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# Topology of real algebraic curves near the non-singular tropical limit

Cédric Le Texier  Universitetet i Oslo, Norway<EMAIL_ADDRESS>
###### Abstract.

In the 1990’s, Itenberg and Haas studied the relations between combinatorial
data in Viro’s patchworking and the topology of the resulting non-singular
real algebraic curves in the projective plane. Using recent results from
Renaudineau and Shaw on real algebraic curves near the non-singular tropical
limit, we continue the study of Itenberg and Haas inside any non-singular
projective toric surface. We give new Haas’ like criteria for patchworking
non-singular real algebraic curves with prescribed number of connected
components, in terms of twisted edges on a non-singular tropical curve. We
then obtain several sufficient conditions for patchworking non-singular real
algebraic curves with only ovals in their real part. One of these sufficient
conditions does not depend on the ambient toric surface. In that case, we
count the number of even and odd ovals of those curves in terms of the dual
subdivision, and construct some new counter-examples to Ragsdale’s conjecture.

###### 2010 Mathematics Subject Classification:

Primary: 14T05, 14P25

###### Contents

  1. 1 Introduction
  2. 2 Tropical curves
    1. 2.1 Tropical curves in ${\mathbb{R}}^{2}$
    2. 2.2 Dual Subdivision
    3. 2.3 Tropical curves as limit of amoebas
    4. 2.4 Tropical curve in a tropical toric surface
  3. 3 Real part of a non-singular tropical curve
    1. 3.1 Combinatorial patchworking
    2. 3.2 Twisted edges on a tropical curve
    3. 3.3 Correspondence between signs and twisted edges
    4. 3.4 Real components and twisted cycles
  4. 4 Prescribed number of real components
    1. 4.1 Exact sequence for real tropical curves
    2. 4.2 Haas conditions for dividing and maximal curves
    3. 4.3 (M-1) curves
    4. 4.4 Dividing (M-2) curves
  5. 5 Ovals in the real part of a tropical curve
    1. 5.1 Partition of dividing configuration of twists
    2. 5.2 Real tropical curves with only ovals
  6. 6 Even and odd ovals
    1. 6.1 Counting in terms of distribution of signs
    2. 6.2 Even circuits
    3. 6.3 Even-free circuits
    4. 6.4 Even-free complete bipartite graphs
    5. 6.5 Counter-examples to Ragsdale conjecture

##  1\. Introduction

The aim of this article is to study the topology of real algebraic curves near
the non-singular tropical limit, in terms of the induced combinatorial data on
the tropical limit, continuing the work started by Itenberg [Ite95] and Haas
[Haa97]. A _non-singular real algebraic curve_ $\mathcal{C}$ in a non-singular
projective toric surface $\mathbb{P}_{\Sigma}$ is a non-singular algebraic
curve with set of real points
$\mathcal{C}({\mathbb{R}})=\mathcal{C}\cap\mathbb{P}_{\Sigma}({\mathbb{R}})$,
for $\mathbb{P}_{\Sigma}({\mathbb{R}})$ non-empty defined by the construction
from Gelfand, Kapranov and Zelevinsky [GKZ14, Theorem 11.5.4].

One of the first results motivating the study of the topology of non-singular
real algebraic curves is the _Harnack-Klein inequality_ [Har76], [Kle73],
which states that the number of connected components of the real part
$\mathcal{C}({\mathbb{R}})$ of a non-singular real algebraic curve
$\mathcal{C}$ of genus $g$ is less than or equal to $g+1$. Moreover, Harnack
gave a construction of non-singular real algebraic curves in $\mathbb{P}^{2}$
reaching this bound for any degree $d$ [Har76].

######  Definition 1.1.

Let $\mathcal{C}$ be a non-singular real algebraic curve of genus $g$. We say
that $\mathcal{C}$ is a _maximal_ curve (or _$M$ -curve_) if its real part
$\mathcal{C}({\mathbb{R}})$ has $g+1$ connected components. More generally, we
say that $\mathcal{C}$ is a _$(M-r)$ -curve_ if its real part
$\mathcal{C}({\mathbb{R}})$ has $g+1-r$ connected components.

Haas ([Haa97]) gave a necessary and sufficient criterion for a real algebraic
curve near the non-singular tropical limit to be maximal, using Viro’s
combinatorial patchworking [Vir80], [Vir84], [Vir01]. This criterion can be
stated in terms of _twisted edges_ on a tropical curve (see Definition 3.9 and
Figure 3(c)). Renaudineau and Shaw ([RS21]) introduced a tool allowing to
compute the number of connected components of a real algebraic curve near the
non-singular tropical limit in terms of twisted edges on the tropical limit,
and recovered Haas result. Using these results, we introduce a new criterion
for a real algebraic curve near the non-singular tropical limit to be a
$(M-1)$-curve. In the following, a _primitive cycle_ is a cycle on a tropical
curve bounding a connected component of the complement, and an _exposed edge_
is an edge of a tropical curve lying in the boundary of an unbounded connected
component of the complement.

######  Theorem 1.2 (Theorem 4.16).

Let $C$ be a non-singular tropical curve in a non-singular projective tropical
toric surface ${\mathbb{T}}\mathbb{P}_{\Sigma}$, and let $T$ be an admissible
set of twisted edges on $C$. A real algebraic curve
$\mathcal{C}\subset\mathbb{P}_{\Sigma}$ near the non-singular tropical limit
$C$ and inducing the set of twisted edges $T$ is a $(M-1)$-curve if and only
if the dual subdivision $\Delta_{C}$ of $C$ contains a complete subgraph
$K_{n}$ on $1\leq n\leq 4$ vertices satisfying the following conditions.

  1. (1)

The non-exposed twisted edges are those dual to the edges of $K_{n}$.

  2. (2)

Every primitive cycle $\gamma$ in $C$ dual to a vertex of $K_{n}$ has an odd
number of twisted edges.

  3. (3)

Every primitive cycle $\gamma$ in $C$ not dual to a vertex of $K_{n}$ has an
even number of twisted edges.

We can also use Theorem 1.2 in order to construct many examples of $(M-r)$
curves for any $r\leq g$, see Corollary 4.21. Indeed, let $C$ be a non-
singular tropical curve given as the gluing of $r$ non-singular tropical
curves $C_{1},\ldots,C_{r}$ at some unbounded edges, such that the set of
twisted edges $T_{i}$ on each $C_{i}$ satisfy Theorem 1.2. Then a real
algebraic curve $\mathcal{C}$ near the non-singular tropical limit $C$ with
set of twisted edges $T_{1}\sqcup\ldots\sqcup T_{r}$ will be a $(M-r)$ curve.
However, we would not be able to construct _dividing curves_ by this process.

######  Definition 1.3.

A non-singular real algebraic curve $\mathcal{C}$ is said to be _dividing_ if
$\mathcal{C}({\mathbb{C}})\backslash\mathcal{C}({\mathbb{R}})$ is
disconnected.

Since the genus of a Riemann surface $S$ is given as the maximal number of
non-intersecting loops on $S$ which does not disconnect $S$, every maximal
curve is dividing. Furthermore, every dividing curve of genus $g$ is a
$(M-2s)$-curve, for $0\leq s\leq\left\lfloor\frac{g}{2}\right\rfloor$. In the
process of proving the criterion for maximal curves, Haas obtained a necessary
and sufficient condition to construct dividing curves near the non-singular
tropical limit [Haa97], again via Viro’s combinatorial patchworking. Using
Haas dividing criterion in addition to the results of [RS21], we give a new
criterion for a real algebraic curve near the non-singular tropical limit to
be a $(M-2)$-dividing curve, in terms of twisted edges on the tropical limit.

######  Theorem 1.4 (Theorem 4.22).

Let $C$ be a non-singular tropical curve in a non-singular projective tropical
toric surface ${\mathbb{T}}\mathbb{P}_{\Sigma}$, and let $T$ be an admissible
dividing set of twisted edges on $C$. A real algebraic curve
$\mathcal{C}\subset\mathbb{P}_{\Sigma}$ near the non-singular tropical limit
$C$ and inducing the set of twisted edges $T$ is a dividing $(M-2)$ curve if
and only if the subgraph of the dual subdivision $\Delta_{C}$ dual to the set
of non-exposed twisted edges is either a complete planar bipartite or a
complete planar tripartite graph.

We can also use Theorem 1.4 in order to construct many examples of dividing
$(M-2s)$ curves for any $s\leq\left\lfloor\frac{g}{2}\right\rfloor$, see
Corollary 4.25, in a similar way as we construct $(M-r)$ curves via Theorem
1.2. The next natural question to ask would be about the possible arrangements
of connected components in the real part of real algebraic curves near the
non-singular tropical limit.

In its 16th problem, Hilbert [Hil00] asked what are the realisable _isotopy
types_ for a non-singular real algebraic curve $\mathcal{C}$ of degree 6 in
the projective plane $\mathbb{P}^{2}$, or said otherwise the realisable
homeomorphism types for the pair
$(\mathbb{P}^{2}({\mathbb{R}}),\mathcal{C}({\mathbb{R}}))$. This problem and
its generalisation to higher degrees has been extensively studied during the
20th century [Hil91], [Rag06], [Pet38], [Gud69], [Rok78], [Vir80], [Ite93],
[Ore02]. In the projective plane case, the classification problem reduces to
the realisable arrangements of _ovals_ and _pseudo-lines_.

######  Definition 1.5.

Let $\mathcal{C}$ be a non-singular real algebraic curve in a non-singular
projective toric surface $\mathbb{P}_{\Sigma}$. A connected component $C_{0}$
of $\mathcal{C}({\mathbb{R}})$ is called an _oval_ if $C_{0}$ divides
$\mathbb{P}_{\Sigma}({\mathbb{R}})$ into two connected components. We say that
an oval $C_{0}$ of $\mathcal{C}({\mathbb{R}})$ is _even_ if $C_{0}$ lies
inside an even number of ovals of $\mathcal{C}({\mathbb{R}})$, and we say that
$C_{0}$ is _odd_ otherwise.

For $\mathcal{C}$ a non-singular real algebraic curve of even degree in
$\mathbb{P}^{2}$, the connected components of the real part
$\mathcal{C}({\mathbb{R}})$ are all ovals. In general, for
$\mathbb{P}_{\Sigma}$ a non-singular projective toric surface distinct from
$\mathbb{P}^{2}$, we need stronger restrictions in order to have only ovals in
the real part of a non-singular real algebraic curve. For
$\mathcal{C}\subset\mathbb{P}_{\Sigma}$ a real algebraic curve near the non-
singular tropical limit $C$ with set of twisted edges $T$, we obtain that if
every twisted edge in $T$ is non-exposed, the real part
$\mathcal{C}({\mathbb{R}})$ consists only of ovals, see Proposition 5.10. In
particular, that condition does not depend on the ambient non-singular
projective toric surface $\mathbb{P}_{\Sigma}$.

In the dividing case, we can partition a set of twisted edges into _multi-
bridges_ , see Definition 5.1 and Proposition 5.2. A multi-bridge $B$ on a
non-singular tropical curve $C$ is a twist-admissible set of bounded edges of
$C$ which are parallel modulo 2, such that the graph $C\backslash B$ is
disconnected and for every proper subset $B^{\prime}\subset B$, the graph
$C\backslash B$ is connected. These multi-bridges form a spanning family of
the ${\mathbb{Z}}_{2}$-vector space $\operatorname{Div}(C)$ of dividing
configuration of twists, Proposition 5.2. If we restrict to configuration of
twists with only non-exposed twisted edges, we obtain a
${\mathbb{Z}}_{2}$-vector subspace
$\operatorname{Circuit}(C)\subset\operatorname{Div}(C)$ generated by
_circuits_ (see Definition 5.1), which are multi-bridges with only non-exposed
edges. An element $T$ of $\operatorname{Circuit}(C)$ satisfies Proposition
5.10, hence we want to count the number of even and odd ovals of real
algebraic curves near the non-singular tropical limit $C$ with set of twisted
edges belonging to $\operatorname{Circuit}(C)$.

Whenever a partition of a set of twisted edges $T$ contains a circuit
$T^{\prime}$ such that its dual $(T^{\prime})^{\vee}\subset\Delta_{C}$
contains an integer point of the dual subdivision $\Delta_{C}$ of $C$ with
both coordinates even, then untwisting the edges of $T^{\prime}$ cannot
decrease the number of even and odd ovals of the real part, see Proposition
6.6. A similar statement in the maximal case was shown by Haas [Haa97, Theorem
10.6.0.5]. In order to construct non-singular real algebraic curves with many
even or many odd ovals, we can then restrict to even-free configurations of
twists. A configuration of twists $T\in\operatorname{Circuit}(C)$ is _even-
free_ if its the dual set of edges $T^{\vee}$ does not contain any integer
point with both coordinates even in $\Delta_{C}\cap{\mathbb{Z}}^{2}$, for
$\Delta$ the Newton polygon of $C$. Note that the even-free circuits generate
again a ${\mathbb{Z}}_{2}$-vector subspace
$\operatorname{EvFreeCirc}(C)\subset\operatorname{Circuit}(C)$.

Itenberg [Ite95] introduced a way to count the number of even and odd ovals of
maximal curves of even degree in $\mathbb{P}^{2}$ near the non-singular
tropical limit, whenever the induced set of twisted edges on the tropical
limit $C$ can be partitioned into even-free _bridges_ , that is multi-bridges
with a single edge of $C$. Haas [Haa97] extended this result to all maximal
curves near the non-singular tropical limit, hence whenever the induced set of
twisted edges can be partitioned into even-free bridges and even-free _double-
bridges_ , that is multi-bridges with exactly two edges of $C$. Note that
Itenberg and Haas’ counts are expressed in terms of _zone decomposition_ of a
Newton polygon (see Definition 6.7), that is a decomposition of the Newton
polygon induced by the set of edges dual to the set of twisted edges. In the
maximal case, these zone decompositions are induced by set of edges dual to
bridges and double-bridges. We extend those counts first to a partial count of
even and odd empty ovals in the case when the tropical limit has set of
twisted edges $T$ belonging to $\operatorname{EvFreeCirc}(C)$, see Proposition
6.11.

In order to control easily the number of connected components, we want to use
a configuration of twists in $\operatorname{EvFreeCirc}(C)$ satisfying
Corollary 4.25. Such a configuration of twists $T$ is dual to a disjoint union
of complete bipartite graphs of the form $K_{2,2l}$ and complete tripartite
graphs $K_{2,2,2}$, see Lemma 6.13. We restrict to disjoint unions of complete
bipartite graphs of the form $K_{2,2l}$ and obtain the following count of even
and odd ovals.

######  Theorem 1.6 (Theorem 6.15).

Let $C$ be a non-singular tropical curve in ${\mathbb{T}}\mathbb{P}_{\Sigma}$
such that its Newton polygon $\Delta$ (dual to $\Sigma$) has every edge of
even lattice length. Let $T:=T_{1}\sqcup\ldots\sqcup
T_{s}\in\operatorname{EvFreeCirc}(C)$ be such that all
$(T_{i})^{\vee}\subset\Delta_{C}$ are disjoints and each $(T_{i})^{\vee}$ is a
complete bipartite graph of the form $K_{2,2l_{i}}$. Let
$Y_{T}:=Y_{1}^{T}\sqcup Y_{0}^{T}$ be the zone partition (see Definition 6.7)
of $\Delta$ induced by $T$ such that $Y_{1}^{T}$ contains the unique zone
$Z^{T}$ meeting the boundary of $\Delta$. Let $p_{j}$ and $n_{j}$ be the
number of even and odd interior integer points in $Y_{j}^{T}$. Then for
$\mathcal{C}\subset\mathbb{P}_{\Sigma}$ a real algebraic curve near the non-
singular tropical limit $C$ with set of twisted edges $T$, the numbers $p$ of
even ovals and $n$ of odd ovals of $\mathcal{C}({\mathbb{R}})$ are

$\displaystyle p$
$\displaystyle=n_{1}+p_{0}+1+\sum\limits_{i=1}^{s}(l_{i}+1),$ $\displaystyle
n$ $\displaystyle=p_{1}+n_{0}+\sum\limits_{i=1}^{s}(l_{i}-1).$

Let $R(k)=\frac{3k^{2}-3k}{2}$. Ragsdale [Rag06] made the conjecture that a
non-singular real algebraic curve $\mathcal{C}$ of degree $2k$ in
$\mathbb{P}^{2}$ has $p\leq R(k)+1$ even ovals and $n\leq R(k)$ odd ovals.
Petrowsky [Pet38] proved that

$|p-n|\leq R(k)+1$

and that there exist some curves reaching this bound for any degree $2k$. He
then conjectured that $p\leq R(k)+1$ and $n\leq R(k)+1$. Viro disproved
Ragsdale’s conjecture (but not Petrowsky’s one) by constructing maximal curves
of degree 8 with $n=R(k)+1$ [Vir80], and Itenberg disproved both Ragsdale and
Petrowsky’s conjectures by constructing via combinatorial patchworking curves
with

$p=R(k)+1+\left\lfloor\frac{(k-3)^{2}+4}{8}\right\rfloor\text{ and
}n=R(k)+\left\lfloor\frac{(k-3)^{2}+4}{8}\right\rfloor.$

Haas [Haa95] gave another construction of counter-examples, again via
combinatorial patchworking, giving rise to

$p=R(k)+1+\left\lfloor\frac{k^{2}-7k+16}{6}\right\rfloor\text{ and
}n=R(k)+\left\lfloor\frac{k^{2}-7k+16}{6}\right\rfloor.$

Brugallé [B+06] constructed a family of non-singular real algebraic curves,
with increasing degree $2k$, such that the family has asymptotically maximal
number of even ovals with respect to the Harnack-Klein and Petrowsky
inequalities, that is

$\lim\limits_{k\rightarrow+\infty}\frac{p}{k^{2}}=\frac{7}{4}.$

Moreover, Renaudineau gave a tropical construction of a family with
asymptotically maximal number of even ovals [Ren17]. Hence one of the
remaining open problem is to determine the maximal number of even ovals
constructible in low degree. Using Theorem 1.6, we construct some new counter-
examples to Ragsdale conjecture.

######  Theorem 1.7 (Theorem 6.19).

There exists a dividing $(M-2\left\lfloor\frac{k-3}{2}\right\rfloor)$ non-
singular real algebraic curve of degree $2k\geq 10$ in $\mathbb{P}^{2}$ with

$R(k)+1+\frac{k^{2}-5k+s(k)}{6}$

even ovals, with $s(k)$ determined by the value of $k$ modulo 6 as follows.

  * •

If $k=0\mod 6$, then $s(k)=0$.

  * •

If $k=1\mod 6$, then $s(k)=10$.

  * •

If $k=2\mod 6$, then $s(k)=8$.

  * •

If $k=3,5\mod 6$, then $s(k)=6$.

  * •

If $k=4\mod 6$, then $s(k)=4$.

For instance, we construct in Example 6.20 a non-singular real algebraic curve
of degree 14 in $\mathbb{P}^{2}$ with 68 even ovals, hence 4 even ovals more
than Ragsdale’s conjecture (see also Figure 11(a) for the construction and
Figure 11(b) for the isotopy type of the real part). As an intermediate step
for the general construction, we construct in Lemma 6.16 a dividing $(M-2)$
non-singular real algebraic curve of degree $2k\geq 10$ in $\mathbb{P}^{2}$
with more even ovals than Ragsdale’s conjecture. As we see in Example 6.18, we
obtain in degree 14 a dividing $(M-2)$ non-singular real algebraic curve with
67 even ovals, hence 3 even ovals more than Ragsdale’s conjecture (see also
Figure 10(b) for the construction).

