D$ (and its copy on $D^{\prime}$) is, the contribution to
the longitudinal component of the resulting dividing set on the $c_{3}$-real
tight solid torus $B_{1}-N$ is zero. Then Lemma 9 assures that
$t(r_{1}r_{2})=-1$. Similar conclusion applies for $B_{2}$. $\Box$

\begin{picture}(4651.0,2214.0)(443.0,-2098.0)\end{picture}

Figure 1. Possible dividing set configurations on a horizontal disk for $m=3$;
here $C_{2}=2$.

One would attempt to get an alternative upper bound for the count in
Proposition 11 as follows. Instead of taking a convex $D$ and its equivariant
copy $D^{\prime}$ one might have considered a pair of horizontal
$c_{1}$-invariant disks $D_{12},D_{34}$ containing the real arcs $r_{1}r_{2}$
and $r_{3}r_{4}$ respectively. The problem here is that it is not apriori true
that $D_{12}$ or $D_{34}$ can be perturbed convex equivariantly. This would be
possible if and only if the Legendrian boundary of each half disk (for example
one of the connected components of $D_{12}-r_{1}r_{2}$) has negative twisting
with respect to the half disk, which is not for granted. Nevertheless if one
assumes convexity, it can be observed that the count is still large and is
related to the meandric numbers.

Another method to find an upper bound for the count could be the slicing
operation in [9]. However the intriguing phenomenon here is that slicing does
not work $c_{1}$-equivariantly, simply because of the nonexistence of
$c_{1}$-real tight basic slices.

######  Proposition 12.

There is no $c_{1}$-real tight basic slice.

Proof: Consider a basic slice $R$ with a $c_{1}$-real contact structure and
with invariant convex inner and outer boundary tori $T_{1}$ and $T_{2}$
respectively. Without loss of generality, we may suppose that $T_{1}$ and
$T_{2}$ are in $c_{1}$-equivariant standard form with boundary slopes $-1$ and
$-2$ respectively and slopes of Legendrian ruling 0 on both.

Take an equivariant pair of disjoint meridional annuli with Legendrian
boundary. Perturb the pair equivariantly relative to their boundary to obtain
two convex annuli $A$, $A^{\prime}$. The dividing set $\Gamma_{A}$ on $A$ has
one boundary parallel arc with end points on $T_{2}$ and two arcs from $T_{1}$
to $T_{2}$. The dividing set $\Gamma_{A^{\prime}}$ is a symmetric copy of
$\Gamma_{A}$.

Keeping track of the dividing set on the boundary of any of the $c_{1}$-real
solid tori bounded by $A$, $A^{\prime}$, $T_{0}$ and $T_{1}$ (after edge
rounding), the possible configurations for the dividing sets all result with
an overtwisted solid torus (See Figure 2). $\Box$

$B$$A$$A^{\prime}$$A^{\prime}$$A$$T_{2}$$T_{2}$

Figure 2. Both dividing set configurations on $A_{0}$ and $A^{\prime}_{0}$
result with OT solid tori. The innermost and outermost annuli are $A$ and
$A^{\prime}$. The annulus in between is a portion of the outer boundary
$T_{2}$ where the dividing curves rotate $-1/4$ turn. The annulus that is a
part of $T_{1}$ is invisible in the figure; on it the dividing curves rotate
$-1/2$ turn.

Now, we prove the following proposition for a genuine equivariant double
slice, that is, one which cannot be peeled-off into two equivariant basic
slices with convex boundary.

######  Proposition 13.

There are exactly two genuine $c_{1}$-real tight double slices up to
equivariant isotopy relative to the equivariant convex boundary. They are
distinguished by the sign decoration on the boundary.

Proof: We assume without loss of generality that the inner and outer boundary
slopes of the double slice $M$ are $-1$ and $-3$ respectively and that the
Legendrian ruling has slope $0$ on both. Such a $c_{1}$-real tight genuine
double slice indeed exists. It is nothing but a union of two usual basic
slices of Honda, the by-pass disks of which are placed symmetrically.

To count them, we proceed as in the previous proof. Keeping track of the
dividing set on the boundary of any of the $c_{1}$-real solid tori bounded by
$A$, $A^{\prime}$, $T_{0}$ and $T_{1}$ (after edge rounding), the possible
configurations for the dividing sets all result with an overtwisted solid
torus except the single one up to rotation (See Figure 3). There we get a
$c_{1}$-tight solid torus with boundary slope $-1$. There are two such tori up
to equivariantly contact isotopy relative to the boundary. These structures
can be distinguished by the sign decoration on the boundary (or equivalently
on, say, $A$).

\begin{picture}(6024.0,5790.0)(544.0,-7823.0)\end{picture}

Figure 3. The only configuration on $A$ and $A^{\prime}$ that does not lead an
OT disk. It gives a $c_{1}$-real $(-1)$-solid torus.

Having said these, fixing any of the two sign decorations on the boundary of
the double slice, there is at most one $c_{1}$-real tight structure on the
double slice, obtained by gluing the two $c_{1}$-real tight solid tori of
slope $-1$. $\Box$

The discrepancy between the contact topological and the $c_{1}$-equivariant
cases is that in the former inside a double thick torus there are convex tori
of every intermediate slope [9, Proposition 4.16], which is not necessarily
true in the latter. Let us note explicitly the following corollary of the two
preceeding propositions.

######  Corollary 14.

$c_{1}$-equivariant bypass attachments come in $c_{1}$-equivariant pairs.
There is no $c_{1}$-equivariant bypass attachment in a $c_{1}$-real tight
genuine double slice.

##  4\. Classification on $S^{3}$ and $\mathbb{R}P^{3}$

###  4.1. Real tight $S^{3}$

######  Theorem 15.

Up to isotopy through real tight structures there is a unique real tight
$S^{3}$ with nonempty real part.

