# Haas’ Theorem revisited

Benoît Bertrand, Erwan Brugallé, and Arthur Renaudineau

###### Abstract

Haas’ theorem describes all patchworkings of a given non-singular plane
tropical curve $C$ giving rise to a maximal real algebraic curve. The space of
such patchworkings is naturally a linear subspace $W_{C}$ of the
$\mathbb{Z}/2\mathbb{Z}$-vector space $\overrightarrow{\Pi}_{C}$ generated by
the bounded edges of $C$, and whose origin is the Harnack patchworking. The
aim of this note is to provide an interpretation of affine subspaces of
$\overrightarrow{\Pi}_{C}$ parallel to $W_{C}$.
To this purpose, we work in the setting of abstract graphs rather than plane
tropical curves. We introduce a topological surface $S_{\Gamma}$ above a
trivalent graph $\Gamma$, and consider a suitable affine space $\Pi_{\Gamma}$
of real structures on $S_{\Gamma}$ compatible with $\Gamma$. We identify the
vector subspace $W_{\Gamma}$ of $\overrightarrow{\Pi}_{\Gamma}$ characterizing
real structures inducing the same action on
$H_{1}(S_{\Gamma},\mathbb{Z}/2\mathbb{Z})$. We then deduce from this statement
another proof of Haas’ original result.

  *     * Keywords. Patchworking; Viro method; Hilbert 16th problem; tropical curves; topology of involutions; maximal curves

2010 Mathematics Subject Classification. 14P25; 14T05

[Français]

Titre. Théorème de Haas revisité Résumé. Le théorème de Haas décrit tous les
patchworks d’une courbe tropicale plane lisse donnée $C$ donnant lieu à une
courbe algébrique réelle maximale. L’espace de ces patchworks est
naturellement un sous-espace linéaire $W_{C}$ du
$\mathbb{Z}/2\mathbb{Z}$-espace vectoriel $\overrightarrow{\Pi}_{C}$ engendré
par les arêtes bornées de $C$, et dont l’origine est le patchwork de Harnack.
Le but de cette note est de donner une interprétation des sous-espaces affines
de $\overrightarrow{\Pi}_{C}$ parallèles à $W_{C}$.
Pour ce faire, nous nous plaçons dans le cadre des graphes abstraits plutôt
que celui des courbes tropicales planes. Nous introduisons une surface
topologique $S_{\Gamma}$ au dessus d’un graphe trivalent $\Gamma$, et
considérons un espace affine adéquat $\Pi_{\Gamma}$ de structures réelles sur
$S_{\Gamma}$ compatibles avec $\Gamma$. Nous identifions le sous-espace
vectoriel $W_{\Gamma}$ de $\overrightarrow{\Pi}_{\Gamma}$ caractérisant des
structures réelles induisant la même action sur
$H_{1}(S_{\Gamma},\mathbb{Z}/2\mathbb{Z})$. Nous déduisons alors de cet énoncé
une autre démonstration du résultat original de Haas.

###### Contents

  1. 1\. Introduction
  2. 2\. Real structures above real graphs
  3. 3\. Action on $H_{1}(S_{\Gamma};\mathbb{Z}/2\mathbb{Z})$ induced by a real structure
  4. 4\. Haas’ Theorem

##  1\. Introduction

Haas classified in his thesis [Haa97] all unimodular combinatorial
patchworkings (see [Vir84, IV96]) producing a real algebraic $M$-curve (i.e. a
non-singular compact real algebraic curve of genus $g$ whose real part has
$g+1$ connected components). In the tropical reformulation of patchworking in
terms of _twist-admissible_ sets of edges of a non-singular plane tropical
curve $C$ given in [BIMS15], the set of all possible patchworkings with this
given underlying tropical curve $C$ is naturally a vector space
$\overrightarrow{\Pi}_{C}$ over $\mathbb{Z}/2\mathbb{Z}$ and Haas’ Theorem can
be interpreted as follows: the space of all such patchworkings producing an
$M$-curve is a well identified and easily described subvector space $W_{C}$ of
$\overrightarrow{\Pi}_{C}$ (see Section 4.A). In this note we address the
question of interpreting affine subspaces of $\overrightarrow{\Pi}_{C}$
parallel to $W_{C}$. To this purpose, it is suitable to follow Klein’s
approach and to work in the framework of abstract real topological surfaces,
i.e. oriented topological surfaces equipped with an orientation-reversing
continuous involution, rather than restricting to real algebraic curves in a
given toric surface (see [Kle76]). Accordingly, we deal with abstract graphs
rather than with plane tropical curves. Given an abstract graph $\Gamma$ with
only 3-valent and 1-valent vertices, we construct a topological surface
$S_{\Gamma}$ decomposed into a union of disks, cylinders, and pairs of pants
(this is just a variation on standard pair of pants decompositions of a
surface). Next, given a continuous involution $\tau:\Gamma\to\Gamma$, we
define _real structures_ above the pair $(\Gamma,\tau)$, which are roughly
real structures on $S_{\Gamma}$ compatible with the decomposition induced by
$\Gamma$ together with $\tau$ (see Section 2.C for precise definitions). The
set of real structures above $(\Gamma,\tau)$ is naturally an affine space
$\Pi_{(\Gamma,\tau)}$ over $\mathbb{Z}/2\mathbb{Z}$ whose direction
$\overrightarrow{\Pi_{(\Gamma,\tau)}}$ has for basis the set of edges of the
quotient graph $\Gamma/\tau$ that are adjacent to two 3-valent vertices (see
Lemma 2.11). As any involution, a real structure on $S_{\Gamma}$ induces an
action on $H_{1}(S_{\Gamma};\mathbb{Z}/2\mathbb{Z})$ and the main result of
this note can be summarized as follows.

_Two real structures above $(\Gamma,\tau)$ induce the same action on
$H_{1}(S_{\Gamma};\mathbb{Z}/2\mathbb{Z})$ if and only if they differ by an
element of a given vector subspace $W_{(\Gamma,\tau)}$ of
$\overrightarrow{\Pi_{(\Gamma,\tau)}}$ having a simple description_.

This statement is proved in Theorem 3.2 after the definition of
$W_{(\Gamma,\tau)}$ at the begining of Section 3. We show in Proposition 3.5
that $W_{(\Gamma,\tau)}$ admits an alternative description which corresponds
to Haas’ description of the above-mentioned vector space $W_{C}$ in the
special case of a non-singular plane tropical curve $C$. The connection of
Theorem 3.2 and Proposition 3.5 with Haas’ Theorem then comes from the fact
that a real topological surface with a non-empty real part is maximal if and
only if the corresponding induced action on the first homology group of the
surface is trivial. In particular, we recover Haas’ Theorem as a corollary of
our main results combined with standard results in tropical geometry (see
Section 4.B). Our main motivation for the present work was the possible
generalisations of Haas’ Theorem in higher dimensions. Generalising a
statement usually first requires to identify the suitable notions coming into
play and allowing a suitable formulation of the original statement. Both in
its original formulation and in its “twist-admissible” tropical reformulation,
Haas’ Theorem involves, sometimes implicitely, several features that a priori
make sense only for curves. As an example, it is based on the existence of
Harnack distribution of signs discovered by Itenberg (see [IV96]), which has
no known analogue yet in higher dimensions. The existence of such Harnack
distribution of signs on $\mathbb{Z}^{2}$ is precisely the fact that naturally
turns the space of all patchworkings of a non-singular tropical curve in
$\mathbb{R}^{2}$ in a vector space rather than an affine space as in the case
of abstract graphs: it provides a canonical patchworking on any non-singular
tropical curve in $\mathbb{R}^{2}$, which in addition produces an $M$-curve.
By focusing on abstract graphs rather than on embedded tropical curves, and on
the action on homology induced by a real structure rather than the number of
connected components of its real part, we place Haas’ Theorem in a wider
perspective. There, it turns out to be a corollary of a more general statement
which seems to us more likely to have a higher dimensional analogue.

#### A few words on the context

Despite the surprising elegance of the complete description of maximal
unimodular patchworkings, Haas’ Theorem unfortunately only ever appeared in
his thesis [Haa97]. We briefly recap its origin, rooted in the first part of
Hilbert’s 16th problem. This latter concerns the classification, up to
isotopy, of all possible mutual positions of the connected components of the
real part of a non-singular real algebraic curve in $\mathbb{RP}^{2}$ of a
given degree $d$. If $d$ is even, every connected component of the real part
of such a curve is called an _oval_ , and bounds a disc which is called the
_interior_ of the oval. One says that an oval is _even_ (resp. _odd_) if it is
contained in the interior of an even (resp. odd) number of ovals. The
following Ragsdale conjecture played an important role in subsequent
developments in real algebraic geometry, and remains one challenging open
question in the case of $M$-curves.

Conjecture (Refined Ragsdale conjecture, [Rag06, Pet33]). For any non-singular
real algebraic curve of even degree $2k$ in $\mathbb{RP}^{2}$ having $p$ even
and $n$ odd ovals, one has

$p\leq\dfrac{3k(k-1)}{2}+1\qquad\mbox{and}\qquad n\leq\dfrac{3k(k-1)}{2}+1.$

A series of counterexamples to this conjecture have been constructed since the
90’s [Ite93, Haa95, Bru06]. Nevertheless, no counterexample is known yet among
$M$-curves. As an application of his classification of maximal unimodular
combinatorial patchworkings, Haas proved in his thesis the following theorem.

Theorem (Haas, [Haa97, Theorem 12.4.0.12 and Proposition 13.5.0.13]). Let $A$
be a non-singular real algebraic $M$-curve in $\mathbb{RP}^{2}$ constructed by
a unimodular combinatorial patchworking. If $A$ has $p$ even and $n$ odd
ovals, then one has

$p\leq\dfrac{3k(k-1)}{2}+1\qquad\mbox{and}\qquad n\leq\dfrac{3k(k-1)}{2}+4.$

Furthermore, such a curve having more than $\dfrac{3k(k-1)}{2}+1$ odd ovals
would have exactly $n=\dfrac{3k(k-1)}{2}+4$ such ovals.

As far as we know, this is the only known result in the direction of Ragsdale
conjecture for maximal curves.

#### Organisation of the paper

We start by defining real structures above an abstract graph in Section 2. In
Section 3, we prove our main statement (Theorem 3.2) and Proposition 3.5 which
relates our definition of $W_{(\Gamma,\tau)}$ to Haas’ description of the
vector space $W_{C}$. We end this paper by recalling in Section 4 the
combinatorial patchworking construction and Haas’ Theorem, and by deducing
this latter from results from Section 3.

#### Acknowledgements

Part of this work has been done during visits of B.B. and A.R. at Centre
Mathématiques Laurent Schwartz, École Polytechnique. We thank this institution
for excellent working conditions. We also thank Ilia Itenberg for fruitful
discussions, as well as anonymous referees for many valuable comments on an
earlier version of this paper.

