Let $t$ be an arbitrary triangulation and consider its family $[t]$. We know by Lemma <ref> that there exists a triangulation in $t'\in[t]$ where all bridging arcs have winding number 0. We will prove that $t'$ contains three arcs $\alpha,\beta,\gamma$ which are the defining arcs of one of our sets, or that a $2\pi$ Dehn twist of $t'$, also a member of $[t]$, contains such three arcs. We begin by proving the existence of an arc $\gamma$. Note that $a(b_m,b_1)$ is always a boundary arc crossing fdl. Thus, since we have a finite number of arcs, we can consider the largest interior arc crossing the fdl, namely an arc connecting vertices $b_{m-k},b_j$ that crosses the fdl and minimizes $m-k-j$. In the case that $\gamma=a(b_m,b_1)$, then we consider $\gamma$ to be the part of the boundary that $a(b_m,b_1)$ is homotopic to. Thus we have found $\gamma_{m-k,j}$. Notice that $\gamma_{m-k,j}$ is invariant by a $2\pi$ Dehn twist. Similarly, we can construct the largest boundary arc crossing the fdl with both endpoints being on the outside boundary component. Suppose this arc is $a(a_l,a_k)$. Next, we prove that there is a bridging arc crossing the fdl. By contradiction, suppose there was no such arc. Consider the arcs $a(b_{m-k},a_l), a(a_k,b_j)$. By the maximality of a triangulation, these two arcs are only not allowed if there is already an arc crossing at least one of them, and such an arc would cross the fdl by construction of $a(b_{m-k},b_j)$ and $a(a_l,a_k)$. Hence, if these are not part of our triangulation, there exists a bridging arc crossing the fdl. If these arcs do exist, we still have the quadrilateral composed of vertices $b_{m-k},a_l,a_k,b_j$, contradicting the fact that we have a triangulation. Thus we must either have $a(b_{m-k},a_k)$ or $a(a_l,b_j)$, both of which are bridging arcs crossing the fdl.

Now, we prove the existence of defining arcs $\alpha,\beta$. First notice that there is a bridging arc with endpoint $b_j$. If not, we could draw an interior arc crossing the fdl given by $a(b_{m-k},b_{j+1})$ contradicting the maximality of $\gamma$. Similarly, we obtain a bridging arc connecting to the vertex $b_{m-k}$. There are now two cases, either there is a bridging arc with an endpoint $b_{m-k}$ crossing the fdl, or there is not. Suppose first that there is. We can pick the one furthest to the right (clockwise direction), that is the arc $a(b_{m-k},a_i)$ maximizing $i$. Denote this arc by $\alpha=\alpha_{m-k}^i$. It is easy to see that we immediately now have $\beta=\beta_{i,j}$ be the arc $a(a_i,b_j)$, as the only way such an arc could not exist would bring forth a contradiction to the maximality of either $\gamma$ or $\alpha$. Notice also that, by the construction of $\gamma$, we have $m-k-j\geq 0$ and $j\geq 0$, so that $m-k\geq j\geq 0$, whence our three arcs are indeed defining arcs. Thus our triangulation contains three defining arcs $\alpha, \beta, \gamma$ and is therefore contained in the set $S_{(a_i,b_{m-k},b_j)}$. Now suppose that there is no bridging arc crossing the fdl with endpoint $b_{m-k}$. Note that we do have a bridging arc with endpoint $b_{m-k}$, so after applying a $2\pi$ Dehn twist this arc becomes a bridging arc with endpoint $b_{m-k}$ that crosses the fdl in another triangulation contained in $[t]$. Thus by our above work we can again find arcs $\alpha,\beta$, such that this triangulation obtain by a Dehn twist is contained in one of our sets.

Next, in the following lemma, we show that the sets $S_{(-,-,-)}$ are disjoint.

$S_{(a_k,b_i,b_j)} = S_{(a_{k'},b_{i'},b_{j'})}$ if and only if $k=k', i= i', j= j'$. Moreover,
$S_{(a_k,b_i,b_j)} \cap S_{a_{k'},b_{i'},b_{j'}} \neq \emptyset \Rightarrow S_{(a_k,b_i,b_j)}=S_{(a_{k'},b_{i'},b_{j'})}$.

The if direction is clear. The only if direction will be shown in the proof of the moreover statement, whose proof we now give. Suppose that $S_{(a_k,b_i,b_j)} \cap S_{a_{k'},b_{i'},b_{j'}} \neq \emptyset$. By construction, we have the arcs $\gamma_{i,j},\gamma_{i',j'}$. These do not intersect if and only if either $i\leq i' \bmod m, j'\leq j\bmod m$ or $i'\leq i \bmod m , j\leq j' \bmod m$. Without loss of generality suppose $i\leq i' \bmod m, j'\leq j \bmod m$, and denote by $\gamma$ the arc $\gamma_{i,j}$. If $i< i'$ then $\alpha_{i'}^k$ would intersect $\gamma$, a contradiction. Thus $i=i'$. Similarly, we obtain $j=j'$ by considering the arc $\beta_{j'}^k$. Thus $\gamma_{i,j}=\gamma_{i',j'}$. Consider now the arcs $\alpha_{i}^k$, $\alpha_{i'}^{k'}=\alpha_{i}^{k'}$, as well as $\beta_{j}^k$, and $\beta_{j'}^{k
}=\beta_{j}^{k'}$. If $k'<k$, then the arcs $\alpha_i^k$ and $\beta_j^{k'}$ would cross. If $k<k'$ the arcs $\alpha_i^{k'}$ and $\beta_j^k$ cross. Thus $k=k'$ and we can conclude that $S_{(a_k,b_i,b_j)} = S_{(a_{k'},b_{i'},b_{j'})}$.

The direct application of the three lemmas above gives Theorem <ref>. We can think of it as a map from the set of all possible triangulations, into our constructed set. In effect, we have shown that this map is injective and surjective, and calculated the cardinality of our sets. Thus the statement of the theorem is clear. We now give proofs of the technical lemmas used in our proofs. We begin with reducing the double summation to the single summation.

This identity is evident by considering how many times the term $C_xC_{n+m-x-1}$ for $0\leq x \leq m-1$ appears in the double summation. Namely, for each $0\leq j \leq x$ it appears precisely once in the inside summation. For $j>x$ the term does not appear in the inside summation. Hence, in total it shows up $x+1$ many times.

It is important to note that in the above proof, by the term $C_xC_{n+m-x-1}$ we mean mean the symbolic term, not other terms that equate to the same value. For example, in the case $n=2,m=4,x=2$, when considering the term $C_2C_{2+4-2-1}$ we do not consider the term $C_3C_{2+4-3-1}$.

Let $a_i\in \mathcal{A},b_j\in \mathcal{B}$. Then up to homeomorphism,
$(p^{-1}(a_i)_x)\cap (p^{-1}(b_j)_x) = \emptyset$. Hence the pre-image of any bridging arc on the annulus can be viewed as a line segment with a finite positive or negative slope on the universal cover.

It is clear that the lifts of two distinct vertices in $\mathcal{A}$ ( respectively in $\mathcal{B}$) intersect trivially. Suppose we have two vertices $a_i \in \mathcal{A}$, $b_j \in \mathcal{B}$ such that there exists lifts of $a_i, b_j$ that share the same $x$-coordinate. Let $E=\big{\{} |(a'_k)_x-(b'_l)_x|\bmod 1 ; a'_k\in p^{-1}(a_k), b'_l\in p^{-1}(b_l) \forall k,l \big{\}}- \{0\}$. This set is finite due to the modular property of $p$. Intuitively, we take the pre-images of $\mathcal{A}$ and $\mathcal{B}$, put them onto a single real line, ignore the overlapping vertices, and consider the distances of the remaining points modulo 1. Put $\epsilon=\frac{1}{3}\text{min}(E)$. Then a clockwise or counter-clockwise $2\pi \epsilon$ Dehn twist of the inner boundary component gives a homeomorphic annulus such that for any $a_i\in \mathcal{A},b_j\in \mathcal{B}$,
$(p^{-1}(a_i)_x)\cap (p^{-1}(b_j)_x) = \emptyset$. This is because under the $2\pi \epsilon$ Dehn twist, any “overlapped" marked points get “un-overlapped", while not “overlapping" any other marked points due to the choice of $\epsilon$. Formally, if $p^{-1}(a_i)_x=p^{-1}(b_j)_x$ for some $i,j$ in the annulus obtained via the $2\pi \epsilon$ Dehn twist, then on the universal cover of the original matrix, $p^{-1}(a_i)_x=p^{-1}(b_j)_x \pm \frac{\epsilon}{3}$, contradicting our choice of $\epsilon$.

§ ALGEBRAIC DESCRIPTION OF FAMILIES

In this section, we will provide an algebraic description of inner equivalence classes in terms of the cluster category of $Q:=Q^{\bm{\varepsilon}}$, a quiver of type $\tilde{\mathbb{A}}_{p-1}$. First, recall the definition of the cluster category $\mathcal{C}_Q=\mathscr{D}/F$ where $\mathscr{D} := \mathscr{D}^b(mod\text{-}\Bbbk Q)$ and $F:\mathscr{D} \rightarrow \mathscr{D}$ is the auto-equivalence defined by $\Sigma \tau^{-1}$. Typically, we choose the representatives of the $F$ orbits to be the unshifted objects along with the once shifted projectives in $\mathscr{D}$. In what follows, the $F$ orbit of objects and morphisms in $\mathcal{C}_Q$ will be important to provide intuition behind the algebraic consequences of performing Dehn twists of the annulus. Thus, when we draw the AR quiver of the cluster category associated to $Q^{\bm{\varepsilon}}$, we will include negatively shifted injectives and irreducible morphisms that are identified with the shifted projectives in the cluster category.

Let $Q$ be the quiver
$$\xymatrix{1 \ar[r]<EMAIL_ADDRESS>& 2 & 3 \ar[l] \ar[r] & 4}$$

In this example, we will construct the parts of the AR quiver of $\mathcal{C}_Q$ containing only the preprojectives, preinjectives, and shifted projectives. Moreover, we will include the negatively shifted injectives and color them to indicate that they are in the same $F$ orbit as the corresponding shifted projective, which is the  typically chosen representative. The reason we are doing this will be explained later in the section. The way we have constructed this AR quiver, and the way we will always construct the AR quiver of $\mathcal{C}_Q$ is as follows. The arrows with positive slope between two string modules indicate the addition (deletion) of a hook (cohook) at the start of the string. The downward sloping arrows between two string modules indicate the addition (deletion) of a hook (cohook) at the end of the string. With this convention, the position of the shifted projectives is uniquely defined.