The article will be organised as follows. We begin by recalling standard
notions about _tropical curves_ in Section 2. We define then in Section 3 the
real part of a tropical curve in terms of _Viro’s combinatorial patchworking_
, and introduce the notion of _twisted cycle_ on a non-singular tropical curve
with twisted edges, which allows us to recover the data of a connected
component of the real part directly on the tropical curve. In Section 4, we
prove Theorem 1.2 and Theorem 1.4 using a reformulation of the results of
Renaudineau and Shaw [RS21] in the case of curves (see Theorem 4.3 and Theorem
4.5). In Section 5.2, we give sufficient conditions for a non-singular real
algebraic curve near the tropical limit to have only ovals in its real part
(Proposition 5.9 , Proposition 5.10). Finally, in Section 6, we prove the
count of even and odd ovals from Theorem 1.6, justifying first the choice of
restrictions to the configurations of twists we consider (Proposition 6.6,
Lemma 6.13), and then we use the count to construct new counter-examples to
Ragsdale’s conjecture (Theorem 1.7).

Acknowledgements. The author would like to thank Kris Shaw and Frédéric Bihan
for their interest, support and useful discussions in this project. The
research of the author is supported by the Trond Mohn Stiftelse (TMS) project
“Algebraic and topological cycles in complex and tropical geometry”.

##  2\. Tropical curves

We recall in this section several definitions in tropical geometry. For more
details and examples, one can read [BIMS15, Section 2], [IMS09, Chapter 1],
[MS09].

###  2.1. Tropical curves in ${\mathbb{R}}^{2}$

A _tropical polynomial in two variables_ is

$P(x,y)=\max\limits_{(i,j)\in A}(a_{i,j}+ix+jy),$

where $A$ is a finite subset of $({\mathbb{Z}}_{\geq 0})^{2}$ and the
coefficients $a_{i,j}$ are in the tropical semi-ring
${\mathbb{T}}=({\mathbb{R}}\cup\\{-\infty\\},\max,+)$. Note that we use the
$``\max"$ operation, while some references in the literature use the $``\min"$
operation instead. Thus, a tropical polynomial is a convex piecewise affine
function, with corner locus

$V_{P}=\\{(x_{0},y_{0})\in{\mathbb{R}}^{2}~{}|~{}\exists(i,j)\neq(k,l),~{}a_{i,j}+ix_{0}+jy_{0}=a_{k,l}+kx_{0}+ly_{0}\\}.$

The set $V_{P}$ is a 1-dimensional polyhedral complex in ${\mathbb{R}}^{2}$.

######  Definition 2.1.

The _weight function_ is defined on the edges $e$ of $V_{P}$ as

$w(e):=\max\limits_{(i,j),(k,l)}(\gcd(|i-k|,|j-l|))$

for all pairs $(i,j)$ and $(k,l)$ such that the value of $P(x,y)$ on $e$ is
given by the corresponding monomials. The _tropical curve_
$C\subset{\mathbb{R}}^{2}$ defined by $P(x,y)$ is the 1-dimensional polyhedral
complex $V_{P}$ equipped with the weight function $w$ on the edges.

###  2.2. Dual Subdivision

Let $P(x,y)=\max\limits_{i,j\in A}(a_{i,j}+ix+jy)$ be a tropical polynomial
defining a tropical curve $C$. The _Newton polygon_ of $P(x,y)$, denoted by
$\Delta(P)$, is the convex lattice polygon given by

$\Delta(P):=\operatorname{Conv}\\{(i,j)\in({\mathbb{Z}}_{\geq
0})^{2}~{}|~{}a_{i,j}\neq-\infty\\}\subset{\mathbb{R}}^{2}.$

The tropical polynomial $P$ also determines a subdivision of $\Delta(P)$, as
follows. Given $(x_{0},y_{0})\in{\mathbb{R}}^{2}$, let

$\Delta(x_{0},y_{0}):=\operatorname{Conv}\\{(i,j)\in({\mathbb{Z}}_{\geq
0})^{2}~{}|~{}P(x_{0},y_{0})=a_{i,j}+ix_{0}+jy_{0}\\}\subset\Delta(P).$

The tropical curve $C$ induces a polyhedral decomposition of
${\mathbb{R}}^{2}$, and the polytope $\Delta(x_{0},y_{0})$ only depends on the
cell $F\ni(x_{0},y_{0})$ of the decomposition given by $C$. Thus, we define
the face $F^{\vee}:=\Delta(x_{0},y_{0})$ dual to the face $F$ of the
polyhedral subdivision of ${\mathbb{R}}^{2}$ induced by $C$, for
$(x_{0},y_{0})\in F$. The _dual subdivision_ $\Delta_{C}$ of a non-singular
tropical curve $C$ is the union

$\bigcup_{F}F^{\vee},$

for $F$ running through the faces of the decomposition of ${\mathbb{R}}^{2}$
induced by $C$.

######  Definition 2.2.

We say that a tropical curve $C$ is _non-singular_ if its dual subdivision
$\Delta_{C}$ is a triangulation with each 2-dimensional cell of Euclidean area
$1/2$.

Equivalently, a tropical curve $C\subset{\mathbb{R}}^{2}$ with Newton polytope
$\Delta$ is non-singular if and only if $C$ has $2\operatorname{Area}(\Delta)$
vertices. In particular, if a tropical curve $C\subset{\mathbb{R}}^{2}$ is
non-singular, every vertex of $C$ is 3-valent and every edge $e$ of $C$ has
weight $w(e)=1$, and all integer points of $\Delta_{C}$ are vertices in the
subdivision.

###  2.3. Tropical curves as limit of amoebas

We want to see that any tropical curve can be approximated by algebraic curves
in $({\mathbb{C}}^{\times})^{2}$, via the logarithm map. Consider the
following map, where $t\in{\mathbb{R}}_{>0}$:

$\displaystyle\operatorname{Log}_{t}:({\mathbb{C}}^{\times})^{2}$
$\displaystyle\rightarrow{\mathbb{R}}^{2}$ $\displaystyle(z,w)$
$\displaystyle\mapsto(\log_{t}|z|,\log_{t}|w|).$

We call the image of a subset $V\subset({\mathbb{C}}^{\times})^{2}$ via
$\operatorname{Log}_{t}$ the _amoeba_ of $V$ in base $t$ (see for instance
[GKZ14]).

######  Theorem 2.3 ([Kap00]).

Let $\mathcal{P}_{t}(z,w)=\sum_{i,j}\alpha_{i,j}(t)z^{i}w^{j}$ be a polynomial
whose coefficients are functions
$\alpha_{i,j}:{\mathbb{R}}\rightarrow{\mathbb{C}}$, and suppose that
$\alpha_{i,j}(t)\sim\beta_{i,j}t^{-a_{i,j}}$ when $t\in{\mathbb{R}}_{>0}$
tends to $0$, with $\beta_{i,j}\in{\mathbb{C}}^{\times}$ and
$a_{i,j}\in{\mathbb{T}}$. If $\mathcal{C}_{t}$ denotes the curve in
$({\mathbb{C}}^{\times})^{2}$ defined by the polynomial $\mathcal{P}_{t}$,
then the amoebas $\operatorname{Log}_{t}(\mathcal{C}_{t})$ converge to the
tropical curve defined by the tropical polynomial
$P(x,y)=\max_{i,j}(a_{i,j}+ix+jy)$ when $t\in{\mathbb{R}}_{>0}$ tends to $0$.

###  2.4. Tropical curve in a tropical toric surface

We refer to [MS09] and [MR09] for more details on tropical toric varieties.
Let $\Sigma$ be a fan in ${\mathbb{R}}^{2}$, and let $\sigma$ be a cone of
$\Sigma$. We denote
$U_{\sigma}:={\mathbb{T}}^{\dim\sigma}\times{\mathbb{R}}^{2-\dim\sigma}$. For
$\tau$ a face of $\sigma$, we denote by $\phi_{\sigma,\tau}$ the embedding
$U_{\tau}\rightarrow U_{\sigma}$ described in [MR09], and let
$U_{\sigma}^{\tau}:=\phi_{\sigma,\tau}(U_{\tau})$.

######  Definition 2.4 ([MR09]).

Let $\Sigma$ be a complete unimodular pointed fan in ${\mathbb{R}}^{2}$. The
_non-singular projective tropical toric surface_ associated to $\Sigma$ is the
topological space

${\mathbb{T}}\mathbb{P}_{\Sigma}:=\bigcup\limits_{\sigma\in\Sigma}U_{\sigma}/\sim$

stratified by the fan $\Sigma$, where $\sim$ is given by
$x\sim\phi_{\sigma,\sigma^{\prime}}(x)$ for all
$\sigma,\sigma^{\prime}\in\Sigma$ and all $x\in U_{\sigma^{\prime}}^{\tau}$.

The stratification on ${\mathbb{T}}\mathbb{P}_{\Sigma}$ is compatible with the
stratification on the usual non-singular projective toric variety
$\mathbb{P}_{\Sigma}$ defined by the fan $\Sigma$. In particular, if $\Delta$
is the 2-dimensional polytope dual to the complete unimodular pointed fan
$\Sigma$, there is a one-to-one correspondence between the $n$-dimensional
strata of ${\mathbb{T}}\mathbb{P}_{\Sigma}$ and the $n$-dimensional strata of
$\Delta$, for $0\leq n\leq 2$. The tropical toric surface
${\mathbb{T}}\mathbb{P}_{\Sigma}$ can be seen as a compactification of the
tropical algebraic torus $({\mathbb{T}}^{\times})^{2}={\mathbb{R}}^{2}$, and
has maps to affine charts ${\mathbb{T}}^{2}$.

######  Definition 2.5.

Let $C^{\prime}$ be a non-singular tropical curve in ${\mathbb{R}}^{2}$ with
Newton polygon $\Delta$. Let ${\mathbb{T}}\mathbb{P}_{\Sigma}$ be the non-
singular projective tropical toric surface such that $\Sigma$ is the dual fan
of $\Delta$. We extend $C^{\prime}$ to a _non-singular tropical curve_ $C$ in
${\mathbb{T}}\mathbb{P}_{\Sigma}$ by compactification of the unbounded edges
of $C^{\prime}$ (in particular, the 1-valent vertices created by the
compactification lie on the 1-dimensional strata of
${\mathbb{T}}\mathbb{P}_{\Sigma}$).

The choice of compactification in Definition 2.5 allows us to work with non-
singular tropical curves $C$ _combinatorially ample_ in a non-singular
projective tropical toric surface ${\mathbb{T}}\mathbb{P}_{\Sigma}$, see
[ARS19, Definition 2.3].

(a) Tropical line in ${\mathbb{T}}\mathbb{P}^{2}$.

(b) Tropical curve of bidegree $(1,1)$ in
${\mathbb{T}}\mathbb{P}^{1}\times{\mathbb{T}}\mathbb{P}^{1}$.

Figure 1. Example 2.6.

######  Example 2.6.

Figure 1(a) represents a tropical line $L$ inside the tropical projective
plane ${\mathbb{T}}\mathbb{P}^{2}$. Similarly, we have on Figure 1(b) a non-
singular tropical curve of bidegree $(1,1)$ on
${\mathbb{T}}\mathbb{P}^{1}\times{\mathbb{T}}\mathbb{P}^{1}$.

######  Definition 2.7.

Let $C^{\prime}$ be a non-singular tropical curve in ${\mathbb{R}}^{2}$, and
let $C$ be its compactification inside the non-singular projective tropical
toric surface ${\mathbb{T}}\mathbb{P}_{\Sigma}$, for $\Sigma$ the fan dual to
the Newton polygon of $C$. We say that an edge $e$ of $C$ is _bounded_ if $e$
is bounded in $C^{\prime}$, and _unbounded_ if $e$ is unbounded in
$C^{\prime}$. The set of all edges of $C$ is denoted $\operatorname{Edge}(C)$,
and the subset of bounded edges is denoted $\operatorname{Edge}^{0}(C)$.

##  3\. Real part of a non-singular tropical curve

Given a non-singular tropical curve $C$ and some additional data, we want to
construct the _real part_ of $C$ with respect to the additional data. One way
to do this is via the combinatorial version of _Viro’s patchworking_ [Vir80].
We will see that in the case of tropical curves, an equivalent way is given in
terms of _amoeba_ and _twisted edges_ , as seen in [BIMS15]. We will then see
that a set of twisted edges on a tropical curve is equivalent, up to an
element of ${\mathbb{Z}}_{2}^{2}$, to the necessary data for combinatorial
patchworking.

###  3.1. Combinatorial patchworking

Let $\mathcal{R}^{2}$ be the group of symmetries in ${\mathbb{R}}^{2}$
generated by the reflections with respect to the coordinate hyperplanes. For
${\mathbb{Z}}_{2}:={\mathbb{Z}}/2{\mathbb{Z}}$, we identify $\mathcal{R}^{2}$
with ${\mathbb{Z}}_{2}^{2}$ as follows. If
$\varepsilon=(\varepsilon_{1},\varepsilon_{2})\in{\mathbb{Z}}_{2}^{2}$, then
$\varepsilon\in\mathcal{R}^{2}$ is the symmetry defined by

$\varepsilon(x,y)=((-1)^{\varepsilon_{1}}x,(-1)^{\varepsilon_{2}}y),$

for $\varepsilon_{1}$ and $\varepsilon_{2}$ seen as integers. Throughout the
article, for $A$ a topological set, we will denote by $A^{*}$ the disjoint
union of symmetric copies

$A^{*}:=\bigsqcup\limits_{\varepsilon\in\mathcal{R}^{2}}\varepsilon(A).$

######  Theorem 3.1 ([GKZ14]).

Let $\mathbb{P}_{\Sigma}$ be a non-singular projective toric surface, and let
$\Delta$ be the 2-dimensional polytope dual to the fan $\Sigma$. The real part
$\mathbb{P}_{\Sigma}({\mathbb{R}})$ of $\mathbb{P}_{\Sigma}$ is homeomorphic
to the quotient of $\Delta^{*}$ under the following identifications: if
$\Gamma$ is a face of $\Delta^{*}$, then for all integer vectors
$(\alpha_{1},\alpha_{2})$ orthogonal to the primitive integer directions in
$\Gamma$, the face $\Gamma$ is identified with $\varepsilon(\Gamma)$, where
$\varepsilon=(\bar{\alpha_{1}},\bar{\alpha_{2}})$ is the class of
$(\alpha_{1},\alpha_{2})$ in $\mathcal{R}^{2}\equiv{\mathbb{Z}}_{2}^{2}$.

Let $C$ be a non-singular tropical curve in a non-singular projective tropical
toric surface ${\mathbb{T}}\mathbb{P}_{\Sigma}$, such that $C$ has Newton
polygon $\Delta$ dual to the fan $\Sigma$. Recall that we have a one-to-one
correspondence between the $n$-dimensional strata of
${\mathbb{T}}\mathbb{P}_{\Sigma}$ and the $n$-dimensional strata of $\Delta$.
We denote by ${\mathbb{R}}\mathbb{P}_{\Sigma}$ the quotient of
${\mathbb{T}}\mathbb{P}_{\Sigma}^{*}$ under the identifications of Theorem
3.1. Let $\Delta_{C}$ be the dual subdivision of $\Delta$ associated to $C$.
Let

$\delta:\Delta_{C}\cap{\mathbb{Z}}^{2}\rightarrow\\{+1,-1\\}$

be a distribution of signs on the integer points of the dual subdivision
$\Delta_{C}$ (see for example Figure 2(a)). We extend $\delta$ to a
distribution of signs on the integer points of the disjoint union of symmetric
copies $\Delta_{C}^{*}$ following the rule: for $v=(v_{1},v_{2})$ an integer
point in $\Delta_{C}^{*}\cap{\mathbb{Z}}^{2}$ and
$\varepsilon=(\varepsilon_{1},\varepsilon_{2})\in\mathcal{R}^{2}$ a symmetry,
we have

$\delta(\varepsilon(v))=(-1)^{\varepsilon_{1}v_{1}+\varepsilon_{2}v_{2}}\delta(v),$

again with $\varepsilon_{1}$ and $\varepsilon_{2}$ seen as integers (see for
example the signs on Figure 2(b)). A face of $\Delta_{C}^{*}$ is called _non-
empty_ if at least two of its vertices have opposite signs. We can now
construct the real part of $C$ with respect to $\delta$ (denoted
${\mathbb{R}}C_{\delta}$) as follows. For every edge $e$ of $C$, let
$e^{\vee}$ be its dual edge in $\Delta_{C}$. The real part
${\mathbb{R}}e_{\delta}$ of $e$ is defined as

${\mathbb{R}}e_{\delta}:=\bigcup\limits_{\varepsilon}\varepsilon(e)\subset({\mathbb{T}}\mathbb{P}_{\Sigma})^{*},$

for $\varepsilon$ running through the symmetries of $\mathcal{R}^{2}$ such
that $\varepsilon(e^{\vee})$ is non-empty.

######  Definition 3.2.

Let $C$ be a non-singular tropical curve in a non-singular projective tropical
toric surface ${\mathbb{T}}\mathbb{P}_{\Sigma}$. The _real part of_ $C$ with
respect to the distribution of signs $\delta$ is the set

${\mathbb{R}}C_{\delta}:=\overline{\bigcup\limits_{e\in\operatorname{Edge}(C)}{\mathbb{R}}e_{\delta}}\subset{\mathbb{R}}\mathbb{P}_{\Sigma}.$

######  Theorem 3.3 (Viro’s patchworking theorem [Vir01]).

Let $(\mathcal{C}_{t})_{t}$ be a family of non-singular real algebraic curves
in the non-singular projective toric surface $\mathbb{P}_{\Sigma}$ converging
to a non-singular tropical curve $C$ (in the sense of Theorem 2.3), such that
$C$ has Newton polygon dual to the fan $\Sigma$. Let $(\mathcal{P}_{t})_{t}$
be a family of polynomials defining $(\mathcal{C}_{t})_{t}$ with distribution
of signs $\delta$. For $t\in{\mathbb{R}}_{>0}$ small enough, there exists a
homeomorphism
$\mathbb{P}_{\Sigma}({\mathbb{R}})\rightarrow{\mathbb{R}}\mathbb{P}_{\Sigma}$
mapping the set of real points $\mathcal{C}_{t}({\mathbb{R}})$ onto
${\mathbb{R}}C_{\delta}$.

######  Definition 3.4.

If $(\mathcal{C}_{t})_{t}$ is a family of non-singular real algebraic curves
satisfying Theorem 3.3 for a couple $(C,\delta)$, we say that for
$t\in{\mathbb{R}}_{>0}$ small enough (in the sense of Theorem 3.3), the curve
$\mathcal{C}_{t}$ is near the tropical limit $(C,\delta)$.

Given a distribution of signs $\delta$ on the dual subdivision $\Delta_{C}$ of
a non-singular tropical curve $C$, [RBB17], [Ren17] and [RS21] define a real
structure directly on $C$. We define this real structure as follows:

######  Definition 3.5.

Let $C$ be a non-singular tropical curve. For $\delta$ a distribution of signs
on the dual subdivision $\Delta_{C}$, a _real phase structure_ on $C$ is a
collection $\mathcal{E}:=\\{\mathcal{E}_{e}\\}_{e\in\operatorname{Edge}(C)}$
of ${\mathbb{Z}}_{2}$-affine spaces defined as

$\mathcal{E}_{e}=\\{\varepsilon\in\mathcal{R}^{2}~{}|~{}\varepsilon(e)\subset{\mathbb{R}}e_{\delta}\\}.$

The couple $(C,\mathcal{E})$ is called a _real tropical curve_.

(a) Distribution of signs $\delta$ on $\Delta_{C}$.

(b) Real part ${\mathbb{R}}C_{\delta}$ on ${\mathbb{R}}\mathbb{P}^{2}$.

(c) Real phase structure on $C$ (Example 3.7).

Figure 2. Example 3.7.

######  Remark 3.6.

We can define directly a real phase structure $\mathcal{E}$ on a non-singular
tropical curve $C$ without defining first a distribution of signs on the dual
subdivision, as described in [RS21, Definition 3.1]. Such a real phase
structure $\mathcal{E}:=\\{\mathcal{E}_{e}\\}_{e}$ must satisfy the two
following properties:

  * •

for $e$ an edge of $C$ and $\overrightarrow{e}\in{\mathbb{Z}}_{2}^{2}$ its
direction modulo 2, we have
$\varepsilon+\varepsilon^{\prime}=\overrightarrow{e}$ for
$\varepsilon,\varepsilon^{\prime}$ the two elements of $\mathcal{E}_{e}$
(using the identification $\mathcal{R}^{2}\equiv{\mathbb{Z}}_{2}^{2}$);

  * •

for $v$ a 3-valent vertex of $C$ incident to an edge $e$, for any element
$\varepsilon\in\mathcal{E}_{e}$, there exists a unique edge $e^{\prime}\neq e$
such that $v$ is incident to $e^{\prime}$ and
$\varepsilon\in\mathcal{E}_{e^{\prime}}$, with $e$ and $e^{\prime}$.