#### Convention and notation

A real topological surface $(S,\tau)$ is an oriented topological surface $S$
equipped with an orientation-reversing continuous involution $\tau:S\to S$,
called a _real structure_. Given a real surface $(S,\tau)$, we denote by
$\mathbb{R}S$ the _real part_ of $S$, i.e. the set of fixed points of $\tau$.
It is a submanifold of $S$ of dimension 1. If $S$ is compact and of genus $g$,
then $\mathbb{R}S$ has at most $g+1$ connected components (see [Kle76]) and we
say that $(S,\tau)$ is _maximal_ if it has $g+1$ connected components. If
$(S,\tau)$ is a compact real surface and $A\subset S$ is a real finite set of
points, we say that $(S\setminus A,\tau)$ is maximal if $(S,\tau)$ is. Given a
finite graph $\Gamma$, we denote by $\text{Vert}(\Gamma)$ the set of its
vertices, by $\text{Vert}^{\infty}(\Gamma)$ the set of its 1-valent vertices,
and we set
$\text{Vert}^{0}(\Gamma)=\text{Vert}(\Gamma)\setminus\text{Vert}^{\infty}(\Gamma)$.
By definition, the valency of a vertex $v\in\text{Vert}(\Gamma)$, denoted by
$\text{val}(v)$, is the number of edges of $\Gamma$ adjacent to $v$. We also
denote by $\text{Edge}(\Gamma)$ the set of edges of $\Gamma$, and by
$\text{Edge}^{0}(\Gamma)$ the set of edges of $\Gamma$ adjacent to two
vertices in $\text{Vert}^{0}(\Gamma)$. Throughout the text we identify a graph
with any of its topological realisations in which edges are open segments.

##  2\. Real structures above real graphs

###  2.A. Surfaces associated to a trivalent graph

We will call _trivalent_ a graph such that any of its vertices is either
$3$-valent or $1$-valent, thus authorizing that it also has leaves. Given a
trivalent graph $\Gamma$, we construct a topological surface $S_{\Gamma}$ as
follows. Recall that a pair of pants is an oriented sphere with three open
disks removed, in particular it has three boundary components with induced
orientation.

  1. 1.

To each vertex $v$ of $\Gamma$, we associate a topological surface $S_{v}$,
which is either a pair of pants if $v$ is 3-valent, or an oriented closed disk
if $v$ is 1-valent. Furthermore, we choose a one to one correspondence between
boundary components of $S_{v}$ and edges of $\Gamma$ adjacent to $v$; the
boundary component corresponding to $e$ is denoted by $\gamma_{v,e}$.

  2. 2.

To each edge $e$ of $\Gamma$, we associate an oriented cylinder $S_{e}$, and a
one to one correspondence between boundary components of $S_{e}$ and vertices
adjacent to $e$; the boundary component corresponding to $v$ is denoted by
$\gamma_{e,v}$.

  3. 3.

To each pair $(v,e)\in\text{Vert}(\Gamma)\times\text{Edge}(\Gamma)$ such that
$e$ is adjacent to $v$, we associate an orientation-reversing homeomorphism
$\phi_{v,e}:\gamma_{v,e}\to\gamma_{e,v}$.

The surface $S_{\Gamma}$ is obtained by gluing all surfaces $S_{v}$ and
$S_{e}$ via the maps $\phi_{v,e}$. It is a closed oriented topological
surface. Clearly, the surface $S_{\Gamma}$ is not uniquely defined by
$\Gamma$, but also depends on the choice of the homeomorphisms $\phi_{v,e}$
and of the surfaces $S_{v}$. However, different surfaces obtained by different
choices are homeomorphic, and such an homeomorphism is canonical up to
composition by Dehn twists along the circles $\gamma_{e,v}$ in $S_{\Gamma}$.

###### Example 2.1

We depicted in Figure 1 a trivalent graph $\Gamma$ and the corresponding
surface $S_{\Gamma}$.

$v_{1}$$v_{2}$$v_{3}$$v_{4}$ |  |  $S_{v_{1}}$$S_{v_{2}}$$S_{v_{3}}$$S_{v_{4}}$
---|---|---
$\Gamma$ |  | $S_{\Gamma}$
Figure 1: A trivalent graph $\Gamma$ and the corresponding surface
$S_{\Gamma}$

###  2.B. The groups $H_{1,0}(S_{\Gamma};\mathbb{Z}/2\mathbb{Z})$ and
$H_{0,1}(S_{\Gamma};\mathbb{Z}/2\mathbb{Z})$

Let $\Gamma$ be a trivalent graph. Each circle $\gamma_{e,v}$ defines an
element in $H_{1}(S_{\Gamma};\mathbb{Z}/2\mathbb{Z})$. If $v$ and $v^{\prime}$
are the two vertices adjacent to $e$, both circles $\gamma_{e,v}$ and
$\gamma_{e,v^{\prime}}$ define the same element in
$H_{1}(S_{\Gamma};\mathbb{Z}/2\mathbb{Z})$, that we denote by $\gamma_{e}$. We
define $H_{1,0}(S_{\Gamma};\mathbb{Z}/2\mathbb{Z})$ to be the subgroup of
$H_{1}(S_{\Gamma};\mathbb{Z}/2\mathbb{Z})$ generated by all the loops
$\gamma_{e}$, and we set

$H_{0,1}(S_{\Gamma};\mathbb{Z}/2\mathbb{Z})=H_{1}(S_{\Gamma};\mathbb{Z}/2\mathbb{Z})/H_{1,0}(S_{\Gamma};\mathbb{Z}/2\mathbb{Z}).$

Let $\alpha_{1},\ldots,\alpha_{g}$ be a basis of
$H_{1}(\Gamma;\mathbb{Z}/2\mathbb{Z})$. To each 1-cycle $\alpha_{i}$, we
associate a class $\gamma_{\alpha_{i}}$ in
$H_{1}(S_{\Gamma};\mathbb{Z}/2\mathbb{Z})$ as follows:

  1. 1.

for each $e\in\text{Edge}(\Gamma)$ such that $e\subset\alpha_{i}$, choose a
$1$-chain $\eta_{e}$ in $S_{e}$ joining the two boundary components of
$S_{e}$. Denote by $\eta_{v,e}$ the boundary point of $\eta_{e}$ contained in
$\gamma_{e,v}$;

  2. 2.

for each $v\in\text{Vert}(\Gamma)$ such that $v\in\alpha_{i}$, choose a
$1$-chain $\eta_{v}$ in $S_{v}$ joining the points $\eta_{v,e}$ and
$\eta_{v,e^{\prime}}$, where $e$ and $e^{\prime}$ are the two edges contained
in $\alpha_{i}$ and adjacent to $v$.

We denote by $\gamma_{\alpha_{i}}\in H_{1}(S_{\Gamma};\mathbb{Z}/2\mathbb{Z})$
the class defined by the union of all $1$-chains $\eta_{e}$ and $\eta_{v}$
above. Further, we extend the map $\alpha_{i}\mapsto\gamma_{\alpha_{i}}$ by
linearity to an injective $\mathbb{Z}/2\mathbb{Z}$-linear map

$\begin{array}[]{ccc}H_{1}(\Gamma;\mathbb{Z}/2\mathbb{Z})&\longrightarrow&H_{1}(S_{\Gamma};\mathbb{Z}/2\mathbb{Z})\\\
\alpha&\longmapsto&\gamma_{\alpha}\end{array}.$

Note that $\gamma_{\alpha}$ is not uniquely defined by $\alpha$, but is well
defined only up to an element in $H_{1,0}(S_{\Gamma};\mathbb{Z}/2\mathbb{Z})$.
In other words, there is a natural and well defined injective
$\mathbb{Z}/2\mathbb{Z}$-linear map (or group morphism)

$H_{1}(\Gamma;\mathbb{Z}/2\mathbb{Z})\longrightarrow
H_{0,1}(S_{\Gamma};\mathbb{Z}/2\mathbb{Z})$

that associates to $\alpha$ the class realised by $\gamma_{\alpha}$ in
$H_{0,1}(S_{\Gamma};\mathbb{Z}/2\mathbb{Z})$. The next lemma shows in
particular that this map is an isomorphism.

###### Example 2.2

We consider the trivalent graph from Example 2.1. We depicted in Figure 2 the
1-cycle $\gamma_{e}$, and two possible 1-cycles $\gamma_{\alpha}$, denoted by
$\gamma_{\alpha}$ and $\gamma^{\prime}_{\alpha}$ (recall that the
decomposition of $S_{\Gamma}$ into the union of the surfaces $S_{v}$ and
$S_{e}$ is depicted in Figure 1).

$e$$\alpha$ |  |  $\gamma_{e}$$\gamma_{\alpha}$$\gamma^{\prime}_{\alpha}$
---|---|---
$\Gamma$ |  | $S_{\Gamma}$
Figure 2: Lifting cycles

###### Lemma 2.3

Let $\Gamma$ be a trivalent graph with $b_{1}(\Gamma)=g$. Then both
$\mathbb{Z}/2\mathbb{Z}$-vector spaces
$H_{1,0}(S_{\Gamma};\mathbb{Z}/2\mathbb{Z})$ and
$H_{0,1}(S_{\Gamma};\mathbb{Z}/2\mathbb{Z})$ have dimension $g$. Furthermore,
the intersection form on $H_{1}(S_{\Gamma};\mathbb{Z}/2\mathbb{Z})$ vanishes
on $H_{1,0}(S_{\Gamma};\mathbb{Z}/2\mathbb{Z})$.

  * Proof.

The group $H_{1,0}(S_{\Gamma};\mathbb{Z}/2\mathbb{Z})$ is clearly contained in
its orthogonal for the intersection form on
$H_{1}(S_{\Gamma};\mathbb{Z}/2\mathbb{Z})$. Since this latter is non-
degenerate, we deduce that $H_{1,0}(S_{\Gamma};\mathbb{Z}/2\mathbb{Z})$ has
dimension at most $g$. Let $U$ be the vector subspace of
$H_{1}(S_{\Gamma};\mathbb{Z}/2\mathbb{Z})$ of dimension $g$ generated by all
classes $\gamma_{\alpha}$ with $\alpha\in
H_{1}(\Gamma;\mathbb{Z}/2\mathbb{Z})$. By construction, the intersection of
$U$ with the orthogonal of $H_{1,0}(S_{\Gamma};\mathbb{Z}/2\mathbb{Z})$ is
trivial, from which we deduce that
$H_{1,0}(S_{\Gamma};\mathbb{Z}/2\mathbb{Z})$ has dimension at least $g$. Hence
$H_{1,0}(S_{\Gamma};\mathbb{Z}/2\mathbb{Z})$ has dimension $g$. Since by
definition, the group $H_{0,1}(S_{\Gamma};\mathbb{Z}/2\mathbb{Z})$ is the
quotient of $H_{1}(S_{\Gamma};\mathbb{Z}/2\mathbb{Z})$ by
$H_{1,0}(S_{\Gamma};\mathbb{Z}/2\mathbb{Z})$, it also has dimension $g$.
$\Box$

We will abusively identify $H_{0,1}(S_{\Gamma};\mathbb{Z}/2\mathbb{Z})$ with
the subgroup of $H_{1}(S_{\Gamma};\mathbb{Z}/2\mathbb{Z})$ generated by all
classes $\gamma_{\alpha}$ with $\alpha\in
H_{1}(\Gamma;\mathbb{Z}/2\mathbb{Z})$. We then have the decomposition

$H_{1}(S_{\Gamma};\mathbb{Z}/2\mathbb{Z})=H_{1,0}(S_{\Gamma};\mathbb{Z}/2\mathbb{Z})\oplus
H_{0,1}(S_{\Gamma};\mathbb{Z}/2\mathbb{Z}).$

Hence the group $H_{0,1}(S_{\Gamma};\mathbb{Z}/2\mathbb{Z})$ viewed as a
subgroup of $H_{1}(S_{\Gamma};\mathbb{Z}/2\mathbb{Z})$ is the image of a
section of the quotient map $H_{1}(S_{\Gamma};\mathbb{Z}/2\mathbb{Z})\to
H_{0,1}(S_{\Gamma};\mathbb{Z}/2\mathbb{Z})$.