& & & & & & \ar[dr] & & & & \\
& {\color{red}\Sigma^{-1} I(1)} \ar[dr] & & 42_3 \ar[dr] \ar[ur] & & & & 13_3 \ar[dr] & & {\color{yellow} \Sigma P(2)} \ar[dr] & \\
{\color{yellow}\Sigma^{-1} I(2)} \ar[ur] \ar[dr] & & 22_1 \ar[ur] \ar[dr] & & & & \ar[ur] \ar[dr] & & 11_1 \ar[ur] \ar[dr] & & {\color{red}\Sigma P(1)} \\
& {\color{blue} \Sigma^{-1} I(3)} \ar[ur] \ar[dr] & & 24_3 \ar[ur] \ar[dr] & & \cdots & & 31_3 \ar[ur] \ar[dr] & & {\color{green}\Sigma P(4)} \ar[ur] \ar[dr] & \\
{\color{green}\Sigma^{-1} I(4)} \ar[ur] \ar[dr] & & 44_1 \ar[ur] \ar[dr] & & & & \ar[ur] \ar[dr] & & 33_1 \ar[ur] \ar[dr] & & {\color{blue}\Sigma P(3)} \\
& {\color{red} \Sigma^{-1} I(1)} \ar[ur] & & 42_3 \ar[dr] \ar[ur] & & & & 13_3 \ar[ur] & & {\color{yellow} \Sigma P(2)} \ar[ur] & \\
& & & & & & \ar[ur] & & &}$$

Let $M,N\in\text{Ob}(\mathcal{C}_Q)$. We say that $M$ connects to $N$ if there exist objects $M',N'\in \text{Ob}(\mathcal{C}_Q) \cup \{\Sigma^{-1} I(j) | j\in Q_0\}$ such that $M'$ and $N'$ are in the $F$ orbits of $M$ and $N$ respectively and there exists an arrow from $M'$ to $N'$ in the AR quiver of $\mathscr{D}$. By a path in $\mathcal{C}_Q$, we mean a sequence of objects $(\dots,M_1,M_2,\dots,M_k,\dots)$ in $\mathcal{C}_Q$ such that for each $i$, $M_i$ connects to $M_{i+1}$. We say there exists a path from $M$ to $N$ in $\mathcal{C}_Q$ if there exists a finite sequence of objects $(M = M_1,M_2,\dots,M_k = N)$ such that for each $i \in \{1,2,3,\dots, k-1\}$, $M_i$ connects to $M_{i+1}$. For example, there is a path from $33_1$ to $44_1$ in $\mathcal{C}_Q$ from Example <ref> given by $(33_1,\Sigma P(4), \Sigma P(1), 44_1)$ where $\Sigma P(1)$ connects to $44_1$ since there is an arrow in the AR quiver of $\mathscr{D}$ from $\Sigma^{-1} I(1)$, which is in the $F$ orbit of $\Sigma P(1)$, to $44_1$. On the other hand, there is no path from $44_1$ to $33_1$.

* By a ray in $\mathcal{C}_Q$, we mean an infinite path $(\dots,M_1,M_2,\dots,M_k,\dots)$ such that for consecutive objects $M_i$ and $M_{i+1}$ in the path, the following hold:

* If $M_i,M_{i+1} \in \mathcal{R}$ are consecutive objects in a path, then the arrow between them has positive slope.

* If $M_i,M_{i+1} \in \mathcal{I} \cup \{\Sigma P(j) | j\in Q_0\}$ are consecutive objects in a path, the arrow between them also has positive slope.

* Otherwise, the arrow from $M_i$ to $M_{i+1}$ has negative slope.

* Dually, a coray in $\mathcal{C}_Q$, is an infinite path $(\dots,M_1,M_2,\dots,M_k,\dots)$ such that for consecutive objects $M_i$ and $M_{i+1}$ in the path, the following hold:

* If $M_i,M_{i+1} \in \mathcal{R}$ are consecutive objects in a path, then the arrow between them has negative slope.

* If $M_i,M_{i+1} \in \mathcal{I} \cup \{\Sigma P(j) | j\in Q_0\}$ are consecutive objects in a path, the arrow between them also has negative slope.

* Otherwise, the arrow from $M_i$ to $M_{i+1}$ has positive slope.

The reason the positive/negative convention switches between the preprojective and preinjective components is that when we identify the shifted projectives with the negatively shifted injectives, we must flip the shifted projectives. In a certain sense, we can intuitively think of the preprojective and preinjective components of the AR quiver of $\mathcal{C}_Q$ as being glued together on a Möbius strip.

In $\mathcal{C}_Q$ from Example <ref>, we have a coray given by
$$(\dots, 11_1, \Sigma P(4), \Sigma P(3), 22_1, 42_3, \dots).$$
This can be seen by identifying $\Sigma P(3)$ with $\Sigma^{-1} I(3)$. We also have a ray given by
$$(\dots, 33_1, \Sigma P(4), \Sigma P(1), 22_1, 24_3, \dots)$$
which can be seen by identifying $\Sigma P(1)$ with $\Sigma^{-1} I(1)$.

We wish to show that elementary twists of the annulus correspond to moving objects up and down rays/corays in $\mathcal{C}_Q$. Recall that $Q$ is a quiver of type $\tilde{\mathbb{A}}_{p-1}$ and let $A_{n,m}$ be the associated annulus with $n$ marked points on the outer boundary and $m$ marked points on the inner boundary, where $n+m = p$. In order to provide an algebraic explanation of the families of clusters, we first need the notion of an elementary Dehn twist of the inner boundary component, by which we mean a $2\pi/m$ Dehn twist of the inner boundary component. Elementary Dehn twists of the outer boundary component are defined analogously. We will call these elementary twists. We have the following result from the proof of Theorem 4.18 in [14].

[[14] Theorem 4.18]

* Let $M$ be preprojective and not projective. Then a clockwise (counter-clockwise) elementary Dehn twist of the inner boundary component corresponds to adding (deleting) a hook at the end of $M$.

* Let $M$ be preprojective and not projective. Then a clockwise (counter-clockwise) elementary Dehn twist of the outer boundary component corresponds to deleting (adding) a hook at the start of $M$.

* Let $M$ be preinjective and not injective. Then a clockwise (counter-clockwise) elementary Dehn twist of the inner boundary component corresponds to deleting (adding) a cohook at the start of $M$.

* Let $M$ be preinjective and not injective. Then a clockwise (counter-clockwise) elementary Dehn twist of the outer boundary component corresponds to adding (deleting) a cohook at the end of $M$.

It follows from Lemma <ref> that elementary twists correspond to irreducible morphisms between the corresponding modules. Following our conventions for constructing the AR quiver of $\mathcal{C}_Q$, we conclude in this case that elementary clockwise (counter-clockwise) twists of the inner boundary component correspond to moving one place forward (backward) in a ray in the cluster category. Analogously, elementary clockwise (counter-clockwise) twists of the outer boundary component correspond to moving one place backward (forward) in a coray in the cluster category.

For completeness, we mention that performing elementary clockwise (counter-clockwise) twists of the inner boundary component correspond to $\tau^{-1}$ ($\tau)$ of the left regular modules. Performing elementary clockwise (counter-clockwise) twists of the outer boundary component correspond to $\tau$ ($\tau^{-1})$ of the right regular modules.

We now need to understand the action of these twists on projective/injective modules and the shifted projectives. Actually, it follows from the proof of Theorem 4.18 in [14] that in the case we have an injective module such that a cohook can be deleted at the start of the string, an elementary clockwise twist of the inner boundary component corresponds to deleting a cohook at the start of the string, hence moving one position forward in a ray. Analogous statements can be made for the other twists and for projectives. Thus it suffices to study the case in which hooks and cohooks can't be added or deleted from the string.

* Let $I(j)$ be the injective module at vertex $j$ such that a cohook can't be deleted at the start (end). Then an elementary clockwise (counter-clockwise) twist of the inner (outer) boundary component sends $I(j)$ to $\Sigma P(j+1 \bmod \, |Q_0|)$ ($\Sigma P(j-1 \bmod \, |Q_0|)$).

* Let $P(j)$ be the projective module at vertex $j$ such that a hook can't be deleted at the the end (start). Then an elementary counter-clockwise (clockwise) twist of the inner (outer) boundary component sends $P(j)$ to $\Sigma P(j-1 \text{mod} \, |Q_0|)$ ($\Sigma P(j+1 \bmod \, |Q_0|)$).

We will show the first part of 1. The proof of the second part of 1 is analogous and 2 is dual to 1. Let $I(j)$ be the injective module at vertex $j$ where $j$ is such that it is not possible to delete a cohook from the start of $I(j)$ and let $a(x,y)$ be the arc corresponding to $I(j)$. Let $l = a(l',l'')$ be the first (left most in the cover) steep arc that $a(x,y)$ crosses and since $I(j)$ is the injective module at $j$, $j = a(j',j'')$ is the last steep arc (right most in the cover) that $a(x,y)$ crosses. Since it is not possible to delete a cohook from the start of $I(j)$, we know that all arrows in the walk corresponding to $I(j)$ must be direct, hence in $Q_1$. Therefore, we conclude that all of the steep arcs that $a(x,y)$ crosses must be clockwise of $l$. Since the steep arcs are bridging, we conclude that they all have an endpoint in common. Suppose this end point lies on the inner boundary component. The proof when this endpoint lies on the outer boundary component is analogous. Moreover, since the steep arcs form a triangulation, we conclude $y$ must be the successor of $j''$ on the outer boundary component. Then a clockwise $2\pi/m$ Dehn twist sends $a(x,y)$ to the arc $a(x',y)$ beginning at the successor of $x$ and ending at the successor of $j''$. Again, since the steep arcs form a triangulation and it is impossible to delete a cohook from the start of $I(j)$, we conclude that $a(x',y)$ must be the steep arc $j+1 \bmod \, |Q_0|$. Thus $I(j)$ gets sent to $\Sigma P(j+1 \bmod \, |Q_0|)$.

It follows from Lemma <ref> that elementary twists give irreducible morphisms in $\mathcal{C}_Q$ between injectives and shifted projectives. Thus in this case elementary clockwise (counter-clockwise) twists of the inner (outer) boundary component move the corresponding object one place forward in a ray (coray) in $\mathcal{C}_Q$. On the other hand, we see that elementary twists of projectives can give shifted projectives. To make this consistent with the connection between elementary twists and irreducible morphisms, notice that $\Sigma^{-1} I(j-1 \bmod \, |Q_0|)$ is in the same $F$ orbit as $\Sigma P(j-1 \bmod \, |Q_0|)$. With this in mind, we can think of an elementary counter-clockwise twist of the inner boundary component as sending $P(j)$ to $\Sigma^{-1} I(j-1 \bmod \, |Q_0|)$, which is one shift backward in a ray in $\mathscr{D}$. Thus by construction, $P(j)$ gets sent one shift back in a ray to $\Sigma P(j-1 \bmod \, |Q_0|)$.  Analogously, an elementary clockwise twist of the outer boundary component sends $P(j)$ one shift back in a coray to $\Sigma P(j+1 \bmod \, |Q_0|)$.