Proof: Recall that up to isotopy through real structures there is a unique
real structure $c_{S^{3}}$ on $S^{3}$ where the fixed point set is an unknot
$L$. Let $H$ be an equivariant contact neighborhood of $L$. By Theorem 3(2),
$H$ is isotopic to the real tight solid torus $\eta_{\,tb(L)}$ with
$tb(L)\in\mathbb{Z}^{-}$. The complement of $H$ in $S^{3}$ is a $c_{3}$-real
tight solid torus with convex boundary of slope $tb(L)$. Thus by Theorem 6 and
Theorem 3(3), $tb(L)$ must be $-1$ and the proof follows. $\Box$

In the above proof we use a genus-1 Heegaard splitting of $S^{3}$. A similar
proof can be produced using a genus-0 Heegaard splitting, in which case we
apply [13, Theorem 2.1] twice.

###  4.2. Real tight $\mathbb{R}P^{3}$

Let us consider $\mathbb{R}P^{3}$ as the link manifold $X_{1}^{+}$ of the
$A_{1}$ singularity $\\{x^{2}-y^{2}+z^{2}=0\\}\subset\mathbb{C}^{3}$, i.e.

$X_{1}^{+}=S^{5}_{\varepsilon}\cap\\{x^{2}-y^{2}+z^{2}=0\\}$

where $S^{5}_{\varepsilon}$ is a small 5-sphere in $\mathbb{C}^{3}$ centered
at 0 with radius $\varepsilon>0$. $X_{1}^{+}$ is naturally a real tight
3-manifold with the real and contact structures being the standard ones
induced respectively by the complex conjugation and the standard symplectic
structure on $\mathbb{C}^{3}$. We recall that up to contact isotopy there is a
unique tight structure on $\mathbb{R}P^{3}$ [9] and up to isotopy through real
structures there is a unique real structure on $\mathbb{R}P^{3}$ with nonempty
real part [8].

######  Theorem 16.

Up to isotopy through real tight structures there is a unique real tight
$\mathbb{R}P^{3}$ with nonempty real part. Any such is equivariantly contact
isotopic to $X_{1}^{+}$.

Proof: The real structure $c$ may be described as follows [8]: in the genus-1
Heegaard splitting of $\mathbb{R}P^{3}$ let $c$ act on the solid tori $H_{1}$
and $H_{2}$ as $c_{2}$ and $c_{4}$ respectively. (This is a type-$B$ real
structure. See the next section for the definition.) The fixed point set of
$c$ is the union of the core circles $L_{i}\subset H_{i}$ and the gluing map
is $\begin{pmatrix}-1&0\\\ 2&1\end{pmatrix}$. We may suppose that $H_{1}$ is
the standard $c_{2}$-real contact neighborhood of $L_{1}$. Thus the dividing
curve on $\partial H_{1}$ has slope $s_{1}=1/m$, $m\in\mathbb{Z}$, by Theorem
6, so that the slope on $\partial H_{2}$ is $s_{2}=-(2m+1)/m$ too. By
Corollary 7 the only value for such a slope on the $c_{4}$-real $H_{2}$ is
$\infty$ and $m=0$. There is a unique such $H_{2}$ which is $c_{4}$-real
tight; that is the standard $c_{4}$-real contact neighborhood of $L_{2}$.
Hence there is at most 1 real tight $\mathbb{R}P^{3}$. Of course $X_{1}^{+}$
is such a manifold. $\Box$

Let us observe that in this unique real tight $\mathbb{R}P^{3}$, the rational
Thurston-Bennequin numbers $tb_{\mathbb{Q}}(L_{1})$ and
$tb_{\mathbb{Q}}(L_{2})$ are both $-1/2$; to see this it is enough to note
that the rational Seifert framings along $L_{1}$ and $L_{2}$ are given by
$(-1,2)$ on both $\partial H_{1}$ and $\partial H_{2}$. This observation is in
accordance with [12] and [1].

Recall that for $L(2,1)$, a type-$B$ real structure is equivariantly isotopic
to a type-$A$. (See the next section.) Thus one could attempt to produce an
alternative proof using a genus-1 Heegaard splitting using $c_{1}$-equivariant
solid tori. Again $s_{1}$ and $s_{2}$ would both be equal to $\infty$ or $-1$.
Since both cases may occur in principle, we would conclude that there are at
most two real tight structures on $\mathbb{R}P^{3}$, which is a weaker result
than the above.

##  5\. Real tight lens spaces of types $A$ and $B$

Here we start to classify the real tight structures on lens spaces $L(p,1)$
and $L(p,p-1)$ with $p>2$ and with nonempty real part.

Let $q=1,p-1$ and $M=(L(p,q),c)$ be a real lens space with nonempty real part.
We consider a genus-1 Heegaard decomposition of $M$ with handlebodies
$H_{1},H_{2}$. By definition this decomposition is a $c$-real Heegaard
decomposition if and only if $c$ exchanges the handlebodies, i.e. $c$ is
isotopic to a real structure of type $C$ or $C^{\prime}$ in the notation of
[8]. (See [13] for an introduction of real Heegaard decompositions.) In that
case the gluing map is an orientation reversing involution of $\partial H_{1}$
hence is strongly equivalent to a linear involution $\phi=\pm\Phi$ where
$\Phi=\begin{pmatrix}-q&q^{\prime}\\\ p&q\end{pmatrix}$; the sign $+$ for type
C and $-$ for type C′ [8, Lemma 4.6].

If $c$ maps each handlebody to itself then the pair $(c|_{H_{1}},c|_{H_{2}})$
is isotopic through real structures to $(c_{1},c_{1})$, $(c_{2},c_{3})$ or
$(c_{2},c_{4})$ which correspond to the real structures of type $A$, $B$ and
$B^{\prime}$ respectively. For $L(2,1)$, all of these 5 types of real
structures are isotopic through real structures. For $L(p,1)$, $p>2$, types
$A$ and $C$ are equivalent and $B,B^{\prime},C^{\prime}$ are equivalent. For
$L(p,p-1)$, $p>2$, types $A,C^{\prime}$ are equivalent and $B,B^{\prime},C$
are equivalent.