###### Remark 2.4

Lemma 2.3 and the filtration of $H_{1}(S_{\Gamma};\mathbb{Z}/2\mathbb{Z})$ by
$H_{1,0}(S_{\Gamma};\mathbb{Z}/2\mathbb{Z})$ and
$H_{0,1}(S_{\Gamma};\mathbb{Z}/2\mathbb{Z})$ may be seen as very particular
cases of [IKMZ16, Theorem 1]. This relation to tropical homology explains our
choice of notation.

###  2.C. Real trivalent graphs and real structures above them

###### Definition 2.5

A _real trivalent graph_ is a pair $(\Gamma,\tau)$, where $\Gamma$ is a
trivalent graph and $\tau:\Gamma\to\Gamma$ a continuous involution (called
real structure), such that the restriction of $\tau$ on any (open) edge of
$\Gamma$ is either the identity or has no fixed points.

The condition on the restriction of $\tau$ on edges of $\Gamma$ is to exclude
the situation when this restriction is a symmetry locally given by
$x\mapsto-x$.

###### Example 2.6

Any trivalent graph $\Gamma$ carries a canonical real structure given by
$\tau=\mathop{\mathrm{Id}}$.

###### Example 2.7

Two real trivalent graphs $(\Gamma,\tau)$ are depicted in Figure 3. The graph
on the left is of genus $2$ whereas the other one is of genus $3$. In both
cases the graph $\Gamma$ is drawn on the plane, and $\tau$ is the axial
symmetry with respect to the line supporting edges $e_{1}$ (resp. $e_{1}$ and
$e_{2}$). Hence $e_{1}$ (resp. $e_{1}$ and $e_{2}$) are exactly the
$\tau$-invariant edges of $\Gamma$.

$\tau$$e_{1}$ |  |  $\tau$$e_{1}$$e_{2}$
---|---|---
a) A real trivalent graph $(\Gamma,\tau)$ of genus $2$ |  |  b) A real trivalent graph $(\Gamma,\tau)$ of genus $3$
Figure 3: Real graphs

Next we define the lifts of $\tau$ to $S_{\Gamma}$ that we consider in this
text. For this purpose, we need to fix the following:

  * •

the closed disk $\SS_{1}=\\{z\in\mathbb{C},\ |z|\leq 1\\}$ equipped with the
complex conjugation $\sigma_{1}:\SS_{1}\to\SS_{1}$;

  * •

a pair of pants $\SS_{3}$;

  * •

two orientation-reversing involutions $\sigma_{3}:\SS_{3}\to\SS_{3}$ and
$\widetilde{\sigma}:\SS_{3}\to\SS_{3}$ such that the fixed locus of
$\sigma_{3}$ is three disjoint segments, and the fixed locus of
$\widetilde{\sigma}$ is a segment (see Figure 4);

$\sigma_{3}$ |  |  $\widetilde{\sigma}$
---|---|---
Figure 4: Two real structures on a pair of pants

  * •

for each vertex $v$ of $\Gamma$, an orientation-preserving homeomorphism
$\psi_{v}:S_{v}\to\SS_{\text{val}(v)}$; if in addition $\tau$ interchanges two
adjacent edges of a $\tau$-invariant vertex $v$, then $\psi_{v}$ is chosen so
that $\widetilde{\sigma}(\psi_{v}(\gamma_{v,e}))=\psi_{v}(\gamma_{v,\tau(e)})$
for any edge $e$ adjacent to $v$.

We say that a continuous orientation-reversing involution
$\tau_{\Gamma}:S_{\Gamma}\to S_{\Gamma}$ _lifts_ $\tau:\Gamma\to\Gamma$ if the
following conditions are satisfied:

  1. 1.

if $\tau$ is locally the identity around $v$, then
${\tau_{\Gamma}}\raisebox{-3.87495pt}{$\big{|}$}_{{}^{S_{v}}}=\psi_{v}^{-1}\circ\sigma_{\text{val}(v)}\circ\psi_{v}$;

  2. 2.

if $v$ is a 3-valent vertex fixed by $\tau$, and if $\tau$ interchanges two
adjacent edges of $v$, then
${\tau_{\Gamma}}\raisebox{-3.87495pt}{$\big{|}$}_{{}^{S_{v}}}=\psi_{v}^{-1}\circ\widetilde{\sigma}\circ\psi_{v}$,

  3. 3.

if $\tau(v)\neq v$, then
${\tau_{\Gamma}}\raisebox{-3.87495pt}{$\big{|}$}_{{}^{S_{v}}}=\psi_{\tau(v)}^{-1}\circ\sigma_{\text{val}(v)}\circ\psi_{v}$.

Note that the three above conditions imply that if $\tau_{\Gamma}$ lifts
$\tau$, then $\tau_{\Gamma}(S_{e})=S_{\tau(e)}$ for each edge of $\Gamma$. We
denote by $G$ the subgroup of the group of homeomorphisms of $S_{\Gamma}$
generated by elements $h:S_{\Gamma}\to S_{\Gamma}$ such that

  * •

$h$ restricts to the identity on
$\displaystyle\bigcup_{v\in\text{Vert}^{0}(\Gamma)}S_{v}$;

  * •

if $e$ is a $\tau$-invariant edge of $\Gamma$, then the restriction of $h$ on
$S_{e}$ is isotopic to a power of the Dehn twist along $\gamma_{e}$;

  * •

if $\tau(e)\neq e$, then the restriction of $h$ on $S_{e}\cup S_{\tau(e)}$ is
isotopic to a power of the Dehn twist along $\gamma_{e}-\gamma_{\tau(e)}$.

Recall that by definition, two real topological surfaces $(S_{1},\tau_{1})$
and $(S_{2},\tau_{2})$ are isomorphic if there exists an orientation-
preserving homeomorphism $\theta:S_{1}\to S_{2}$ such that
$\theta\circ\tau_{1}\circ\theta^{-1}=\tau_{2}$. In particular, if
$(\Gamma,\tau)$ is a real trivalent graph, and if two lifts $\tau_{\Gamma,1}$
and $\tau_{\Gamma,2}$ of $\tau$ are conjugated by an element of $G$, then the
real topological surfaces $(S_{\Gamma},\tau_{\Gamma_{1}})$ and
$(S_{\Gamma},\tau_{\Gamma_{2}})$ are isomorphic.

###### Definition 2.8

A _real structure above_ a real trivalent graph $(\Gamma,\tau)$ is a lift of
$\tau$ considered up to conjugation by an element of $G$.

Clearly, the set of lifts of $\tau$ for a real trivalent graph $(\Gamma,\tau)$
depends on all the choices made above. However, it follows from Proposition
2.11 below that the set of real structures above $(\Gamma,\tau)$ only depends
on $(\Gamma,\tau)$.

###### Lemma 2.9

Let $\mathcal{S}_{2}$ be the cylinder $\\{z\in\mathbb{C}\>\>|\>\>1\leq|z|\leq
2\\}$, and $\mathop{\mathrm{conj}}$ be the complex conjugation on
$\mathbb{C}$. Denote by $T$ the Dehn twist of $\mathcal{S}_{2}$ along the
circle of radius $\frac{3}{2}$ given by

$\begin{array}[]{cccc}T:&\mathcal{S}_{2}&\longrightarrow&\mathcal{S}_{2}\\\
&z&\longmapsto&e^{2i\pi(|z|-1)}z\end{array}.$

Then, up to isotopy and conjugation by a homeomorphism
$\theta:\mathcal{S}_{2}\to\mathcal{S}_{2}$ restricting to the identity on
$\partial\mathcal{S}_{2}$, the only real structures on $\mathcal{S}_{2}$
restricting to $\mathop{\mathrm{conj}}$ on $\partial\mathcal{S}_{2}$ are
$\mathop{\mathrm{conj}}$ and $T\circ\mathop{\mathrm{conj}}$.

The real part of the two real structures $\mathop{\mathrm{conj}}$ and
$T\circ\mathop{\mathrm{conj}}$ differ by a “half” Dehn twist, see Figure 5.

|  |
---|---|---
$z\mapsto\overline{z}$ |  | $z\mapsto e^{2i\pi(|z|-1)}\overline{z}$
Figure 5: The real part of two real structures on the annulus
$\\{z\in\mathbb{C}\>\>|\>\>1\leq|z|\leq 2\\}$

  * Proof.

Let $\tau_{1}:\mathcal{S}_{2}\to\mathcal{S}_{2}$ be an orientation-reversing
continuous involution that restricts to $\mathop{\mathrm{conj}}$ on
$\partial\mathcal{S}_{2}$. The map $\tau_{1}\circ\mathop{\mathrm{conj}}$ is an
orientation-preserving continuous involution that restricts to the identity on
$\partial\mathcal{S}_{2}$. Hence it is, up to a composition by an isotopy
restricting to the identity on $\partial\mathcal{S}_{2}$, equal to $T^{k}$ for
some integer $k$. We compute

$T\circ\mathop{\mathrm{conj}}\circ
T^{-1}(z)=e^{4i\pi(|z|-1)}\overline{z}=T^{2}\circ\mathop{\mathrm{conj}}.$

Hence there exists a homeomorphism $\theta$ as in the lemma such that
$\tau_{1}=\theta\circ\mathop{\mathrm{conj}}\circ\theta^{-1}$ if $k$ is even,
and such that $\tau_{1}=\theta\circ
T\circ\mathop{\mathrm{conj}}\circ\theta^{-1}$ if $k$ is odd. If there would
exist a homeomorphism $\theta$ as in the lemma such that
$\mathop{\mathrm{conj}}=\theta\circ
T\circ\mathop{\mathrm{conj}}\circ\theta^{-1}$, it would map the real part of
$T\circ\mathop{\mathrm{conj}}$ to the real part of $\mathop{\mathrm{conj}}$.
Since these two real parts differ by a “half” Dehn twist, this is impossible.
$\Box$

###### Lemma 2.10

Let $\mathcal{S}^{\prime}_{2}$ be the cylinder
$\\{z\in\mathbb{C}\>\>|\>\>\frac{1}{4}\leq|z|\leq\frac{1}{2}\\}$, let
$\mathcal{S}^{\prime\prime}_{2}$ be the cylinder
$\\{z\in\mathbb{C}\>\>|\>\>2\leq|z|\leq 4\\}$. Denote respectively by $T_{1}$
and $T_{2}$ the Dehn twists of $\mathcal{S}^{\prime}_{2}$ and
$\mathcal{S}^{\prime\prime}_{2}$ given by

$\begin{array}[]{cccc}T_{1}:&\mathcal{S}^{\prime}_{2}&\longrightarrow&\mathcal{S}^{\prime}_{2}\\\
&z&\longmapsto&e^{2i\pi(1-\frac{1}{2|z|})}z\end{array}\quad\mbox{and}\quad\begin{array}[]{cccc}T_{2}:&\mathcal{S}^{\prime\prime}_{2}&\longrightarrow&\mathcal{S}^{\prime\prime}_{2}\\\
&z&\longmapsto&e^{2i\pi(\frac{|z|}{2}-1)}z\end{array}.$

Then, up to isotopy and conjugation by a homeomorphism
$\theta:\mathcal{S}^{\prime}_{2}\cup\mathcal{S}^{\prime\prime}_{2}\to\mathcal{S}^{\prime}_{2}\cup\mathcal{S}^{\prime\prime}_{2}$
restricting to the identity on
$\partial(\mathcal{S}^{\prime}_{2}\cup\mathcal{S}^{\prime\prime}_{2})$, the
only real structures on
$\mathcal{S}^{\prime}_{2}\cup\mathcal{S}^{\prime\prime}_{2}$ restricting to
$\frac{1}{\mathop{\mathrm{conj}}}$ on
$\partial(\mathcal{S}^{\prime}_{2}\cup\mathcal{S}^{\prime\prime}_{2})$ are
$\frac{1}{\mathop{\mathrm{conj}}}$ and $T_{1}\circ
T_{2}\circ\frac{1}{\mathop{\mathrm{conj}}}$.