Finally, the last case to consider is how the action of elementary twists affects the steep arcs, equivalently, the shifted projectives.

Let $\Sigma P(j)$ be a shifted projective.

* An elementary clockwise twist of the inner boundary component sends $\Sigma P(j)$ to either $\Sigma P(j+1 \bmod \, |Q_0|)$ or to $P(j+1 \bmod \, |Q_0|))$.

* An elementary counter-clockwise twist of the inner boundary component sends $\Sigma P(j)$ to either $\Sigma P(j-1 \bmod \, |Q_0|)$ or to $I(j-1 \bmod \, |Q_0|))$.

* An elementary clockwise twist of the outer boundary component sends $\Sigma P(j)$ to either $\Sigma P(j+1 \bmod \, |Q_0|)$ or to $I(j+1 \bmod \, |Q_0|))$.

* An elementary counter-clockwise twist of the outer boundary component sends $\Sigma P(j)$ to either $\Sigma P(j-1 \bmod \, |Q_0|)$ or to $P(j-1 \bmod \, |Q_0|))$.

We will prove 1. The proof of 2 is analogous to 1 and 3 and 4 are dual. Consider the object $\Sigma P(j)$ which corresponds to a steep arc labeled $j$. If an elementary clockwise twist of the inner boundary component sends the steep arc to another steep arc, then the new steep arc must have label $j+1 \bmod \, |Q_0|$ from our labeling convention. Moreover, the new steep arc must be counter-clockwise of the steep arc labeled $j$. We conclude that there is an arrow in $\mathcal{C}_Q$ from $\Sigma P(j)$ to $\Sigma P(j+1 \bmod \, |Q_0|)$, so this elementary twist corresponds to moving $\Sigma P(j)$ one position forward in a ray in $\mathcal{C}_Q$.

On the other hand, suppose that an elementary twist of the steep arc $j$ gives a non-steep bridging arc. Then this arc must cross all steep arcs that are clockwise of $j$. The first steep arc the twist of $j$ crosses is labeled $j+1 \bmod \, |Q_0|$. Suppose that the last steep arc that the twist of $j$ crosses is labeled $k = a(x,y)$. Consider the steep arc labeled $k+1\bmod |Q_0|$ and for a contradiction, suppose this arc is clockwise of $k$. Then it must be of the form $a(x,y')$ with $y'$ the successor of $y$ on the outer boundary component or $a(x',y)$ with $x$ the successor of $x'$ on the inner boundary component. In the first case, this contradicts the fact that $k$ is the last steep arc crossed, and in the second case, $a(x',y)$ would cross the steep arc labeled $j$ contradicting the fact that we began with a triangulation. We conclude that the steep arc labeled $k+1\bmod |Q_0|$ must be counter-clockwise of $k$ and therefore, there is an arrow in $Q$ pointing from $k+1$ to $k$. We conclude that the module associated to the twist of $j$ is precisely the projective $P(j+1 \bmod \, |Q_0|)$.

In light of Lemma <ref>, we see that in the case of steep arcs, elementary clockwise twists of the inner (outer) boundary component send shifted projectives one position forward in a ray (backward in a coray) in $\mathcal{C}_Q$. Analagously, elementary counter-clockwise twists of the inner (outer) boundary component send shifted projectives one position backward in a ray (forward in a coray) in $\mathcal{C}_Q$.

Combining Remarks <ref>, <ref>, and <ref>, we have the following proposition.

Let $M \in \text{Ob}(\mathcal{C}_Q)$ that is not regular and let $\alpha$ be its corresponding arc on the annulus $A_{Q}$. We have the following:

* A clockwise (counter-clockwise) elementary Dehn twist of the inner boundary component sends $M$ one position forward (backward) in a ray in $C_Q$.

* A clockwise (counter-clockwise) elementary Dehn twist of the outer boundary component sends $M$ one position forward (backward) in a coray in $C_Q$.

We are now ready to provide a description of inner equivalence classes of clusters in terms of $\mathcal{C}_Q$. The proof of the following theorem follows from Proposition <ref>

Let $C$ and $C'$ be two clusters in $\mathcal{C}_Q$ and label their corresponding triangulations of the annulus $A_Q = A_{n,m}$ by $T$ and $T'$ respectively. Suppose that for $z\in\mathbb{N}$, the triangulation $T'$ is attained from $T$ by performing $z$ $2\pi$ clockwise Dehn twists of the inner boundary component of the annulus. Then the cluster $C'$ has all the same regular objects as $C$ and each non-regular object $M'$ in $C'$ is attained from the non-regular object $M$ of $C$ by moving $M$  $z\cdot m$ positions forward in any ray containing $M$.

Consider the quiver $Q$ from Example <ref>

$$\xymatrix{1 \ar[r]<EMAIL_ADDRESS>& 2 & 3 \ar[l] \ar[r] & 4}$$

Then a triangulation of $A_Q$ by steep arcs that gives this orientation is

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[color=rgb, 255:red, 0; green, 0; blue, 0   ,draw opacity=1 ]   (378.44,83.7) .. controls (368.44,96.18) and (355.44,115.18) .. (325.22,104.62) ;

(291,43.4) node [anchor=north west][inner sep=0.75pt]    $1$;

(337,39.4) node [anchor=north west][inner sep=0.75pt]    $2$;

(344,108.4) node [anchor=north west][inner sep=0.75pt]    $3$;

(295,108.4) node [anchor=north west][inner sep=0.75pt]    $4$;

Consider the following triangulations that are inner equivalent.

every picture/.style=line width=0.75pt

(184.38,88.7) .. controls (184.38,77.14) and (194.16,67.77) .. (206.22,67.77) .. controls (218.28,67.77) and (228.06,77.14) .. (228.06,88.7) .. controls (228.06,100.25) and (218.28,109.62) .. (206.22,109.62) .. controls (194.16,109.62) and (184.38,100.25) .. (184.38,88.7)(153,88.7) .. controls (153,59.81) and (176.83,36.39) .. (206.22,36.39) .. controls (235.61,36.39) and (259.44,59.81) .. (259.44,88.7) .. controls (259.44,117.58) and (235.61,141) .. (206.22,141) .. controls (176.83,141) and (153,117.58) .. (153,88.7) ;

(206.22,67.77) ;
[shift=(206.22,67.77), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(206.22,109.62) ;
[shift=(206.22,109.62), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(259.44,88.7) ;
[shift=(259.44,88.7), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(153,88.7) ;
[shift=(153,88.7), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(414.38,88.7) .. controls (414.38,77.14) and (424.16,67.77) .. (436.22,67.77) .. controls (448.28,67.77) and (458.06,77.14) .. (458.06,88.7) .. controls (458.06,100.25) and (448.28,109.62) .. (436.22,109.62) .. controls (424.16,109.62) and (414.38,100.25) .. (414.38,88.7)(383,88.7) .. controls (383,59.81) and (406.83,36.39) .. (436.22,36.39) .. controls (465.61,36.39) and (489.44,59.81) .. (489.44,88.7) .. controls (489.44,117.58) and (465.61,141) .. (436.22,141) .. controls (406.83,141) and (383,117.58) .. (383,88.7) ;

(436.22,67.77) ;
[shift=(436.22,67.77), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(436.22,109.62) ;
[shift=(436.22,109.62), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(489.44,88.7) ;
[shift=(489.44,88.7), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(383,88.7) ;
[shift=(383,88.7), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(277.44,89.39) – (367.44,89.39) ;
[shift=(369.44,89.39), rotate = 180] [color=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]    (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29)   ;
[shift=(275.44,89.39), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]    (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29)   ;

[color=rgb, 255:red, 248; green, 231; blue, 28   ,draw opacity=1 ]   (153,88.7) .. controls (169.44,66.39) and (186.44,58.39) .. (206.22,67.77) ;

[color=rgb, 255:red, 208; green, 2; blue, 27   ,draw opacity=1 ]   (153,88.7) .. controls (171.44,109.39) and (189.44,115.39) .. (206.22,109.62) ;

[color=rgb, 255:red, 0; green, 255; blue, 0   ,draw opacity=1 ]   (206.22,67.77) .. controls (227.44,53.39) and (249.44,71.39) .. (259.44,88.7) ;

[color=rgb, 255:red, 22; green, 48; blue, 226   ,draw opacity=1 ]   (206.22,67.77) .. controls (287.44,59.39) and (207.44,184.39) .. (153,88.7) ;

[color=rgb, 255:red, 22; green, 48; blue, 226   ,draw opacity=1 ]   (383,88.7) .. controls (399.44,66.39) and (416.44,58.39) .. (436.22,67.77) ;

[color=rgb, 255:red, 208; green, 2; blue, 27   ,draw opacity=1 ]   (383,88.7) .. controls (441.44,-6.61) and (519.44,121.39) .. (436.22,109.62) ;

[color=rgb, 255:red, 248; green, 231; blue, 28   ,draw opacity=1 ]   (383,88.7) .. controls (394.38,45.25) and (459.44,23.39) .. (475.44,71.39) .. controls (491.44,119.39) and (444.44,132.39) .. (423.44,125.39) .. controls (402.44,118.39) and (374.44,70.39) .. (436.22,67.77) ;

[color=rgb, 255:red, 0; green, 255; blue, 0   ,draw opacity=1 ]   (436.22,67.77) .. controls (329.44,59.39) and (434.44,207.39) .. (489.44,88.7) ;

The triangulation on the left corresponds to the cluster
$$(I(3) = S(3)) \oplus \Sigma P(1) \oplus \Sigma P(2) \oplus \Sigma P(4)$$
and the one on the right corresponds to
$$(P(2) = S(2)) \oplus (24_3 = P(3)) \oplus (P(4) = S_4) \oplus \Sigma P(1).$$
Below is the first cluster depicted in the cluster category.