By [RS21, Remark 3.8], a real phase structure defined this way determines (up
to multiplication by $(-1)$) a distribution of signs $\delta$ on the dual
subdivision $\Delta_{C}$.

######  Example 3.7.

Let $C$ be a non-singular tropical curve of degree 2 in
${\mathbb{T}}\mathbb{P}^{2}$ as pictured in Figure 2(c). The distribution of
signs $\delta$ on the dual subdivision $\Delta_{C}$ is pictured in Figure
2(a). Extending the distribution of signs to the symmetric copies of
$\Delta_{C}$, we obtain the real part ${\mathbb{R}}C_{\delta}$ pictured in
color in Figure 2(b). We can recover the corresponding real phase structure
$\mathcal{E}$ on $C$, pictured in color in Figure 2(c), by looking for each
edge $e$ of $C$ its non-empty symmetric copies
$\varepsilon(e),\varepsilon^{\prime}(e)$. Notice that the real phase structure
$\mathcal{E}$ can be constructed directly using Remark 3.6, and we recover (up
to multiplication by $(-1)$) the distribution of signs $\delta$ from the data
of edges $e$ satisfying $(0,0)\in\mathcal{E}_{e}$.

######  Notation 3.8.

From now on, we will write equivalently ${\mathbb{R}}C_{\mathcal{E}}$ or
${\mathbb{R}}C_{\delta}$ for the real part of a non-singular tropical curve
$C$, as the distribution of signs $\delta$ and $-\delta$ determine the same
real phase structure $\mathcal{E}$, and so the same real part of $C$ (we refer
again to [RS21, Remark 3.8] for more details). Similarly, we will say that a
non-singular real algebraic curve is _near the tropical limit_
$(C,\mathcal{E})$ if $\delta$ determines the real phase structure
$\mathcal{E}$.

###  3.2. Twisted edges on a tropical curve

######  Definition 3.9.

Let $(C,\mathcal{E})$ be a non-singular real tropical curve. A bounded edge
$e$ of $C$ is said to be _twisted_ if for any $\varepsilon\in\mathcal{E}_{e}$,
the two edges $e_{1},e_{2}$ of $C$ adjacent to $e$ such that
$\varepsilon\in\mathcal{E}_{e_{1}}$ and $\varepsilon\in\mathcal{E}_{e_{2}}$
lie on distinct sides of the affine line containing $e$ (see for instance
Figure 3(c)). Otherwise, the edge $e$ is said to be _non-twisted_ , and
satisfies the fact that for any $\varepsilon\in\mathcal{E}_{e}$, the two edges
$e_{1},e_{2}$ of $C$ adjacent to $e$ such that
$\varepsilon\in\mathcal{E}_{e_{1}},\varepsilon\in\mathcal{E}_{e_{2}}$ lie on
the same side of the affine line containing $e$ (see for instance Figure
3(b)).

(a) Real phase structure around a vertex.

(b) Real phase structure around a non-twisted bounded edge.

(c) Real phase structure around a twisted bounded edge.

Figure 3. Properties of real phase structure from Remark 3.6 and Definition
3.9.

We can define the twisted edges on a non-singular tropical curve in terms of
_amoebas_ , see for instance [BIMS15, Section 3.2] and [RBB17, Section 4.A].
For an edge $e$ of $C$, we denote by
$\overrightarrow{e}\in{\mathbb{Z}}^{2}_{2}$ the direction modulo 2 of $e$. Not
any subset $T$ of $\operatorname{Edge}^{0}(C)$ may arise as the set of twisted
edges. The possible configurations of twists must satisfy the following
condition.

######  Definition 3.10 ([BIMS15],[RBB17]).

Let $C$ be a non-singular tropical curve. Let $T$ be a subset of
$\operatorname{Edge}^{0}(C)$. We say that $T$ is _admissible_ , or _twist-
admissible_ , if for any cycle $\gamma$ of $C$, we have

(1) $\sum_{e\in\gamma\cap T}\overrightarrow{e}=\overrightarrow{0}\mod 2.$

We denote by $\operatorname{Adm}(C)$ the set of all twist-admissible subsets
of bounded edges $T\subset\operatorname{Edge}^{0}(C)$.

The set $\operatorname{Edge}^{0}(C)$ of all bounded edges of $C$ can be given
a ${\mathbb{Z}}_{2}$-vector space structure by identification with
${\mathbb{Z}}_{2}^{|\operatorname{Edge}^{0}(C)|}$, so that a canonical basis
is given by the vectors identified with single bounded edges of $C$.

######  Definition 3.11.

Let $T=\\{e_{1},\ldots,e_{r}\\}$ be a subset of bounded edges of $C$. The set
$T$ can be seen as the sum $\sum_{i=1}^{r}e_{i}\in\operatorname{Edge}^{0}(C)$,
with $\operatorname{Edge}^{0}(C)$ equipped with its ${\mathbb{Z}}_{2}$-vector
space structure. We call the space-theoretic description of $T$ a
_configuration of twists_ on $C$.

The set $\operatorname{Adm}(C)$ has a ${\mathbb{Z}}_{2}$-vector space
structure induced from $\operatorname{Edge}^{0}(C)$ given by the $2g$ linear
conditions from Equation 1, for $g$ the number of cycles of $C$ bounding a
connected component of the complement.

###  3.3. Correspondence between signs and twisted edges

We recall the correspondence between combinatorial patchworking and admissible
sets of twisted edges. Let $C$ be a non-singular tropical curve with Newton
polytope $\Delta$. For each $e\in\operatorname{Edge}^{0}(C)$, denote by
$v_{1}^{e}$ and $v_{2}^{e}$ the two vertices of the edge $e^{\vee}$ dual to
$e$ in the dual subdivision $\Delta_{C}$. The edge $e^{\vee}$ is contained in
exactly two 2-dimensional simplices of $\Delta_{C}$. Let $v_{3}^{e}$ and
$v_{4}^{e}$ denote the vertices of these two simplices which are distinct from
$v_{1}^{e},v_{2}^{e}$. Let $\delta$ be a distribution of signs on the integer
points in $\Delta_{C}\cap{\mathbb{Z}}^{2}$. The following proposition is
described in [BIMS15, Remark 3.9]

######  Proposition 3.12 ([BIMS15]).

Let $e\in\operatorname{Edge}^{0}(C)$.

  * •

If the coordinates modulo 2 of $v_{3}^{e}$ and $v_{4}^{e}$ are distinct, then
the edge $e$ is twisted if and only if
$\delta(v_{1}^{e})\delta(v_{2}^{e})\delta(v_{3}^{e})\delta(v_{4}^{e})=+1$.

  * •

If the coordinates modulo 2 of $v_{3}^{e}$ and $v_{4}^{e}$ coincide, then the
edge $e$ is twisted if and only if $\delta(v_{3}^{e})\delta(v_{4}^{e})=-1$.

In particular, for every twist-admissible set $T$ of bounded edges on $C$,
there exists a distribution of signs $\delta$ on the dual subdivision
$\Delta_{C}$ satisfying Proposition 3.12. Therefore, by Viro’s Patchworking
Theorem (Theorem 3.3), there exists a family of real algebraic curve
$(\mathcal{C}_{t})_{t>1}$ in $({\mathbb{C}}^{\times})^{2}$ which converges to
$C$ in the sense of Theorem 2.3 and such that the corresponding set of twisted
edges is $T$. This fact is stated in [BIMS15, Theorem 3.4] as a reformulation
of Viro’s Patchworking Theorem.

From Proposition 3.12, we can define a canonical distribution of signs on the
dual subdivision $\Delta_{C}$ (which does not depend on the subdivision) such
that no bounded edge of $C$ is twisted.

######  Definition 3.13 ([Ite95]).

The _Harnack distribution of signs_ $\delta_{\emptyset}$ on the dual
subdivision $\Delta_{C}$ of a non-singular tropical curve $C$ is given as:

  * •

for $v=(v_{1},v_{2})\in\Delta_{C}\cap{\mathbb{Z}}^{2}$ such that both $v_{1}$
and $v_{2}$ are even, we assign $\delta_{\emptyset}(v)=-1$;

  * •

for $v=(v_{1},v_{2})\in\Delta_{C}\cap{\mathbb{Z}}^{2}$ such that at least one
$v_{i}$ is odd, we assign $\delta_{\emptyset}(v)=+1$.

We say that a distribution of signs $\delta$ on $\Delta_{C}$ is of _Harnack
type_ if there exists $\varepsilon\in\mathcal{R}^{2}$ a symmetry such that for
every $v\in\Delta_{C}\cap{\mathbb{Z}}^{2}$, we have
$\delta(v)=\delta_{\emptyset}(\varepsilon(v))$. We say that a distribution of
signs $\delta$ on $\Delta_{C}$ is of _inverse Harnack type_ if there exists
$\varepsilon\in\mathcal{R}^{2}$ a symmetry such that for every
$v\in\Delta_{C}\cap{\mathbb{Z}}^{2}$, we have
$\delta(v)=-\delta_{\emptyset}(\varepsilon(v))$.

By Proposition 3.12, every distribution of signs of Harnack type or inverse
Harnack type on $\Delta_{C}$ induces an empty set of twisted edges on $C$.
Moreover, any distribution of signs inducing an empty set of twisted edges is
either of Harnack or inverse Harnack type.

###  3.4. Real components and twisted cycles

Given a non-singular real tropical curve $(C,\mathcal{E})$ in a non-singular
projective tropical toric surface ${\mathbb{T}}\mathbb{P}_{\Sigma}$, we want
to find a description of the connected components of
${\mathbb{R}}C_{\mathcal{E}}$ from the data of $C$ and the admissible set of
twisted edges $T$ on $C$ induced by $\mathcal{E}$. The following construction
is inspired from the notion of _left-right cycle_ in graph theory, see [GR13].
Let $C$ be a non-singular tropical curve in a non-singular projective tropical
toric surface ${\mathbb{T}}\mathbb{P}_{\Sigma}$ and let $T$ be a subset of
bounded edges on $C$. Choose a starting vertex $v$ on $C$, as well as one of
its incident edges $e$. Let $S$ be an alternating sequence of vertices and
edges of $C$ given by walking along the left side of $C$ from $(v,e)$ until
reaching an edge $e^{\prime}\in T$, then walk along the right side until
reaching another edge $e^{\prime\prime}\in T$, and so on, until reaching again
the vertex $v$ of $C$. Whenever the walk reaches a 1-valent vertex of $C$,
turn around that vertex and continue the walk.

######  Definition 3.14.

A _closed twisted walk_ starting from $(v,e)$ on a non-singular tropical curve
$C$ equipped with a subset of bounded edges $T$ is the alternating sequence
$S$ of vertices and edges encountered during the process above. A _twisted
cycle_ on $(C,T)$ is an equivalence class of closed twisted walks under cyclic
permutation and reversal.

######  Example 3.15.

Let $C$ be a non-singular tropical curve of degree 2 in
${\mathbb{T}}\mathbb{P}^{2}$, as represented in black in Figure 4. Let
$\mathcal{E}$ be a real phase structure on $C$ inducing the admissible set of
twisted edges $T=\\{e_{1},e_{2}\\}$, for $e_{1}$ the unique bounded edge of
direction $(1,1)$ and $e_{2}$ the unique bounded edge of direction $(0,1)$. A
closed twisted walk on $C$ starting at the couple $(v,e)$, with $v$ the
uppermost 1-valent vertex and $e$ its unique incident edge, is represented in
blue on Figure 4. The unique twisted cycle on $(C,\mathcal{E})$ is given as
the equivalence class of this closed twisted walk.

Figure 4. Closed walk inducing a twisted cycle from Example 3.15.

Given a twisted cycle $\rho$ on a non-singular real tropical curve
$(C,\mathcal{E})$ (hence a twisted cycle with respect to the admissible set of
twisted edges $T$ induced by $\mathcal{E}$) inside a non-singular projective
tropical toric surface ${\mathbb{T}}\mathbb{P}_{\Sigma}$, we can construct its
real part ${\mathbb{R}}\rho_{\mathcal{E}}$ as follows:

  * •

For every two consecutive edges $e,e^{\prime}$ in $\rho$ incident to a vertex
$v\in\rho$ which is 3-valent in $C$, there exists a unique
$\varepsilon\in\mathcal{R}^{2}$ such that
$\mathcal{E}_{e}\cap\mathcal{E}_{e^{\prime}}=\\{\varepsilon\\}$. We then say
that $(e,\varepsilon)$ and $(e^{\prime},\varepsilon)$ belong to
$(\rho,\mathcal{E})$.

  * •

For every edge $e^{\prime\prime}\in\rho$ incident to a vertex
$v^{\prime\prime}\in\rho$ which is 1-valent in $C$, by construction of twisted
cycles $e^{\prime\prime}$ appears twice consecutively in $\rho$. Then for
$\mathcal{E}_{e^{\prime\prime}}=\\{\varepsilon^{\prime},\varepsilon^{\prime\prime}\\}$,
we say that $(e^{\prime\prime},\varepsilon^{\prime})$ and
$(e^{\prime\prime},\varepsilon^{\prime\prime})$ belong to
$(\rho,\mathcal{E})$.

######  Definition 3.16.

The _real part_ ${\mathbb{R}}\rho_{\mathcal{E}}$ of the twisted cycle $\rho$
is given as

${\mathbb{R}}\rho_{\mathcal{E}}:=\overline{\bigcup\limits_{(e,\varepsilon)\subset(\rho,\mathcal{E})}\varepsilon(e)}\subset{\mathbb{R}}\mathbb{P}_{\Sigma}.$

######  Proposition 3.17.

Let $(C,\mathcal{E})$ be a non-singular real tropical curve in a non-singular
projective tropical toric surface ${\mathbb{T}}\mathbb{P}_{\Sigma}$. The real
part ${\mathbb{R}}\rho_{\mathcal{E}}$ of a twisted cycle $\rho$ in
$(C,\mathcal{E})$ is a connected component of ${\mathbb{R}}C_{\mathcal{E}}$,
and every connected component of ${\mathbb{R}}C_{\mathcal{E}}$ is given as the
real part of a twisted cycle on $(C,\mathcal{E})$. In particular, the set of
chains

$\left\\{[{\mathbb{R}}\rho_{\mathcal{E}}]\subset
C_{1}({\mathbb{R}}C_{\mathcal{E}};{\mathbb{Z}}_{2})~{}|~{}\rho\text{ twisted
cycle on }(C,\mathcal{E})\right\\}$

induces a basis of $H_{1}({\mathbb{R}}C_{\mathcal{E}};{\mathbb{Z}}_{2})$.

###### Proof.

Let $\rho$ be a twisted cycle in $(C,\mathcal{E})$. By definition of a twisted
cycle, the underlying graph of $\rho$ is connected, hence by construction its
real part ${\mathbb{R}}\rho_{\mathcal{E}}$ is a connected graph. Again by
construction, every vertex of ${\mathbb{R}}\rho_{\mathcal{E}}$ is in the
boundary of exactly two edges, hence for $[{\mathbb{R}}\rho_{\mathcal{E}}]$
the 1-dimensional chain in
$C_{1}({\mathbb{R}}C_{\mathcal{E}};{\mathbb{Z}}_{2})$ induced by the real part
${\mathbb{R}}\rho_{\mathcal{E}}$, the boundary chain
$\partial[{\mathbb{R}}\rho_{\mathcal{E}}]$ is trivial in
$C_{0}({\mathbb{R}}C_{\mathcal{E}};{\mathbb{Z}}_{2})$. Then the chain
$[{\mathbb{R}}\rho_{\mathcal{E}}]$ induces a class (also called
$[{\mathbb{R}}\rho_{\mathcal{E}}]$) of the homology group
$H_{1}({\mathbb{R}}C_{\mathcal{E}},{\mathbb{Z}}_{2})$. Since
${\mathbb{R}}C_{\mathcal{E}}$ is 1-dimensional, the class
$[{\mathbb{R}}\rho_{\mathcal{E}}]$ is non-trivial in
$H_{1}({\mathbb{R}}C_{\mathcal{E}},{\mathbb{Z}}_{2})$. Then since
${\mathbb{R}}\rho_{\mathcal{E}}$ is connected, it is a connected component of
the real part ${\mathbb{R}}C_{\mathcal{E}}$.

Now let $c$ be a connected component of ${\mathbb{R}}C_{\mathcal{E}}$. Let
$\varphi:{\mathbb{R}}C_{\mathcal{E}}\rightarrow C$ be the absolute value map.
The map $\varphi$ sends any two consecutive edges of $c$ to two (not
necessarily distinct) adjacent edges of $C$. Therefore the map $\varphi$ sends
$c$ to a cycle $\rho$ on $C$ (in the sense of equivalence class of a closed
walk under cyclic permutation and reversal). If $c$ is inflected locally
around an edge $e$, then a closed walk inducing the cycle $\rho$ must change
side at the edge $\varphi(e)$ in $C$, since $C$ induces a convex polyhedral
subdivision of ${\mathbb{T}}\mathbb{P}_{\Sigma}$. Similarly, if $c$ is locally
convex around an edge $e$, then a closed walk inducing the cycle $\rho$ goes
along the edge $\varphi(e)$ in $C$ without changing side. Therefore $\rho$ is
a twisted cycle on $(C,\mathcal{E})$. ∎

##  4\. Prescribed number of real components

###  4.1. Exact sequence for real tropical curves

Let $C$ be a non-singular tropical curve in a non-singular projective tropical
toric surface ${\mathbb{T}}\mathbb{P}_{\Sigma}$. The _tangent cosheaf_
${\mathcal{F}}_{1}$ on $C$ is defined as follows. For $e$ an edge of $C$, we
set ${\mathcal{F}}_{1}(e):={\mathbb{Z}}_{2}\langle\overrightarrow{e}\rangle$,
with $\overrightarrow{e}$ the direction modulo 2 of $e$. For $v$ a 3-valent
vertex of $C$, we set ${\mathcal{F}}_{1}(v):={\mathbb{Z}}_{2}^{2}$, and for
$v^{\prime}$ a 1-valent vertex of $C$, we set
${\mathcal{F}}_{1}(v^{\prime}):=0$. We refer to [RS21] for more details, and
to [IKMZ19] for the initial notion of $p$-multitangent cosheaves
${\mathcal{F}}_{p}$ with ${\mathbb{Q}}$-coefficients. The _cellular chain
complex_ $C_{\bullet}(C;{\mathcal{F}}_{1})$ is given by

$\displaystyle\bigoplus\limits_{e\in\operatorname{Edge}(C)}{\mathcal{F}}_{1}(e)=:C_{1}(C;{\mathcal{F}}_{1})$
$\displaystyle\rightarrow
C_{0}(C;{\mathcal{F}}_{1}):=\bigoplus\limits_{v\in\operatorname{Vertex}(C)}{\mathcal{F}}_{1}(v)$
$\displaystyle e\otimes\overrightarrow{e}$ $\displaystyle\mapsto(\partial
e)\otimes\overrightarrow{e}.$

Then the homology groups $H_{q}(C;{\mathcal{F}}_{1})$ are given as
$H_{q}(C_{\bullet}(C;{\mathcal{F}}_{1}))$. The homology group
$H_{1}(C;{\mathcal{F}}_{1})$ is generated by the class $[C]$ induced by the
chain

$\sum_{e\in\operatorname{Edge}(C)}e\otimes\overrightarrow{e}\subset
C_{1}(C;{\mathcal{F}}_{1}).$

The group $H_{0}(C;{\mathcal{F}}_{1})$ has dimension $g$, for $g:=\dim
H_{1}(C;{\mathbb{Z}}_{2})$ the number of _primitive cycles_ on the non-
singular tropical curve $C$.