  * Proof.

The proof is similar to the proof of Lemma 2.9. Let
$\tau_{1}:\mathcal{S}^{\prime}_{2}\cup\mathcal{S}^{\prime\prime}_{2}$ be an
orientation-reversing continuous involution that restricts to
$\frac{1}{\mathop{\mathrm{conj}}}$ on
$\partial(\mathcal{S}^{\prime}_{2}\cup\mathcal{S}^{\prime\prime}_{2})$. The
map $\tau_{1}\circ\frac{1}{\mathop{\mathrm{conj}}}$ is an orientation-
preserving continuous involution that restricts to the identity on
$\partial(\mathcal{S}^{\prime}_{2}\cup\mathcal{S}^{\prime\prime}_{2})$. Hence
it is, up to a composition by an isotopy restricting to the identity on
$\partial(\mathcal{S}^{\prime}_{2}\cup\mathcal{S}^{\prime\prime}_{2})$, equal
to $T_{1}^{k}\circ T_{2}^{l}$ for some integers $k$ and $l$. We extend the
Dehn twists $T_{1}$ and $T_{2}$ to $\mathbb{C}$ by the identity respectively
outside $\mathcal{S}^{\prime}_{2}$ and $\mathcal{S}^{\prime\prime}_{2}$. We
compute

${\left(T_{1}^{k}\circ
T_{2}^{l}\circ\frac{1}{\mathop{\mathrm{conj}}}\right)}^{2}(z)=ze^{2i\pi(l-k)(\frac{1}{2|z|}-1)}$

if $z\in\mathcal{S}^{\prime}_{2}$ and

${\left(T_{1}^{k}\circ
T_{2}^{l}\circ\frac{1}{\mathop{\mathrm{conj}}}\right)}^{2}(z)=ze^{2i\pi(l-k)(\frac{|z|}{2}-1)}$

if $z\in\mathcal{S}^{\prime\prime}_{2}$. Hence the map $(T_{1}^{k}\circ
T_{2}^{l})\circ\frac{1}{\mathop{\mathrm{conj}}}$ is an involution on
$\mathcal{S}^{\prime}_{2}\cup\mathcal{S}^{\prime\prime}_{2}$ if and only if
$k=l$. We compute

$(T_{1}\circ T_{2})\circ\frac{1}{\mathop{\mathrm{conj}}}\circ(T_{1}^{-1}\circ
T_{2}^{-1})=(T_{1}^{2}\circ T_{2}^{2})\circ\frac{1}{\mathop{\mathrm{conj}}},$

and the lemma is proved as Lemma 2.9. $\Box$

The following proposition is the main observation that allows one to compare
two different real structures above a given real trivalent graph
$(\Gamma,\tau)$, or in other words, to make sense of the difference
$\tau_{\Gamma,1}-\tau_{\Gamma,2}$ of two real structures in
$\Pi_{(\Gamma,\tau)}$. Let $\Gamma/\tau$ be the quotient of the graph $\Gamma$
by $\tau$, i.e. edges and vertices exchanged by $\tau$ are identified.

###### Proposition 2.11

Let $(\Gamma,\tau)$ be a real trivalent graph. Let $e_{1},\ldots,e_{k}$ be the
edges in $\text{Edge}^{0}(\Gamma/\tau)$, Then the set $\Pi_{(\Gamma,\tau)}$ of
real structures above $(\Gamma,\tau)$ is naturally an affine space with
direction

$\overrightarrow{\Pi_{(\Gamma,\tau)}}=\mathbb{Z}/2\mathbb{Z}e_{1}\oplus\mathbb{Z}/2\mathbb{Z}e_{2}\oplus\ldots\oplus\mathbb{Z}/2\mathbb{Z}e_{k}.$

Furthermore the set of fixed points of such a real structure $\tau_{\Gamma}$
is well defined up to isotopy and Dehn twists on cylinders corresponding to
$\tau$-invariant edges in $\text{Edge}^{0}(\Gamma)$.

  * Proof.

Let $e$ be a $\tau$-invariant edge in $\text{Edge}^{0}(\Gamma)$. It follows
from Lemma 2.9 that there exist exactly two possibilities for the restriction
of a real structure on $S_{e}$, that are given, up to isotopy and conjugating
by $T$, by $\mathop{\mathrm{conj}}$ and $T\circ\mathop{\mathrm{conj}}$ on the
cylinder $S_{e}=\\{z\in\mathbb{C}\>\>|\>\>1\leq|z|\leq 2\\}$. Similarly, for
each pair of cylinders $\\{S_{e},S_{\tau(e)}\\}$ with
$e\in\text{Edge}^{0}(\Gamma)$ such that $\tau(e)\neq e$, it follows from Lemma
2.10 that there are exactly two possibilities for the restriction of a real
structure on $S_{e}\cup S_{\tau(e)}$, which differ by a composition with the
Dehn twist along $\gamma_{e}-\gamma_{\tau(e)}$. $\Box$

###### Example 2.12

On Figure 6 we show the fixed point set of two different real structures
lifting the involution $\tau$ on the real graph of genus 2 depicted on Figure
3a.

|  |
---|---|---
Figure 6: Two real structures above the same graph $(\Gamma,\tau)$ of genus
$2$ of Example 2.7

###### Remark 2.13

In the case when $\tau=\mathop{\mathrm{Id}}$, the data of a real structure
$\tau_{\Gamma}:S_{\Gamma}\to S_{\Gamma}$ above $(\Gamma,\mathop{\mathrm{Id}})$
is equivalent to the data of the ribbon structure on $\Gamma$ given by
$S_{\Gamma}/\tau_{\Gamma}$. Furthermore, two real structures differ along an
edge $e$ of $\Gamma$ if and only if the two ribbon structures differ by a half
twist along $e$.

##  3\. Action on $H_{1}(S_{\Gamma};\mathbb{Z}/2\mathbb{Z})$ induced by a real
structure

We identify in this section a vector subspace $W_{(\Gamma,\tau)}$ of
$\overrightarrow{\Pi_{(\Gamma,\tau)}}$ that characterises all real structures
$\tau_{\Gamma}:S_{\Gamma}\to S_{\Gamma}$ inducing the same map
$\tau_{\Gamma*}:H_{1}(S_{\Gamma};\mathbb{Z}/2\mathbb{Z})\to
H_{1}(S_{\Gamma};\mathbb{Z}/2\mathbb{Z})$. Next lemma ensures that the action
on $H_{1}(S_{\Gamma};\mathbb{Z}/2\mathbb{Z})$ induced by a real structure
above a real trivalent graph is well defined.

###### Lemma 3.1

Let $(\Gamma,\tau)$ be a real trivalent graph and $\tau_{\Gamma}:S_{\Gamma}\to
S_{\Gamma}$ be a lift of $\tau$. Then the induced involution
$\tau_{\Gamma*}:H_{1}(S_{\Gamma};\mathbb{Z}/2\mathbb{Z})\to
H_{1}(S_{\Gamma};\mathbb{Z}/2\mathbb{Z})$ only depends on the real structure
class of $\tau_{\Gamma}$.

  * Proof.

It follows from the proof of Lemmas 2.9 and 2.10 that any other representative
of the real structure class of $\tau_{\Gamma}$ is obtained from this latter by
a finite sequence of compositions with either an isotopy or by an even power
of a Dehn twist. Since both types of maps induce a trivial action on
$H_{1}(S_{\Gamma};\mathbb{Z}/2\mathbb{Z})$, the result follows. $\Box$

Given $(\Gamma,\tau)$ a real trivalent graph, we denote by
$\pi:\Gamma\to\Gamma/\tau$ the quotient map. Recall that by Proposition 2.11,
an edge in $\text{Edge}^{0}(\Gamma)$ defines a vector in
$\overrightarrow{\Pi_{(\Gamma,\tau)}}$. There is a natural bilinear map

$\begin{array}[]{cccc}\mu:&H_{1}(\Gamma;\mathbb{Z}/2\mathbb{Z})\times
H_{1}(\Gamma;\mathbb{Z}/2\mathbb{Z})&\longrightarrow&\overrightarrow{\Pi_{(\Gamma,\tau)}}\end{array}$

that associates to two 1-cycles $\alpha$ and $\beta$ the sum of the vectors in
$\overrightarrow{\Pi_{(\Gamma,\tau)}}$ defined by the edges contained in the
support of the 1-chain $\pi(\alpha\cap\beta)$. We denote by
$\langle.,.\rangle$ the standard bilinear form on
$\overrightarrow{\Pi_{(\Gamma,\tau)}}$ defined by

$\left\langle\sum_{e\in\text{Edge}^{0}(\Gamma/\tau)}u_{e}e,\sum_{e\in\text{Edge}^{0}(\Gamma/\tau)}v_{e}e\right\rangle=\sum_{e\in\text{Edge}^{0}(\Gamma/\tau)}u_{e}v_{e}.$

We define the following vector subspace $W_{(\Gamma,\tau)}$ of
$\overrightarrow{\Pi_{(\Gamma,\tau)}}$:

$W_{(\Gamma,\tau)}=\left\\{w\in\overrightarrow{\Pi_{(\Gamma,\tau)}}\>\>|\>\>\forall\alpha,\beta\in
H_{1}(\Gamma;\mathbb{Z}/2\mathbb{Z}),\>\>\langle
w,\mu(\alpha,\beta)\rangle=0\right\\}.$

By definition, we have

$W_{(\Gamma,\tau)}=\bigcap_{\alpha,\beta\in
H_{1}(\Gamma;\mathbb{Z}/2\mathbb{Z})}Ker\left(\mu^{*}(\alpha,\beta)\right),$

where $\mu^{*}(\alpha,\beta)$ denotes the linear map dual to the vector
$\mu(\alpha,\beta)$.

###### Theorem 3.2

Let $(\Gamma,\tau)$ be a real trivalent graph, and let
$\tau_{\Gamma,1}:S_{\Gamma}\to S_{\Gamma}$ and $\tau_{\Gamma,2}:S_{\Gamma}\to
S_{\Gamma}$ be two real structures above $(\Gamma,\tau)$. Then we have

$\tau_{\Gamma,1*}=\tau_{\Gamma,2*}\Longleftrightarrow\tau_{\Gamma,1}-\tau_{\Gamma,2}\in
W_{(\Gamma,\tau)}.$

  * Proof.