Letting $X$ stand for one of the types $A$, $B$, $B^{\prime}$, $C$ or
$C^{\prime}$, we denote by $l_{X}(p,q)$ the number of real tight structures on
the real lens space $L(p,q)$ of type $X$ up to isotopy through real tight
structures.

Our main classification results for real tight lens spaces are

######  Theorem 17.

(a) $l_{B}(p,q)>0$ if and only if $q=1$ or $q=p-1$.
(b) $l_{B}(p,p-1)=1$. This unique real tight structure is given by the
isolated surface singularity $A^{+}_{p-1}$.
(c) $l_{B}(p,1)=1$ for $p$ even; $l_{B}(p,1)\leq 1$ for $p$ odd.

######  Theorem 18.

(a) $l_{A}(p,p-1)\geq 1$. The existent real tight structure is given by the
isolated surface singularity $A^{-}_{p-1}$.
(b) $p-1\leq l_{A}(p,1)\leq C_{p-2}+1$.

Here $A^{\pm}_{p-1}=\\{\pm x^{p}-y^{2}+z^{2}=0\\}\subset\mathbb{C}^{3}$.

We prove these theorems by first justifying in Section 5.1 the upper bounds
involved in the given (in)equalities and then in Section 5.2 we explicitly
construct distinct real tight lens spaces via contact equivariant surgery to
prove the validity of lower bounds. See Figure 4 for the upper and lower
bounds and the claimed equivariant surgery diagrams for each case.

In Section 6 we construct real open books for the case $q=1$. That
construction displays in particular real contact Heegaard decompositions of
higher genera.

Following that observation we ask in Section 7 if there is any real contact
Heegaard decomposition of genus 1 at all. We answer that question negatively
for some cases and for the remaining cases we give upper bounds for their
count.

In Section 8 we show that all the examples for the case $q=p-1$ in Figure 4
are real Milnor fillable.

Figure 4.

###  5.1. Upper bounds.

Here we prove that the upper bounds stated in Theorems 17-18 are valid. First
we do that for Theorem 17.

######  Lemma 19.

For $q=1,p-1$, $l_{B}(p,q)\leq 1$. Otherwise $l_{B}(p,q)=0$.

Proof: Assume $p>2$. Similar to the proof of Theorem 16 we consider the real
decomposition $(H,c_{2};H^{\prime},c_{j})$ for the lens space $L(p,1)$. If $p$
is even then $j=2$, otherwise $j=3$ or $4$. By Theorem 6 we have $s=1/k$,
$k\in\mathbb{Z}$ for the slope of $H$ so that $s^{\prime}=-(kp+1)/k$. Again by
Theorem 6 this latter slope is possible only if $kp+1=\pm 1$ or $k=0$. Since
$p\neq 2$, the only possible solution is when $k=0$ and $s=s^{\prime}=\infty$.
For $p$ even such $H^{\prime}$ is unique by Theorem 6 and Remark 2. For $p$
odd, since there is no $c_{3}$-real tight solid torus with slope $\infty$, $j$
must be 4 and such $H^{\prime}$ is unique.

As for the lens space $L(p,p-1)$, the same argument gives us the slope
$s_{2}=(kp+p-1)/(kp+p-k-2)$ and the single possible situation, occurring for
$kp+p-1=\pm 1$. This has a unique solution $k=-1$ and that holds for any $p$.
Thus we obtain $s=s^{\prime}=-1$. If $p$ is even, then $j=2$ and there is a
unique such $H^{\prime}$ with slope $-1$. If $p$ is odd, since there is no
$c_{4}$-real tight solid torus with slope $-1$, $j$ must be 3 and there is a
unique such $H^{\prime}$.

With similar considerations one can easily see that in order to get
$s_{1},s_{2}\in\\{0,1/k:k\in\mathbb{Z}\\}$ $q$ must be either $1$ or $p-1$.
$\Box$

The following lemma justifies the upper bound in Theorem 18(b).

######  Lemma 20.

$l_{A}(p,1)\leq C_{p-2}+1$.

Proof: We consider the decomposition $(H,c_{1};H^{\prime},c_{1})$ for the lens
space $L(p,1)$. We assume $s=1/k$, $k\in\mathbb{Z},k\leq 0$. Recall that $H$
can be thickened to a solid torus of slope $\infty$ by simply peeling off
basic slices from $H^{\prime}$ (see the proof of [9, Proposition 4.17]). By
Proposition 13 and its corollary, two consecutive basic slices can be arranged
to form a $c_{1}$-equivariant geniune double slice. Thus in the
$c_{1}$-equivariant case, $H$ can be thickened to a $c_{1}$-equivariant solid
torus of slope either $\infty$ or $-1$ depending on $k$ being even or odd
respectively, leading $s^{\prime}=\infty$ or $p-1$ respectively. In the former
case, there is a unique $c_{1}$-real tight $H^{\prime}$ relative fixed
boundary (Proposition 10) while in the latter there are $C_{p-2}$ (Proposition
11). Hence $C_{p-2}+1$ is an upper bound. $\Box$

###  5.2. Lower bounds: construction via equivariant contact surgery.

Here we show that the lower bounds in Theorems 17-18 are valid. We do that by
constructing explicitly distinct examples via equivariant contact surgery.

Let $K$ be a $c_{1}$-invariant Legendrian knot in the standard real tight
$S^{3}$. On the boundary of a standard equivariant contact neighborhood $U$ of
$K$, let us denote the longitude determined by the contact framing by
$\lambda$ and a meridian by $\mu$, as usual. With respect to $\lambda$ the
slope on $\partial U$ is $\infty$. For an equivariant $+1$\- and respectively
$-1$-surgery along $K$, the $c_{1}$-tight solid torus to be glued back must
have the boundary slope $+1$ and $\infty$ respectively, both of which exist
(Proposition 10). This discussion proves that $c_{1}$-real contact $(\pm
1)$-surgery exists and is well-defined. In the notation of [16] this is an
equivariant surgery of type $1_{1}$, gluing back a $c_{1}$-solid torus. (For a
more general discussion see [16].)