######  Definition 4.1.

Let $C$ be a non-singular tropical curve in a non-singular projective tropical
toric surface ${\mathbb{T}}\mathbb{P}_{\Sigma}$. A cycle $\gamma$ of the non-
singular tropical curve $C$ is called _primitive_ if it bounds a connected
component of ${\mathbb{T}}\mathbb{P}_{\Sigma}\backslash C$.

Let $\\{\gamma_{1},\ldots,\gamma_{g}\\}$ be the set of primitive cycles on
$C$. The set of classes

$\\{[\gamma_{i}^{*}];i=1,\ldots,g\\}$

form a basis of $H_{0}(C;{\mathcal{F}}_{1})$, for $[\gamma_{i}^{*}]$ induced
by the chain

$\sum\limits_{e\in\operatorname{Edge}(\gamma_{i})}v_{e}\otimes\overrightarrow{e}\subset
C_{0}(C;{\mathcal{F}}_{1}),$

with $v_{e}$ the first vertex of the edge $e$ after choosing a cyclic order on
$\gamma_{i}$.

Using these notions of tropical homology and several results from [RS21], we
will be able to determine the number of connected components of the real part
of a non-singular real tropical curve.

######  Definition 4.2.

Let $(C,\mathcal{E})$ be a non-singular real tropical curve, and let $g$ be
the number of primitive cycles on $C$. We say that $(C,\mathcal{E})$ is
_maximal_ if its real part ${\mathbb{R}}C_{\mathcal{E}}$ consists of $g+1$
connected components. More generally, we say that $(C,\mathcal{E})$ is a
_$(M-r)$_ real tropical curve if its real part ${\mathbb{R}}C_{\mathcal{E}}$
consists of $g+1-r$ connected components.

######  Theorem 4.3 ([RS21]).

Let $(C,\mathcal{E})$ be a non-singular real tropical curve in a non-singular
projective tropical toric surface ${\mathbb{T}}\mathbb{P}_{\Sigma}$. We have
the following long exact sequence:

${0}$${H_{1}(C;{\mathcal{F}}_{1})}$${H_{1}({\mathbb{R}}C_{\mathcal{E}};{\mathbb{Z}}_{2})}$${H_{1}(C;{\mathbb{Z}}_{2})}$${H_{0}(C;{\mathcal{F}}_{1})}$${H_{0}({\mathbb{R}}C_{\mathcal{E}};{\mathbb{Z}}_{2})}$${H_{0}(C;{\mathbb{Z}}_{2})}$${0.}$$\scriptstyle{\partial}$

In particular, the number of connected components of
${\mathbb{R}}C_{\mathcal{E}}$ is equal to $\dim\ker\partial+1$, or said
otherwise, the non-singular real tropical curve $(C,\mathcal{E})$ is a
$(M-r)$-real tropical curve for $r:=\operatorname{rank}\partial$.

###### Proof.

We apply [RS21, Proposition 4.10] for $p=0$ and for $p=1$ to the case of non-
singular real tropical curves. ∎

For $C$ a non-singular tropical curve with primitive cycles
$\\{\gamma_{1},\ldots,\gamma_{g}\\}$ and $T$ a subset of bounded edges of $C$,
we introduce a $g\times g$ symmetric matrix with
${\mathbb{Z}}_{2}$-coefficients keeping track of the (mod 2)-intersection
numbers $|\gamma_{i}\cap\gamma_{j}\cap T|\mod 2$ for $i,j=1,\ldots,g$.

######  Definition 4.4.

Let $C$ be a non-singular tropical curve, let
$\\{\gamma_{1},\ldots,\gamma_{g}\\}$ be the primitive cycles on $C$, and let
$T$ be a subset of bounded edges on $C$. Seeing the primitive cycles as
subsets of bounded edges of $C$, we define the $g\times g$ symmetric matrix
$A_{T}$ with ${\mathbb{Z}}_{2}$-coefficients as

$A_{T}:=\left(|\gamma_{i}\cap\gamma_{j}\cap T|\mod 2\right)_{i,j=1,\ldots,g}.$

For $(C,\mathcal{E})$ a non-singular real tropical curve in a non-singular
projective tropical toric surface ${\mathbb{T}}\mathbb{P}_{\Sigma}$, we want
to be able to compute the connecting homomorphism $\partial$ of Theorem 4.3 in
terms of the admissible set of twisted edges $T$ induced by $\mathcal{E}$. We
have a non-degenerate pairing:

$\langle\cdot,\cdot\rangle:H_{0}(C;{\mathcal{F}}_{1})\times
H_{1}(C;{\mathbb{Z}}_{2})\rightarrow{\mathbb{Z}}_{2}$

induced from the pairing on integral homology groups for non-singular tropical
curves in [Sha11] (a similar non-degenerate pairing defined between tropical
homology and cohomology groups is also defined in [BIMS15] (Section 7.8) and
[MZ14] (Section 3.2)). Renaudineau and Shaw [RS21] used this pairing in order
to recover the whole map $\partial$ as follows.

######  Theorem 4.5 ([RS21]).

Let $(C,\mathcal{E})$ be a non-singular real tropical curve with admissible
set of twisted edges $T$. Let $\\{\gamma_{1},\ldots,\gamma_{g}\\}$ be the
primitive cycles on $C$, seen as subsets of bounded edges of $C$. Then

$\langle\partial[\gamma_{i}],[\gamma_{j}]\rangle=|\gamma_{i}\cap\gamma_{j}\cap
T|\mod 2.$

In particular, we have $\dim\ker A_{T}=\dim\ker\partial$, for $\partial$ the
connecting homomorphism in Theorem 4.3.

The theorem above is a reformulation of [RS21, Theorem 7.2] using the non-
degenerate pairing formulation used in the proof of [RS21, Theorem 7.5]. From
now on, the connecting homomorphism from Theorem 4.5 will be denoted
$\partial_{T}$, as it is determined by the admissible set of twisted edges $T$
on $(C,\mathcal{E})$.

###  4.2. Haas conditions for dividing and maximal curves

######  Definition 4.6.

We say that a non-singular real tropical curve $(C,\mathcal{E})$ is _dividing_
if $(C,\mathcal{E})$ is the tropical limit of a family of non-singular real
algebraic curves $(\mathcal{C}_{t})_{t}$ such that for $t\in{\mathbb{R}}_{>0}$
small enough, the curve $\mathcal{C}_{t}$ is dividing.

Haas gave a necessary and sufficient condition to construct dividing curves
via combinatorial patchworking, which translates in terms of real tropical
curves as follows:

######  Theorem 4.7 ([Haa97]).

Let $(C,\mathcal{E})$ be a non-singular real tropical curve with admissible
set of twisted edges $T$. Then $(C,\mathcal{E})$ is dividing if and only if
for any cycle $\gamma$ in $C$ (seen as a subset of bounded edges of $C$), we
have

(2) $|\gamma\cap T|=0\mod 2,$

######  Definition 4.8.

A twist-admissible set $T$ of bounded edges of $C$ is _dividing_ if Equation 2
is satisfied for all cycles $\gamma$ in $C$. We denote by
$\operatorname{Div}(C)\subset\operatorname{Adm}(C)$ the set of all dividing
sets of bounded edges on $C$.

######  Remark 4.9.

Let $g$ be the number of primitive cycles of a non-singular tropical curve
$C$. The linear conditions from Equation 2 imply that the set
$\operatorname{Div}(C)$ has a ${\mathbb{Z}}_{2}$-vector space structure
induced from $\operatorname{Adm}(C)$. In particular, the
${\mathbb{Z}}_{2}$-vector space $\operatorname{Div}(C)$ has codimension $l$ in
$\operatorname{Adm}(C)$, for $l\leq g$ the number of primitive cycles $\gamma$
of $C$ such that $\gamma$ contains at least one edge of direction $(1,0)\mod
2$, one edge of direction $(0,1)\mod 2$ and one edge of direction $(1,1)\mod
2$.

######  Definition 4.10.

Let $C$ be a non-singular tropical curve in ${\mathbb{R}}^{2}$. We say that an
edge $e$ of $C$ is _exposed_ if $e$ lies on the boundary of an unbounded
connected component of ${\mathbb{R}}^{2}\backslash C$. For $C^{\prime}$ the
compactification of $C$ in a non-singular projective tropical toric surface
${\mathbb{T}}\mathbb{P}_{\Sigma}$, we say that an edge $e$ of $C^{\prime}$ is
exposed if the corresponding edge of $C$ is exposed. We denote by
$\operatorname{Exp}(C)$ the set of all exposed edges on $C$.

Haas gave a necessary and sufficient condition to construct $M$-curves via
combinatorial patchworking [Haa97, Theorem 7.3.0.10], which can be translated
in terms of real tropical curves as follows.

Figure 5. Tropical curve with maximal set of twisted edges in Example 4.14

######  Theorem 4.11 ([Haa97]).

Let $(C,\mathcal{E})$ be a non-singular real tropical curve in a non-singular
projective tropical toric surface ${\mathbb{T}}\mathbb{P}_{\Sigma}$ with
admissible set of twisted edges $T$. Then $(C,\mathcal{E})$ is maximal if and
only if

  1. (1)

the set of twisted edges $T$ is dividing;

  2. (2)

every edge of $T$ is exposed.

Note that Item 1 of Theorem 4.11 comes from the fact that every maximal real
algebraic curve is dividing.

######  Definition 4.12.

We say that a twist-admissible subset $T$ of bounded edges of a non-singular
tropical curve $C$ is _maximal_ if $T$ satisfies Item 1 and Item 2 in Theorem
4.11. We denote by $\operatorname{Max}(C)$ the set of all maximal subsets $T$
of bounded edges of $C$.

######  Remark 4.13.

Let $C$ be a non-singular tropical curve with $g$ primitive cycles. Item 2 of
Theorem 4.11 imply $l$ linear conditions on the space $\operatorname{Div}(C)$,
for $l\leq\binom{g}{2}$ the number of pairs of distinct primitive cycles of
$C$ with non-empty intersection. Then the set $\operatorname{Max}(C)$ has a
${\mathbb{Z}}_{2}$-vector space structure induced from
$\operatorname{Div}(C)$, and is of codimension $l$ in $\operatorname{Div}(C)$.
The same set without twist-admissibility condition (Equation 1) has a
${\mathbb{Z}}_{2}$-vector space structure induced from
$\operatorname{Edge}^{0}(C)$, as shown in [RBB17].

######  Example 4.14.

Every non-singular real tropical curve $(C,\mathcal{E})$ with empty set of
twisted edges is maximal. In Figure 5, the edges marked by black dots on the
tropical curve form an example of non-empty maximal set of bounded edges.
Indeed, all these edges are exposed, and the unique cycle in the tropical
curve contains an even number of twisted edges.

###  4.3. (M-1) curves

We want to give several new combinatorial conditions, starting by $(M-1)$
curves. Before stating the theorem, we need the following definition:

(a) Complete graph $K_{1}$ and exposed twisted edges.

(b) Complete graph $K_{2}$ and exposed twisted edges.

Figure 6. Example 4.18.

######  Definition 4.15.

Let $C$ be a non-singular tropical curve in a non-singular projective tropical
toric surface ${\mathbb{T}}\mathbb{P}_{\Sigma}$, and let $T$ be a set of
bounded edges on $C$. The _dual graph of non-exposed twisted edges_
$\Gamma_{T}$ is defined as follows.

  * •

The set of vertices $\operatorname{Vertex}(\Gamma_{T})$ of $\Gamma_{T}$ is the
set of primitive cycles of $C$.

  * •

The set of edges $\operatorname{Edge}(\Gamma_{T})$ of $\Gamma_{T}$ is given by
the pairs of distinct primitive cycles $\gamma\neq\gamma^{\prime}$ such that
$\gamma\cap\gamma^{\prime}\cap T\neq\emptyset$ (seeing the primitive cycles as
subsets of bounded edges of $C$).

######  Theorem 4.16 (Theorem 1.2).

Let $(C,\mathcal{E})$ be a non-singular real tropical curve in a non-singular
projective tropical toric surface ${\mathbb{T}}\mathbb{P}_{\Sigma}$, with
admissible set of twisted edges $T$. Then $(C,\mathcal{E})$ is a $(M-1)$ real
tropical curve if and only if:

  1. (1)

the dual graph of non-exposed twisted edges $\Gamma_{T}$ consists of a
complete subgraph $K_{n}\subset\Gamma_{T}$ on $1\leq n\leq 4$ vertices plus
isolated vertices;

  2. (2)

every primitive cycle $\gamma$ in $\operatorname{Vertex}(K_{n})$, seen as a
subset of bounded edges of $C$, satisfies

$|\gamma\cap T|=1\mod 2;$

  3. (3)

every primitive cycle $\gamma$ in
$\operatorname{Vertex}(\Gamma_{T})\backslash\operatorname{Vertex}(K_{n})$,
seen as a subset of bounded edges of $C$, satisfies

$|\gamma\cap T|=0\mod 2.$

######  Remark 4.17.

Since a non-singular tropical curve $C$ in a non-singular projective tropical
toric surface ${\mathbb{T}}\mathbb{P}_{\Sigma}$ is a planar graph, the dual
graph of non-exposed twisted edges cannot contain a complete subgraph $K_{n}$
on $n\geq 5$ vertices by Kuratowski’s Theorem [Kur30].

(a) Complete graph $K_{3}$ and exposed twists.

(b) Complete graph $K_{4}$.

Figure 7. Example 4.18.

######  Example 4.18.

Figures 6 and 7 represent examples of tropical curves with set of twisted
edges (marked by black dots) such that their dual graph of non-exposed twisted
edges consists of a complete graph $K_{n},1\leq n\leq 4$, represented in plain
blue. In particular, the graph $K_{1}$ in Figure 6(a) is the single blue
point. The dotted blue edges are dual to the exposed twisted edges. All those
sets of twisted edges give rise to $(M-1)$ real tropical curves by Theorem
4.16. Note that we do not represent all the edges of the tropical curve in
Figure 7(b).

In order to prove the theorem, we recall several standard definitions in graph
theory (see for instance [GR13]).

######  Definition 4.19.

Let $\Gamma$ be a graph (without loop and without multiple edges) with
$\operatorname{Vertex}(\Gamma)=\\{v_{1},\ldots,v_{r}\\}$. The _adjacency
matrix_ $N$ of $\Gamma$ is the $r\times r$-matrix with coefficients in
${\mathbb{Z}}_{2}$ such that $N_{ij}=1$ if the vertices $v_{i}$ and $v_{j}$
are adjacent, and $N_{ij}=0$ otherwise (in particular, the diagonal
coefficients of $N$ are all zero as $\Gamma$ has no loop). The _valency matrix
mod 2_ of $\Gamma$ is the $r\times r$ diagonal matrix $D$ with coefficients
$D_{ii}$ given as the valency of $v_{i}$ modulo 2. The _Laplacian mod 2_ of
$\Gamma$ is the matrix $Q_{\Gamma}:=D+N$ (in particular it is a symmetric
$r\times r$-matrix with coefficients in ${\mathbb{Z}}_{2}$).

###### Proof of Theorem 4.16.

Let $\\{\gamma_{1},\ldots,\gamma_{g}\\}$ be the primitive cycles of $C$. Let
$Q_{\Gamma_{T}}$ be the Laplacian mod 2 of the dual graph of non-exposed
twisted edges $\Gamma_{T}$ with respect to
$\\{\gamma_{1},\ldots,\gamma_{g}\\}$. Seeing the primitive cycles as subsets
of bounded edges of $C$, let $D_{T}$ be the $g\times g$ diagonal matrix with
${\mathbb{Z}}_{2}$-coefficients

$(D_{T})_{ii}=|\gamma_{i}\cap T\cap\operatorname{Exp}(C)|\mod 2.$

Then the matrix $A_{T}$ is given by $A_{T}=Q_{\Gamma_{T}}+D_{T}$. Now
$(C,\mathcal{E})$ is a $(M-1)$ real tropical curve if and only if
$\operatorname{rank}A_{T}=1$ by Theorem 4.5. A symmetric $g\times g$-matrix of
rank 1 with coefficients in ${\mathbb{Z}}_{2}$ is the product of a vector
$z\in{\mathbb{Z}}_{2}^{g}$ with its transpose. Up to permutation of its
coefficients, we can write

$z^{t}=(1,\ldots,1,0,\ldots,0).$

Therefore, up to renumbering the primitive cycles $\gamma_{i}$, the matrix
$A_{T}$ has the form

$\left(\NiceMatrix 1&\Cdots 10\Cdots 0\\\
\Vdots\Ddots\Vdots\Vdots\Ddots\Vdots\\\ 1\Cdots 10\Cdots 0\\\ 0\Cdots 00\Cdots
0\\\ \Vdots\Ddots\Vdots\Vdots\Ddots\Vdots\\\ 0\Cdots 00\Cdots 0\\\ \right).$

Let $Q=D+N$ be the diagonal bloc of $A_{T}$ with only non-zero coefficients,
with $D$ the diagonal matrix of $Q$. We obtain the following facts:

  1. (1)

the matrix $N$ is the adjacency matrix of a complete graph $K_{n}$ (with
$1\leq n\leq 4$ by Remark 4.17), since all its non-diagonal coefficients are
non-zero;

  2. (2)

every primitive cycle in
$\operatorname{Vertex}(\Gamma_{T})\backslash\operatorname{Vertex}(K_{n})$ is
isolated, since the non-diagonal coefficients of $A_{T}$ not belonging to the
minor $Q$ are all zero;

  3. (3)

every primitive cycle $\gamma$ in $\operatorname{Vertex}(K_{n})$ satisfies
$|\gamma\cap T|=1\mod 2$, since the diagonal coefficients of $Q$ are all non-
zero;

  4. (4)

every primitive cycle $\gamma$ in
$\operatorname{Vertex}(\Gamma_{T})\backslash\operatorname{Vertex}(K_{n})$
satisfies $|\gamma\cap T|=0\mod 2$, since the diagonal coefficients of $A_{T}$
not belonging to the minor $Q$ are all zero. $\blacksquare$

∎

######  Definition 4.20.

Let $C$ be a non-singular tropical curve, and let $T,T^{\prime}$ be two twist-
admissible set of bounded edges on $C$. We say that $T$ and $T^{\prime}$ are
_cycle-disjoint_ if for every primitive cycle $\gamma$ on $C$, we have

$\displaystyle\gamma\cap T\neq\emptyset$ $\displaystyle\Rightarrow\gamma\cap
T^{\prime}=\emptyset$ $\displaystyle\text{ and }\gamma\cap
T^{\prime}\neq\emptyset$ $\displaystyle\Rightarrow\gamma\cap T=\emptyset.$

######  Corollary 4.21.

Let $(C,\mathcal{E})$ be a non-singular real tropical curve with admissible
set of twisted edges $T=T_{1}\sqcup\ldots\sqcup T_{r}$, such that the dual
graphs of non-exposed twisted edges $\Gamma_{T_{i}}$ are all disjoint and they
all satisfy Theorem 4.16. Then $(C,\mathcal{E})$ is a $(M-r)$ real tropical
curve which is non-dividing.

###### Proof.

By assumption, the $T_{i}$’s are all cycle-disjoint, hence there exists an
ordering of the primitive cycles of $C$ such that the matrix $A_{T}$
satisfying this ordering is bloc-diagonal, with each non-zero bloc being the
non-zero bloc of the matrix $A_{T_{i}}$ satisfying the same ordering of
primitive cycles. Then

$\operatorname{rank}A_{T}=\sum_{i=1}^{r}\operatorname{rank}A_{T_{i}}=r$

by Theorem 4.16, hence $(C,\mathcal{E})$ is a $(M-r)$ real tropical curve by
Theorem 4.5. Since every bloc has non-zero diagonal, we obtain by Theorem 4.7
that $(C,\mathcal{E})$ is non-dividing. ∎

###  4.4. Dividing (M-2) curves

Let $(C,\mathcal{E})$ be a non-singular dividing real tropical curve. By
Corollary 4.21, the set of twisted edges on $(C,\mathcal{E})$ cannot be given
as cycle-disjoint union of $(M-1)$ configurations of twists, that is a cycle-
disjoint union of configurations of twists satisfying Theorem 4.16. This
motivates the following result.