Clearly, both maps $\tau_{\Gamma,1*}$ and $\tau_{\Gamma,2*}$ have the same
restriction to $H_{1,0}(S_{\Gamma};\mathbb{Z}/2\mathbb{Z})$. Hence according
to Lemma 2.3, we are left to show that

$\forall\alpha\in
H_{1}(\Gamma;\mathbb{Z}/2\mathbb{Z}),\>\>\tau_{\Gamma,1*}(\gamma_{\alpha})=\tau_{\Gamma,2*}(\gamma_{\alpha})\Longleftrightarrow\tau_{\Gamma,1}-\tau_{\Gamma,2}\in
W_{(\Gamma,\tau)}.$

Given a vector $w\in\overrightarrow{\Pi_{(\Gamma,\tau)}}$, we denote by
$\mathcal{E}(w)$ the set of edges $e\in\text{Edge}^{0}(\Gamma)$ such that
$\langle w,\pi(e)\rangle=1$. Choosing other representatives of the real
structure class of $\tau_{\Gamma,1}$ and $\tau_{\Gamma,2}$ if necessary, we
may assume that these latter coincide on

$\displaystyle
S_{\Gamma}\setminus\bigcup_{e\in\mathcal{E}(\tau_{\Gamma,1}-\tau_{\Gamma,2})}S_{e},$

and differ by a composition with a Dehn twist along $\gamma_{e}$ for all edges
$e$ in $\mathcal{E}(\tau_{\Gamma,1}-\tau_{\Gamma,2})$. Hence we have

$\tau_{\Gamma,1*}(\gamma_{\alpha})=\tau_{\Gamma,2*}(\gamma_{\alpha})+\sum_{e\in\mathcal{E}(\tau_{\Gamma,1}-\tau_{\Gamma,2})\cap\alpha}\gamma_{e}.$
(3.1)

Since the intersection form is non-degenerate on
$H_{1}(S_{\Gamma};\mathbb{Z}/2\mathbb{Z})$, we have

$\tau_{\Gamma,1*}(\gamma_{\alpha})-\tau_{\Gamma,2*}(\gamma_{\alpha})=0\Longleftrightarrow\forall\gamma\in
H_{1}(S_{\Gamma};\mathbb{Z}/2\mathbb{Z}),\
(\tau_{\Gamma,1*}(\gamma_{\alpha})-\tau_{\Gamma,2*}(\gamma_{\alpha}))\cdot\gamma=0,$

where $a\cdot b\in\mathbb{Z}/2\mathbb{Z}$ stands for the intersection product
of two classes $a$ and $b$ in $H_{1}(S_{\Gamma};\mathbb{Z}/2\mathbb{Z})$.
Combining Relation $(\ref{equ:diff action})$ and Lemma 2.3, we obtain

$\tau_{\Gamma,1*}(\gamma_{\alpha})-\tau_{\Gamma,2*}(\gamma_{\alpha})=0\Longleftrightarrow\forall\beta\in
H_{1}(\Gamma;\mathbb{Z}/2\mathbb{Z}),\
\sum_{e\in\mathcal{E}(\tau_{\Gamma,1}-\tau_{\Gamma,2})\cap\alpha}\gamma_{e}\cdot\gamma_{\beta}=0.$

By construction we have

$\sum_{e\in\mathcal{E}(\tau_{\Gamma,1}-\tau_{\Gamma,2})\cap\alpha}\gamma_{e}\cdot\gamma_{\beta}=\langle\tau_{\Gamma,1}-\tau_{\Gamma,2},\
\mu(\alpha,\beta)\rangle,$

so the result is proved. $\Box$

###### Example 3.3

Let us consider the real trivalent graph $(\Gamma,\tau)$ of genus 3 depicted
in Figure 3b, with two $\tau$-invariant edges $e_{1}$ and $e_{2}$. There
exists 16 real structures above $(\Gamma,\tau)$, distributed into $8$ parallel
affine subspaces of $\Pi_{(\Gamma,\tau)}$ with direction
$W_{(\Gamma,\tau)}=\mathbb{Z}/2\mathbb{Z}\pi(e_{1}+e_{2})$. Note that the real
part of the real topological surface $(S_{\Gamma},\tau_{\Gamma})$ is composed
of two circles for any element $\tau_{\Gamma}$ of $\Pi_{(\Gamma,\tau)}$.
Furthermore, all possible real structures above $(\Gamma,\tau)$ produce
exactly two real isomorphism types of real topological surfaces:
$(S_{\Gamma},\tau_{\Gamma})$ is of type111Recall that a connected real
topological surface $(S,\tau)$ is of type I if $S\setminus\mathbb{R}S$ is
disconnected, and of type II otherwise. I for $4$ out of these $8$ families of
real structures, and of type II for the $4$ remaining families.

###### Corollary 3.4

Let $(\Gamma,\tau)$ be a real trivalent graph, and let
$\tau_{\Gamma,1}:S_{\Gamma}\to S_{\Gamma}$ and $\tau_{\Gamma,2}:S_{\Gamma}\to
S_{\Gamma}$ be two real structures above $(\Gamma,\tau)$. Assume that
$(S_{\Gamma},\tau_{\Gamma,1})$ is a maximal real topological surface. Then
$(S_{\Gamma},\tau_{\Gamma,2})$ is also maximal if and only if the difference
$\tau_{\Gamma,1}-\tau_{\Gamma,2}$ lies in $W_{(\Gamma,\tau)}$.

  * Proof.

Recall that a real topological surface $(S,\tau_{S})$ with
$\mathbb{R}S\neq\emptyset$ is maximal if and only if
$\tau_{S*}=\mathop{\mathrm{Id}}$, see [BR90, Proposition 5.4.9]. Hence we have
$\tau_{\Gamma,1*}=\mathop{\mathrm{Id}}$. Furthermore, since
$(S_{\Gamma},\tau_{\Gamma,1})$ has a non-empty real part, the trivalent graph
$\Gamma$ has at least one $\tau$-invariant edge, which in its turn implies
that any real structure above $(\Gamma,\tau)$ has a non-empty real part. Then,
by Theorem 3.2, $\tau_{\Gamma,2*}=\mathop{\mathrm{Id}}$ if and only if
$\tau_{\Gamma,1}-\tau_{\Gamma,2}\in W_{(\Gamma,\tau)}$, that is to say
$(S_{\Gamma},\tau_{\Gamma,2})$ is maximal if and only if
$\tau_{\Gamma,1}-\tau_{\Gamma,2}$ lies in $W_{(\Gamma,\tau)}$. $\Box$

Next proposition gives an alternative description of the space
$W_{(\Gamma,\tau)}$ in terms of disconnecting edges and disconnecting pairs of
edges. Recall that edges are always considered open. Given a real connected
trivalent graph $(\Gamma,\tau)$, we define the following sets:

  * •

$\text{Edge}^{0,(1)}(\Gamma,\tau)$ is the set of disconnecting edges of
$\Gamma/\tau$ in $\text{Edge}^{0}(\Gamma/\tau)$, i.e. edges
$e\in\text{Edge}^{0}(\Gamma/\tau)$ such that $\Gamma/\tau\setminus e$ is not
connected;

  * •

$\text{Edge}^{0,(2)}(\Gamma,\tau)$ is the set of pairs
$\\{\pi(e),\pi(e^{\prime})\\}\subset\text{Edge}^{0}(\Gamma/\tau)$, where
$\\{e,e^{\prime}\\}\subset\text{Edge}^{0}(\Gamma)$ is such that
$\pi(e),\pi(e^{\prime})\notin\text{Edge}^{0,(1)}(\Gamma,\tau)$ and
$\Gamma\setminus\\{e,e^{\prime}\\}$ is not connected.

###### Proposition 3.5

Let $(\Gamma,\tau)$ be a real trivalent graph. Then we have

$W_{(\Gamma,\tau)}=\left(\bigoplus_{e\in\text{Edge}^{0,(1)}(\Gamma,\tau)}\mathbb{Z}/2\mathbb{Z}e\right)\oplus\mathrm{Span}\left\\{e+e^{\prime}\mid\\{e,e^{\prime}\\}\in\mathrm{Edge}^{0,(2)}(\Gamma,\tau)\right\\}.$

  * Proof.

Let us denote by $V$ the $\mathbb{Z}/2\mathbb{Z}$-vector space on the right
hand side of the equality stated in the proposition. Hence we want to show
that $W_{(\Gamma,\tau)}=V$. Step 1: reduction to the case
$\tau=\mathop{\mathrm{Id}}$. The map $\pi:\Gamma\to\Gamma/\tau$ induces a
linear map
$\widetilde{\pi}:\overrightarrow{\Pi_{(\Gamma,\mathop{\mathrm{Id}})}}\to\overrightarrow{\Pi_{(\Gamma,\tau)}}$.
In order to avoid confusion, we denote by $\mu_{\Gamma}$ the map
$H_{1}(\Gamma;\mathbb{Z}/2\mathbb{Z})\times
H_{1}(\Gamma;\mathbb{Z}/2\mathbb{Z})\to\overrightarrow{\Pi_{(\Gamma,\mathop{\mathrm{Id}})}}$,
and we keep the notation $\mu:H_{1}(\Gamma;\mathbb{Z}/2\mathbb{Z})\times
H_{1}(\Gamma;\mathbb{Z}/2\mathbb{Z})\to\overrightarrow{\Pi_{(\Gamma,\tau)}}$.
Recall that

$W_{(\Gamma,\mathop{\mathrm{Id}})}=\left\\{w\in\overrightarrow{\Pi_{(\Gamma,\mathop{\mathrm{Id}})}}\>\>|\>\>\forall\alpha,\beta\in
H_{1}(\Gamma;\mathbb{Z}/2\mathbb{Z}),\>\>\langle
w,\mu_{\Gamma}(\alpha,\beta)\rangle=0\right\\}.$

Let
$\overrightarrow{\Pi_{(\Gamma,\mathop{\mathrm{Id}})}}^{sym}\subset\overrightarrow{\Pi_{(\Gamma,\mathop{\mathrm{Id}})}}$
be the subvector space consisting of vectors
$w\in\overrightarrow{\Pi_{(\Gamma,\mathop{\mathrm{Id}})}}$ fixed by the
natural map induced by $\tau$ on
$\overrightarrow{\Pi_{(\Gamma,\mathop{\mathrm{Id}})}}$. Note that the
difference between two real structures above $(\Gamma,\tau)$ can always be
expressed as such an element. Given
$w\in\overrightarrow{\Pi_{(\Gamma,\mathop{\mathrm{Id}})}}^{sym}$, we write
$w=w^{fix}+w^{even}$ where $w^{fix}$ is supported by the edges of $\Gamma$
fixed by $\tau$ and $w^{even}$ is supported by pairs of edges exchanged by
$\tau$. Define the surjective linear map
$\bar{\pi}:\overrightarrow{\Pi_{(\Gamma,\mathop{\mathrm{Id}})}}^{sym}\to\overrightarrow{\Pi_{(\Gamma,\tau)}}$
by $\bar{\pi}(w)=\widetilde{\pi}(w^{fix})+\widetilde{\pi}(w^{1})$, where
$w^{even}=w^{1}+\tau(w^{1})$ and the supports of $w^{1}$ and $\tau(w^{1})$ are
disjoint. The vector $w^{1}$ is not unique, however $\tilde{\pi}(w^{1})$ does
clearly not depend on the particular choice of $w^{1}$. Let us prove that
$W_{(\Gamma,\tau)}=\bar{\pi}\left(W_{(\Gamma,\mathop{\mathrm{Id}})}\cap\overrightarrow{\Pi_{(\Gamma,\mathop{\mathrm{Id}})}}^{sym}\right).$
Let $w\in\overrightarrow{\Pi_{(\Gamma,\mathop{\mathrm{Id}})}}^{sym}$ and let
$(\alpha,\beta)\in H_{1}(\Gamma;\mathbb{Z}/2\mathbb{Z})\times
H_{1}(\Gamma;\mathbb{Z}/2\mathbb{Z})$. Write
$\mu_{\Gamma}(\alpha,\beta)=\mu_{\Gamma}^{fix}+\mu_{\Gamma}^{even}+\mu_{\Gamma}^{odd}$,
where $\mu_{\Gamma}^{odd}$ is supported by edges $e\in\alpha\cap\beta$ such
that $\tau(e)\notin\alpha\cap\beta$ and $\mu_{\Gamma}^{fix}$ and
$\mu_{\Gamma}^{even}$ are defined as above (in general, the decomposition is
not unique). We obviously have