It follows that the $c_{1}$-equivariant contact surgeries in Figure 4 (and in
Figures 5, 7-9) are well-defined. We would like to emphasize here that in each
of those diagrams, since the real involution exchanges the signs on the
boundary of a standard neighborhood of the unknots and since the involution is
equivariantly isotopic to the identity, the sign choice does not alter the
final manifold up to $c_{1}$-equivariant contact isotopy.

Similarly let $K$ be a $c_{4}$-invariant Legendrian knot in the standard real
tight $S^{3}$. A contact $\pm 1$ equivariant surgery along $K$, exists and is
unique. This is an equivariant surgery of type $4_{2}$, gluing back a
$c_{2}$-solid torus [16].

If $K$ and $K^{\prime}$ constitute an equivariant pair of disjoint Legendrian
knots, an equivariant pair of contact surgeries performed along $K$ and
$K^{\prime}$ is always possible. The upper right diagrams in Figure 4 contains
such.

Proof of lower bounds in Theorems 17-18. Recall that the only real structure
on a lens space having quotient $S^{3}$ is of type $A$ [8]. Hence all the
diagrams in Figures 7-9 determine real tight structures of type $A$,
justifying the bottom left part of Figure 4. Their count proves the lower
bound in Theorem 18(b).

Similarly, the types of real structures obtained via the remaining equivariant
contact surgery diagrams is determined as seen in Figure 4. Their existence
proves all the remaining lower bounds. $\Box$

######  Remark 3.

Note that the contact surgery diagram in Figure 5 determines a real tight
structure of type $A$ on $L(p,p-1)$ too when $p$ is even. We have not any
other tool yet to distinguish this real tight structure from that given in top
left diagram in Figure 4.

Figure 5.  Type $A$ real tight structure on $L(p,p-1)$, $p$ even.

##  6\. Construction via real open books for $q=1$

In this section we first recall the definition of a real open book
decomposition [15]. Then we explicitly construct real open books for the real
tight contact structures on lens spaces $L(p,1)$ with $p>2$ and with nonempty
real part.

An abstract real open book decomposition is a triple $(S,f,c)$ where $S$ is a
compact surface with boundary, $c$ a real structure on $S$ and $(S,f)$ is an
abstact open book with monodromy $f$ satisfying $f\circ c=c\circ f^{-1}$. A
positive real stabilization of the abstract real open book $(S,f,c)$ is the
abstract real open book
$(S^{\prime},\tilde{f}\circ\sigma,\tilde{c}\circ\sigma)$ where
$S^{\prime}=S\cup H$ and $H$ is either a 1-handle with $c$-invariant attaching
region or a union of two 1-handles with their attaching regions interchanged
by $c$. The new monodromy $\tilde{f}$ is the natural extension of $f$ over
$S^{\prime}$. Positive real stabilizations come in types I-IX (see [15]).

Any of the $p-1$ tight contact structures on $L(p,1)$ can be obtained via a
Legendrian surgery, i.e. a contact $(-1)$-surgery along a Legendrian unknot
$L$ with ${tb}(L)=p-1$. The diagrams in Figures 7-9 reveal that those surgery
operations can be made equivariant. As noted at the end of Section 5.2 they
all give type-$A$ real structures. Here we construct real open book
decompositions $(S,f_{i},A)$ (with abuse for $A$ in the notation) for each of
those real tight lens spaces $(L(p,1),\xi_{i},A)$, $i=1,\ldots,p-1$. The real
pages are to be disks with $p-1$ holes. For this we start with the real open
book $(D,{\tt id},c)$ of the real tight $(S^{3},\xi_{st},c_{st})$ where the
real page $D$ is a disk (Figure 6). We then perform positive real
stabilizations (when necessary) in order to obtain the page on which the
required Legendrian knot would reside.

Figure 6. Real open book $(D,{\tt id},c)$ of the real tight
$(S^{3},\xi_{st},c_{st})$.

###  6.1. Real open books for $L(5,1)$.

As an illustration of the general case we construct real open book
decompositions $(S,f_{i},A)$ for each real tight $(L(5,1),\xi_{i},A)$,
$i=1,\ldots,4$, where the page $S$ is a disk with 4 holes.

$L(5,1)$ has two universally tight contact structures. Surgery diagram for one
of the universally tight contact structures, say $\xi_{1}$, and the real open
book supporting the real tight $(L(5,1),\xi_{1},A)$ are given in Figure 7. The
monodromy $f_{1}$ of the real open book is
$f_{1}=t_{a_{1}}t_{a_{2}}t_{a_{3}}t_{a_{4}}t_{L}$ where each $t_{a_{i}}$ is a
Dehn twist along the belt circle $a_{i}$. Note that in this case we perform
two sets of positive real stabilizations of type III (i.e. two pairs of
symmetric stabilizations; see [15]) on the real open book $(D,{\tt id},A)$ of
the real tight $S^{3}$.

Figure 7. Real open book $(S,f_{1},A)$ of real tight $(L(5,1),\xi_{1},A)$.

Surgery diagram for the other universally tight contact structure, say
$\xi_{4}$, on the lens space $L(5,1)$ and the real open book supporting the
real tight $(L(5,1),\xi_{4},A)$ are given in Figure 8. Note that in this case
we perform first two positive real stabilizations of type II on the real open
book $(D,{\tt id},c)$ of the real tight $S^{3}$. Next, we add a type-III
positive real stabilization to this open book. The monodromy $f_{1}$ of the
real open book is $f_{4}=t_{a_{1}}t_{a_{2}}t_{a_{3}}t_{a_{4}}t_{L}$ where the
curves $a_{i}$ and the Legendrian knot $L$ are as in Figure 8.