######  Theorem 4.22 (Theorem 1.4).

Let $(C,\mathcal{E})$ be a non-singular real tropical curve in a non-singular
projective tropical toric surface ${\mathbb{T}}\mathbb{P}_{\Sigma}$, with
dividing set of twisted edges $T$. Then $(C,\mathcal{E})$ is a dividing
$(M-2)$ real tropical curve if and only if the dual graph of non-exposed
twisted edges $\Gamma_{T}$ consists of

  1. (1)

either a complete planar bipartite graph and isolated vertices;

  2. (2)

or a complete planar tripartite graph and isolated vertices.

######  Remark 4.23.

Since a non-singular tropical curve $C$ in a non-singular projective tropical
toric surface ${\mathbb{T}}\mathbb{P}_{\Sigma}$ is a planar graph, by [Kur30],
the dual graph of non-exposed twisted edges $\Gamma_{T}$ cannot contain a
complete bipartite subgraph $K_{3,3}$ on two sets of 3 vertices.

######  Example 4.24.

The underlying non-singular tropical curve in Figure 8 is of bidegree $(4,4)$
in ${\mathbb{T}}\mathbb{P}^{1}\times{\mathbb{T}}\mathbb{P}^{1}$. In Figure
8(a), we have a set of twisted edges (marked by black dots) with dual graph of
non-exposed twisted edges (represented in plain blue) consisting of a complete
bipartite graph $K_{2,2}$ and isolated vertices. Notice that there are no
exposed twisted edges in this case. In Figure 8(b), we have a set of twisted
edges (marked by black dots) with dual graph of non-exposed twisted edges
(represented in plain blue) consisting of a complete tripartite graph
$K_{2,2,1}$ and isolated vertices. The edges dual to exposed twisted edges are
represented in dotted blue. Those two sets of twisted edges give rise to
dividing $(M-2)$ real tropical curves by Theorem 4.22.

(a) Complete bipartite dual graph of non-exposed twisted edges $K_{2,2}$.

(b) Complete tripartite dual graph of non-exposed twisted edges $K_{2,2,1}$.

Figure 8. Example 4.24.

###### Proof of Theorem 4.22.

Let $\\{\gamma_{1},\ldots,\gamma_{g}\\}$ be the set of primitive cycles of
$C$. Let $Q_{\Gamma_{T}}$ be the Laplacian mod 2 of the dual graph of non-
exposed twisted edges $\Gamma_{T}$ with respect to the ordering in
$\\{\gamma_{1},\ldots,\gamma_{g}\\}$. Seeing the primitive cycles as subsets
of bounded edges of $C$, let $D_{T}$ be the $g\times g$ diagonal matrix with
${\mathbb{Z}}_{2}$-coefficients

$(D_{T})_{ii}=|\gamma_{i}\cap T\cap\operatorname{Exp}(C)|\mod 2.$

Then the matrix $A_{T}$ is given by $A_{T}=Q_{\Gamma_{T}}+D_{T}$. The non-
singular real tropical curve $(C,\mathcal{E})$ is a $(M-2)$ real tropical
curve if and only if $\operatorname{rank}A_{T}=2$ by Theorem 4.5. This is
equivalent to the following condition on the column space
$\operatorname{Col}(A_{T})$:

$\operatorname{Col}(A_{T})={\mathbb{Z}}_{2}\langle
v_{1},v_{2}\rangle\subset{\mathbb{Z}}_{2}^{g},$

with $v_{1},v_{2}$ linearly independent. Since $T$ is a dividing set of
twisted edges, by Theorem 4.7 the diagonal coefficients of $A_{T}$ are zero.
Up to renumbering the primitive cycles, we can then assume that
$(A_{T})_{1,2}=(A_{T})_{2,1}=1$. We can then set $v_{1}:=(A_{T})_{\bullet,1}$
and $v_{2}:=(A_{T})_{\bullet,2}$. Again since the diagonal coefficients of
$A_{T}$ are zero, for any $j\neq j^{\prime}$ such that
$(A_{T})_{\bullet,j}=(A_{T})_{\bullet,j^{\prime}}$ as a vector of
${\mathbb{Z}}_{2}^{g}$, the primitive cycles $\gamma_{j},\gamma_{j^{\prime}}$
satisfy $\gamma_{j}\cap\gamma_{j^{\prime}}\cap T=\emptyset$. Similarly, by the
condition $(A_{T})_{1,2}=(A_{T})_{2,1}=1$, for any $j\neq j^{\prime}$ such
that

$(A_{T})_{\bullet,j}\neq(A_{T})_{\bullet,j^{\prime}},\quad(A_{T})_{\bullet,j}\neq
0\text{ and }(A_{T})_{\bullet,j^{\prime}}\neq 0$

as vectors of ${\mathbb{Z}}_{2}^{g}$, the primitive cycles $\gamma_{j}$ and
$\gamma_{j^{\prime}}$ satisfy $\gamma_{j}\cap\gamma_{j^{\prime}}\cap
T\neq\emptyset$.

We now have two possible cases:

  * •

either no column of $A_{T}$ is equal to the vector $v_{1}+v_{2}$, and then the
minor $M$ of $A_{T}$ given by all the non-zero rows and columns is the
adjacency matrix of a complete bipartite graph $K_{m,n}$;

  * •

or there exists a column of $A_{T}$ equal to $v_{1}+v_{2}$, and then the minor
$M$ of $A_{T}$ given by all the non-zero rows and columns is the adjacency
matrix of a complete tripartite graph $K_{l,m,n}$.

Adding the minor $M$ to the minor of $D_{T}$ given by the same rows and
columns as $M$, we obtain the Laplacian mod 2 of the graph $K_{m,n}$
(respectively $K_{l,m,n}$) as a minor of $Q_{\Gamma_{T}}$, hence $K_{m,n}$
(respectively $K_{l,m,n}$) is a subgraph of $\Gamma_{T}$, and $\Gamma_{T}$ has
only isolated vertices outside this subgraph. ∎

######  Corollary 4.25.

Let $(C,\mathcal{E})$ be a non-singular real tropical curve with admissible
set of twisted edges $T=T_{1}\sqcup\ldots\sqcup T_{s}$, such that the dual
graphs of non-exposed twisted edges $\Gamma_{T_{i}}$ are all disjoint and they
all satisfy Theorem 4.22. Then $(C,\mathcal{E})$ is a dividing $(M-2s)$ real
tropical curve.

###### Proof.

By assumption, the $T_{i}$’s are all cycle-disjoint, hence there exists an
ordering of the primitive cycles of $C$ such that the matrix $A_{T}$
satisfying this ordering is bloc-diagonal, with each non-zero bloc being the
non-zero bloc of the matrix $A_{T_{i}}$ satisfying the same ordering of
primitive cycles. Then

$\operatorname{rank}A_{T}=\sum_{i=1}^{s}\operatorname{rank}A_{T_{i}}=2s$

by Theorem 4.22, hence $(C,\mathcal{E})$ is a $(M-2s)$ real tropical curve by
Theorem 4.5. Since every bloc has zero diagonal, we obtain by Theorem 4.7 that
$(C,\mathcal{E})$ is dividing. ∎

##  5\. Ovals in the real part of a tropical curve

###  5.1. Partition of dividing configuration of twists

We want to describe a set of generators of the ${\mathbb{Z}}_{2}$-vector space
$\operatorname{Div}(C)$, in order to formulate relative topology criteria for
dividing real tropical curves in the next sections.

######  Definition 5.1.

Let $C$ be a non-singular tropical curve in a non-singular projective tropical
toric surface ${\mathbb{T}}\mathbb{P}_{\Sigma}$. A _multi-bridge_ $T$ on $C$
is a dividing set of twisted edges on $C$ with common direction modulo 2, such
that $C\backslash T$ is disconnected and for any proper subset
$T^{\prime}\subsetneq T$, the graph $C\backslash T^{\prime}$ is connected. A
_circuit_ $T$ on $C$ is a multi-bridge on $C$ such that every edge of $T$ is
non-exposed.

######  Proposition 5.2.

Let $C$ be a non-singular tropical curve in a non-singular projective tropical
toric surface ${\mathbb{T}}\mathbb{P}_{\Sigma}$. Then the set of all multi-
bridges on $C$ spans the ${\mathbb{Z}}_{2}$-vector space
$\operatorname{Div}(C)$. Moreover, if every primitive cycle of $C$ has at most
two edges of each possible direction modulo 2, then the set of all multi-
bridges form a basis of $\operatorname{Div}(C)$.

###### Proof.

Let $T\in\operatorname{Div}(C)$. Assume $e_{1}$ is an edge of $T$ that does
not belong to a primitive cycle of $C$, for $T$ seen as a subset of bounded
edges of $C$. The edge $e_{1}$ is then a multi-bridge, and the identification
$\operatorname{Edge}^{0}(C)\simeq{\mathbb{Z}}_{2}^{|\operatorname{Edge}^{0}(C)|}$
sends $e_{1}$ to the configuration of twists $T_{1}\in\operatorname{Div}(C)$.
Hence the sum $T+T_{1}$ is a configuration of twists
$\widehat{T_{1}}\in\operatorname{Div}(C)$, since $\operatorname{Div}(C)$ is a
${\mathbb{Z}}_{2}$-space, such that $T_{1}$ and $\widehat{T_{1}}$ are disjoint
seen as subsets of bounded edges of $C$. Therefore, we can restrict to the
case where all the edges of $T$ lie on primitive cycles of $C$.

Assume $e_{2}$ is an edge of $T$ lying on a primitive cycle $\gamma$ of $C$,
for $T$ seen as a subset of bounded edges of $C$. By Equation 1 and Equation
2, there exists an edge $e_{3}$ distinct from $e_{2}$ with
$\overrightarrow{e_{2}}=\overrightarrow{e_{3}}\mod 2$ in the set $\gamma\cap
T$, for $\gamma$ and $T$ seen as subsets of bounded edges of $C$. If both
$e_{2}$ and $e_{3}$ are exposed, the set $\\{e_{2},e_{3}\\}$ is a multi-bridge
on $C$, and for $T_{2}:=e_{2}+e_{3}\in\operatorname{Div}(C)$ the associated
configuration of twists, the sum $T+T_{2}$ is a configuration of twists
$\widehat{T_{2}}\in\operatorname{Div}(C)$ such that $T_{2}$ and
$\widehat{T_{2}}$ are disjoint seen as subsets of bounded edges of $C$. If at
least one of the two edges $e_{2},e_{3}$ is non-exposed (say $e_{3}$), we use
again Equation 1 and Equation 2 on the unique primitive cycle
$\gamma^{\prime}$ distinct from $\gamma$ containing $e_{3}$, and so on, until
we cannot add a new edge in this manner (if $e_{2}$ is non-exposed, one has to
add edges in the same way as for $e_{3}$). The resulting subset of bounded
edges $\\{e_{2},\ldots,e_{k}\\}$ is a multi-bridge, and for
$T_{3}:=\sum_{i=2}^{k}e_{i}\in\operatorname{Div}(C)$ the corresponding
configuration of twists, the sum $T+T_{3}$ is a configuration of twists
$\widehat{T_{3}}\in\operatorname{Div}(C)$ such that $T_{3}$ and
$\widehat{T_{3}}$ are disjoint seen as subsets of bounded edges of $C$.
Therefore, any configuration of twists $T\in\operatorname{Div}(C)$ is given as
a sum of configuration of twists associated to multi-bridges on $C$. Moreover,
if every primitive cycle of $C$ has at most two edges of each possible
direction modulo 2, any two distinct multi-bridges on $C$ are disjoint, hence
the configurations of twists associated to the multi-bridges on $C$ form a
basis of $\operatorname{Div}(C)$. ∎

######  Corollary 5.3.

Let $C$ be a non-singular tropical curve in a non-singular projective tropical
toric surface ${\mathbb{T}}\mathbb{P}_{\Sigma}$. Then the set of all circuits
on $C$ spans a ${\mathbb{Z}}_{2}$-vector subspace of $\operatorname{Div}(C)$.

######  Definition 5.4.

Let $C$ be a non-singular tropical curve in ${\mathbb{T}}\mathbb{P}_{\Sigma}$.
We denote by $\operatorname{Circuit}(C)$ the ${\mathbb{Z}}_{2}$-vector
subspace of $\operatorname{Div}(C)$ spanned by the circuits on $C$.

###  5.2. Real tropical curves with only ovals

For $(C,\mathcal{E})$ a non-singular real tropical curve in the non-singular
tropical toric surface ${\mathbb{T}}\mathbb{P}_{\Sigma}$, we say that a
connected component ${\mathbb{R}}\rho_{\mathcal{E}}$ of the real part
${\mathbb{R}}C_{\mathcal{E}}$ is an _oval_ if ${\mathbb{R}}\rho_{\mathcal{E}}$
divides ${\mathbb{R}}\mathbb{P}_{\Sigma}$ into two connected components.

######  Lemma 5.5.

Let $(C,\mathcal{E})$ be a non-singular real tropical curve in a non-singular
tropical projective toric surface ${\mathbb{T}}\mathbb{P}_{\Sigma}$. Let
$\rho$ be a twisted cycle on $(C,\mathcal{E})$ which does not contain any
1-valent vertex of $C$. Then the real part ${\mathbb{R}}\rho_{\mathcal{E}}$ is
an oval.

###### Proof.

Since $\rho$ does not contain any 1-valent vertex of $C$, the real part
${\mathbb{R}}\rho_{\mathcal{E}}$ of $\rho$ is strictly contained in an orthant
of the real algebraic torus
$({\mathbb{R}}^{\times})^{2}\subset{\mathbb{R}}\mathbb{P}_{\Sigma}$. Since
${\mathbb{R}}\rho_{\mathcal{E}}$ is homeomorphic to the circle $S^{1}$, it
must then divide that orthant into two connected components. Therefore
${\mathbb{R}}\rho_{\mathcal{E}}$ is an oval. ∎

By definition, the homology class $[{\mathbb{R}}\rho_{\mathcal{E}}]$ of an
oval is trivial in the homology group of the ambient space
$H_{1}({\mathbb{R}}\mathbb{P}_{\Sigma};{\mathbb{Z}}_{2})$. Then if all the
connected components of the real part ${\mathbb{R}}C_{\mathcal{E}}$ of a non-
singular real tropical curve $(C,\mathcal{E})$ are ovals, we obtain that the
homology class $[{\mathbb{R}}C_{\mathcal{E}}]$, given as the sum of homology
class of the connected components, is trivial in
$H_{1}({\mathbb{R}}\mathbb{P}_{\Sigma};{\mathbb{Z}}_{2})$.

If we only assume that the class $[{\mathbb{R}}C_{\mathcal{E}}]$ is trivial in
$H_{1}({\mathbb{R}}\mathbb{P}_{\Sigma};{\mathbb{Z}}_{2})$, we have the
following equivalent condition on the Newton polygon of $C$.

######  Proposition 5.6.

Let $(C,\mathcal{E})$ be a non-singular real tropical curve in
${\mathbb{T}}\mathbb{P}_{\Sigma}$, and let $\Delta$ be the Newton polygon of
$C$ dual to the fan $\Sigma$. The homology class
$[{\mathbb{R}}C_{\mathcal{E}}]$ is trivial in
$H_{1}({\mathbb{R}}\mathbb{P}_{\Sigma};{\mathbb{Z}}_{2})$ if and only if every
edge of $\Delta$ has even lattice length.

######  Definition 5.7.

A non-singular real tropical curve $(C,\mathcal{E})$ in
${\mathbb{T}}\mathbb{P}_{\Sigma}$ is said to be of _strict even degree_ if
every edge of the associated Newton polygon $\Delta$ has even lattice length
(in particular, the non-singular real tropical curve $(C,\mathcal{E})$
satisfies Proposition 5.6).

###### Proof of Proposition 5.6.

The homology class $[{\mathbb{R}}C_{\mathcal{E}}]$ is trivial in
$H_{1}({\mathbb{R}}\mathbb{P}_{\Sigma};{\mathbb{Z}}_{2})$ if and only if
${\mathbb{R}}C_{\mathcal{E}}$ intersects each 1-dimensional face of the
stratification of ${\mathbb{R}}\mathbb{P}_{\Sigma}$ an even number of times.
Now there is a one-to-one correspondence between the 1-valent vertices of $C$
on a 1-dimensional stratum of ${\mathbb{T}}\mathbb{P}_{\Sigma}$ and the
intersection points of ${\mathbb{R}}C_{\mathcal{E}}$ with the corresponding
1-dimensional stratum of ${\mathbb{R}}\mathbb{P}_{\Sigma}$. Therefore the
homology class $[{\mathbb{R}}C_{\mathcal{E}}]$ is trivial in
$H_{1}({\mathbb{R}}\mathbb{P}_{\Sigma};{\mathbb{Z}}_{2})$ if and only if the
non-singular tropical curve $C$ intersect each 1-dimensional stratum of
${\mathbb{T}}\mathbb{P}_{\Sigma}$ in an even number of points. The latter
condition can be translated into the desired Newton polygon condition. ∎

######  Corollary 5.8.

Let $(C,\mathcal{E})$ be a non-singular real tropical curve in
${\mathbb{T}}\mathbb{P}_{\Sigma}$. If all the connected components of the real
part ${\mathbb{R}}C_{\mathcal{E}}$ are ovals, then $(C,\mathcal{E})$ is of
strict even degree in ${\mathbb{T}}\mathbb{P}_{\Sigma}$.

We want to study the sufficient conditions for ${\mathbb{R}}C_{\mathcal{E}}$
to contain only ovals, in two distinct directions. The first idea comes from
the well known fact that a non-singular real algebraic curve of even degree in
$\mathbb{P}^{2}$ has real part consisting only of ovals.

We denote by $\Delta_{d}$ the 2-dimensional lattice polytope given as the
convex hull of the set $\\{(0,0),(d,0),(0,d)\\}\subset{\mathbb{R}}^{2}_{\geq
0}$. We will say that a lattice polytope $\Delta$ is a _lattice transformation
of_ a lattice polytope $\Delta^{\prime}$ if $\Delta$ is given as the image of
$\Delta^{\prime}$ via the composition of a lattice translation with a linear
map in $\operatorname{{\mathbf{G}L}}(2,{\mathbb{Z}})$.

######  Proposition 5.9.

Let $(C,\mathcal{E})$ be a non-singular real tropical curve of strict even
degree in ${\mathbb{T}}\mathbb{P}_{\Sigma}$. Assume that the Newton polygon
$\Delta$ of $C$, dual to the fan $\Sigma$, is a lattice transformation of some
$\Delta_{2k}$, for $k\geq 1$. Then all the connected components of the real
part ${\mathbb{R}}C_{\mathcal{E}}$ are ovals.

###### Proof.

Assume first that $C$ is of even degree in ${\mathbb{T}}\mathbb{P}^{2}$, so
that the Newton polygon of $C$ is the lattice polytope $\Delta_{2k}$ for some
$k\geq 1$. The homology group
$H_{1}({\mathbb{R}}\mathbb{P}^{2};{\mathbb{Z}}_{2})$ is isomorphic to
${\mathbb{Z}}_{2}$, thus the real part ${\mathbb{R}}C_{\mathcal{E}}$ contains
at most one pseudo-line, as they would otherwise intersect. By assumption on
the Newton polygon, for each 1-dimensional stratum $\tau$ of
${\mathbb{T}}\mathbb{P}^{2}$, we have $|C\cap\tau|=0\mod 2$. Now there is a
one-to-one correspondence between the 1-valent vertices of $C$ on a
1-dimensional stratum of ${\mathbb{T}}\mathbb{P}_{\Sigma}$ and the
intersection points of ${\mathbb{R}}C_{\mathcal{E}}$ with the corresponding
1-dimensional stratum of ${\mathbb{R}}\mathbb{P}_{\Sigma}$. Therefore the
homology class $[{\mathbb{R}}C_{\mathcal{E}}]$ is trivial in
$H_{1}({\mathbb{R}}\mathbb{P}^{2};{\mathbb{Z}}_{2})$. Thus, the real part
${\mathbb{R}}C_{\mathcal{E}}$ does not contain a pseudo-line, hence every
connected component of ${\mathbb{R}}C_{\mathcal{E}}$ must be an oval.