$\langle w,\mu_{\Gamma}(\alpha,\beta)\rangle=\langle
w^{fix},\mu_{\Gamma}^{fix}\rangle+\langle w^{even},\mu_{\Gamma}^{odd}\rangle.$

Similarily, we have
$\mu(\alpha,\beta)=\widetilde{\pi}(\mu_{\Gamma}^{fix})+\widetilde{\pi}(\mu_{\Gamma}^{odd})$,
and

$\langle\bar{\pi}(w),\mu(\alpha,\beta)\rangle=\langle\bar{\pi}(w^{fix}),\widetilde{\pi}(\mu_{\Gamma}^{fix})\rangle+\langle\bar{\pi}(w^{even}),\widetilde{\pi}(\mu_{\Gamma}^{odd})\rangle.$

Thus $\langle
w,\mu_{\Gamma}(\alpha,\beta)\rangle=\langle\bar{\pi}(w),\mu(\alpha,\beta)\rangle$,
which implies that
$W_{\Gamma,\tau}=\bar{\pi}\left(W_{\Gamma,\mathop{\mathrm{Id}}}\cap\overrightarrow{\Pi_{(\Gamma,\mathop{\mathrm{Id}})}}^{sym}\right)$
as announced. Assuming that Proposition 3.5 is known in the case of
$\tau=\mathop{\mathrm{Id}}$ and computing
$W_{(\Gamma,\mathop{\mathrm{Id}})}\cap\overrightarrow{\Pi_{(\Gamma,\mathop{\mathrm{Id}})}}^{sym}$,
it is easy to see that it remains to show that

$\text{Edge}^{0,(1)}(\Gamma,\tau)=\pi\left(\text{Edge}^{0,(1)}(\Gamma,\mathop{\mathrm{Id}})\right)\bigcup\pi\left(\left\\{\\{e,e^{\prime}\\}\subset\text{Edge}^{0,(2)}(\Gamma,\mathop{\mathrm{Id}})\,|\
\tau(e)=e^{\prime},\textrm{ and }\tau(e^{\prime})=e\right\\}\right)$

and

$\displaystyle\text{Edge}^{0,(2)}(\Gamma,\tau)$ $\displaystyle=$
$\displaystyle\pi\left(\left\\{\\{e,e^{\prime}\\}\subset\text{Edge}^{0,(2)}(\Gamma,\mathop{\mathrm{Id}})\,|\
\tau(e)\neq e^{\prime}\right\\}\right)$ $\displaystyle=$
$\displaystyle\pi\left(\left\\{\\{e,e^{\prime}\\}\subset\text{Edge}^{0,(2)}(\Gamma,\mathop{\mathrm{Id}})\,|\
\tau(e)=e\textrm{ and }\tau(e^{\prime})=e^{\prime}\right\\}\right)$
$\displaystyle\cup\
\pi\left(\left\\{\\{e,e^{\prime}\\}\subset\text{Edge}^{0,(2)}(\Gamma,\mathop{\mathrm{Id}})\,|\
\tau(e)\neq e,\>\tau(e^{\prime})\neq e^{\prime}\textrm{ and }\tau(e)\neq
e^{\prime}\right\\}\right).$

These equalities follows from the fact that the quotient map
$\pi:\Gamma\to\Gamma/\tau$ induces a surjective map
$\pi_{*}:H_{1}(\Gamma;\mathbb{Z}/2\mathbb{Z})\to
H_{1}(\Gamma/\tau;\mathbb{Z}/2\mathbb{Z})$.

So from now on, we assume that $\tau=\mathop{\mathrm{Id}}$, and we use the
shorter notation

$W_{\Gamma}=W_{(\Gamma,\tau)},\qquad\text{Edge}^{0,(1)}(\Gamma)=\text{Edge}^{0,(1)}(\Gamma,\mathop{\mathrm{Id}}),\quad\mbox{and}\quad\text{Edge}^{0,(2)}(\Gamma)=\text{Edge}^{0,(2)}(\Gamma,\mathop{\mathrm{Id}}).$

Step 2: $V$ is contained in $W_{\Gamma}$. An edge in
$\text{Edge}^{0,(1)}(\Gamma)$ is not contained in the support of any cycle in
$H_{1}(\Gamma;\mathbb{Z}/2\mathbb{Z})$. In particular we have

$\bigoplus_{e\in\text{Edge}^{0,(1)}(\Gamma)}\mathbb{Z}/2\mathbb{Z}e\subset
W_{\Gamma}.$

We claim that a pair of edges
$\\{e,e^{\prime}\\}\subset\text{Edge}^{0}(\Gamma)$ is contained in
$\text{Edge}^{0,(2)}(\Gamma)$ if and only if the two following conditions are
satisfied:

    1. 1.

there exists a cycle $\gamma\in H_{1}(\Gamma;\mathbb{Z}/2\mathbb{Z})$
containing both $e$ and $e^{\prime}$;

    2. 2.

for any theta subgraph $\Theta$ of $\Gamma$ containing $e$ and $e^{\prime}$,
the graph $\Theta\setminus\\{e,e^{\prime}\\}$ is not connected, see Figure 7a
and b.

Indeed, condition $(1)$ is equivalent to the fact that neither $e$ nor
$e^{\prime}$ is in $\text{Edge}^{0,(1)}(\Gamma)$. Next, a pair
$\\{e,e^{\prime}\\}$ satisfying condition $(1)$ is in
$\text{Edge}^{0,(2)}(\Gamma)$ if and only if any path in
$\Gamma\setminus\\{e\\}$ joining the two vertices adjacent to $e$ contains
$e^{\prime}$. This is equivalent to condition $(2)$.

$e$$e^{\prime}$ |  |  $e$$e^{\prime}$
---|---|---
a) $\Theta\setminus\\{e,e^{\prime}\\}$ connected |  | b) $\Theta\setminus\\{e,e^{\prime}\\}$ disconnected
Figure 7:

It follows from the above claim that for any
$\\{e,e^{\prime}\\}\in\text{Edge}^{0,(2)}(\Gamma)$ and any $\gamma\in
H_{1}(\Gamma;\mathbb{Z}/2\mathbb{Z})$, we have

$e\subset\gamma\Longleftrightarrow e^{\prime}\subset\gamma.$

This implies that

$\mathrm{Span}\left\\{e+e^{\prime}\mid\\{e,e^{\prime}\\}\in\text{Edge}^{0,(2)}(\Gamma)\right\\}\subset
W_{\Gamma}$

and so $V\subset W_{\Gamma}$. We define the following vector subspace of
$\overrightarrow{\Pi_{(\Gamma,\tau)}}$:

$U=\left(\bigoplus_{e\in\text{Edge}^{0,(1)}(\Gamma,\tau)}\mathbb{Z}/2\mathbb{Z}e\right)\oplus\mathrm{Span}\left\\{e\mid\exists
e^{\prime}\mbox{ with
}\\{e,e^{\prime}\\}\in\text{Edge}^{0,(2)}(\Gamma,\tau)\right\\}.$

Step 3: $W_{\Gamma}$ is contained in $U$. Let $e\notin U$. It means exactly
that $e\in\text{Edge}^{0}(\Gamma)\setminus\text{Edge}^{0,(1)}(\Gamma)$ and
that $\Gamma\setminus\\{e,e^{\prime}\\}$ is connected for any
$e^{\prime}\in\text{Edge}^{0}(\Gamma)\setminus\text{Edge}^{0,(1)}(\Gamma)$.
Denote by $v$ and $v^{\prime}$ the two vertices of $\Gamma$ adjacent to $e$.
By Menger’s Theorem [BR12, Theorem 3.6.11], there exist two paths $c_{1}$ and
$c_{2}$ in $\Gamma\setminus\\{e\\}$ joining $v$ and $v^{\prime}$, and whose
intersection is reduced to $\\{v,v^{\prime}\\}$. Hence $e$ is the only edge
common to the two cycles $c_{1}\cup e$ and $c_{2}\cup e$ in
$H_{1}(\Gamma;\mathbb{Z}/2\mathbb{Z})$. By definition of $W_{\Gamma}$, we have
$\langle w,e\rangle=0$ for any $w\in W_{\Gamma}$, and so $e\notin W_{\Gamma}$.
We conclude that $W_{\Gamma}\subset U$. Step 4: $W_{\Gamma}$ is contained in
$V$. By Step 3 above, an element $\widetilde{w}$ of $W_{\Gamma}$ can be
written in the (non-unique) form

$\widetilde{w}=w_{0}+e_{1}+\ldots+e_{k}+(e_{k+1}+e^{\prime}_{k+1})+\ldots+(e_{m}+e^{\prime}_{m}),$

where

    * –

$\displaystyle
w_{0}\in\bigoplus_{e\in\text{Edge}^{0,(1)}(\Gamma)}\mathbb{Z}/2\mathbb{Z}e$;

    * –

$\\{e_{i},e^{\prime}_{i}\\}\in\text{Edge}^{0,(2)}(\Gamma)$ if
$i\in\\{k+1,\ldots,m\\}$;

    * –

for any $i\in\\{1,\ldots,k\\}$, there exists an edge $e^{\prime}_{i}$ of
$\Gamma$ such that $\\{e_{i},e^{\prime}_{i}\\}\in\text{Edge}^{0,(2)}(\Gamma)$;

    * –

$e_{i}\neq e_{j}$ and $\\{e_{i},e_{j}\\}\notin\text{Edge}^{0,(2)}(\Gamma)$ for
any pair $\\{i,j\\}\subset\\{1,\ldots,k\\}$.

Recall that $V$ is a sub-vector space of $W_{\Gamma}$ by Step 2, hence
$w=w_{0}+e_{1}+\ldots+e_{k}$ is an element of $W_{\Gamma}$. If $k=0$, then $w$
is in $V$, and so is $\widetilde{w}$. Assume now that $k>0$. In this case $w$
is an element of $W_{\Gamma}$ such that for any edge
$e\in\text{Edge}^{0}(\Gamma)\setminus\text{Edge}^{0,(1)}(\Gamma)$ with
$\langle w,e\rangle=1$, we have $\langle w,e^{\prime}\rangle=0$ for any edge
$e^{\prime}$ with $\\{e,e^{\prime}\\}\in\text{Edge}^{0,(2)}(\Gamma)$. The rest
of the proof consists in proving by contradiction that such an element cannot
exist. Let $\alpha\in H_{1}(\Gamma;\mathbb{Z}/2\mathbb{Z})$ be a cycle
containing an edge in the support of $w$. Let us denote by
$e_{1},\ldots,e_{k}$ all edges belonging to the intersection of the support of
$w$ and $\alpha$, enumerated in a cyclic order induced by $\alpha$. By
definition of $W_{\Gamma}$, we have $\langle w,\mu(\alpha,\alpha)\rangle=0$,
i.e. $k$ is even. Let us also denote by $u_{i}$ the connected component of
$\alpha\setminus\\{e_{1},\ldots,e_{k}\\}$ adjacent to the edges $e_{i}$ and
$e_{i+1}$ (where the indices are taken modulo $k$). By assumption on $w$, for
each $i$ there exists a path $c_{i}$ in $\Gamma$ joining the two connected
components of $\alpha\setminus\\{e_{i},e_{i+1}\\}$, and such that
$c_{i}\cap\alpha$ is reduced to the two endpoints of $c_{i}$. Denote by
$\sigma(i)\in\mathbb{Z}/k\mathbb{Z}$ the integer such that $u_{i}\cup
u_{\sigma(i)}$ contains the two endpoints of $c_{i}$. Hence
$\sigma:\mathbb{Z}/k\mathbb{Z}\to\mathbb{Z}/k\mathbb{Z}$ is an involution with
no fixed points. The two endpoints of $c_{i}$ divide $\alpha$ into two
connected components. Let $u$ be one of these latter. Since $w\in W_{\Gamma}$,
we have $\langle w,\mu(\alpha,u\cup c_{i}))\rangle=0$, that is to say
$\sigma(i)=i\mod 2$ (see Figure 8a).