& & & & & & \ar[dr] & & & & \\
& \Sigma^{-1} I(1) \ar[dr] & & 42_3 \ar[dr] \ar[ur] & & & & 13_3 \ar[dr] & &  {\color{green}\Sigma P(2)} \ar[dr] & \\
\Sigma^{-1} I(2) \ar[ur] \ar[dr] & & 22_1 \ar[ur] \ar[dr] & & & & \ar[ur] \ar[dr] & & 11_1 \ar[ur] \ar[dr] & & {\color{yellow}\Sigma P(1)} \\
& \Sigma^{-1} I(3) \ar[ur] \ar[dr] & & 24_3 \ar[ur] \ar[dr] & & \cdots & & 31_3 \ar[ur] \ar[dr] & & {\color{red}\Sigma P(4)} \ar[ur] \ar[dr] & \\
\Sigma^{-1} I(4) \ar[ur] \ar[dr] & & 44_1 \ar[ur] \ar[dr] & & & & \ar[ur] \ar[dr] & & {\color{blue}33_1} \ar[ur] \ar[dr] & & \Sigma P(3) \\
& \Sigma^{-1} I(1) \ar[ur] & & 42_3 \ar[dr] \ar[ur] & & & & 13_3 \ar[ur] & & {\color{green}\Sigma P(2)} \ar[ur] & \\
& & & & & & \ar[ur] & & &}$$

To attain the second cluster, we move each object forward two positions in their corresponding ray as follows.

& & & & & & \ar[dr] & & & & \\
& \Sigma^{-1} I(1) \ar[dr] & & 42_3 \ar[dr] \ar[ur] & & & & 13_3 \ar[dr] & &  \Sigma P(2) \ar[dr] & \\
\Sigma^{-1} I(2) \ar[ur] \ar[dr] & & {\color{red} 22_1} \ar[ur] \ar[dr] & & & & \ar[ur] \ar[dr] & & 11_1 \ar[ur] \ar[dr] & & {\color{blue}\Sigma P(1)} \\
& \Sigma^{-1} I(3) \ar[ur] \ar[dr] & & {\color{yellow} 24_3} \ar[ur] \ar[dr] & & \cdots & & 31_3 \ar[ur] \ar[dr] & & \Sigma P(4) \ar[ur] \ar[dr] & \\
\Sigma^{-1} I(4) \ar[ur] \ar[dr] & & {\color{green} 44_1} \ar[ur] \ar[dr] & & & & \ar[ur] \ar[dr] & & 33_1 \ar[ur] \ar[dr] & & \Sigma P(3) \\
& \Sigma^{-1} I(1) \ar[ur] & & 42_3 \ar[dr] \ar[ur] & & & & 13_3 \ar[ur] & & \Sigma P(2) \ar[ur] & \\
& & & & & & \ar[ur] & & &}$$

§ EXAMPLES AND TABLE OF COUNTS OF FAMILIES

We begin this section with Example <ref>, which is a table providing a count of the number of families of triangulations of the annulus $A_{n,m}$ for a few small values of $n$ and $m$.

\[\begin{array}{c|cccccc}
\tikz{\node[below left, inner sep=1pt] (m) {m};%
\node[above right,inner sep=1pt] (n) {n};%
\draw (m.north west|-n.north west) -- (m.south east-|n.south east);}
& 1 & 2 & 3 & 4 & 5 & 6 \\
\hline
1 & 1 & 4 & 15 & 56 & 210 & 792\\
2 & 4 & 18 & 72 & 280 & 1080 & 4158\\
3 & 15 & 72 & 300 & 1200 & 4725 & 18480\\
4 & 56 & 280 & 1200 & 4900 & 19600 & 77616\\
5 & 210 & 1080 & 4725 & 19600 & 79380 & 317520\\
6 & 792 & 4158 & 18480 & 77616 & 317520 & 1280664\\
\end{array}\]

Now in Example <ref>, we have a depiction of all the sets $S_{(-,-,-)}$ for $A_{2,3}$.

every picture/.style=line width=0.75pt

(26.71,46) .. controls (26.71,37.9) and (33.28,31.34) .. (41.38,31.34) .. controls (49.48,31.34) and (56.05,37.9) .. (56.05,46) .. controls (56.05,54.11) and (49.48,60.67) .. (41.38,60.67) .. controls (33.28,60.67) and (26.71,54.11) .. (26.71,46)(4.71,46) .. controls (4.71,25.75) and (21.13,9.33) .. (41.38,9.33) .. controls (61.63,9.33) and (78.05,25.75) .. (78.05,46) .. controls (78.05,66.26) and (61.63,82.68) .. (41.38,82.68) .. controls (21.13,82.68) and (4.71,66.26) .. (4.71,46) ;

(15.81,20.26) ;
[shift=(15.81,20.26), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(15.81,20.26), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(41.38,31.34) ;
[shift=(41.38,31.34), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(41.38,31.34), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(56.05,46) ;
[shift=(56.05,46), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(56.05,46), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(67.39,71.93) ;
[shift=(67.39,71.93), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(67.39,71.93), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

[color=rgb, 255:red, 255; green, 125; blue, 0   ,draw opacity=1 ]   (15.81,20.26) .. controls (37.31,15.4) and (42.51,30.14) .. (41.38,31.34) ;

[color=rgb, 255:red, 0; green, 0; blue, 255   ,draw opacity=1 ]   (15.81,20.26) .. controls (14.76,15.58) and (15,71) .. (35.14,59.11) ;

[line width=2.25]    (82.39,46.44) – (114.45,46.44) ;
[shift=(118.45,46.44), rotate = 180] [color=rgb, 255:red, 0; green, 0; blue, 0   ][line width=2.25]    (17.49,-5.26) .. controls (11.12,-2.23) and (5.29,-0.48) .. (0,0) .. controls (5.29,0.48) and (11.12,2.23) .. (17.49,5.26)   ;

(35.14,59.11) ;
[shift=(35.14,59.11), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(35.14,59.11), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(147.74,46) .. controls (147.74,37.9) and (154.3,31.34) .. (162.4,31.34) .. controls (170.51,31.34) and (177.07,37.9) .. (177.07,46) .. controls (177.07,54.11) and (170.51,60.67) .. (162.4,60.67) .. controls (154.3,60.67) and (147.74,54.11) .. (147.74,46)(125.73,46) .. controls (125.73,25.75) and (142.15,9.33) .. (162.4,9.33) .. controls (182.66,9.33) and (199.08,25.75) .. (199.08,46) .. controls (199.08,66.26) and (182.66,82.68) .. (162.4,82.68) .. controls (142.15,82.68) and (125.73,66.26) .. (125.73,46) ;

(136.83,20.26) ;
[shift=(136.83,20.26), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(136.83,20.26), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(162.4,31.34) ;
[shift=(162.4,31.34), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(162.4,31.34), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(177.07,46) ;
[shift=(177.07,46), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(177.07,46), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(188.41,71.93) ;
[shift=(188.41,71.93), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(188.41,71.93), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

[color=rgb, 255:red, 255; green, 125; blue, 0   ,draw opacity=1 ]   (136.83,20.26) .. controls (161.97,8.47) and (189.54,18.35) .. (177.07,46) ;

[color=rgb, 255:red, 0; green, 0; blue, 255   ,draw opacity=1 ]   (136.83,20.26) .. controls (135.5,28) and (126,62.5) .. (156.16,59.11) ;

(156.16,59.11) ;
[shift=(156.16,59.11), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(156.16,59.11), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

[line width=2.25]    (202.37,46.44) – (234.44,46.44) ;
[shift=(238.44,46.44), rotate = 180] [color=rgb, 255:red, 0; green, 0; blue, 0   ][line width=2.25]    (17.49,-5.26) .. controls (11.12,-2.23) and (5.29,-0.48) .. (0,0) .. controls (5.29,0.48) and (11.12,2.23) .. (17.49,5.26)   ;

(269.65,46.35) .. controls (269.65,38.25) and (276.22,31.68) .. (284.32,31.68) .. controls (292.42,31.68) and (298.99,38.25) .. (298.99,46.35) .. controls (298.99,54.45) and (292.42,61.02) .. (284.32,61.02) .. controls (276.22,61.02) and (269.65,54.45) .. (269.65,46.35)(247.65,46.35) .. controls (247.65,26.1) and (264.07,9.68) .. (284.32,9.68) .. controls (304.57,9.68) and (320.99,26.1) .. (320.99,46.35) .. controls (320.99,66.6) and (304.57,83.02) .. (284.32,83.02) .. controls (264.07,83.02) and (247.65,66.6) .. (247.65,46.35) ;

(258.75,20.6) ;
[shift=(258.75,20.6), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(258.75,20.6), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(284.32,31.68) ;
[shift=(284.32,31.68), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(284.32,31.68), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(298.99,46.35) ;
[shift=(298.99,46.35), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(298.99,46.35), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(310.33,72.27) ;
[shift=(310.33,72.27), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(310.33,72.27), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

[color=rgb, 255:red, 255; green, 125; blue, 0   ,draw opacity=1 ]   (258.75,20.6) .. controls (298.28,8.56) and (313.02,32.74) .. (311.63,51.12) .. controls (310.24,69.5) and (292.5,78) .. (278.08,59.46) ;

[color=rgb, 255:red, 0; green, 0; blue, 255   ,draw opacity=1 ]   (258.75,20.6) .. controls (257,28) and (251.5,66.5) .. (278.08,59.46) ;

(278.08,59.46) ;
[shift=(278.08,59.46), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(278.08,59.46), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(26.54,145.7) .. controls (26.54,137.6) and (33.11,131.03) .. (41.21,131.03) .. controls (49.31,131.03) and (55.88,137.6) .. (55.88,145.7) .. controls (55.88,153.8) and (49.31,160.37) .. (41.21,160.37) .. controls (33.11,160.37) and (26.54,153.8) .. (26.54,145.7)(4.53,145.7) .. controls (4.53,125.45) and (20.95,109.03) .. (41.21,109.03) .. controls (61.46,109.03) and (77.88,125.45) .. (77.88,145.7) .. controls (77.88,165.96) and (61.46,182.37) .. (41.21,182.37) .. controls (20.95,182.37) and (4.53,165.96) .. (4.53,145.7) ;

(15.63,119.96) ;
[shift=(15.63,119.96), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(15.63,119.96), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(34.96,158.81) ;
[shift=(34.96,158.81), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(34.96,158.81), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(41.21,131.03) ;
[shift=(41.21,131.03), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(41.21,131.03), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(61.72,175.45) ;
[shift=(61.72,175.45), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(61.72,175.45), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

[color=rgb, 255:red, 255; green, 125; blue, 0   ,draw opacity=1 ]   (15.63,119.96) .. controls (37.13,115.1) and (42.33,129.84) .. (41.21,131.03) ;

[color=rgb, 255:red, 0; green, 0; blue, 255   ,draw opacity=1 ]   (15.63,119.96) .. controls (11.66,128.99) and (12,161) .. (27,168) .. controls (42,175) and (65.5,166.5) .. (55.88,145.7) ;