Figure 8. Real open book $(S,f_{4},A)$ of real tight $(L(5,1),\xi_{4},A)$.

The real open books for the remaining two virtually overtwisted contact
structures, say $\xi_{2}$ and $\xi_{3}$, are given in Figure 9. Note that in
each of these cases the real open book is constructed from the real open book
in the intermediate stage of the previous paragraph by adding one positive
real stabilization of type III appropriately.

Figure 9. Real open book $(S,f_{j},A)$ of real tight $(L(5,1),\xi_{j},A)$:
$(a)j=2$; $(b)j=3$.

###  6.2. Real open books for $L(p,1)$ for $p$ odd.

The construction in the preceding section generalizes in a straightforward
manner to the lens spaces $L(p,1)$, $p$ odd. The real open books supporting
the two real universally tight structures are given in Figure 10. The real
open book on the left (resp. on the right) corresponds to the real universally
tight contact structure coming from a Legendrian surgery along a Legendrian
knot having all its $p-2$ zigzags on its right (resp. on its left). In the
former case all real stabilizations are of type III while in the latter case
two type-II positive real stabilizations are needed too.

Figure 10. Real open book decomposition of the universally tight real
$L(p,1)$’s when $p$ is odd.

The real open books of the remaining virtually overtwisted real tight lens
spaces are given in Figure 11. The real open book on the left of Figure 11
corresponds to a Legendrian knot $L$ where $L$ has odd number of zigzags on
its left and even number of zigzags on its right (compare to
$(L(5,1),\xi_{2},A)$ in Section 6.1). The real open book on the right of
Figure 11 corresponds to a Legendrian knot $L$ where $L$ has even number of
zigzags on its left and odd number of zigzags on its right (compare to
$(L(5,1),\xi_{3},A)$ in Section 6.1).

Figure 11. Real open book of real virtually overtwisted tight $L(p,1)$, $p$ is
odd.

###  6.3. Real open books for $L(p,1)$ for $p$ even.

The analogue of the construction of real open books for $L(p,1)$ for $p$ odd
extends to lens spaces $L(p,1)$ for $p$ even as expected. The real open books
supporting the two real universally tight lens spaces $L(p,1)$ for $p$ even
are given in Figure 12.

Figure 12. Real open book of the universally tight real $L(p,1)$’s when $p$ is
even.

The real open book supporting the virtually overtwisted real tight lens spaces
$L(p,1)$ for $p$ even is given in Figure 13.

Figure 13. Real open book of real virtually overtwisted tight $L(p,1)$, $p$ is
even. The Legendrian knot $L$ on the left/right has odd/even number of zigzags
on its left and even/odd number of zigzags on its right.

##  7\. Classification for type $C$ and $C^{\prime}$

In the previous section, all the real open books that appeared determine real
contact Heegaard decompositions of genera greater than 1. When the lens space
is given as a real Heegaard decomposition of genus 1 (i.e. exactly in case of
types $C$ or $C^{\prime}$), it might be the case that the Heegaard surface
cannot be made equivariantly convex. We denote by $l^{*}_{X}(p,q)$ the number
of real tight structures on the real lens space $L(p,q)$ of type
$X=C,C^{\prime}$ up to equivariant contact isotopy with the condition that the
lens space assumes a real contact Heegaard decomposition of genus 1. We ask
what $l^{*}_{X}(p,q)$ equals for $q=1,p-1$ and if that number is nonzero at
all. At the end of this section we come up with a negative result:
$l^{*}_{C^{\prime}}(p,1)=0$ and $l^{*}_{C}(p,p-1)=0$ (Theorem 22).

We start with determining upper bounds for the remaining cases.

######  Proposition 21.

For $p>2$,
(a) $\displaystyle
l^{*}_{C^{\prime}}(p,p-1)\leq\left\\{\begin{array}[]{ll}4,&p\mbox{ even};\\\
6,&p\mbox{ odd.}\end{array}\right.$
(b) $\displaystyle l^{*}_{C}(p,1)\leq\left\\{\begin{array}[]{ll}p,&p\mbox{
even};\\\ 2p-2,&p\mbox{ odd.}\end{array}\right.$

Proof: Consider an abstract genus-1 $C$-real Heegaard decomposition
$(H,H^{\prime};c)$ of the lens space $M=L(p,q)$. Set $T=\partial H$.
(Sometimes we view $H$ embedded in $M$.) Since we are interested in computing
$l_{X}^{*}$, we assume $T$ is convex in $M$. $T$ may be assumed to be in
standard form as a convex torus. The real part (and the characteristic
foliation) on $T$ and the invariant dividing set $\Gamma$ are determined as
linear sets by the eigenvector of $\phi$ that corresponds to the eigenvalue
$+1$ and $-1$ respectively. For type $C$ and $q=1$ the $(-1)$-eigenvector of
$\phi$ is $(2,-p)$ and the slope of $T$ is

$s=-p/2=\left\\{\begin{array}[]{ll}\left[-p/2\right],&p\mbox{ even};\\\
\left[-(p+1)/2,-2\right],&p\mbox{ odd.}\end{array}\right.$

For type $C^{\prime}$ and $q=p-1$ the $(-1)$-eigenvector is $(2-p,p)$ and

$s=p/(2-p)=\left\\{\begin{array}[]{ll}-(p/2)/(-1+p/2)=\left[-2,-2,\ldots,-2\right],&p\mbox{
even};\\\ \left[-2,\ldots,-2,-3\right],&p\mbox{ odd.}\end{array}\right.$

The number of connected components of $\Gamma$ is an even positive integer; we
denote this number by $\\#\Gamma$. If we show that $\\#\Gamma=2$ then the
proof follows. Indeed in that case, for type $C$ and $q=1$ we read off from
the slope expressions above that the number of tight contact structures on the
solid torus $H$ up to contact isotopy relative boundary is $-p/2$ for $p$ even
and $((p+1)/2-1)\cdot 2=p-1$ for $p$ odd [9]. For type $C^{\prime}$ and
$q=p-1$, we have that number $2$ for $p$ even and $3$ for $p$ odd. Note that
$H^{\prime}$ is an exact copy of $H$. Since, as we have already noted in
Proposition 10, the sign assignment on the regions of convex $T$ determines a
priori distinct real tight structures on $M$, the numbers above for the
possible tight structures must be multiplied by 2.