$e_{1}$$e_{2}$$\alpha$$c_{i}$$c_{i}\cup u$ |  |  $\beta_{1}$$\beta_{2}$
---|---|---
a) |  | b)
Figure 8:

Furthermore, we claim that for any $j$, we have

$j\in\\{i+1,\ldots,\sigma(i)-1\\}\Longleftrightarrow\sigma(j)\in\\{i+1,\ldots,\sigma(i)-1\\}.$

It is enough to prove the claim in the case when $j=i+1$. If
$\sigma(i+1)\notin\\{i+1,\ldots,\sigma(i)-1\\}$, one easily constructs two
cycles $\beta_{1}$ and $\beta_{2}$ in $H_{1}(\Gamma;\mathbb{Z}/2\mathbb{Z})$
such that $\langle w,\mu(\beta_{1},\beta_{2}))\rangle=1$ (see Figure 8b),
which contradicts that $w\in W_{\Gamma}$. Hence the map $\sigma$ induces an
involution on $\\{2,\ldots,\sigma(1)-1\\}$ with no fixed points. However the
cardinal of this latter set is odd, so such a fixed-point free involution
cannot exist. $\Box$

Recall that a graph is said to be _$k$ -edge connected_ if it remains
connected after removing any set of $l<k$ edges.

###### Corollary 3.6

Let $(\Gamma,\tau)$ be a real trivalent graph. If $\Gamma/\tau$ is $3$-edge
connected, then $W_{(\Gamma,\tau)}$ is the trivial vector space. In particular
there exists at most one real structure $\tau_{\Gamma}$ above $(\Gamma,\tau)$
for which $(S_{\Gamma},\tau_{\Gamma})$ is maximal.

##  4\. Haas’ Theorem

Here we explain how the results from the previous section specialise to Haas’
Theorem in the particular case of non-singular tropical curves in
$\mathbb{R}^{2}$. We first give in Section 4.A a tropical formulation of
Viro’s combinatorial patchworking, and state Haas’ Theorem classifying
combinatorial patchworkings producing $M$-curves. We prove this latter in
Section 4.B. We assume that the reader has a certain acquaintance with
tropical geometry. We refer to [BIMS15] for an introduction to tropical
geometry at the level needed here, as well as a more detailed exposition of
combinatorial patchworking and Haas’ Theorem.

###  4.A. Combinatorial patchworking and Haas’ Theorem

Here we present the reformulation of the particular case of unimodular
combinatorial patchworking in terms of twist-admissible sets of edges of a
non-singular plane tropical curve given in [BIMS15]. If $e$ is an edge of a
tropical curve in $\mathbb{R}^{2}$, we denote by
$(x_{e},y_{e})\in\mathbb{Z}^{2}$ a primitive direction vector of the line
supporting $e$ (note that $(x_{e},y_{e})$ is well defined up to sign, however
this does not play a role in what follows).

###### Definition 4.1

Let $C$ be a non-singular tropical curve in $\mathbb{R}^{2}$. A subset $T$ of
$\text{Edge}^{0}(C)$ is called twist-admissible if for any cycle $\gamma$ of
$C$, we have

$\sum_{e\in\gamma\cap T}(x_{e},y_{e})=0\quad\text{mod }2.$ (4.2)

Given $T$ a twist-admissible subset of edges of a non-singular tropical curve
$C$ in $\mathbb{R}^{2}$, perform the following operations:

  * (a)

at each vertex of $C$, draw three arcs as depicted in Figure 9a;

  * (b)

for each edge $e\in\text{Edge}^{0}(C)$ adjacent to the vertices $v$ and
$v^{\prime}$, join the two corresponding arcs at $v$ to the corresponding ones
for $v^{\prime}$ in the following way: if $e\notin T$, then join these arcs as
depicted in Figure 9b; if $e\in T$, then join these arcs as depicted in Figure
9c; denote by $\mathcal{P}$ the obtained collection of arcs;

  * (c)

choose arbitrarily an arc of $\mathcal{P}$ and a pair of signs for it;

  * (d)

associate pairs of signs to all arcs of $\mathcal{P}$ using the following
rule: given $e\in\text{Edge}(C)$ the pairs of signs of the two arcs of
$\mathcal{P}$ corresponding to $e$ differ by a factor
$((-1)^{x_{e}},(-1)^{y_{e}})$, see Figure 9b and c; (Note that the
compatibility condition (4.2) precisely means that this rule is consistent.)

|  $(\varepsilon_{1},\varepsilon_{2})$$((-1)^{x_{e}}\varepsilon_{1},(-1)^{y_{e}}\varepsilon_{2})$ |  $(\varepsilon_{1},\varepsilon_{2})$$((-1)^{x_{e}}\varepsilon_{1},(-1)^{y_{e}}\varepsilon_{2})$
---|---|---
a) | b) $e\notin T$ | c) $e\in T$
Figure 9: A patchworking of a non-singular tropical curve

  * (e)

map each arc $A$ of $\mathcal{P}$ to $(\mathbb{R}^{\times})^{2}$ by
$(x,y)\mapsto(\varepsilon_{1}\exp(x),\varepsilon_{2}\exp(y))$, where
$(\varepsilon_{1},\varepsilon_{2})$ is the pair of signs associated to $A$.
Denote by $C_{T}$ the curve in $(\mathbb{R}^{\times})^{2}$ which is the union
of these images over all arcs of $\mathcal{P}$.

Note that all possible choices at step (c) above produce the same curve
$C_{T}$ up to the action of $(\mathbb{Z}/2\mathbb{Z})^{2}$ by axial symmetries
$(z,w)\mapsto(\pm z,\pm w)$. Viro combinatorial patchworking Theorem [Vir01]
may be reformulated in terms of twist-admissible sets as follows.

######  Theorem 4.2 (Viro)

Let $T$ be a twist-admissible subset of edges of a non-singular tropical curve
$C$ in $\mathbb{R}^{2}$. Then there exists a real algebraic curve in
$(\mathbb{C}^{\times})^{2}$ with the same Newton polygon as $C$, and whose
real part in $(\mathbb{R}^{\times})^{2}$ is isotopic to $C_{T}$.

###### Remark 4.3

It is possible to produce an equation for the real algebraic curve whose
existence is attested by Theorem 4.2, see [BIMS15, Remark 3.9].

###### Example 4.4

One may choose $T$ to be empty as the empty set clearly satisfies (4.2). This
corresponds to Harnack patchworking mentioned in the introduction. The
resulting curve corresponds to the construction of _simple Harnack curves_
described in [Mik00] via Harnack distribution of signs, see [IV96].
Furthermore, the isotopy type of $C_{T}$ in $(\mathbb{R}^{\times})^{2}$, up to
axial symmetries, only depends on the Newton polygon of $C$, see Proposition
4.11.

###### Example 4.5

Let us consider the non-singular tropical curve $C$ of degree 6 depicted on
Figure 10. We equip $C$ with three different twist-admissible collection of
edges in Figures 11, 12, and 13. In each case we depict the isotopy types of
the real part of the corresponding real algebraic curve in both
$(\mathbb{R}^{\times})^{2}$ and $\mathbb{RP}^{2}$. Note that these are the
only isotopy types of maximal real sextics in $\mathbb{RP}^{2}$.

---
Figure 10: A tropical sextic |  |
---|---|---
Figure 11: Simple Harnack sextic |  |
---|---|---
Figure 12: Gudkov’s sextic |  |
---|---|---
Figure 13: Hilbert’s sextic

Given a non-singular tropical curve $C$, Haas’ classified in [Haa97] all
twist-admissible sets producing an $M$-curve. At that time the formalism of
tropical geometry did not exist yet, and the original formulation of Haas’
theorem is dual to the one we present here in Theorem 4.6. We say that a
twist-admissible set of edges $T$ of $C$ is _maximal_ if it satisfies the two
following conditions:

  1. 1.

any cycle in $H_{1}(C;\mathbb{Z}/2\mathbb{Z})$ contains an even number of
edges in $T$;

  2. 2.

for any edge $e\in T$, either $C\setminus e$ is disconnected, or there exists
an edge $e^{\prime}\in T$ such that $C\setminus e$ and $C\setminus e^{\prime}$
are connected, but $C\setminus(e\cup e^{\prime})$ is disconnected.

######  Theorem 4.6 (Haas)

Let $C$ be a non-singular tropical curve in $\mathbb{R}^{2}$, and let $T$ be a
twist-admissible set of edges of $C$. Then a real algebraic curve whose
existence is asserted by Theorem 4.2 is maximal if and only if $T$ is maximal.

###### Example 4.7

The empty collection is maximal, hence we recover the existence, for each non-
singular tropical curve, of a canonical maximal patchworking mentioned in the
introduction.

###  4.B. Proof of Haas’ Theorem

Here we deduce Theorem 4.6 from Theorem 3.2 and Proposition 3.5. First, we
introduce some standard notations. The coordinatewise argument and $\log$ maps
are defined by:

$\begin{array}[]{cccc}{\text{Arg}}:&(\mathbb{C}^{\times})^{2}&\longrightarrow&(\mathbb{R}/2\pi\mathbb{Z})^{2}\\\
&(z,w)&\longmapsto&\left(\text{arg}(z),\text{arg}(w)\right)\end{array}\quad\mbox{and}\quad\begin{array}[]{cccc}{\text{Log}}:&(\mathbb{C}^{\times})^{2}&\longrightarrow&\mathbb{R}^{2}\\\
&(z,w)&\longmapsto&\left(\log|z|,\log|w|\right)\end{array}.$

Note that the image of the map Arg is canonically identified with any fiber of
the map Log. We also define the following self-diffeomorphism of
$(\mathbb{C}^{\times})^{2}$ fot $t>1$

$\begin{array}[]{cccc}H_{t}:&(\mathbb{C}^{\times})^{2}&\longrightarrow&(\mathbb{C}^{\times})^{2}\\\
&(z,w)&\longmapsto&\displaystyle\left(|z|^{\frac{1}{\log(t)}}\frac{z}{|z|},|w|^{\frac{1}{\log(t)}}\frac{w}{|w|}\right)\end{array}.$

Given a complex polynomial $P(z,w)=\sum a_{i,j}z^{i}w^{j}$ and
$\Delta\subset\mathbb{R}^{2}$, we define

$P^{\Delta}(z,w)=\sum_{(i,j)\in\Delta}a_{i,j}z^{i}w^{j}.$

The _(closed) coamoeba_ of the algebraic curve $X$ with equation $P(z,w)=0$,
denoted by $\mathcal{C}\mathcal{A}(P)$, is defined as the topological closure
in $(\mathbb{R}/2\pi\mathbb{Z})^{2}$ of the set ${\text{Arg}}(X)$.