[line width=2.25]    (82.21,146.14) – (116.36,146.14) ;
[shift=(120.36,146.14), rotate = 180] [color=rgb, 255:red, 0; green, 0; blue, 0   ][line width=2.25]    (17.49,-5.26) .. controls (11.12,-2.23) and (5.29,-0.48) .. (0,0) .. controls (5.29,0.48) and (11.12,2.23) .. (17.49,5.26)   ;

(148.87,145.7) .. controls (148.87,137.6) and (155.44,131.03) .. (163.54,131.03) .. controls (171.64,131.03) and (178.21,137.6) .. (178.21,145.7) .. controls (178.21,153.8) and (171.64,160.37) .. (163.54,160.37) .. controls (155.44,160.37) and (148.87,153.8) .. (148.87,145.7)(126.87,145.7) .. controls (126.87,125.45) and (143.29,109.03) .. (163.54,109.03) .. controls (183.79,109.03) and (200.21,125.45) .. (200.21,145.7) .. controls (200.21,165.96) and (183.79,182.37) .. (163.54,182.37) .. controls (143.29,182.37) and (126.87,165.96) .. (126.87,145.7) ;

(137.96,119.96) ;
[shift=(137.96,119.96), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(137.96,119.96), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(163.54,131.03) ;
[shift=(163.54,131.03), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(163.54,131.03), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(178.21,145.7) ;
[shift=(178.21,145.7), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(178.21,145.7), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(189.55,171.62) ;
[shift=(189.55,171.62), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(189.55,171.62), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(157.3,158.81) ;
[shift=(157.3,158.81), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(157.3,158.81), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(381.54,46.17) .. controls (381.54,38.07) and (388.11,31.5) .. (396.21,31.5) .. controls (404.31,31.5) and (410.88,38.07) .. (410.88,46.17) .. controls (410.88,54.27) and (404.31,60.84) .. (396.21,60.84) .. controls (388.11,60.84) and (381.54,54.27) .. (381.54,46.17)(359.54,46.17) .. controls (359.54,25.92) and (375.96,9.5) .. (396.21,9.5) .. controls (416.47,9.5) and (432.88,25.92) .. (432.88,46.17) .. controls (432.88,66.42) and (416.47,82.84) .. (396.21,82.84) .. controls (375.96,82.84) and (359.54,66.42) .. (359.54,46.17) ;

(370.64,20.42) ;
[shift=(370.64,20.42), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(370.64,20.42), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(396.21,31.5) ;
[shift=(396.21,31.5), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(396.21,31.5), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(410.88,46.17) ;
[shift=(410.88,46.17), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(410.88,46.17), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(422.22,72.09) ;
[shift=(422.22,72.09), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(422.22,72.09), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

[line width=2.25]    (437.22,46.61) – (469.28,46.61) ;
[shift=(473.28,46.61), rotate = 180] [color=rgb, 255:red, 0; green, 0; blue, 0   ][line width=2.25]    (17.49,-5.26) .. controls (11.12,-2.23) and (5.29,-0.48) .. (0,0) .. controls (5.29,0.48) and (11.12,2.23) .. (17.49,5.26)   ;

(389.97,59.28) ;
[shift=(389.97,59.28), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(389.97,59.28), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(502.57,46.17) .. controls (502.57,38.07) and (509.14,31.5) .. (517.24,31.5) .. controls (525.34,31.5) and (531.91,38.07) .. (531.91,46.17) .. controls (531.91,54.27) and (525.34,60.84) .. (517.24,60.84) .. controls (509.14,60.84) and (502.57,54.27) .. (502.57,46.17)(480.57,46.17) .. controls (480.57,25.92) and (496.99,9.5) .. (517.24,9.5) .. controls (537.49,9.5) and (553.91,25.92) .. (553.91,46.17) .. controls (553.91,66.42) and (537.49,82.84) .. (517.24,82.84) .. controls (496.99,82.84) and (480.57,66.42) .. (480.57,46.17) ;

(491.66,20.42) ;
[shift=(491.66,20.42), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(491.66,20.42), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(517.24,31.5) ;
[shift=(517.24,31.5), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(517.24,31.5), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(531.91,46.17) ;
[shift=(531.91,46.17), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(531.91,46.17), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(543.25,72.09) ;
[shift=(543.25,72.09), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(543.25,72.09), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(511,59.28) ;
[shift=(511,59.28), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(511,59.28), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

[line width=2.25]    (557.2,46.61) – (589.27,46.61) ;
[shift=(593.27,46.61), rotate = 180] [color=rgb, 255:red, 0; green, 0; blue, 0   ][line width=2.25]    (17.49,-5.26) .. controls (11.12,-2.23) and (5.29,-0.48) .. (0,0) .. controls (5.29,0.48) and (11.12,2.23) .. (17.49,5.26)   ;

(624.49,46.52) .. controls (624.49,38.42) and (631.05,31.85) .. (639.16,31.85) .. controls (647.26,31.85) and (653.82,38.42) .. (653.82,46.52) .. controls (653.82,54.62) and (647.26,61.19) .. (639.16,61.19) .. controls (631.05,61.19) and (624.49,54.62) .. (624.49,46.52)(602.48,46.52) .. controls (602.48,26.27) and (618.9,9.85) .. (639.16,9.85) .. controls (659.41,9.85) and (675.83,26.27) .. (675.83,46.52) .. controls (675.83,66.77) and (659.41,83.19) .. (639.16,83.19) .. controls (618.9,83.19) and (602.48,66.77) .. (602.48,46.52) ;

(613.58,20.77) ;
[shift=(613.58,20.77), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(613.58,20.77), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(639.16,31.85) ;
[shift=(639.16,31.85), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(639.16,31.85), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(653.82,46.52) ;
[shift=(653.82,46.52), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(653.82,46.52), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(665.16,72.44) ;
[shift=(665.16,72.44), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(665.16,72.44), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(632.91,59.63) ;
[shift=(632.91,59.63), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(632.91,59.63), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(41.25,92.25) node  $S_{(a_1,b_3,b_1)}$;

(163.25,92.25) node  $S_{(a_1,b_3,b_2)}$;

(283.25,92.25) node  $S_{(a_1,b_3,b_3)}$;

(40.25,192.25) node  $S_{(a_1,b_2,b_1)}$;

(162.25,192.25) node  $S_{(a_1,b_2,b_2)}$;

(395.25,92.25) node  $S_{(a_2,b_3,b_1)}$;

(517.25,92.25) node  $S_{(a_2,b_3,b_2)}$;

(637.25,92.25) node  $S_{(a_2,b_3,b_3)}$;

(55.88,145.7) ;
[shift=(55.88,145.7), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(55.88,145.7), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

[color=rgb, 255:red, 0; green, 0; blue, 255   ,draw opacity=1 ]   (137.96,119.96) .. controls (133.99,128.99) and (134.33,161) .. (149.33,168) .. controls (164.33,175) and (187.83,166.5) .. (178.21,145.7) ;

[color=rgb, 255:red, 255; green, 125; blue, 0   ,draw opacity=1 ]   (137.96,119.96) .. controls (186.5,107) and (180.5,139.5) .. (178.21,145.7) ;

[color=rgb, 255:red, 0; green, 0; blue, 255   ,draw opacity=1 ]   (422.22,172.09) .. controls (433.72,162.59) and (424.5,118.5) .. (398,116) .. controls (371.5,113.5) and (369.27,144.57) .. (373,158.5) .. controls (376.73,172.43) and (410.65,183.49) .. (410.88,146.17) ;

[color=rgb, 255:red, 255; green, 125; blue, 0   ,draw opacity=1 ]   (396.21,31.5) .. controls (417.71,26.65) and (422.5,47) .. (422.22,72.09) ;

[color=rgb, 255:red, 255; green, 125; blue, 0   ,draw opacity=1 ]   (531.91,46.17) .. controls (548,48.5) and (543.5,66) .. (543.25,72.09) ;

[color=rgb, 255:red, 0; green, 0; blue, 255   ,draw opacity=1 ]   (665.16,72.44) .. controls (676.66,62.94) and (667.44,18.85) .. (640.94,16.35) .. controls (614.44,13.85) and (608.44,60.35) .. (632.91,59.63) ;

[color=rgb, 255:red, 255; green, 125; blue, 0   ,draw opacity=1 ]   (632.91,59.63) .. controls (633.5,69.5) and (640.5,78) .. (665.16,72.44) ;

(381.54,146.17) .. controls (381.54,138.07) and (388.11,131.5) .. (396.21,131.5) .. controls (404.31,131.5) and (410.88,138.07) .. (410.88,146.17) .. controls (410.88,154.27) and (404.31,160.84) .. (396.21,160.84) .. controls (388.11,160.84) and (381.54,154.27) .. (381.54,146.17)(359.54,146.17) .. controls (359.54,125.92) and (375.96,109.5) .. (396.21,109.5) .. controls (416.47,109.5) and (432.88,125.92) .. (432.88,146.17) .. controls (432.88,166.42) and (416.47,182.84) .. (396.21,182.84) .. controls (375.96,182.84) and (359.54,166.42) .. (359.54,146.17) ;

(370.64,120.42) ;
[shift=(370.64,120.42), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(370.64,120.42), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(396.21,131.5) ;
[shift=(396.21,131.5), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(396.21,131.5), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(410.88,146.17) ;
[shift=(410.88,146.17), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(410.88,146.17), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(422.22,172.09) ;
[shift=(422.22,172.09), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(422.22,172.09), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

[line width=2.25]    (437.22,147.61) – (469.28,147.61) ;
[shift=(473.28,147.61), rotate = 180] [color=rgb, 255:red, 0; green, 0; blue, 0   ][line width=2.25]    (17.49,-5.26) .. controls (11.12,-2.23) and (5.29,-0.48) .. (0,0) .. controls (5.29,0.48) and (11.12,2.23) .. (17.49,5.26)   ;

(389.97,159.28) ;
[shift=(389.97,159.28), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(389.97,159.28), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(503.57,146.17) .. controls (503.57,138.07) and (510.14,131.5) .. (518.24,131.5) .. controls (526.34,131.5) and (532.91,138.07) .. (532.91,146.17) .. controls (532.91,154.27) and (526.34,160.84) .. (518.24,160.84) .. controls (510.14,160.84) and (503.57,154.27) .. (503.57,146.17)(481.57,146.17) .. controls (481.57,125.92) and (497.99,109.5) .. (518.24,109.5) .. controls (538.49,109.5) and (554.91,125.92) .. (554.91,146.17) .. controls (554.91,166.42) and (538.49,182.84) .. (518.24,182.84) .. controls (497.99,182.84) and (481.57,166.42) .. (481.57,146.17) ;