Now let us discuss $\\#\Gamma$. Consider a Legendrian core $K$ of $H$ and its
equivariant copy $K^{\prime}$ in $H^{\prime}$. A sufficiently small standard
neighborhood $\nu(K)\subset H$ of $K$ with convex boundary has $\\#\Gamma=2$
(same situation for the equivariant copy). Suppose there is a bypass disk $D$
on $\partial\nu(K)$. Let $D\cap\mathrm{fix}(c)\neq\emptyset$. Then $D$ can be
deformed so that the intersection contains a single point and further it can
be made smaller such that it avoids the real point. Finally, if still $D\cap
T\neq\emptyset$ then $T$ can be pushed further (and $H$, $H^{\prime}$ and the
real structure are modified accordingly) so that the intersection becomes
empty. Now we can peel off a basic slice in $M-\nu(K)$ staying in $H$.
Equivariantly we peel off a basic slice in $M-\nu(K_{2})$ staying in
$H^{\prime}$ too. Thus, as in [9], we guarantee that the inner boundary of the
closure of $H-(\nu(K)\cup\mbox{ (basic slices)})$ (and of the equivariant
copy) is convex with $\\#\Gamma=2$. In this manner peeling of basic slices
equivariantly we obtain the remaining part
$P=M-(\nu(K)\cup\nu(K^{\prime})\cup\;\mathrm{(all\;basic\;slices)}$, which is
a $c$-real tight manifold with convex boundary. We may suppose that $P\cong
T^{2}\times[0,2]$; $c(T_{1})=T_{1}$, reversing the orientation;
$c(T_{t})=T_{2-t}$, reversing the orientation; $T_{0}$ and $T_{2}$ are convex
with $\\#\Gamma_{j}=2$. Moreover $s_{2}=s_{0}$; in fact it is obvious that
$dist(s_{2},s_{0})\leq 1$. If it is 1, there is a bypass disk on $T_{0}$ not
touching $T$ as above and a symmetric bypass disk on $T_{2}$. Then the
surfaces $T^{\prime}_{0}$ and $T^{\prime}_{2}$ obtained after bypass
attachment have slopes $s_{2}$ and $s_{0}$ respectively. However
$T^{\prime}_{2}$ is between $T^{\prime}_{0}$ and $T_{2}$, both of which have
slope $s_{2}$. This contradicts with the fact that all the thick tori in this
setup are minimally twisting [9].

Now for a contradiction let us assume $\\#\Gamma=\\#\Gamma_{T_{1}}=2u>2$. In
the words of [9], this $T^{2}\times[0,1]$ is a thick torus increasing the
torus division number $\\#\Gamma/2$. Of course, this thick torus has its
equivariant copy $T^{2}\times[1,2]$ in $H^{\prime}$.

We will observe that such an equivariant tight thick torus
$T=(T^{2}\times[0,2],c,\xi)$ that alters the torus division number may not
exist. Let the fixed point set be parallel to $f_{1}=(a,b)$ on $T_{1}$; thus
$c(f_{1})=f_{1}$. Furthermore we assume that the slope of the characteristic
foliation on every $T_{k}$ is $b/a$. We denote by $A$ the equivariant convex
annulus bounded by $f_{0}\cup f_{2}$, containing $f_{1}$. Now, since
$s_{0}=s_{1}$, the cardinality $|\Gamma_{T_{j}}\cap A|$ are the same for
$j=0,2$ and 2 more for $j=1$. Since $\xi$ is nonrotative, $\Gamma_{A}$ must
have at least two dividing arcs which go across from $T_{0}$ to $T_{1}$ and
therefore from $T_{2}$ to $T_{1}$ (see [9]). Then there must be a dividing arc
$\gamma$ on $A$ parallel to $A\cap T_{1}$. the curve $\gamma$ and its
symmetric copy $c(\gamma)$ bound an overtwisted disk on $A$ together, which
contradicts the fact that $\xi$ is tight (see Figure 14). $\Box$

$T_{0}$$T_{1}$$T_{2}$$A$

Figure 14. The proof that $\\#\Gamma=2$ on the equivariant convex Heegaard
torus.

###  7.1. Twisting numbers.

One way to distinguish the real tight structures (up to equivariant
contactomophisms) is to calculate the rational twisting number
$tb_{\mathbb{Q}}$ of the Legendrian real parts. Here we compute that number
explicitly for the real contact structures for types $B$, $C$ and $C^{\prime}$
in all existing cases.

(type $B$) Consider the proof of Lemma 19 and let $q=p-1$. When $p$ is even,
the fixed point set is the union of the cores $L_{1}$ and $L_{2}$ of $H_{1}$
and $H_{2}$ respectively. Otherwise fixed point set is $L_{1}$. For each real
knot in each case, one can compute that $tb_{\mathbb{Q}}=-1/p$. Recall that
the slopes $s_{1}$ and $s_{2}$ are $-1$. Now, for instance, when $p$ is even,
$pL_{1}$ bounds a topological disk $D$ since $(1-p,p)\mapsto(1,0)$ under the
gluing map. Hence the rational framing on $L_{1}$ provided by $D$ is
$p(p-1)/p^{2}=(p-1)/p$. Therefore $tb_{\mathbb{Q}}(L_{2})=(p-1)/p-1=-1/p$.
Similar computation in case $p$ odd.