###### Example 4.8

If $P(z,w)$ is a real binomial whose Newton segment $\Delta(P)$ has integer
length 1, then the real part of the real algebraic curve defined by $P(z,w)$
intersects two quadrants $Q_{1}$ and $Q_{2}$ of $(\mathbb{R}^{\times})^{2}$.
If $u$ denotes a primitive integer vector normal to $\Delta(P)$, then the
coamoeba $\mathcal{C}\mathcal{A}(P)$ is the geodesic in
$(\mathbb{R}/2\pi\mathbb{Z})^{2}$ with direction $u$ and passing through
${\text{Arg}}(Q_{1})$ and ${\text{Arg}}(Q_{2})$. On Figure 14a we depicted the
coamoeba of a line given by the equation $z+aw=0$ with $a>0$. It joins the
points $(0,\pi)$ and $(\pi,0)$.

$(\pi,0)$$(0,0)$$(0,\pi)$$(\pi,0)$$(0,\pi)$ |  |  $(\pi,0)$$(0,0)$$(0,\pi)$$(\pi,0)$$(0,\pi)$
---|---|---
a) Coamoeba of a real line |  | b) Coamoeba of a real line
defined by $z+aw$ with $a>0$ |  | defined by $z+aw+b$ with $a>0$ and $b<0$
Figure 14:

######  Example 4.9 (See [Mik05, Proposition 6.11 and Lemma 8.19])

If $P(z,w)$ is a real trinomial whose Newton triangle $\Delta(P)$ has
Euclidean area $\frac{1}{2}$, then the real part of the real algebraic curve
defined by $P(z,w)$ intersects three quadrants $Q_{1},Q_{2}$, and $Q_{3}$ of
$(\mathbb{R}^{\times})^{2}$. If $u_{1},u_{2},$ and $u_{3}$ denote primitive
integer vectors normal to the three edges of $\Delta(P)$, then the coamoeba
$\mathcal{C}\mathcal{A}(P)$ is the union of the two triangles with vertices
${\text{Arg}}(Q_{1})$, ${\text{Arg}}(Q_{2})$, and ${\text{Arg}}(Q_{3})$, and
whose sides are geodesics with direction $u_{1}$, $u_{2}$, and $u_{3}$. In
particular $\mathcal{C}\mathcal{A}(P)$ is a (degenerate) pair of pants. On
Figure 14b we represented the coamoeba of a line given by the equation
$z+aw+b=0$ with $a>0$ and $b<0$.

Recall (see for example [BIMS15, Section 2.2]) that to any tropical curve $C$
in $\mathbb{R}^{2}$ is associated a dual subdivision of its Newton polygon. We
denote respectively by $\Delta_{v}$ and $\Delta_{e}$ the polygon dual to the
vertex $v$ and the edge $e$ of $C$. Next statement is proved in [Mik05,
Section 6].

######  Theorem 4.10 (Mikhalkin)

Let $C$ be a tropical curve in $\mathbb{R}^{2}$ defined by some tropical
polynomial ${\text{``}}\sum b_{i,j}x^{i}y^{j}{\text{''}}$. Then given a
collection $a_{i,j}$ of non-zero complex numbers, the image by $H_{t}$ of the
algebraic curve in $(\mathbb{C}^{\times})^{2}$ defined by the complex
polynomial

$P_{t}(z,w)=\sum a_{i,j}t^{-b_{i,j}}z^{i}w^{j}$

converges when $t\to+\infty$, for the Hausdorff metric on compact subsets of
$(\mathbb{C}^{\times})^{2}$, to a subset $V_{\infty}$ that can be described as
follows:

  * •

${\text{Log}}(V_{\infty})=C$;

  * •

for any vertex $v$ of $C$, we have ${\text{Log}}^{-1}(v)\cap
V_{\infty}=\mathcal{C}\mathcal{A}(P_{1}^{\Delta_{v}})$;

  * •

for each edge $e$ of $C$, we have ${\text{Log}}^{-1}(e)\cap
V_{\infty}=e\times\mathcal{C}\mathcal{A}(P_{1}^{\Delta_{e}})$ .

If furthermore all the $a_{i,j}$’s are real numbers, then the real part of the
real algebraic curve defined by $P_{t}(z,w)$ converges when $t\to+\infty$, for
the Hausdorff metric on compact subsets of $(\mathbb{R}^{\times})^{2}$, to
$V_{\infty}\cap(\mathbb{R}^{\times})^{2}$.

Note that one can recover Theorem 4.2 by combining Theorem 4.10 together with
Examples 4.8 and 4.9. In particular, an equation of a simple Harnack curve is
given by next proposition. We define the function
$\varepsilon:\mathbb{Z}^{2}\to\\{\pm 1\\}$ by $\varepsilon(i,j)=1$ if both $i$
and $j$ are even, and by $\varepsilon(i,j)=-1$ otherwise. Recall that the
Viro’s patchworking construction and the definition of a twist admissible set
of edges of a non-singular tropical curve are given in Section 4.A.

######  Proposition 4.11 (Itenberg, see [IV96] or [BIMS15, Remark 3.9])

Let $C$ be a non-singular tropical curve in $\mathbb{R}^{2}$ defined by some
tropical polynomial ${\text{``}}\sum b_{i,j}x^{i}y^{j}{\text{''}}$, and let

$P_{t}(z,w)=\sum\varepsilon(i,j)t^{-b_{i,j}}z^{i}w^{j},\quad
V_{\infty}=\lim_{t\to+\infty}H_{t}\left(\\{P_{t}=0\\}\right).$

Then up to axial symmetries, the real part of $V_{\infty}$ is isotopic in
$(\mathbb{R}^{\times})^{2}$ to the curve $C_{\emptyset}$ constructed out of
$C$ and the empty twist admissible set. In particular, the real algebraic
curve defined by $P_{t}(z,w)$ with $t$ large enough is maximal. Furthermore,
the isotopy type of its real part in $(\mathbb{R}^{\times})^{2}$, up to axial
symmetries, only depends on the Newton polygon of $C$.

We are now ready to deduce Haas’ Theorem from what is discussed above.

  * Proof of Theorem 4.6.

Let $C$ be a non-singular tropical curve in $\mathbb{R}^{2}$ defined by the
tropical polynomial ${\text{``}}\sum b_{i,j}x^{i}y^{j}{\text{''}}$, and let
$\nu:\mathbb{Z}^{2}\to\\{\pm 1\\}$ be some function. We define

$P_{t}(z,w)=\sum\varepsilon(i,j)t^{-b_{i,j}}z^{i}w^{j},\quad
V_{\infty}^{0}=\lim_{t\to+\infty}H_{t}\left(\\{P_{t}=0\\}\right),$

and

$R_{t}(z,w)=\sum\nu(i,j)t^{-b_{i,j}}z^{i}w^{j},\quad
V_{\infty}=\lim_{t\to+\infty}H_{t}\left(\\{R_{t}=0\\}\right).$

We equipped both $V_{\infty}$ and $V_{\infty}^{0}$ with the real structure
coming from the restriction of the complex conjugation on
$(\mathbb{C}^{\times})^{2}$. Compactifying $C$ by gluing a point to each
unbounded edge $C$, we obtain a graph $\Gamma$. The map
${\text{Log}}:V_{\infty}^{0}\to C$ induces a pair of pants decomposition of
the topological surface $V_{\infty}^{0}$. To each unbounded edge of $C$
corresponds an unbounded cylinder of $V_{\infty}^{0}$. A surface $S_{\Gamma}$
as in Section 2.A is obtained by gluing a disk to each such unbounded cylinder
of $V_{\infty}^{0}$, and we have the identities

$S_{v}={\text{Log}}^{-1}(v)\cap V^{0}_{\infty}\ \ \forall
v\in\text{Vert}(C),\quad\mbox{and}\quad S_{e}={\text{Log}}^{-1}(e)\cap
V^{0}_{\infty}\ \ \forall e\in\text{Edge}(C),$

up to considering degenerate pairs of pants for $S_{v}$ instead of usual ones
in the construction of $S_{\Gamma}$ in Section 2.A. All previous definitions
and results are easily seen to hold with this benign substitution.
Furthermore, the real structure on $V_{\infty}^{0}$ induces a real structure
$\tau_{\Gamma,1}$ above the real graph $(\Gamma,\mathop{\mathrm{Id}})$. Since
$C$ in non-singular, for any vertex $v$ of $C$ there exists
$(\mu_{1},\mu_{2})\in\\{\pm 1\\}^{2}$ such that

$R_{t}^{\Delta_{v}}(\mu_{1}z,\mu_{2}w)=\pm P_{t}^{\Delta_{v}}(z,w).$

In particular, the map $s_{v}:(z,w)\mapsto(\mu_{1}z,\mu_{2}w)$ induces a real
homeomorphism (i.e. commuting with real structures)

$\zeta_{v}:{\text{Log}}^{-1}(v)\cap V_{\infty}\to S_{v}.$

Let $\zeta:V_{\infty}\to V_{\infty}^{0}$ be a (not necessarily real)
homeomorphism restricting to $\zeta_{v}$ on ${\text{Log}}^{-1}(v)\cap
V_{\infty}$ for each vertex $v$ of $C$. Note that $\zeta$ is well defined up
to isotopy and conjugation by a finite product of Dehn twists along cylinders
$S_{e}$ with $e\in\text{Edge}^{0}(\Gamma)$. Furthermore, given an edge
$e\in\text{Edge}^{0}(\Gamma)$ adjacent to two vertices $v$ and $v^{\prime}$,
the two maps $s_{v}$ and $s_{v^{\prime}}$ coincide if and only if, up to an
isotopy restricting on the identity on $\partial S_{e}$, the restriction of
$\zeta$ to ${\text{Log}}^{-1}(e)\cap V_{\infty}$ is a real map. The real
structure on $V_{\infty}$ induces, via the map $\zeta$, a real structure on
$V_{\infty}^{0}$, which in its turn induces a real structure $\tau_{\Gamma,2}$
above the real graph $(\Gamma,\mathop{\mathrm{Id}})$. Let $T$ be the set of
edges for which
$\zeta\raisebox{-3.87495pt}{$\big{|}$}_{{}^{{\text{Log}}^{-1}(e)\cap
V_{\infty}}}$ is not a real map. By Lemma 2.9, we have

$\tau_{\Gamma,2}-\tau_{\Gamma,1}=\sum_{e\in T}e.$

On the other hand, by [BIMS15, Theorem 3.4 and Remark 3.9], the set $T$ is
twist-admissible and the real part of the algebraic curve in
$(\mathbb{C}^{\times})^{2}$ with equation $R_{t}(z,w)=0$ is isotopic in
$(\mathbb{R}^{\times})^{2}$ to the curve $C_{T}$ for $t$ large enough. Hence
by Corollary 3.4 and Proposition 3.5 applied to $\tau_{\Gamma,1}$ and
$\tau_{\Gamma,2}$, the real algebraic curve in $R_{t}(z,w)=0$ for $t$ large
enough is maximal if and only if $T$ is maximal. To finish the proof of Haas’
Theorem, it is enough to notice that for any set $T$ of twist-admissible edges
of $C$, there exists a function $\nu:\mathbb{Z}^{2}\to\\{\pm 1\\}$ as above
such that the real part of the algebraic curve in $(\mathbb{C}^{\times})^{2}$
with equation

$\sum\nu(i,j)t^{-b_{i,j}}z^{i}w^{j}=0$

is isotopic in $(\mathbb{R}^{\times})^{2}$ to the curve $C_{T}$ for $t$ large
enough, see [BIMS15, Remark 3.9]. $\Box$

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