Assume now that $\Delta$ is a lattice transformation of some $\Delta_{2k}$.
The lattice transformation induces a homeomorphism between
$({\mathbb{R}}\mathbb{P}_{\Sigma},{\mathbb{R}}C_{\mathcal{E}})$ and
$({\mathbb{R}}\mathbb{P}^{2},{\mathbb{R}}C_{\mathcal{E}^{\prime}}^{\prime})$,
where $C^{\prime}$ is a non-singular tropical curve of even degree in
${\mathbb{T}}\mathbb{P}^{2}$. As we saw above, the connected components of
${\mathbb{R}}C_{\mathcal{E}^{\prime}}^{\prime}$ are all ovals. Therefore all
the connected components of ${\mathbb{R}}C_{\mathcal{E}}$ are ovals. ∎

The second idea we want to explore is to consider the configurations of twists
preserving the “all ovals” property no matter the ambient tropical toric
surface.

######  Proposition 5.10.

Let $(C,\mathcal{E})$ be a non-singular real tropical curve of strict even
degree in ${\mathbb{T}}\mathbb{P}_{\Sigma}$. Assume that every edge in the set
of twisted edges $T$ on $C$ is non-exposed. Then all the connected components
of the real part ${\mathbb{R}}C_{\mathcal{E}}$ are ovals.

###### Proof.

By assumption, there exists a unique twisted cycle $\rho$ on $(C,\mathcal{E})$
intersecting the 1-dimensional strata of ${\mathbb{T}}\mathbb{P}_{\Sigma}$.
Since every edge of the Newton polygon $\Delta$ has even lattice length, the
homology class $[{\mathbb{R}}C_{\mathcal{E}}]$ is trivial in
$H_{1}({\mathbb{R}}\mathbb{P}_{\Sigma};{\mathbb{Z}}_{2})$ by Proposition 5.6.
By Lemma 5.5, every twisted cycle in $(C,\mathcal{E})$ distinct from $\rho$
has real part an oval. We can write the homology class
$[{\mathbb{R}}\rho_{\mathcal{E}}]$ as the sum of homology classes (with
${\mathbb{Z}}_{2}$-coefficients)
$[{\mathbb{R}}C_{\mathcal{E}}]+\sum_{i=1}^{r}[{\mathbb{R}}(\rho_{i})_{\mathcal{E}}]$,
with each $\rho_{i}$ being a twisted cycle on $(C,\mathcal{E})$ distinct from
$\rho$. Since all the summands are trivial in
$H_{1}({\mathbb{R}}\mathbb{P}_{\Sigma};{\mathbb{Z}}_{2})$, the class
$[{\mathbb{R}}\rho_{\mathcal{E}}]$ is trivial in
$H_{1}({\mathbb{R}}\mathbb{P}_{\Sigma};{\mathbb{Z}}_{2})$, hence
${\mathbb{R}}\rho_{\mathcal{E}}$ is an oval. ∎

######  Example 5.11.

A non-singular real tropical curve $(C,\mathcal{E})$ of strict even degree in
${\mathbb{T}}\mathbb{P}_{\Sigma}$ has real part only ovals if its set of
twisted edges $T$ is given as cycle-disjoint union of sets of twisted edges of
the same form as in Figure 7(b).

######  Corollary 5.12.

Let $(C,\mathcal{E})$ be a non-singular real tropical curve of strict even
degree in ${\mathbb{T}}\mathbb{P}_{\Sigma}$ with configuration of twists
$T\in\operatorname{Circuit}(C)$. Then all the connected components of the real
part ${\mathbb{R}}C_{\mathcal{E}}$ are ovals.

##  6\. Even and odd ovals

Let $(C,\mathcal{E})$ be a non-singular real tropical curve in
${\mathbb{T}}\mathbb{P}_{\Sigma}$ with real part consisting only of ovals. Let
${\mathbb{R}}\rho_{\mathcal{E}}$ be such an oval. We say that
${\mathbb{R}}\rho_{\mathcal{E}}$ is _even_ if ${\mathbb{R}}\rho_{\mathcal{E}}$
is contained in an even number of discs in ${\mathbb{R}}\mathbb{P}_{\Sigma}$
bounded by another oval of ${\mathbb{R}}C_{\mathcal{E}}$, and otherwise we say
that ${\mathbb{R}}\rho_{\mathcal{E}}$ is _odd_.

###  6.1. Counting in terms of distribution of signs

We want to count the number of even and odd ovals in
${\mathbb{R}}C_{\mathcal{E}}$ based on a distribution of signs $\delta$
associated to $\mathcal{E}$ on the dual subdivision $\Delta_{C}$ of $C$.
Recall that a non-singular tropical curve $C$ is of _strict even degree_ in
${\mathbb{T}}\mathbb{P}_{\Sigma}$ if all the edges of the Newton polygon
$\Delta$ of $C$ (dual to the fan $\Sigma$) are of even lattice length.

######  Lemma 6.1.

Let $C$ be a non-singular tropical curve of strict even degree in
${\mathbb{T}}\mathbb{P}_{\Sigma}$, and let $\delta$ be a distribution of signs
on $\Delta_{C}$. Assume that the connected components of
${\mathbb{R}}C_{\delta}$ are all ovals. Any two connected components
$v^{\vee},(v^{\prime})^{\vee}\subset({\mathbb{T}}\mathbb{P}_{\Sigma}\backslash
C)^{*}$ are contained in the same number modulo 2 of ovals of
${\mathbb{R}}C_{\delta}$ if and only if the integer points
$v,v^{\prime}\in\Delta_{C}^{*}\cap{\mathbb{Z}}^{2}$ dual to
$v^{\vee},(v^{\prime})^{\vee}$ satisfy $\delta(v)=\delta(v^{\prime})$.

###### Proof.

Let $(\mathcal{C}_{t})$ be a family of non-singular real algebraic curves
converging to the tropical limit $(C,\delta)$. For $t\in{\mathbb{R}}_{>0}$
small enough, the curve $\mathcal{C}_{t}$ is a non-singular real algebraic
curve such that the pair
$(\mathbb{P}_{\Sigma}({\mathbb{R}}),\mathcal{C}_{t}({\mathbb{R}}))$ is
homeomorphic to $({\mathbb{R}}\mathbb{P}_{\Sigma},{\mathbb{R}}C_{\delta})$ by
Theorem 3.3. In particular, we have a one-to-one correspondence between the
connected components of
$\mathbb{P}_{\Sigma}({\mathbb{R}})\backslash\mathcal{C}_{t}({\mathbb{R}})$ and
the connected components of
${\mathbb{R}}\mathbb{P}_{\Sigma}\backslash{\mathbb{R}}C_{\delta}$. Let $f_{t}$
be a defining polynomial of $\mathcal{C}_{t}$, such that the signs of the
monomials of $f_{t}$ induces the distribution of signs $\delta$ on
$\Delta_{C}\cap{\mathbb{Z}}^{2}$. Let $D_{t}$ be a connected component of
$\mathbb{P}_{\Sigma}({\mathbb{R}})\backslash\mathcal{C}_{t}({\mathbb{R}})$ and
$v^{\vee}$ the corresponding connected component of
${\mathbb{R}}\mathbb{P}_{\Sigma}\backslash{\mathbb{R}}C_{\delta}$. Let
$v\in\Delta_{C}^{*}\cap{\mathbb{Z}}^{2}$ be the integer point dual to
$v^{\vee}$. By definition of $\delta$, we have $f_{t}|_{D_{t}}>0$ if and only
if $\delta(v)>0$, and similarly $f_{t}|_{D_{t}}<0$ if and only if
$\delta(v)<0$. The value of $f_{t}(x)$ along a path in
$\mathbb{P}_{\Sigma}({\mathbb{R}})$ intersecting
$\mathcal{C}_{t}({\mathbb{R}})$ transversely changes of sign at each
intersection of the path with an oval. Therefore, any two connected components
$v^{\vee},(v^{\prime})^{\vee}\subset({\mathbb{T}}\mathbb{P}_{\Sigma}\backslash
C)^{*}$ are contained in the same number of ovals modulo 2 of
${\mathbb{R}}C_{\delta}$ if and only if the dual integer points
$v,v^{\prime}\in\Delta_{C}^{*}\cap{\mathbb{Z}}^{2}$ satisfy
$\delta(v)=\delta(v^{\prime})$. ∎

######  Corollary 6.2.

Let $C$ be a non-singular tropical curve of strict even degree in
${\mathbb{T}}\mathbb{P}_{\Sigma}$, and let $\delta$ and $\delta^{\prime}$ be
distributions of signs on $\Delta_{C}$ such that all the connected components
of the real parts ${\mathbb{R}}C_{\delta}$ and
${\mathbb{R}}C_{\delta^{\prime}}$ are ovals. Assume that there exists a
connected component $v_{0}^{\vee}$ of $({\mathbb{T}}\mathbb{P}^{2}\backslash
C)^{*}$ contained in the same number modulo 2 of ovals in
${\mathbb{R}}C_{\delta}$ and ${\mathbb{R}}C_{\delta^{\prime}}$. Assume that
$\delta(v_{0})=\delta^{\prime}(v_{0})$, with
$v_{0}\in\Delta_{C}^{*}\cap{\mathbb{Z}}^{2}$ the integer point dual to
$v_{0}^{\vee}$. Then a connected component $v^{\vee}$ in
$({\mathbb{T}}\mathbb{P}^{2}\backslash C)^{*}$ is contained in the same number
modulo 2 of ovals of ${\mathbb{R}}C_{\delta}$ and
${\mathbb{R}}C_{\delta^{\prime}}$ if and only if
$\delta(v)=\delta^{\prime}(v)$, with $v\in\Delta_{C}^{*}\cap{\mathbb{Z}}^{2}$
the integer point dual to $v^{\vee}$.

###### Proof.

Let $s$ be the number modulo 2 of ovals of ${\mathbb{R}}C_{\delta}$ containing
$v_{0}^{\vee}$. By assumption, the number $s$ is also the number modulo 2 of
ovals of ${\mathbb{R}}C_{\delta^{\prime}}$ containing $v_{0}^{\vee}$. By Lemma
6.1 a connected component $v^{\vee}$ is contained in $s$ ovals modulo 2 in
${\mathbb{R}}C_{\delta}$ if and only if $\delta(v)=\delta(v_{0})$. Again by
Lemma 6.1, a connected component $v^{\vee}$ is contained in $s$ ovals modulo 2
in ${\mathbb{R}}C_{\delta^{\prime}}$ if and only if
$\delta^{\prime}(v)=\delta^{\prime}(v_{0})$. Therefore a connected component
$v^{\vee}$ is contained in the same number modulo 2 of ovals of
${\mathbb{R}}C_{\delta}$ and ${\mathbb{R}}C_{\delta^{\prime}}$ if and only if
$\delta(v)=\delta^{\prime}(v)$. ∎

In order to use Lemma 6.1 and Corollary 6.2, we show that there exists a
distinguished twisted cycle in $(C,\mathcal{E})$ with real part an even oval,
called the _special oval_ in reference to [Haa97].

######  Lemma 6.3.

Let $(C,\mathcal{E})$ be a non-singular real tropical curve of strict even
degree in ${\mathbb{T}}\mathbb{P}_{\Sigma}$ such that all the edges of the set
of twisted edges $T$ on $C$ are non-exposed. Then there exists a unique
twisted cycle $\rho_{0}$ on $(C,\mathcal{E})$ containing some unbounded edges
of $C$ and such that the real part ${\mathbb{R}}(\rho_{0})_{\mathcal{E}}$ is
an oval of ${\mathbb{R}}C_{\mathcal{E}}$ not contained in any other oval.

######  Definition 6.4.

Let $(C,\mathcal{E})$ be a non-singular real tropical curve satisfying the
assumptions of Lemma 6.3. The unique twisted cycle $\rho_{0}$ in
$(C,\mathcal{E})$ satisfying the conditions of Lemma 6.3 is called the
_special twisted cycle_ , and the real part
${\mathbb{R}}(\rho_{0})_{\mathcal{E}}$ is called the _special oval_.

Note that we showed that the real part of the special twisted cycle is an oval
in the proof of Proposition 5.10.

###### Proof of Lemma 6.3.

Since all the edges of $T$ are non-exposed, a closed twisted walk starting at
a one-valent vertex $v$ of $C$ does not meet any twisted edge. In particular,
such a closed twisted walk goes only through exposed edges, and goes through
every unbounded edge of $C$ twice. Hence there exists a unique twisted cycle
$\rho_{0}$ on $(C,\mathcal{E})$ going through all the unbounded edges of $C$.
In that case, all ovals distinct from ${\mathbb{R}}(\rho_{0})_{\mathcal{E}}$
are strictly contained in an orthant of the real algebraic torus
$({\mathbb{R}}^{\times})^{2}\subset{\mathbb{R}}\mathbb{P}_{\Sigma}$ while the
oval ${\mathbb{R}}(\rho_{0})_{\mathcal{E}}$ intersects several orthants. Hence
the real part ${\mathbb{R}}(\rho_{0})_{\mathcal{E}}$ cannot be contained in
any other oval of ${\mathbb{R}}C_{\mathcal{E}}$, therefore $\rho_{0}$ is the
unique special twisted cycle of $(C,\mathcal{E})$. ∎

###  6.2. Even circuits

We consider the difference of number of even and odd ovals between two
distinct real parts of a common non-singular real tropical curve, so that the
corresponding sets of twisted edges differ by an _even_ circuit.

######  Definition 6.5.

Let $v=(v_{1},v_{2})\in\Delta_{C}\cap{\mathbb{Z}}^{2}$ be an integer point of
the dual subdivision of a non-singular tropical curve $C$. We say that $v$ is
_even_ if both $v_{1}$ and $v_{2}$ are even, and we say that $v$ is _odd_
otherwise. A multi-bridge $B$ on $C$ is said to be _even_ if the dual set
$B^{\vee}\subset\Delta_{C}$ contains an even integer point, and otherwise we
say that $B$ is _even-free_.

Recall that $\operatorname{Circuit}(C)$ denotes the ${\mathbb{Z}}_{2}$-vector
subspace of $\operatorname{Div}(C)$ such that a configuration of twists in
$\operatorname{Circuit}(C)$ is given as a sum of circuits in $C$. We denote by
$\operatorname{EvCirc}(C)\subset\operatorname{Circuit}(C)$ the
${\mathbb{Z}}_{2}$-vector subspace spanned by even circuits. From now on, for
$(C,\mathcal{E})$ a non-singular real tropical curve, we denote by
$p_{\mathcal{E}}$ and $n_{\mathcal{E}}$ the number of even and odd ovals in
${\mathbb{R}}C_{\mathcal{E}}$.

######  Proposition 6.6.

Let $(C,\mathcal{E})$ be a non-singular real tropical curve of strict even
degree in ${\mathbb{T}}\mathbb{P}_{\Sigma}$. Assume that the set of twisted
edges $T$ induced by $\mathcal{E}$ belongs to $\operatorname{Circuit}(C)$. Let
$T^{\prime}\in\operatorname{EvCirc}(C)$ such that $T\cap
T^{\prime}=\emptyset$, and let $\mathcal{E}^{\prime}$ be a real phase
structure on $C$ inducing the configuration of twists
$T+T^{\prime}\in\operatorname{Circuit}(C)$. Then we have
$p_{\mathcal{E}^{\prime}}\leq p_{\mathcal{E}}$ and
$n_{\mathcal{E}^{\prime}}\leq n_{\mathcal{E}}$.

In order to prove Proposition 6.6, we will need the following definition.

######  Definition 6.7.

Let $C$ be a non-singular tropical curve in a non-singular projective tropical
toric surface ${\mathbb{T}}\mathbb{P}_{\Sigma}$, and let $T$ be an admissible
set of twisted edges on $C$. The _zone decomposition_ $\Delta_{T}$ of the
Newton polygon $\Delta$ of $C$ is the subdivision of $\Delta$ induced by the
set of edges $T^{\vee}$ dual to the edges of $T$. We can write

$\Delta_{T}:=\bigsqcup_{i=0}^{k}Z_{i}^{T}/\sim,$

where $Z_{i}^{T}$ is a 2-dimensional face of $\Delta_{T}$ and $\sim$ is the
gluing along the edges of $T^{\vee}$. The 2-dimensional faces $Z_{i}^{T}$ are
called _zones_ [Haa97]. A _zone partition_ $Y_{T}:=Y_{0}^{T}\sqcup Y_{1}^{T}$
is a partition of the zones of $\Delta_{T}$ such that any two zones
intersecting in a 1-dimensional set belong to distinct $Y_{j}^{T}$.

###### Proof of Proposition 6.6.

We want to construct an injective homomorphism

$\phi:H_{1}({\mathbb{R}}C_{\mathcal{E}^{\prime}};{\mathbb{Z}}_{2})\rightarrow
H_{1}({\mathbb{R}}C_{\mathcal{E}};{\mathbb{Z}}_{2})$

such that the image by $\phi$ of the class of an oval in
${\mathbb{R}}C_{\mathcal{E}^{\prime}}$ of parity $s\in\\{\text{even, odd}\\}$
is given as the sum of classes of some ovals in ${\mathbb{R}}C_{\mathcal{E}}$
of parity $s$. By definition of multi-bridge, the graph $C\backslash
T^{\prime}$ has exactly two connected components
$C_{0}^{\prime},C_{1}^{\prime}$. Let $C_{i}:=\overline{C_{i}^{\prime}\cup
T^{\prime}}$ for $i=0,1$. Up to action of $\mathcal{R}^{2}$, we can assume
that $\mathcal{E}$ and $\mathcal{E}^{\prime}$ are equal on $C_{1}$. Said
otherwise, we can assume that
${\mathbb{R}}(C_{1})_{\mathcal{E}}={\mathbb{R}}(C_{1})_{\mathcal{E}^{\prime}}$.
Let $\varepsilon\in{\mathbb{Z}}_{2}^{2}$ be the direction modulo 2 of the
edges of $T^{\prime}$. Seeing $\varepsilon$ as a symmetry in
$\mathcal{R}^{2}$, we get that
${\mathbb{R}}(C_{0})_{\mathcal{E}}=\varepsilon({\mathbb{R}}(C_{0})_{\mathcal{E}^{\prime}})$.

Let $Z_{i}$ be the zone of $\Delta_{T^{\prime}}$ containing the edges dual to
edges of $C_{i}^{\prime}$, for $\Delta_{T^{\prime}}$ the zone decomposition
induced by $T^{\prime}$. By Lemma 6.3, there exists a connected component
$v_{0}^{\vee}$ of $({\mathbb{T}}\mathbb{P}_{\Sigma}\backslash C)^{*}$ lying
neighbourly outside special ovals of both ${\mathbb{R}}C_{\mathcal{E}}$ and
${\mathbb{R}}C_{\mathcal{E}^{\prime}}$. Up to renumbering the zones of
$\Delta_{T^{\prime}}$, we can assume that the integer point $v_{0}$ dual to
$v_{0}^{\vee}$ belong to the union of symmetric copies
$Z_{1}^{*}\cap{\mathbb{Z}}^{2}$. We can choose two distributions of signs
$\delta$ and $\delta^{\prime}$ on $\Delta_{C}\cap{\mathbb{Z}}^{2}$ inducing
the real phase structures $\mathcal{E}$ and $\mathcal{E}^{\prime}$ and such
that $\delta=\delta^{\prime}$ on $Z_{1}\cap{\mathbb{Z}}^{2}$. In particular,
we obtain $\delta(v_{0})=\delta^{\prime}(v_{0})$, hence the assumptions of
Corollary 6.2 are satisfied.