(type $C^{\prime}$) In this case we set off by assuming the existence of a
real contact Heegaard decomposition of genus 1. The gluing map of the Heegaard
decomposition is $\Phi$. We observe that $\Phi((-q,p))=(1,0)$ so that $(-q,p)$
bounds an oriented horizontal disk $D_{2}$ in $H_{2}$. Set $g=gcd(q-1,p)$ and
$h=gcd(q+1,p)$. The fixed point set of the real structure is one or two copies
of $K=\frac{1}{g}(q-1,-p)$ on $T=\partial H_{1}$ depending on $p$ being odd or
even respectively. Moreover the equivariant dividing curve $\Gamma$ on $H_{1}$
is parallel to $\frac{1}{h}(q+1,-p)$ and $\\#\Gamma=2$. Then we have
$gK=\partial(D_{2}\cup D_{1})$, with $D_{1}$ being the meridional disk in
$H_{1}$. Let $S$ denote the surface obtained from $D_{2}\cup D_{1}$ by adding
a band at each crossing of $D_{2}\cap D_{1}$ connecting $D_{2}$ to $D_{1}$ so
that the orientations are respected. We see that there are
$(1,0)\cdot(-q,p)=p$ bands in $S$, which is minus the difference $(T,S)$
between framings coming from $T$ and $S$ along $gK$ as well. Now, the twisting
of the contact framing along $K$ with respect to $T$ is
$(K,T)=\frac{1}{2}(K\cdot\Gamma)=-2p/gh$. As a result we have

$tb_{\mathbb{Q}}(K)=tb_{\mathbb{Q}}(K,S)=\frac{1}{g^{2}}(T,S)-(K,T)=p/g^{2}-2p/gh.$

In particular when $q=p-1$ and $p$ is odd we have $h=p$, $g=1$ so that
$tb_{\mathbb{Q}}(K)=p-2$. For the case $q=p-1$ and $p$ is even we have $h=p$
and $g=2$ so that $tb_{\mathbb{Q}}(K)=p/4-1$.

(type $C$) We assume the existence of a real contact Heegaard decomposition of
genus 1. The gluing map is $-\Phi$ this time. Then $(q,-p)$ bounds $D_{2}$;
$K=\frac{1}{h}(q+1,-p)$ and $\Gamma$ is parallel to $\frac{1}{g}(q-1,-p)$.
Then we have $hK=\partial(D_{2}\cup D_{1})$. As above $S$ is obtained from
$D_{2}\cup D_{1}$ by adding a band at each crossing respecting the
orientations and there are $p$ bands again. The twisting of the contact
framing along $K$ with respect to $T$ remains the same:
$(K,T)=-K\cdot\Gamma=-2p/gh$. As a result we have

$tb_{\mathbb{Q}}(K)=\frac{1}{h^{2}}(T,S)-(K,T)=p/h^{2}-2p/gh.$

In particular when $q=1$ and $p$ is odd we have $tb_{\mathbb{Q}}(K)=p-2$. For
the case $q=1$ and $p$ is even we have $tb_{\mathbb{Q}}(K)=p/4-1$.

In case $q=p-1$ and $p$ is odd we have $tb_{\mathbb{Q}}(K)=1/p-2$. For the
case $q=1$ and $p$ is even we have $tb_{\mathbb{Q}}(K)=1/p-1$.

There is a unique real tight $L(p,q)$ of type $B$ for $q=1,p-1$. If
$l^{*}_{C}(p,p-1)$ were positive (or if $l^{*}_{C^{\prime}}(p,1)$ were
positive) then the twisting computation would agree. Since they do not, except
for $p=2$, we conclude

######  Theorem 22.

Suppose $p>2$, $q=1,p-1$. For a $B$-real tight $L(p,q)$, a real contact
Heegaard decomposition must have genus greater than $1$; i.e.
$l^{*}_{C^{\prime}}(p,1)=0$ and $l^{*}_{C}(p,p-1)=0$. Equivalently, an
invariant Heegaard torus in a $B$-real tight $L(p,q)$ cannot be made convex
equivariantly.

In the case of type $A$ we content ourselves with the following result:

######  Proposition 23.

There is no real contact Heegaard decomposition of genus $1$ for a $A$-real
tight $L(p,1)$ induced from a real open book decomposition.

###### Proof.

Suppose that there is such. Then the real open book has annular pages and its
monodromy is ${t_{a}}^{\pm p}$, where $t_{a}$ is a right handed Dehn twist
about the core circle $a$ in the annulus. The real open book supports a real
$L(p,1)$ if and only if the monodromy is ${t_{a}}^{-p}$. However in this case
the contact structure is overtwisted. ∎

##  8\. Construction for $q=p-1$ via surface singularities

Here we show that all the real tight lens spaces given by surgery diagrams for
$L(p,p-1)$ in the first row of Figure 4 are real contact link manifolds of
isolated real algebraic surface singularities. Meanwhile note that all of the
surgery diagrams for $L(p,1)$ in the second row of Figure 4 is virtually
overtwisted except two of them. Since universally tightness is necessary for
Milnor fillability [11], these virtually overtwisted ones are not link
manifolds of isolated real algebraic surface singularities.

We consider the family of surface singularities $A^{\pm}_{p-1}=\\{\pm
x^{p}-y^{2}+z^{2}=0\\}\subset\mathbb{C}^{3}$ and the link manifolds
$X^{\pm}=A^{\pm}_{p-1}\cap S^{5}_{\epsilon}$. We supply these manifolds with
the complex conjugation as the real structure and consider the induced real
tight structure $c^{\pm}$ on $X^{\pm}$. Then $X^{\pm}$ is a real tight lens
space $L(p,p-1)$. It is straightforward to see that for each $p$ these two
contact manifolds are ambiently contact isotopic and as real manifolds they
are equivariantly diffeomorphic. However these manifolds turn out to be
distinct up to equivariant contact isotopy.