(492.66,120.42) ;
[shift=(492.66,120.42), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(492.66,120.42), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(518.24,131.5) ;
[shift=(518.24,131.5), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(518.24,131.5), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(532.91,146.17) ;
[shift=(532.91,146.17), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(532.91,146.17), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(544.25,172.09) ;
[shift=(544.25,172.09), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(544.25,172.09), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(512,159.28) ;
[shift=(512,159.28), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(512,159.28), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(395.25,192.25) node  $S_{(a_2,b_2,b_1)}$;

(518.25,192.25) node  $S_{(a_2,b_2,b_2)}$;

[color=rgb, 255:red, 0; green, 0; blue, 255   ,draw opacity=1 ]   (422.22,72.09) .. controls (433.72,62.59) and (424.5,18.5) .. (398,16) .. controls (371.5,13.5) and (365.5,60) .. (389.97,59.28) ;

[color=rgb, 255:red, 255; green, 125; blue, 0   ,draw opacity=1 ]   (396.21,131.5) .. controls (417.71,126.65) and (422.5,147) .. (422.22,172.09) ;

[color=rgb, 255:red, 0; green, 0; blue, 255   ,draw opacity=1 ]   (543.25,72.09) .. controls (554.75,62.59) and (545.52,18.5) .. (519.02,16) .. controls (492.52,13.5) and (486.52,60) .. (511,59.28) ;

[color=rgb, 255:red, 255; green, 125; blue, 0   ,draw opacity=1 ]   (532.91,146.17) .. controls (549,148.5) and (544.5,166) .. (544.25,172.09) ;

[color=rgb, 255:red, 0; green, 0; blue, 255   ,draw opacity=1 ]   (544.25,172.09) .. controls (555.75,162.59) and (546.52,118.5) .. (520.02,116) .. controls (493.52,113.5) and (491.29,144.57) .. (495.02,158.5) .. controls (498.76,172.43) and (532.68,183.49) .. (532.91,146.17) ;

(381.57,247.17) .. controls (381.57,239.07) and (388.14,232.5) .. (396.24,232.5) .. controls (404.34,232.5) and (410.91,239.07) .. (410.91,247.17) .. controls (410.91,255.27) and (404.34,261.84) .. (396.24,261.84) .. controls (388.14,261.84) and (381.57,255.27) .. (381.57,247.17)(359.57,247.17) .. controls (359.57,226.92) and (375.99,210.5) .. (396.24,210.5) .. controls (416.49,210.5) and (432.91,226.92) .. (432.91,247.17) .. controls (432.91,267.42) and (416.49,283.84) .. (396.24,283.84) .. controls (375.99,283.84) and (359.57,267.42) .. (359.57,247.17) ;

(370.66,221.42) ;
[shift=(370.66,221.42), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(370.66,221.42), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(396.24,232.5) ;
[shift=(396.24,232.5), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(396.24,232.5), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(410.91,247.17) ;
[shift=(410.91,247.17), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(410.91,247.17), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(422.25,273.09) ;
[shift=(422.25,273.09), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(422.25,273.09), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(390,260.28) ;
[shift=(390,260.28), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(390,260.28), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(396.25,293.25) node  $S_{(a_2,b_1,b_1)}$;

[color=rgb, 255:red, 0; green, 0; blue, 255   ,draw opacity=1 ]   (422.25,273.09) .. controls (433.75,263.59) and (425.05,216.5) .. (398.02,217) .. controls (371,217.5) and (370.5,265) .. (390,270) .. controls (409.5,275) and (437.5,236.5) .. (396.24,232.5) ;

[color=rgb, 255:red, 255; green, 125; blue, 0   ,draw opacity=1 ]   (396.24,232.5) .. controls (422,226) and (422.52,248) .. (422.25,273.09) ;

(27.54,246.7) .. controls (27.54,238.6) and (34.11,232.03) .. (42.21,232.03) .. controls (50.31,232.03) and (56.88,238.6) .. (56.88,246.7) .. controls (56.88,254.8) and (50.31,261.37) .. (42.21,261.37) .. controls (34.11,261.37) and (27.54,254.8) .. (27.54,246.7)(5.53,246.7) .. controls (5.53,226.45) and (21.95,210.03) .. (42.21,210.03) .. controls (62.46,210.03) and (78.88,226.45) .. (78.88,246.7) .. controls (78.88,266.96) and (62.46,283.37) .. (42.21,283.37) .. controls (21.95,283.37) and (5.53,266.96) .. (5.53,246.7) ;

(16.63,220.96) ;
[shift=(16.63,220.96), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(16.63,220.96), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(35.96,259.81) ;
[shift=(35.96,259.81), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(35.96,259.81), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(42.21,232.03) ;
[shift=(42.21,232.03), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(42.21,232.03), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(62.72,276.45) ;
[shift=(62.72,276.45), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(62.72,276.45), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

[color=rgb, 255:red, 255; green, 125; blue, 0   ,draw opacity=1 ]   (16.63,220.96) .. controls (38.13,216.1) and (43.33,230.84) .. (42.21,232.03) ;

[color=rgb, 255:red, 0; green, 0; blue, 255   ,draw opacity=1 ]   (16.63,220.96) .. controls (8,256) and (29.5,279) .. (56,270) .. controls (82.5,261) and (69,220) .. (42.21,232.03) ;

(41.25,293.25) node  $S_{(a_1,b_1,b_1)}$;

(56.88,246.7) ;
[shift=(56.88,246.7), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(56.88,246.7), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

Finally, we end this section with Example <ref> which gives a complete list of representatives of inner equivalence classes of triangulations for $A_{2,2}$.

every picture/.style=line width=0.75pt

every picture/.style=line width=0.75pt

every picture/.style=line width=0.75pt

(63.03,100.69) .. controls (63.03,89.71) and (72.15,80.81) .. (83.4,80.81) .. controls (94.66,80.81) and (103.78,89.71) .. (103.78,100.69) .. controls (103.78,111.67) and (94.66,120.58) .. (83.4,120.58) .. controls (72.15,120.58) and (63.03,111.67) .. (63.03,100.69)(33.21,100.69) .. controls (33.21,73.24) and (55.68,50.99) .. (83.4,50.99) .. controls (111.13,50.99) and (133.6,73.24) .. (133.6,100.69) .. controls (133.6,128.15) and (111.13,150.4) .. (83.4,150.4) .. controls (55.68,150.4) and (33.21,128.15) .. (33.21,100.69) ;

(83.4,50.99) ;
[shift=(83.4,50.99), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(83.4,50.99), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(103.48,100.69) ;
[shift=(103.48,100.69), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(103.48,100.69), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(83.4,150.4) ;
[shift=(83.4,150.4), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(83.4,150.4), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

[color=rgb, 255:red, 255; green, 125; blue, 0   ,draw opacity=1 ]   (83.4,50.99) .. controls (111.77,63.28) and (105.03,99.07) .. (103.48,100.69) ;

[color=rgb, 255:red, 0; green, 0; blue, 255   ,draw opacity=1 ]   (83.4,50.99) .. controls (81.98,44.64) and (54.96,66.67) .. (63.33,100.69) ;

(63.33,100.69) ;
[shift=(63.33,100.69), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(63.33,100.69), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

[color=rgb, 255:red, 0; green, 0; blue, 0   ,draw opacity=1 ]   (103.48,100.69) .. controls (115.19,124.95) and (84.95,148.78) .. (83.4,150.4) ;

[color=rgb, 255:red, 0; green, 0; blue, 0   ,draw opacity=1 ]   (63.33,100.69) .. controls (54.96,126.98) and (84.95,148.78) .. (83.4,150.4) ;

(173.53,100.69) .. controls (173.53,89.71) and (182.65,80.81) .. (193.9,80.81) .. controls (205.16,80.81) and (214.28,89.71) .. (214.28,100.69) .. controls (214.28,111.67) and (205.16,120.58) .. (193.9,120.58) .. controls (182.65,120.58) and (173.53,111.67) .. (173.53,100.69)(143.71,100.69) .. controls (143.71,73.24) and (166.18,50.99) .. (193.9,50.99) .. controls (221.63,50.99) and (244.1,73.24) .. (244.1,100.69) .. controls (244.1,128.15) and (221.63,150.4) .. (193.9,150.4) .. controls (166.18,150.4) and (143.71,128.15) .. (143.71,100.69) ;

(193.9,50.99) ;
[shift=(193.9,50.99), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(193.9,50.99), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(213.98,100.69) ;
[shift=(213.98,100.69), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(213.98,100.69), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(193.9,150.4) ;
[shift=(193.9,150.4), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(193.9,150.4), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

[color=rgb, 255:red, 255; green, 125; blue, 0   ,draw opacity=1 ]   (193.9,50.99) .. controls (222.27,63.28) and (215.53,99.07) .. (213.98,100.69) ;

[color=rgb, 255:red, 0; green, 0; blue, 255   ,draw opacity=1 ]   (193.9,50.99) .. controls (192.48,44.64) and (165.46,66.67) .. (173.83,100.69) ;

(173.83,100.69) ;
[shift=(173.83,100.69), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(173.83,100.69), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

[color=rgb, 255:red, 0; green, 0; blue, 0   ,draw opacity=1 ]   (193.9,50.99) .. controls (230.86,63.59) and (236.51,99.39) .. (227.6,117.9) .. controls (218.69,136.41) and (177.62,145.94) .. (173.53,100.69) ;

[color=rgb, 255:red, 0; green, 0; blue, 0   ,draw opacity=1 ]   (173.83,100.69) .. controls (165.46,126.98) and (195.45,148.78) .. (193.9,150.4) ;

(283.53,100.19) .. controls (283.53,89.21) and (292.65,80.31) .. (303.9,80.31) .. controls (315.16,80.31) and (324.28,89.21) .. (324.28,100.19) .. controls (324.28,111.17) and (315.16,120.08) .. (303.9,120.08) .. controls (292.65,120.08) and (283.53,111.17) .. (283.53,100.19)(253.71,100.19) .. controls (253.71,72.74) and (276.18,50.49) .. (303.9,50.49) .. controls (331.63,50.49) and (354.1,72.74) .. (354.1,100.19) .. controls (354.1,127.65) and (331.63,149.9) .. (303.9,149.9) .. controls (276.18,149.9) and (253.71,127.65) .. (253.71,100.19) ;

(303.9,50.49) ;
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[shift=(303.9,50.49), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(323.98,100.19) ;
[shift=(323.98,100.19), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(323.98,100.19), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(303.9,149.9) ;
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[shift=(303.9,149.9), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

[color=rgb, 255:red, 255; green, 125; blue, 0   ,draw opacity=1 ]   (303.9,50.49) .. controls (332.27,62.78) and (325.53,98.57) .. (323.98,100.19) ;