Every edge of $T^{\prime}$ is dual to an edge of $\Delta_{C}$ containing an
even vertex. We denote by $C_{i,j}^{\prime}$ the connected components of
$C\backslash(T\cup T^{\prime})$, such that each $C_{i,j}^{\prime}$ is
contained in the connected component $C_{i}^{\prime}$ of $C\backslash
T^{\prime}$. Let $Z_{i,j}$ be the zone in $\Delta_{T+T^{\prime}}$ containing
the edges dual to edges of $C_{i,j}^{\prime}$ for any $i,j$. Since the
distributions of signs $\delta$ and $\delta^{\prime}$ are equal on
$Z_{1}\cap{\mathbb{Z}}^{2}$, we obtain that for any $Z_{i,j}$ in the zone
decomposition $\Delta_{T+T^{\prime}}$, the distributions of signs $\delta$ and
$\delta^{\prime}$ are either both of Harnack type or both of inverse Harnack
type on $Z_{i,j}\cap{\mathbb{Z}}^{2}$. Since
${\mathbb{R}}(C_{0})_{\mathcal{E}}=\varepsilon({\mathbb{R}}(C_{0})_{\mathcal{E}^{\prime}})$,
we obtain in particular that for any integer point $v$ in the union of
symmetric copies $Z_{0}^{*}\cap{\mathbb{Z}}^{2}$, the equality
$\delta(v)=\delta^{\prime}(\varepsilon(v))$ is satisfied.

We start the construction of the homomorphism $\phi$ by considering the ovals
given as real part of twisted cycles that do not meet the set of twisted edges
$T^{\prime}$. Every twisted cycle of $(C,\mathcal{E}^{\prime})$ contained in a
connected component $C_{i}^{\prime}$ of $C\backslash T^{\prime}$, for $i=1,2$,
can be identified with the twisted cycle in $(C,\mathcal{E})$ with the same
edge support. If such a twisted cycle $\rho$ is contained in $C_{1}^{\prime}$,
we can also identify the real parts ${\mathbb{R}}\rho_{\mathcal{E}}$ and
${\mathbb{R}}\rho_{\mathcal{E}^{\prime}}$, and if $\rho$ is contained in
$C_{0}^{\prime}$, we have
${\mathbb{R}}\rho_{\mathcal{E}}=\varepsilon({\mathbb{R}}\rho_{\mathcal{E}^{\prime}})$.

Assume that $\rho$ is a twisted cycle of $(C,\mathcal{E}^{\prime})$ and
$(C,\mathcal{E})$ which is contained in $C_{1}^{\prime}$. Since the
distribution of signs $\delta$ and $\delta^{\prime}$ are equal on
$Z_{1}\cap{\mathbb{Z}}^{2}$, by Corollary 6.2 the ovals
${\mathbb{R}}\rho_{\mathcal{E}}$ and ${\mathbb{R}}\rho_{\mathcal{E}^{\prime}}$
are contained in the same number modulo 2 of ovals of
${\mathbb{R}}C_{\mathcal{E}}$ and ${\mathbb{R}}C_{\mathcal{E}^{\prime}}$,
respectively. Therefore we set
$\phi[{\mathbb{R}}\rho_{\mathcal{E}^{\prime}}]=[{\mathbb{R}}\rho_{\mathcal{E}}]$
for every twisted cycle $\rho$ of $(C,\mathcal{E}^{\prime})$ contained in the
connected component $C_{1}^{\prime}$.

Assume now that $\rho$ is a twisted cycle of $(C,\mathcal{E}^{\prime})$ and
$(C,\mathcal{E})$ contained in $C_{0}^{\prime}$. Since
${\mathbb{R}}(C_{0})_{\mathcal{E}}=\varepsilon({\mathbb{R}}(C_{0})_{\mathcal{E}^{\prime}})$,
the oval ${\mathbb{R}}\rho_{\mathcal{E}}$ of ${\mathbb{R}}C_{\mathcal{E}}$ is
obtained from the oval ${\mathbb{R}}\rho_{\mathcal{E}^{\prime}}$ of
${\mathbb{R}}C_{\mathcal{E}^{\prime}}$ by applying the symmetry
$\varepsilon\in\mathcal{R}^{2}$. Then since
$\delta(v)=\delta^{\prime}(\varepsilon(v))$ for every integer point $v\in
Z_{0}^{*}\cap{\mathbb{Z}}^{2}$, by Corollary 6.2 the ovals
${\mathbb{R}}\rho_{\mathcal{E}}$ and ${\mathbb{R}}\rho_{\mathcal{E}^{\prime}}$
are contained in the same number modulo 2 of ovals of
${\mathbb{R}}C_{\mathcal{E}}$ and ${\mathbb{R}}C_{\mathcal{E}^{\prime}}$,
respectively. Therefore we set
$\phi[{\mathbb{R}}\rho_{\mathcal{E}^{\prime}}]=[{\mathbb{R}}\rho_{\mathcal{E}}]$
for every twisted cycle $\rho$ of $(C,\mathcal{E}^{\prime})$ contained in the
connected component $C_{0}^{\prime}$.

Let finally $\rho$ be a twisted cycle in $(C,\mathcal{E}^{\prime})$ going
through an edge of $T^{\prime}$. By assumption on $T^{\prime}$, the edges of
$\rho\cap T^{\prime}$ lie in the boundary of connected components of
${\mathbb{T}}\mathbb{P}_{\Sigma}\backslash C$ dual to even integer points in
$\Delta_{C}\cap{\mathbb{Z}}^{2}$. Recall that $\delta=\delta^{\prime}$ on
$Z_{1}^{*}\cap{\mathbb{Z}}^{2}$ and $\delta=\delta^{\prime}\circ\varepsilon$
on $Z_{0}^{*}\cap{\mathbb{Z}}^{2}$. Let
${\mathbb{R}}(\rho^{\prime})_{\mathcal{E}}$ be the real part of some twisted
cycle $\rho^{\prime}$ of $(C,\mathcal{E})$ satisfying $(\rho^{\prime}\cap
C_{1})\subset\rho$, and let ${\mathbb{R}}(\rho^{\prime\prime})_{\mathcal{E}}$
is the real part of some twisted cycle $\rho^{\prime\prime}$ of
$(C,\mathcal{E})$ satisfying $(\rho^{\prime\prime}\cap C_{0})\subset\rho$.
Then for each even integer point $v\in\Delta_{C}\cap{\mathbb{Z}}^{2}$, there
exists a symmetric copy $\varepsilon(v^{\vee})$ of the connected component
$v^{\vee}\subset({\mathbb{T}}\mathbb{P}_{\Sigma}\backslash C)$ such that
$\varepsilon(v^{\vee})$ lies either neighbourly inside or neighbourly outside
${\mathbb{R}}\rho_{\mathcal{E}^{\prime}},{\mathbb{R}}(\rho^{\prime})_{\mathcal{E}}$
and $\varepsilon({\mathbb{R}}(\rho^{\prime\prime})_{\mathcal{E}})$. Since
$\delta,\delta^{\prime}$ and $\delta^{\prime}\circ\varepsilon$ are equal on
the integer points $(T^{\prime})^{\vee}\cap{\mathbb{Z}}^{2}$, by Corollary
6.2, the oval ${\mathbb{R}}\rho_{\mathcal{E}^{\prime}}$ is contained in the
same number modulo 2 of ovals in ${\mathbb{R}}C_{\mathcal{E}^{\prime}}$ as the
number modulo 2 of ovals of ${\mathbb{R}}C_{\mathcal{E}}$ containing
${\mathbb{R}}(\rho^{\prime})_{\mathcal{E}}$ and
${\mathbb{R}}(\rho^{\prime\prime})_{\mathcal{E}}$. Therefore we set
$\phi[{\mathbb{R}}\rho_{\mathcal{E}^{\prime}}]=\Gamma\in
H_{1}({\mathbb{R}}C_{\mathcal{E}};{\mathbb{Z}}_{2})$, for $\rho$ any twisted
cycle of $(C,\mathcal{E}^{\prime})$ going through an edge of $T^{\prime}$ and
for $\Gamma$ the sum of classes
$[{\mathbb{R}}(\rho^{\prime\prime\prime})_{\mathcal{E}}]$ of ovals of
${\mathbb{R}}C_{\mathcal{E}}$ with associated twisted cycles
$\rho^{\prime\prime\prime}$ satisfying $(\rho^{\prime\prime\prime}\cap
C_{i})\subset\rho$ for some $i$.

Then the homomorphism $\phi$ is injective and preserve the parity of ovals,
hence we obtain $p_{\mathcal{E}^{\prime}}\leq p_{\mathcal{E}}$ and
$n_{\mathcal{E}^{\prime}}\leq n_{\mathcal{E}}$. ∎

######  Remark 6.8.

In the case ${\mathbb{T}}\mathbb{P}_{\Sigma}={\mathbb{T}}\mathbb{P}^{2}$, we
can relax the assumptions of Proposition 6.6 so that
$T\in\operatorname{Div}(C)$ and $T^{\prime}$ belong to the
${\mathbb{Z}}_{2}$-vector subspace generated by even multi-bridges, using
Proposition 5.9. However, one needs to be consider the (not necessarily
unique) special oval in this case.

######  Remark 6.9.

We obtain in Proposition 6.6 a result similar to Haas’ result for maximal
curves in $\mathbb{P}^{2}$ obtained via combinatorial patchworking [Haa97,
Theorem 10.6.0.5]. These two results justify that in order to obtain examples
of non-singular real tropical curves with many even or odd ovals, we can
restrict to the case of even-free configuration of twists. Since all the duals
of configurations of twists satisfying Theorem 4.16 contain an even point, we
can then restrict to even-free multi-bridges.

###  6.3. Even-free circuits

Recall that a multi-bridge $B$ is _even-free_ if every integer point in
$B^{\vee}\cap{\mathbb{Z}}^{2}$ is odd. We denote by $\operatorname{EvFree}(C)$
the ${\mathbb{Z}}_{2}$-vector subspace of $\operatorname{Div}(C)$ generated by
even-free multi-bridges, and we denote by $\operatorname{EvFreeCirc}(C)$ the
${\mathbb{Z}}_{2}$-vector subspace of $\operatorname{Circuit}(C)$ generated by
even-free circuits.

######  Definition 6.10.

Let $T\in\operatorname{Circuit}(C)$ a configuration of twists on a non-
singular tropical curve $C$. The unique zone $Z^{T}$ of $\Delta_{T}$ meeting
the boundary edges of the Newton polygon $\Delta$ of $C$ is called the
_special zone_ [Haa97].

######  Proposition 6.11.

Let $(C,\mathcal{E})$ be a non-singular real tropical curve of strict even
degree in ${\mathbb{T}}\mathbb{P}_{\Sigma}$, with set of twisted edges
$T\in\operatorname{EvFreeCirc}(C)$. Let $Y_{T}:=Y_{1}^{T}\sqcup Y_{0}^{T}$ be
a zone partition such that $Y_{1}^{T}$ contains the special zone $Z^{T}$. Let
$p_{j}$ and $n_{j}$ be the number of even and odd interior integer points in
$Y_{j}^{T}$. Then

$\displaystyle p_{\mathcal{E}}$ $\displaystyle\geq n_{1}+p_{0}+1$
$\displaystyle n_{\mathcal{E}}$ $\displaystyle\geq p_{1}+n_{0}.$

Moreover, the real part ${\mathbb{R}}C_{\mathcal{E}}$ has at least
$n_{1}+p_{0}$ empty even ovals and $p_{1}+n_{0}$ empty odd ovals.

###### Proof.

Let $(v_{0})^{\vee}$ be a connected component of
$({\mathbb{T}}\mathbb{P}_{\Sigma}\backslash C)^{*}$ lying neighbourly outside
the real part ${\mathbb{R}}(\rho_{0})_{\mathcal{E}}$ of the special twisted
cycle $\rho_{0}$. Since ${\mathbb{R}}(\rho_{0})_{\mathcal{E}}$ is a special
oval, the connected component $v_{0}^{\vee}$ lies outside every oval of
${\mathbb{R}}C_{\mathcal{E}}$. Let $\delta$ be a distribution of signs on
$\Delta_{C}$ inducing the real phase structure $\mathcal{E}$. Then by Lemma
6.1, a connected component
$v^{\vee}\subset({\mathbb{T}}\mathbb{P}^{2}\backslash C)^{*}$ is contained in
an even number of ovals of ${\mathbb{R}}C_{\mathcal{E}}$ if and only if
$\delta(v)=\delta(v_{0})$, for $v\in\Delta_{C}^{*}\cap{\mathbb{Z}}^{2}$ the
integer point dual to $v^{\vee}$.

Let $Z_{1}^{T}$ and $Z_{0}^{T}$ be two adjacent zones in the zone
decomposition $\Delta_{T}$, that is the intersection $Z_{1}^{T}\cap Z_{0}^{T}$
is 1-dimensional. Let $v^{\prime}$ be an even integer point in $Z_{1}^{T}$ and
let $v^{\prime\prime}$ be an even integer point in $Z_{0}^{T}$. Since
$T\in\operatorname{EvFreeCirc}(C)$, the distribution of signs $\delta$ must
satisfy $\delta(v^{\prime})\neq\delta(v^{\prime\prime})$. In particular, the
distribution of signs $\delta$ is of Harnack type on
$Z_{1}^{T}\cap{\mathbb{Z}}^{2}$ if and only if $\delta$ is of inverse Harnack
type on $Z_{0}^{T}\cap{\mathbb{Z}}^{2}$. Therefore we can assume that $\delta$
is of Harnack type on each zone in $Y_{1}^{T}$ and of inverse Harnack type on
each zone in $Y_{0}^{T}$.

For each integer point $v\in\Delta_{C}\cap{\mathbb{Z}}^{2}$ contained in the
interior of a zone of $\Delta_{T}$, there exists a twisted cycle $\rho_{v}$ in
$(C,\mathcal{E})$ bounding the dual connected component
$v^{\vee}\subset{\mathbb{T}}\mathbb{P}_{\Sigma}\backslash C$. Moreover, there
exists a unique symmetric copy
$\varepsilon(v)\in\Delta_{C}^{*}\cap{\mathbb{Z}}^{2}$ of $v$ such that the
oval ${\mathbb{R}}(\rho_{v})_{\mathcal{E}}$ bounds the symmetric copy
$\varepsilon(v^{\vee})\subset({\mathbb{T}}\mathbb{P}_{\Sigma}\backslash
C)^{*}$. Then the oval ${\mathbb{R}}(\rho_{v})_{\mathcal{E}}$ is even if and
only if $\delta(\varepsilon(v))=\delta(v_{0})$.

We obtain the four following cases.

  * •

If the point $v$ is even and contained in a zone of $Y_{1}^{T}$, then
${\mathbb{R}}(\rho_{v})_{\mathcal{E}}$ is an odd oval.

  * •

If the point $v$ is odd and contained in a zone of $Y_{1}^{T}$, then
${\mathbb{R}}(\rho_{v})_{\mathcal{E}}$ is an even oval.

  * •

If the point $v$ is even and contained in a zone of $Y_{0}^{T}$, then
${\mathbb{R}}(\rho_{v})_{\mathcal{E}}$ is an even oval.

  * •

If the point $v$ is odd and contained in a zone of $Y_{0}^{T}$, then
${\mathbb{R}}(\rho_{v})_{\mathcal{E}}$ is an odd oval.

Hence we get

$\displaystyle p_{\mathcal{E}}$ $\displaystyle\geq n_{1}+p_{0}$ $\displaystyle
n_{\mathcal{E}}$ $\displaystyle\geq p_{1}+n_{0}.$

Since the special twisted cycle $\rho_{0}$ cannot be obtained as a twisted
cycle of the form $\rho_{v}$, we obtain then $p_{\mathcal{E}}\geq
n_{1}+p_{0}+1$. The “Moreover” statement comes from the fact that no oval of
${\mathbb{R}}C_{\mathcal{E}}$ can be contained in an oval of the form
${\mathbb{R}}(\rho_{v})_{\mathcal{E}}$, since they each bound a single
connected component of $({\mathbb{T}}\mathbb{P}_{\Sigma}\backslash C)^{*}$. ∎

######  Remark 6.12.

If ${\mathbb{T}}\mathbb{P}_{\Sigma}={\mathbb{T}}\mathbb{P}^{2}$, we can relax
the assumption $T\in\operatorname{EvFreeCirc}(C)$ to
$T\in\operatorname{EvFree}(C)$ in Proposition 6.11, by Proposition 5.9.
However, one needs to consider the (not necessarily unique) special twisted
cycle in that case.

###  6.4. Even-free complete bipartite graphs

For $(C,\mathcal{E})$ a non-singular real tropical curve of strict even degree
in ${\mathbb{T}}\mathbb{P}_{\Sigma}$ with configuration of twists $T$
belonging to $\operatorname{EvFreeCirc}(C)$, we want to count the number of
even and odd ovals of ${\mathbb{R}}C_{\mathcal{E}}$ not already counted by
Proposition 6.11. We restrict first to a case similar to the configuration of
twists Itenberg and Haas used to construct counter-examples to Ragsdale
conjecture ([Ite93], [Haa95], [Ite01]).

######  Lemma 6.13.

Let $(C,\mathcal{E})$ be a non-singular dividing $(M-2)$ real tropical curve
in ${\mathbb{T}}\mathbb{P}_{\Sigma}$, such that its set of twisted edges $T$
belong to $\operatorname{Circuit}(C)$. Then $T$ is dual to either a complete
bipartite graph $T^{\vee}\subset\Delta_{C}$ of the form $K_{2,2l}$ for some
$l\geq 1$, or to a complete tripartite graph of the form $K_{2,2,2}$.

###### Proof.

By assumption, all the edges of $T$ are non-exposed. Thus, by Theorem 4.22, we
get that $T^{\vee}$ is either a complete planar bipartite or a complete planar
tripartite graph. By Remark 4.23, the graph $T^{\vee}$ cannot contain a
subgraph of the form $K_{3,3}$. Hence $T^{\vee}$ is of the form

  1. (1)

either $K_{1,n}$, for some $n\geq 1$;

  2. (2)

or $K_{2,n}$, for some $n\geq 1$;

  3. (3)

or $K_{1,1,n}$, for some $n\geq 1$;

  4. (4)

or $K_{1,2,2}$;

  5. (5)

or $K_{2,2,2}$.

A complete bipartite graph $K_{1,n}$ is a tree, hence it does not contain any
cycle. Thus $T^{\vee}$ cannot satisfy Item 1.

If the dual graph of non-exposed twisted edges $\Gamma_{T}$ is a graph
$K_{1,1,n}$ (plus isolated vertices), then by twist-admissibility condition
(Equation 1) the set of twisted edges must contain some exposed edges. Hence
$T^{\vee}$ cannot satisfy Item 3.

Similarly, as we can see in Figure 8(b), if the dual graph of non-exposed
twisted edges $\Gamma_{T}$ is a graph $K_{1,2,2}$ (plus isolated vertices),
then by twist-admissibility condition (Equation 1) the set of twisted edges
must contain some exposed edges. Hence $T^{\vee}$ cannot satisfy Item 4.

Therefore, $T^{\vee}$ must satisfy either Item 2 or Item 5. Now if $T^{\vee}$
satisfies Item 2, then $n$ must be even, as otherwise by twist-admissibility
condition (Equation 1) the set of twisted edges must contain some exposed
edges. Thus in the case of Item 2, the graph $T^{\vee}$ must additionally be
of the form $K_{2,2l}$ for some $l\geq 1$. ∎

######  Lemma 6.14.

Let $(C,\mathcal{E})$ be a non-singular real tropical curve of strict even
degree in ${\mathbb{T}}\mathbb{P}_{\Sigma}$, such that its set of twisted
edges $T$ belong to $\operatorname{EvFreeCirc}(C)$ and is dual to a complete
bipartite graph $T^{\vee}\subset\Delta_{C}$ of the form $K_{2,2l}$, for some
$l\geq 1$. Let $Y_{T}:=Y_{1}^{T}\sqcup Y_{0}^{T}$ be the zone partition such
that $Y_{1}^{T}$ contains the special zone. There are exactly $l+1$ twisted
cycles meeting $T$ with real part an even oval and $l-1$ twisted cycles
meeting $T$ with real part an odd oval.