Using the techniques in [12] we will distinguish between these real tight lens
spaces. In particular we will match $tb_{\mathbb{Q}}$ of the real parts of
$X^{\alpha,\beta,\gamma}$ with those computed in Section 7.1.

Figure 15. The two real structures on the link of $A_{p-1}$, $p$ odd.

We consider the minimal resolution (dual) graph $G$ of $A^{\pm}_{p-1}$; it is
a linear graph of length $(p-1)$ with $(-2)$-vertices (see Figure 15). Any
involution on $G$ describes a real structure on the plumbed 4-manifold hence
on the lens space; conversely any real structure on $A^{\pm}_{p-1}$ induces an
involution on $G$. One way to construct $G$ is to consider the resolution of
$A^{\pm}_{p-1}$ as a double covering branched along a resolved curve.
Inspecting the positive and negative sides of the real part of the resolved
curve in appropriate charts, as in [12, Proposition 1], we deduce that the
induced involution of the real structure $c^{-}$ on $G$ fixes all vertices in
$G$ and that induced by $c^{+}$ acts as a symmetry with respect to the
midpoint (see Figure 15)

Having said these, let $p=2k+1$. The fixed point set $S^{\pm}=fix(c^{\pm})$ in
the resolved complex surface is a oriented real surface. The real part
$L^{\pm}=X^{\pm}\cap S^{\pm}$ of $X^{\pm}$ is a connected Legendrian knot. The
genera of $S^{-}$ and $S^{+}$ are $(p-1)/2$ and 0 respectively. Now the
calculation for $L^{-}$ is easy:

$tb_{\mathbb{Q}}(L^{-})=-\chi(S^{-})=p-1-2+1=p-2.$

Figure 16. Blown up $A_{6}$.

For $L^{+}$, we blow up the resolved complex surface once more. One has to
look more closely to the point of blowing-up and the equivariant setting. We
blow up at the point of intersection of the equators of the equivariant pair
of exceptional divisors in the very middle. The complex conjugation maps those
equators onto each other without fixing any point. Thus the conjugation
extends over the new exceptional divisor as a $\pi$-rotation with respect to
the equatorial plane so that its equator (which intersects each of the
previous equators at a single point) is equivariant without any fixed point.
So we obtain a real variety $\tilde{A}$ with the resolution graph as in Figure
16. Now the real surface $\tilde{S}^{+}\subset\tilde{A}$ is a punctured
$\mathbb{R}P^{2}$ with boundary $L^{+}$. The real circle $E$ on the
$(-1)$-sphere is 1-sided in $\tilde{S}^{+}$ in and bounds a disk in $X^{+}$.
The contribution of this disk to $tb_{\mathbb{Q}}(L^{+})$ is easy to determine
using [12, Proposition 9 and Theorem 1]. Thus we get

$\displaystyle tb_{\mathbb{Q}}(L^{+})$
$\displaystyle=-\chi(S^{+}-E)+(-1+2\cdot[-3,\stackrel{{\scriptstyle
k}}{{\widehat{-2,\ldots,-2}}}]^{-1})$
$\displaystyle=0-1+2(-3-\frac{k-1}{-k})^{-1}=-\frac{1}{2k+1}=-1/p.$

Figure 17. Real parts $L_{i}^{\pm}$ bound the real surfaces in
$\tilde{A}_{p-1}$. For $p=2k$, the former has genus $k$ and the latter has
genus $0$.

Now let $p=2k$. The real structure $c^{-}$ fixes all vertices in the minimal
resolution graph; $c^{+}$ is the mirror symmetry with respect to the middle
vertex. The real part of $X^{\pm}$ consists of two circles. Let us denote an
arbitrarily chosen one by $L^{\pm}$ (the calculation below applies to both
components). The genus of $S^{-}$ is $k-1$ while $S^{+}$ is an annulus.
Following the idea in [12, Section 5.1] and using Figure 17 we can compute the
rational twistings as follows

$tb_{\mathbb{Q}}(L^{-})=-\chi(S^{-}_{1})-k\cdot\frac{1}{2}=k-1-\frac{k}{2}=\frac{k}{2}-1=\frac{p}{4}-1$

and

$tb_{\mathbb{Q}}(L^{+})=-\chi(S^{+}_{1})+\frac{1}{4}(-2+2\cdot\frac{k-1}{k})=0-\frac{1}{2k}=-\frac{1}{p}.$

We note that each rational Thurston-Bennequin number that we have obtained
above is maximal for the corresponding knot type. For this we refer, for
example, [3, Theorem 1.3].

######  Remark 4.

We would have obtained another real tight $L(p,p-1)$ if we had extended the
real structure over the middle exceptional sphere in a non-canonical manner,
namely as a $\pi$-rotation in a way that its equator has exactly two fixed
points. This would give a different equivariant contact surgery diagram.

Comparing with the surgery diagrams obtained and with the values of
$tb_{\mathbb{Q}}$ in Section 7.1 and taking into account this last remark, we
conclude

######  Theorem 24.

The unique real tight lens space of type $B$ in case $q=p-1$ is exactly the
link manifold $A^{+}_{p-1}$ of the singularity $\\{+x^{p}-y^{2}+z^{2}=0\\}$.
The link manifold $A^{-}_{p-1}$ of the singularity
$\\{-x^{p}-y^{2}+z^{2}=0\\}$ is a real tight $L(p,p-1)$ of type $A$, which is
depicted in the upper left diagram in Figure 4. The diagram in Figure 5 is
obtained from $A^{+}_{p-1}$ by simply modifying the real structure in the
middle 2-handle.

Acknowledgment. The research for this paper was started during a Research in
Pairs stay at IMBM (İstanbul Center for Mathematical Sciences) in January
2020. We thank IMBM for its support, and for creating an inspiring
environment. SO was partially supported by TÜBİTAK grant 119F411. FÖ thanks
Nermin Salepci for the fruitful discussions that took place during his visit
at Institut Camille Jordan, Université Lyon I.

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