[color=rgb, 255:red, 0; green, 0; blue, 255   ,draw opacity=1 ]   (303.9,50.49) .. controls (302.48,44.14) and (275.46,66.17) .. (283.83,100.19) ;

(283.83,100.19) ;
[shift=(283.83,100.19), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(283.83,100.19), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

[color=rgb, 255:red, 0; green, 0; blue, 0   ,draw opacity=1 ]   (323.98,100.19) .. controls (335.69,124.45) and (305.45,148.28) .. (303.9,149.9) ;

[color=rgb, 255:red, 0; green, 0; blue, 0   ,draw opacity=1 ]   (302.2,49.33) .. controls (263.76,59.19) and (263.6,92.9) .. (273.6,118.4) .. controls (283.6,143.9) and (323.18,152.42) .. (323.98,100.19) ;

(63.03,221.19) .. controls (63.03,210.21) and (72.15,201.31) .. (83.4,201.31) .. controls (94.66,201.31) and (103.78,210.21) .. (103.78,221.19) .. controls (103.78,232.17) and (94.66,241.08) .. (83.4,241.08) .. controls (72.15,241.08) and (63.03,232.17) .. (63.03,221.19)(33.21,221.19) .. controls (33.21,193.74) and (55.68,171.49) .. (83.4,171.49) .. controls (111.13,171.49) and (133.6,193.74) .. (133.6,221.19) .. controls (133.6,248.65) and (111.13,270.9) .. (83.4,270.9) .. controls (55.68,270.9) and (33.21,248.65) .. (33.21,221.19) ;

(83.4,171.49) ;
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(103.48,221.19) ;
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[shift=(103.48,221.19), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(83.4,270.9) ;
[shift=(83.4,270.9), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(83.4,270.9), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

[color=rgb, 255:red, 255; green, 125; blue, 0   ,draw opacity=1 ]   (83.4,171.49) .. controls (133.8,185.4) and (125.67,211.13) .. (116.8,236.9) .. controls (107.93,262.67) and (44.93,268) .. (63.33,221.19) ;

[color=rgb, 255:red, 0; green, 0; blue, 255   ,draw opacity=1 ]   (83.4,171.49) .. controls (81.98,165.14) and (54.96,187.17) .. (63.33,221.19) ;

(63.33,221.19) ;
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[shift=(63.33,221.19), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

[color=rgb, 255:red, 0; green, 0; blue, 0   ,draw opacity=1 ]   (63.03,221.19) .. controls (62.27,185.33) and (99.93,179.67) .. (114.6,208.33) .. controls (129.27,237) and (66.27,277) .. (61.27,220.67) ;

[color=rgb, 255:red, 0; green, 0; blue, 0   ,draw opacity=1 ]   (63.33,221.19) .. controls (33.27,239.33) and (84.95,269.28) .. (83.4,270.9) ;

(172.7,220.86) .. controls (172.7,209.88) and (181.82,200.98) .. (193.07,200.98) .. controls (204.32,200.98) and (213.44,209.88) .. (213.44,220.86) .. controls (213.44,231.84) and (204.32,240.74) .. (193.07,240.74) .. controls (181.82,240.74) and (172.7,231.84) .. (172.7,220.86)(142.88,220.86) .. controls (142.88,193.41) and (165.35,171.16) .. (193.07,171.16) .. controls (220.79,171.16) and (243.27,193.41) .. (243.27,220.86) .. controls (243.27,248.31) and (220.79,270.57) .. (193.07,270.57) .. controls (165.35,270.57) and (142.88,248.31) .. (142.88,220.86) ;

(193.07,171.16) ;
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[shift=(193.07,171.16), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(213.15,220.86) ;
[shift=(213.15,220.86), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(213.15,220.86), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(193.07,270.57) ;
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[shift=(193.07,270.57), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

[color=rgb, 255:red, 255; green, 125; blue, 0   ,draw opacity=1 ]   (193.07,171.16) .. controls (225.13,188.33) and (233.47,219.73) .. (226.13,237.4) .. controls (218.8,255.07) and (154.6,267.67) .. (172.99,220.86) ;

[color=rgb, 255:red, 0; green, 0; blue, 255   ,draw opacity=1 ]   (193.07,171.16) .. controls (191.65,164.81) and (164.63,186.83) .. (172.99,220.86) ;

(172.99,220.86) ;
[shift=(172.99,220.86), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(172.99,220.86), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

[color=rgb, 255:red, 0; green, 0; blue, 0   ,draw opacity=1 ]   (172.7,220.86) .. controls (171.93,185) and (205.47,181.07) .. (220.13,209.73) .. controls (234.8,238.4) and (175.93,276.67) .. (170.93,220.33) ;

[color=rgb, 255:red, 0; green, 0; blue, 0   ,draw opacity=1 ]   (192.2,169.4) .. controls (160.64,181.88) and (156.8,202.12) .. (155.13,217.4) .. controls (153.46,232.68) and (170.39,261.9) .. (191.13,263.33) .. controls (211.88,264.77) and (221.66,256.14) .. (231.8,241.73) .. controls (241.94,227.32) and (242.64,190.66) .. (195.13,173.33) ;

(392.53,101.19) .. controls (392.53,90.21) and (401.65,81.31) .. (412.9,81.31) .. controls (424.16,81.31) and (433.28,90.21) .. (433.28,101.19) .. controls (433.28,112.17) and (424.16,121.08) .. (412.9,121.08) .. controls (401.65,121.08) and (392.53,112.17) .. (392.53,101.19)(362.71,101.19) .. controls (362.71,73.74) and (385.18,51.49) .. (412.9,51.49) .. controls (440.63,51.49) and (463.1,73.74) .. (463.1,101.19) .. controls (463.1,128.65) and (440.63,150.9) .. (412.9,150.9) .. controls (385.18,150.9) and (362.71,128.65) .. (362.71,101.19) ;

(412.9,51.49) ;
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[shift=(412.9,51.49), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(432.98,101.19) ;
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[shift=(432.98,101.19), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(412.9,150.9) ;
[shift=(412.9,150.9), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(412.9,150.9), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

[color=rgb, 255:red, 255; green, 125; blue, 0   ,draw opacity=1 ]   (412.9,51.49) .. controls (441.27,63.78) and (434.53,99.57) .. (432.98,101.19) ;

[color=rgb, 255:red, 0; green, 0; blue, 255   ,draw opacity=1 ]   (412.9,51.49) .. controls (411.48,45.14) and (384.46,67.17) .. (392.83,101.19) ;

(392.83,101.19) ;
[shift=(392.83,101.19), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(392.83,101.19), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

[color=rgb, 255:red, 0; green, 0; blue, 0   ,draw opacity=1 ]   (432.98,101.19) .. controls (446,141.67) and (390.29,134.19) .. (384,106) .. controls (377.71,77.81) and (394.96,59.98) .. (411.53,50) ;

[color=rgb, 255:red, 0; green, 0; blue, 0   ,draw opacity=1 ]   (411.53,50) .. controls (343.67,80) and (372.87,140.67) .. (413.2,140.33) .. controls (453.53,140) and (484,87) .. (415.87,51) ;

(502.53,101.19) .. controls (502.53,90.21) and (511.65,81.31) .. (522.9,81.31) .. controls (534.16,81.31) and (543.28,90.21) .. (543.28,101.19) .. controls (543.28,112.17) and (534.16,121.08) .. (522.9,121.08) .. controls (511.65,121.08) and (502.53,112.17) .. (502.53,101.19)(472.71,101.19) .. controls (472.71,73.74) and (495.18,51.49) .. (522.9,51.49) .. controls (550.63,51.49) and (573.1,73.74) .. (573.1,101.19) .. controls (573.1,128.65) and (550.63,150.9) .. (522.9,150.9) .. controls (495.18,150.9) and (472.71,128.65) .. (472.71,101.19) ;

(522.9,51.49) ;
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[shift=(522.9,51.49), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(542.98,101.19) ;
[shift=(542.98,101.19), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(542.98,101.19), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(522.9,150.9) ;
[shift=(522.9,150.9), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(522.9,150.9), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

[color=rgb, 255:red, 255; green, 125; blue, 0   ,draw opacity=1 ]   (522.9,51.49) .. controls (551.27,63.78) and (544.53,99.57) .. (542.98,101.19) ;

[color=rgb, 255:red, 0; green, 0; blue, 255   ,draw opacity=1 ]   (522.9,51.49) .. controls (521.48,45.14) and (494.46,67.17) .. (502.83,101.19) ;

(502.83,101.19) ;
[shift=(502.83,101.19), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(502.83,101.19), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

[color=rgb, 255:red, 0; green, 0; blue, 0   ,draw opacity=1 ]   (502.83,101.19) .. controls (497.53,156) and (556.53,127) .. (558.87,102.67) .. controls (561.2,78.33) and (547.87,61.67) .. (522.9,51.49) ;

[color=rgb, 255:red, 0; green, 0; blue, 0   ,draw opacity=1 ]   (521.53,50) .. controls (453.67,80) and (482.87,140.67) .. (523.2,140.33) .. controls (563.53,140) and (599.53,85.67) .. (525.87,51) ;

(282.63,221.19) .. controls (282.63,210.21) and (291.75,201.31) .. (303,201.31) .. controls (314.26,201.31) and (323.38,210.21) .. (323.38,221.19) .. controls (323.38,232.17) and (314.26,241.08) .. (303,241.08) .. controls (291.75,241.08) and (282.63,232.17) .. (282.63,221.19)(252.81,221.19) .. controls (252.81,193.74) and (275.28,171.49) .. (303,171.49) .. controls (330.73,171.49) and (353.2,193.74) .. (353.2,221.19) .. controls (353.2,248.65) and (330.73,270.9) .. (303,270.9) .. controls (275.28,270.9) and (252.81,248.65) .. (252.81,221.19) ;

(303,171.49) ;
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[shift=(303,171.49), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(323.08,221.19) ;
[shift=(323.08,221.19), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(323.08,221.19), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(303,270.9) ;
[shift=(303,270.9), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(303,270.9), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

[color=rgb, 255:red, 255; green, 125; blue, 0   ,draw opacity=1 ]   (303,171.49) .. controls (331.37,183.78) and (324.63,219.57) .. (323.08,221.19) ;

[color=rgb, 255:red, 0; green, 0; blue, 255   ,draw opacity=1 ]   (303,171.49) .. controls (251.13,190.67) and (266.8,229.33) .. (277.8,243.67) .. controls (288.8,258) and (337.47,259.67) .. (323.08,221.19) ;

(282.93,221.19) ;
[shift=(282.93,221.19), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(282.93,221.19), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