###### Proof.

By Theorem 4.22, we get that $(C,\mathcal{E})$ is a non-singular dividing
$(M-2)$ real tropical curve, thus ${\mathbb{R}}C_{\mathcal{E}}$ has $g-1$
connected components, for $g$ the number of primitive cycles on $C$. By
assumption on $T$, there are $g-2l-2$ primitive cycles $\gamma$ on $C$ such
that $\gamma\cap T=\emptyset$ (with $\gamma$ seen as a set of bounded edges),
so that there are $g-2l-2$ twisted cycles on $(C,\mathcal{E})$ bounding a
single connected component of ${\mathbb{T}}\mathbb{P}_{\Sigma}\backslash C$.
By Lemma 6.3, there is a unique twisted cycle on $(C,\mathcal{E})$ going
through all unbounded edges of $C$, and this twisted cycle does not meet $T$.
Thus, there are exactly $g-2-(g-2l-2)=2l$ twisted cycles on $(C,\mathcal{E})$
meeting $T$. Their real parts are partitioned into even and odd ovals as
follows. Since $T^{\vee}$ is a graph of the form $K_{2,2l}$ and all edges of
$T$ are non-exposed, we get that for each zone $Z$ in the zone decomposition
$\Delta_{T}$, distinct from the special zone, there exists a unique twisted
cycle $\rho_{Z}$ on $(C,\mathcal{E})$ with edge support dual to the boundary
$\partial Z$.

Since every edge of $T$ is non-exposed, the real part
${\mathbb{R}}(\rho_{Z})_{\mathcal{E}}$ is contained in a single orthant
$\varepsilon$ of ${\mathbb{R}}\mathbb{P}_{\Sigma}$. Moreover, the two integer
points $v,v^{\prime}\in\partial Z\cap{\mathbb{Z}}^{2}$ such that
$\varepsilon(v^{\vee})$ and $\varepsilon((v^{\prime})^{\vee})$ lie neighbourly
inside ${\mathbb{R}}(\rho_{Z})_{\mathcal{E}}$ are exactly the integer points
of $\partial Z$ belonging to the set of $2l$ vertices of $K_{2,2l}$, for
$v^{\vee},(v^{\prime})^{\vee}\subset{\mathbb{T}}\mathbb{P}_{\Sigma}\backslash
C$ the connected components dual to $v$ and $v^{\prime}$.

Let $\delta$ be a distribution of signs on $\Delta_{C}\cap{\mathbb{Z}}^{2}$
inducing the real phase structure $\mathcal{E}$, and let
$v_{0}^{\vee}\in({\mathbb{T}}\mathbb{P}_{\Sigma}\backslash C)^{*}$ a connected
component lying outside every oval of ${\mathbb{R}}C_{\mathcal{E}}$. Since $T$
is even-free, we can assume that $\delta$ is of Harnack type on every zone of
$Y_{1}^{T}$ and of inverse Harnack type on every zone of $Y_{0}^{T}$, see the
proof of Proposition 6.11.

If $Z$ belongs to $Y_{0}^{T}$, we get that

$\delta(v_{0})\neq\delta(\varepsilon(v))\text{ and
}\delta(v_{0})\neq\delta(\varepsilon(v^{\prime})),$

so that ${\mathbb{R}}(\rho_{Z})_{\mathcal{E}}$ is an even oval by Lemma 6.1.
Similarly, if $Z$ belongs to $Y_{1}^{T}$, we get that

$\delta(v_{0})=\delta(\varepsilon(v))=\delta(\varepsilon(v^{\prime})),$

so that ${\mathbb{R}}(\rho_{Z})_{\mathcal{E}}$ is an odd oval by Lemma 6.1.

The last twisted cycle $\rho_{Z^{T}}$ has edge support dual the edges in
$\partial Z^{T}\cap K_{2,2l}$, for $Z^{T}$ the special zone in $\Delta_{T}$.
The two integer points $v,v^{\prime}\in\partial Z^{T}\cap{\mathbb{Z}}^{2}$
such that $\varepsilon(v^{\vee})$ and $\varepsilon((v^{\prime})^{\vee})$ lie
neighbourly inside ${\mathbb{R}}(\rho_{Z})_{\mathcal{E}}$ are exactly the
integer points of $\partial Z$ belonging to the set of $2$ vertices of
$K_{2,2l}$. We get that

$\delta(v_{0})=\delta(\varepsilon(v))=\delta(\varepsilon(v^{\prime})),$

so that ${\mathbb{R}}(\rho_{Z})_{\mathcal{E}}$ is an even oval by Lemma 6.1.

Therefore, we just need to know the number of zones in each $Y_{j}^{T}$,
distinct from the special zone, to count the number of even and odd ovals
coming from twisted cycles meeting $T$. There are $l$ zones in $Y_{0}^{T}$,
which together with the special zone $Z^{T}$ give rise to $l+1$ even ovals.
There are $l-1$ zones in $Y_{1}^{T}$ distinct from $Z^{T}$, which give rise to
$l-1$ odd ovals. ∎

######  Theorem 6.15.

Let $(C,\mathcal{E})$ be a non-singular dividing real tropical curve of strict
even degree in ${\mathbb{T}}\mathbb{P}_{\Sigma}$. Assume that the set of
twisted edges $T$ on $C$ induced by $\mathcal{E}$ is of the form
$T:=T_{1}\sqcup\ldots\sqcup T_{s}\in\operatorname{EvFreeCirc}(C)$ such all
$T_{i}$’s are cycle-disjoints and each $T_{i}$ is dual to a complete bipartite
graph of the form $K_{2,2l_{i}}$. Let $Y_{T}:=Y_{1}^{T}\sqcup Y_{0}^{T}$ be
the zone partition such that $Y_{1}^{T}$ contains the special zone $Z^{T}$.
Let $p_{j}$ and $n_{j}$ be the number of even and odd interior integer points
in $Y_{j}^{T}$. Then

$\displaystyle p_{\mathcal{E}}$
$\displaystyle=n_{1}+p_{0}+1+\sum\limits_{i=1}^{s}(l_{i}+1)$ $\displaystyle
n_{\mathcal{E}}$ $\displaystyle=p_{1}+n_{0}+\sum\limits_{i=1}^{s}(l_{i}-1).$

###### Proof.

By Proposition 6.11, the real part ${\mathbb{R}}C_{\mathcal{E}}$ has
$n_{1}+p_{0}+1$ even ovals and $p_{1}+n_{0}$ odd ovals coming from twisted
cycles not meeting $T$. By Lemma 6.14 applied on each $T_{i}$ (this is
possible since all $T_{i}$’s are disjoint and cycle-disjoint), there are
$l_{i}+1$ even ovals and $l_{i}-1$ odd ovals coming from twisted cycles
meeting $T_{i}$. Therefore we obtain the result. ∎

###  6.5. Counter-examples to Ragsdale conjecture

(a) Itenberg’s zone decomposition in degree 10.

(b) Subdivision inside Itenberg’s configuration.

Figure 9. Example 6.17.

Recall that for a non-singular real algebraic curve $\mathcal{C}$ of degree
$d=2k$ in $\mathbb{P}^{2}$, Ragsdale’s conjecture states that the number of
even ovals of $\mathcal{C}({\mathbb{R}})$ is less than or equal to $R(k)+1$,
and the number of odd ovals of $\mathcal{C}({\mathbb{R}})$ is less or equal to
$R(k)$, for $R(k):=\frac{3k(k-1)}{2}$ the number of odd interior integer
points in the lattice polygon $\Delta_{2k}$. We want to use the results of
preceding sections in order to construct new counter-examples to Ragsdale’s
conjecture, in a similar manner as Itenberg [Ite93], [Ite01] and Haas [Haa95].

We first construct intermediate counter-examples coming from dividing $(M-2)$
real tropical curve $(C,\mathcal{E})$ of degree $2k$ in
${\mathbb{T}}\mathbb{P}^{2}$, for $k\geq 5$, with set of twisted edges dual to
a specific complete bipartite graph.

Let $t$ be an integer satisfying

$0\leq t\leq\left\lfloor\frac{k-5}{2}\right\rfloor.$

Let $m$ be an integer satisfying

$1\leq m\leq\left\lfloor\frac{2k-1-4t}{6}\right\rfloor.$

Let $K_{2,4m}$ be a complete bipartite graph in $\Delta_{2k}$ with two sets of
vertices of the following form.

  * •

the set of $4m$ vertices is of the form

$\\{B_{i}=(b_{i},3+4t),i=1,\ldots,4m\\}\subset\Delta_{2k}\cap{\mathbb{Z}}^{2}$

with $b_{i+1}=b_{i}+1$ if $i=1\mod 2$ and $b_{i+1}=b_{i}+2$ if $i=0\mod 2$.

  * •

The set of $2$ vertices is of the form

$\\{A_{1}=(a_{1},6+4t),A_{2}=(a_{2},4t)\\}\subset\Delta_{2k}\cap{\mathbb{Z}}^{2},$

such that $a_{1},a_{2}$ are odd and

$\displaystyle a_{1}+b_{1}$ $\displaystyle\neq 0\mod 3,\quad a_{1}+b_{4m}\neq
0\mod 3$ $\displaystyle a_{2}+b_{1}$ $\displaystyle\neq 0\mod 3,\quad
a_{2}+b_{4m}\neq 0\mod 3.$

The _length_ $L$ of a configuration $K_{2,4m}$ as described above is the
difference $L:=b_{4m}-b_{1}=4+6(m-1)$. Note that the condition $1\leq
m\leq\left\lfloor\frac{2k-1-4t}{6}\right\rfloor$ is given so that the length
of $K_{2,4m}$ is less than or equal to the horizontal length of $\Delta_{2k}$
at height $3+4t$. Said otherwise, we have $L\leq 2k-3-4t$.

(a) A zone decomposition in degree 14.

(b) Subdivision inside the graph $K_{2,8}$.

Figure 10. Example 6.18

######  Lemma 6.16.

Let $(C,\mathcal{E})$ be a non-singular tropical curve of degree $2k$ in
${\mathbb{T}}\mathbb{P}^{2}$, such that the dual graph $T^{\vee}$ is a
complete bipartite graph $K_{2,4m}$ satisfying the conditions above. Then
$(C,\mathcal{E})$ is a dividing $(M-2)$ real tropical curve with real part
${\mathbb{R}}C_{\mathcal{E}}$ having $R(k)+2m$ even ovals.

###### Proof.

Let $\Gamma$ be the subgraph of $T^{\vee}$ dual to the set of non-exposed
twisted edges of $(C,\mathcal{E})$. Then $\Gamma$ is either a complete
bipartite graph $K_{2,4m}$, or a complete bipartite graph $K_{2,4m-1}$, or a
complete bipartite graph $K_{1,4m}$, or a complete bipartite graph
$K_{1,4m-1}$. By Theorem 4.22, we obtain that $(C,\mathcal{E})$ is a dividing
$(M-2)$ real tropical curve.

Assume first that $\Gamma$ is a complete bipartite graph $K_{2,4m}$. In this
case, we will be able to apply Theorem 6.15, and we will show afterwards that
the count remains unchanged in the other cases.

Let $Y$ be the union of all zones of $\Delta_{T}$ except the special zone
$Z^{T}$. We can assume that the distribution of signs $\delta$ on the dual
subdivision $\Delta_{C}$ is of Harnack type $Y_{1}^{T}$, for
$Y_{T}=Y_{1}^{T}\sqcup Y_{0}^{T}$ the zone partition such that the special
zone $Z^{T}$ belong to $Y_{1}^{T}$. The Newton polygon $\Delta_{2k}$ has
$R(k)$ odd interior integer points, all lying in the closure of the union of
zones $Y_{1}^{T}$. There are $4m+2$ odd integer points lying on the graph
$K_{2,4l}$, hence we obtain $n_{1}=R(k)-4m-2$ with the notation from Theorem
6.15. Moreover, there are $p_{0}=4m$ even integer points in $Y_{0}^{T}$, with
the notation from Theorem 6.15. By Proposition 6.11, we obtain already that

$p_{\mathcal{E}}\geq n_{1}+p_{0}+1=R(k)-1.$

The remaining even ovals to count come from the twisted cycles going through
an edge of $T$. Since $T^{\vee}$ is a complete bipartite graph of the form
$K_{2,4m}$, we obtain by Theorem 6.15 that the real part
${\mathbb{R}}C_{\mathcal{E}}$ has $R(k)-1+(2m+1)=R(k)+2m$ even ovals.

Now, if $T^{\vee}$ meets the boundary of $\Delta_{2k}$ in one or two of its
vertices, the number of connected component of ${\mathbb{R}}C_{\mathcal{E}}$
is unchanged as we saw earlier, and all those connected components are ovals
by Proposition 5.9. Then there are either $4m$ or $4m+1$ odd integer points
lying on the subgraph $\Gamma$, so that ${\mathbb{R}}C_{\mathcal{E}}$ has
either $R(k)-4m$ or $R(k)-4m-1$ even ovals coming from twisted cycles with
edge support a primitive cycle of $C$. In the proof of Lemma 6.14, the special
zone and the zones in $Y_{0}^{T}$ give rise to even ovals. If one of the
vertices of $T^{\vee}$ lying on the boundary of $\Delta_{2k}$ belongs to the
set of $2$ vertices of the bipartite structure of $T^{\vee}$, then the even
oval coming from the special zone is the same as the special oval. If one of
the vertices of $T^{\vee}$ lying on the boundary of $\Delta_{2k}$ belongs to
the set of $4m$ vertices of the bipartite structure of $T^{\vee}$, then the
even oval coming from the zone of $Y_{0}^{T}$ meeting $\Delta_{2k}$ on their
boundaries is the same as the special oval. Said otherwise, there is a unique
twisted cycle on $(C,\mathcal{E})$ going through every unbounded edge of $C$
and through the edges of $T$ dual to the “exterior edges” of the graph
$K_{2,4m}$, so that the real part of this twisted cycle cannot be contained in
any other oval of ${\mathbb{R}}C_{\mathcal{E}}$, hence it is an even oval.

∎

######  Example 6.17.

Let $(C,\mathcal{E})$ be a non-singular real tropical curve of degree 10 in
${\mathbb{T}}\mathbb{P}^{2}$ such that the zone decomposition $\Delta_{T}$ is
induced by a complete bipartite graph $K_{2,4}$ as given in Figure 9(a). The
dual graph of non-exposed twisted edges $\Gamma_{T}$ consists of a complete
bipartite graph $K_{1,4}$ plus isolated vertices, and every cycle of $C$ has
an even number of twists, so by Theorem 4.22 the real tropical curve
$(C,\mathcal{E})$ is a dividing $(M-2)$ real tropical curve. The integer point
at the center of the subdivision given in Figure 10(b) is
$(3,3)\in\Delta_{C}\cap{\mathbb{Z}}^{2}$, and the extremal points are
$(3,0),(3,6),(1,3)$ and $(5,3)$. This configuration of twists is Itenberg’s
counter-example to Ragsdale conjecture [Ite93]. By Lemma 6.16, the real part
${\mathbb{R}}C_{\mathcal{E}}$ has $R(5)+2=32$ even ovals, hence one even oval
more than Ragsdale bound in degree 10.

$\begin{array}[]{|l|c|c|c| }\hline\cr&x_{t}\equiv 0\mod 3&x_{t}\equiv 1\mod
3&x_{t}\equiv 2\mod 3\\\ \hline\cr P_{t}&(3,4t+6)&(3,4t+6)&(3,4t+6)\\\
P_{t}^{\prime}&(3,4t)&(3,4t)&(3,4t)\\\ E_{t}^{1}&(1,4t+3)&(1,4t+3)&(1,4t+3)\\\
E_{t}^{4l}&(x_{t}-4,4t+3)&(x_{t}-2,4t+3)&(x_{t},4t+3)\\\ \hline\cr\end{array}$
Table 1. Case $t$ even. $\begin{array}[]{|l|c|c|c| }\hline\cr&x_{t}\equiv
0\mod 3&x_{t}\equiv 1\mod 3&x_{t}\equiv 2\mod 3\\\ \hline\cr
P_{t}&(x_{t}-3,4t+6)&(x_{t}-4,4t+6)&(x_{t}-5,4t+6)\\\
P_{t}^{\prime}&(x_{t}+3,4t)&(x_{t}+2,4t)&(x_{t}+1,4t)\\\
E_{t}^{1}&(1,4t+3)&(1,4t+3)&(1,4t+3)\\\
E_{t}^{4l}&(x_{t}-4,4t+3)&(x_{t}-2,4t+3)&(x_{t},4t+3)\\\ \hline\cr\end{array}$
Table 2. Case $t$ odd.

######  Example 6.18.

Let $(C,\mathcal{E})$ be a non-singular real tropical curve of degree 14 in
${\mathbb{T}}\mathbb{P}^{2}$ such that the zone decomposition $\Delta_{T}$ is
induced by a complete bipartite graph $K_{2,8}$ as given in Figure 10(a). The
dual graph of non-exposed twisted edges $\Gamma_{T}$ consists of a complete
bipartite graph $K_{1,7}$ plus isolated vertices, and every cycle of $C$ has
an even number of twists, so by Theorem 4.22 the non-singular real tropical
curve $(C,\mathcal{E})$ is a dividing $(M-2)$ real tropical curve. The integer
point at the center of the subdivision given in Figure 10(b) is
$(5,3)\in\Delta_{C}\cap{\mathbb{Z}}^{2}$, and the extremal points are
$(5,0),(5,6),(0,3)$ and $(10,3)$. By Lemma 6.16, the real part
${\mathbb{R}}C_{\mathcal{E}}$ has $R(7)+4=67$ even ovals, hence 3 even oval
more than Ragsdale’s bound.

Let us now describe a construction of counter-examples of Ragsdale conjecture
in degree $2k\geq 10$, by taking a disjoint union of complete bipartite graphs
of the form $K_{2,4m_{t}}$ in $\Delta_{2k}$ satisfying the conditions of Lemma
6.16, for

$t\in\left\\{0,\ldots,\left\lfloor\frac{k-5}{2}\right\rfloor\right\\},$

in a similar fashion as Itenberg’s construction in [Ite01]. Let
$x_{t}=2k-(4t+3)$, so that the integer points $(x_{t},4t+3)$ lie on the
boundary $\partial\Delta_{2k}$. Each bipartite graph $K_{2,4l_{t}}$ will have
its set of $4m_{t}$ vertices lying on the horizontal line at height $4t+3$, so
that the length $L_{t}$ of $K_{2,4m_{t}}$ will be the maximal integer
satisfying $L_{t}\leq x_{t}$ and $L_{t}=4\mod 6$.

Since $x_{t}=1\mod 2$, we can determine suitable coordinates for the extremal
vertices $B_{1}^{t},B_{4m_{t}}^{t}$ of $K_{2,4l_{t}}$ in terms $x_{t}\mod 3$
instead of $x_{t}\mod 6$. We give in Tables 1 and 2 the coordinates of the
extremal points for our construction, where Table 1 correspond to the case
where $t=0\mod 2$ and Table 2 correspond to the case where $t=1\mod 2$.

######  Theorem 6.19 (Theorem 1.7).

Let $(C,\mathcal{E})$ be a non-singular dividing tropical curve of degree
$2k\geq 10$ in ${\mathbb{T}}\mathbb{P}^{2}$ with zone decomposition
$\Delta_{T}$ induced by the disjoint union of graphs

$\bigsqcup\limits_{t=0}^{\left\lfloor\frac{k-5}{2}\right\rfloor}K_{2,4m_{t}}$

with vertices described by Table 1 and Table 2. Then $(C,\mathcal{E})$ is a
dividing $(M-2\left\lfloor\frac{k-3}{2}\right\rfloor)$ real tropical curve
with real part ${\mathbb{R}}C_{\mathcal{E}}$ having

$R(k)+1+\frac{k^{2}-5k+s(k)}{6}$

even ovals, with $0\leq s(k)\leq 10$ determined by the value of $k\mod 6$. The