[color=rgb, 255:red, 0; green, 0; blue, 0   ,draw opacity=1 ]   (323.38,221.19) .. controls (324.13,179) and (276.47,190) .. (273.13,217) .. controls (269.8,244) and (324.8,267) .. (323.08,221.19) ;

[color=rgb, 255:red, 0; green, 0; blue, 0   ,draw opacity=1 ]   (323.08,221.19) .. controls (357.87,234.73) and (304.55,269.28) .. (303,270.9) ;

(392.7,221.53) .. controls (392.7,210.55) and (401.82,201.65) .. (413.07,201.65) .. controls (424.32,201.65) and (433.44,210.55) .. (433.44,221.53) .. controls (433.44,232.51) and (424.32,241.41) .. (413.07,241.41) .. controls (401.82,241.41) and (392.7,232.51) .. (392.7,221.53)(362.88,221.53) .. controls (362.88,194.08) and (385.35,171.82) .. (413.07,171.82) .. controls (440.79,171.82) and (463.27,194.08) .. (463.27,221.53) .. controls (463.27,248.98) and (440.79,271.23) .. (413.07,271.23) .. controls (385.35,271.23) and (362.88,248.98) .. (362.88,221.53) ;

(413.07,171.82) ;
[shift=(413.07,171.82), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(413.07,171.82), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(433.15,221.53) ;
[shift=(433.15,221.53), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(433.15,221.53), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(413.07,271.23) ;
[shift=(413.07,271.23), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(413.07,271.23), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(392.99,221.53) ;
[shift=(392.99,221.53), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(392.99,221.53), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

[color=rgb, 255:red, 0; green, 0; blue, 0   ,draw opacity=1 ]   (412.2,170.07) .. controls (380.64,182.54) and (369.53,205.73) .. (369.87,220.4) .. controls (370.2,235.07) and (390.39,262.56) .. (411.13,264) .. controls (431.88,265.44) and (441.66,256.81) .. (451.8,242.4) .. controls (461.94,227.99) and (463.04,188.06) .. (415.53,170.73) ;

[color=rgb, 255:red, 0; green, 0; blue, 0   ,draw opacity=1 ]   (433.74,221.53) .. controls (434.49,179.33) and (386.83,190.33) .. (383.49,217.33) .. controls (380.16,244.33) and (435.16,267.33) .. (433.44,221.53) ;

[color=rgb, 255:red, 255; green, 125; blue, 0   ,draw opacity=1 ]   (413.66,171.82) .. controls (442.02,184.11) and (435.28,219.91) .. (433.74,221.53) ;

[color=rgb, 255:red, 0; green, 0; blue, 255   ,draw opacity=1 ]   (413.66,171.82) .. controls (369.87,192.07) and (377.45,229.67) .. (388.45,244) .. controls (399.45,258.33) and (448.12,260) .. (433.74,221.53) ;

(63.03,351.76) .. controls (63.03,340.78) and (72.15,331.88) .. (83.4,331.88) .. controls (94.66,331.88) and (103.78,340.78) .. (103.78,351.76) .. controls (103.78,362.74) and (94.66,371.64) .. (83.4,371.64) .. controls (72.15,371.64) and (63.03,362.74) .. (63.03,351.76)(33.21,351.76) .. controls (33.21,324.31) and (55.68,302.06) .. (83.4,302.06) .. controls (111.13,302.06) and (133.6,324.31) .. (133.6,351.76) .. controls (133.6,379.21) and (111.13,401.47) .. (83.4,401.47) .. controls (55.68,401.47) and (33.21,379.21) .. (33.21,351.76) ;

(83.4,302.06) ;
[shift=(83.4,302.06), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(83.4,302.06), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(103.48,351.76) ;
[shift=(103.48,351.76), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(103.48,351.76), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(83.4,401.47) ;
[shift=(83.4,401.47), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(83.4,401.47), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

[color=rgb, 255:red, 255; green, 125; blue, 0   ,draw opacity=1 ]   (83.4,401.47) .. controls (121.3,382.8) and (105.03,350.14) .. (103.48,351.76) ;

[color=rgb, 255:red, 0; green, 0; blue, 255   ,draw opacity=1 ]   (83.4,401.47) .. controls (197.3,350.8) and (50.8,276.8) .. (63.33,351.76) ;

(63.33,351.76) ;
[shift=(63.33,351.76), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(63.33,351.76), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

[color=rgb, 255:red, 0; green, 0; blue, 0   ,draw opacity=1 ]   (103.48,351.76) .. controls (110.8,428.3) and (-13.2,356.3) .. (83.4,302.06) ;

[color=rgb, 255:red, 0; green, 0; blue, 0   ,draw opacity=1 ]   (63.33,351.76) .. controls (47.3,331.3) and (84.95,300.43) .. (83.4,302.06) ;

(173.53,351.76) .. controls (173.53,340.78) and (182.65,331.88) .. (193.9,331.88) .. controls (205.16,331.88) and (214.28,340.78) .. (214.28,351.76) .. controls (214.28,362.74) and (205.16,371.64) .. (193.9,371.64) .. controls (182.65,371.64) and (173.53,362.74) .. (173.53,351.76)(143.71,351.76) .. controls (143.71,324.31) and (166.18,302.06) .. (193.9,302.06) .. controls (221.63,302.06) and (244.1,324.31) .. (244.1,351.76) .. controls (244.1,379.21) and (221.63,401.47) .. (193.9,401.47) .. controls (166.18,401.47) and (143.71,379.21) .. (143.71,351.76) ;

(193.9,302.06) ;
[shift=(193.9,302.06), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(193.9,302.06), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(213.98,351.76) ;
[shift=(213.98,351.76), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(213.98,351.76), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(193.9,401.47) ;
[shift=(193.9,401.47), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(193.9,401.47), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(173.83,351.76) ;
[shift=(173.83,351.76), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(173.83,351.76), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(283.53,351.26) .. controls (283.53,340.28) and (292.65,331.38) .. (303.9,331.38) .. controls (315.16,331.38) and (324.28,340.28) .. (324.28,351.26) .. controls (324.28,362.24) and (315.16,371.14) .. (303.9,371.14) .. controls (292.65,371.14) and (283.53,362.24) .. (283.53,351.26)(253.71,351.26) .. controls (253.71,323.81) and (276.18,301.56) .. (303.9,301.56) .. controls (331.63,301.56) and (354.1,323.81) .. (354.1,351.26) .. controls (354.1,378.71) and (331.63,400.97) .. (303.9,400.97) .. controls (276.18,400.97) and (253.71,378.71) .. (253.71,351.26) ;

(303.9,301.56) ;
[shift=(303.9,301.56), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(303.9,301.56), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(323.98,351.26) ;
[shift=(323.98,351.26), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(323.98,351.26), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(303.9,400.97) ;
[shift=(303.9,400.97), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(303.9,400.97), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(283.83,351.26) ;
[shift=(283.83,351.26), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(283.83,351.26), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(63.03,472.26) .. controls (63.03,461.28) and (72.15,452.38) .. (83.4,452.38) .. controls (94.66,452.38) and (103.78,461.28) .. (103.78,472.26) .. controls (103.78,483.24) and (94.66,492.14) .. (83.4,492.14) .. controls (72.15,492.14) and (63.03,483.24) .. (63.03,472.26)(33.21,472.26) .. controls (33.21,444.81) and (55.68,422.56) .. (83.4,422.56) .. controls (111.13,422.56) and (133.6,444.81) .. (133.6,472.26) .. controls (133.6,499.71) and (111.13,521.97) .. (83.4,521.97) .. controls (55.68,521.97) and (33.21,499.71) .. (33.21,472.26) ;

(83.4,422.56) ;
[shift=(83.4,422.56), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(83.4,422.56), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(103.48,472.26) ;
[shift=(103.48,472.26), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(103.48,472.26), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(83.4,521.97) ;
[shift=(83.4,521.97), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(83.4,521.97), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(63.33,472.26) ;
[shift=(63.33,472.26), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(63.33,472.26), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(172.7,471.93) .. controls (172.7,460.95) and (181.82,452.05) .. (193.07,452.05) .. controls (204.32,452.05) and (213.44,460.95) .. (213.44,471.93) .. controls (213.44,482.91) and (204.32,491.81) .. (193.07,491.81) .. controls (181.82,491.81) and (172.7,482.91) .. (172.7,471.93)(142.88,471.93) .. controls (142.88,444.48) and (165.35,422.22) .. (193.07,422.22) .. controls (220.79,422.22) and (243.27,444.48) .. (243.27,471.93) .. controls (243.27,499.38) and (220.79,521.63) .. (193.07,521.63) .. controls (165.35,521.63) and (142.88,499.38) .. (142.88,471.93) ;

(193.07,422.22) ;
[shift=(193.07,422.22), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(193.07,422.22), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(213.15,471.93) ;
[shift=(213.15,471.93), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(213.15,471.93), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(193.07,521.63) ;
[shift=(193.07,521.63), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(193.07,521.63), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(172.99,471.93) ;
[shift=(172.99,471.93), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(172.99,471.93), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

[color=rgb, 255:red, 0; green, 0; blue, 0   ,draw opacity=1 ]   (172.7,471.93) .. controls (171.93,436.07) and (205.47,432.13) .. (220.13,460.8) .. controls (234.8,489.47) and (175.93,527.73) .. (170.93,471.4) ;

(392.53,352.26) .. controls (392.53,341.28) and (401.65,332.38) .. (412.9,332.38) .. controls (424.16,332.38) and (433.28,341.28) .. (433.28,352.26) .. controls (433.28,363.24) and (424.16,372.14) .. (412.9,372.14) .. controls (401.65,372.14) and (392.53,363.24) .. (392.53,352.26)(362.71,352.26) .. controls (362.71,324.81) and (385.18,302.56) .. (412.9,302.56) .. controls (440.63,302.56) and (463.1,324.81) .. (463.1,352.26) .. controls (463.1,379.71) and (440.63,401.97) .. (412.9,401.97) .. controls (385.18,401.97) and (362.71,379.71) .. (362.71,352.26) ;

(412.9,302.56) ;
[shift=(412.9,302.56), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(412.9,302.56), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(432.98,352.26) ;
[shift=(432.98,352.26), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(432.98,352.26), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(412.9,401.97) ;
[shift=(412.9,401.97), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;
[shift=(412.9,401.97), rotate = 0] [color=rgb, 255:red, 0; green, 0; blue, 0   ][fill=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]      (0, 0) circle [x radius= 3.35, y radius= 3.35]   ;

(392.83,352.26) ;
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§ HOMEOMORPHISM BETWEEN AN ANNULUS AND A CYLINDER