Dedicated to the memory of Andrzej Skowroński.

Given a Jacobian algebra arising from the punctured disk, we show that all non-split extensions can be found using the tagged arcs and skein relations previously developed in cluster algebras theory. Our geometric interpretation can be used to find non-split extensions over other Jacobian algebras arising form surfaces with punctures. We show examples in type $D$ and in a punctured surface.

§ INTRODUCTION

Jacobian algebras arising from surfaces were defined by Labardini-Fragoso in [15] building in the works [12] and [13]. The reader can also find an overview of Labardini's work on Jacobian algebras here [16]. Given a compact Riemann surface with some disk removed, in order to create boundaries, and adding marked points in each boundary component and in the interior of the surface, we can define a triangulation. A triangulation is a finite set of curves (up to isotopy) splitting the surface into (ideal) triangles. A marked point in the interior of the surface is called a puncture. The simplest punctured surface is the once-punctured disk. This surface is associated to the cluster category of type $D$, see [23]. A triangulation of the punctured disk defines a cluster-tilted algebra of type $D$ [6, 5].  More generally, a punctured surface defines a cluster category [22], and a triangulation of the surface defines a Jacobian algebra in the sense of [15].

Cluster categories associated to unpunctured surfaces were defined by Brüstle-Zhang [8].
For the unpunctured version, Jacobian algebras arising from surface triangulations are gentle algebras and were studied by Assem et. all in [1], where string modules are defined by arcs that do not belong to the triangulation. Gentle algebras and their module categories are very well understood. Building on these works, and on their knowledge on gentle algebras and cluster algebras arising from unpunctured surfaces, Canakci and Schroll [10] study non-split extensions in module categories of gentle algebras arising from surface triangulations. They show how to find non-split extensions and, as a consequence, they find non-split triangles in the cluster category. The non-split extensions and triangles arise from skein relations over the cluster algebra, this is viewed as a geometric operation that smooths arc crossings in the interior of the surface. See in the next figure the pair of arcs $\alpha$ and $\beta$ crossing. In our convention $\alpha^+ \beta$ means walk along $\alpha$ and turn right following $\beta$ before reaching the crossing point, and $\alpha^- \beta$ is the same but turning left.

Skein relations were used to study bases of cluster algebras arising from surfaces (and to prove the Fomin-Zelevinsky positivity conjecture over said algebras). See [19, 18]. In the trivial coefficients case, skein relations are well understood, and formulas on cluster algebras arising from surfaces with punctures are known.

See in the next figure the rule $\alpha^+ \beta$ and $\alpha^- \beta$ in the punctured setting. Note that in one case we see a self-intersection. We use cluster algebra equations to find a different presentation. In this work we use skein relations for cluster algebras over a once punctured surface to find all non-split extensions over a cluster tilted algebra of type $D$. We use these extensions to find triangles in the cluster category and then we see when these triangles induce non-split extensions over Jacobian algebras.

The structure of the paper goes as follows. After finding all non split extensions for a particular triangulation of the punctured disk in Section <ref>, we recall skein relations and other cluster algebra formulas we will need in Section <ref>. We see how the extensions in Section <ref> and skein relations are related in Section <ref>. Then we present our main results in Section <ref>.

(Theorem <ref>)
All non-trivial triangles with indecomposable extreme terms in the cluster category of type $D$ can be obtained via (punctured) skein relations.
The triangles have the form

\[ X_\alpha \to X_C \to X_\beta \to X_{\alpha} [1], \]

where $C$ is a multicurve in $\alpha^+ \beta$.

We will use this theorem to find explicit non-split extensions over Jacobian algebras. As is the case of the proposition.

(Proposition <ref>) Let $\alpha$ and $\beta$ be arcs on the (punctured) disk such that $e(\alpha, \beta) \geq 1$. Let $ X_\alpha \to X_C \to X_\beta \to X_{\alpha} [1] $ be a non-split triangles with indecomposable extremes in the cluster category, where $C \in \alpha^+ \beta$, and let $\mathcal{T}$ be a triangulation.
Then there is a non-split extension

\[ 0 \to M_{\alpha} \to M_{C} \to M_{\beta} \to 0 \]

if and only if $d(C) = d(\{ \alpha, \beta\})$,  where $d$ denotes the total number of  crossings of the multicurve with the triangulation.

We conclude with a list of examples in Section <ref>.

§ TRIANGLES IN THE CLUSTER CATEGORY OF TYPE $D$

In this section we find all the non-split extensions for a particular orientation of a quiver $Q$ of type $D$. That is, all the non split extensions with indecomposable extremes in the category of representations $\rep Q$. Then we use those extensions to find all the non-split triangles in the cluster category of type $D$. The reader is referred to [3, 24] for basic definitions that will be used in this section.

§.§ Non-split extensions over the hereditary algebra

Let $H$ be the hereditary algebra $ K Q$ where $Q$ is some orientation of the $D_n$ quiver ($n \geq 4$)

n [dash]rd

n-2 [dash]r     ⋯[dash]r     1

n-1 [dash]ru

The algebra $H$ is hereditary and representation-finite. A basis for all non-split extensions between indecomposable modules, when $H$ is as above or it is of type $A$ (i.e. $Q$ is a linear graph) was given in [7]. Moreover almost all non-split extensions were described in the article.
In the next lemma we find another non-split extension.

From now on let $Q$ be the type $D_n$ quiver, taking an orientation for the graph above where all the arrows point right, and consider $K$ an algebraically closed field of $\mathrm{char} K \neq 2$.

There exist a non-split extension between indecomposable representations in $\mmod H$ given in Figure <ref>.

The non-split extension of inecomposable representations is presented in the next figure, where $1 < r < s < i < n-1$ and $I_n$ is the $n \times n$ identity matrix.

[->][line width=1pt] (-1.25,3.75) – (-0.4,4.75);
[->][line width=1pt] (-1.4,6.25) – (-0.4,5.25);
[->][line width=1pt] (0.4,5) – (1.25,5);
[->][line width=1pt] (3.05,5) – (4,5);
[->][line width=1pt] (4.55,5) – (5.5,5)
node[pos=0.5,above] $1$;
[->][line width=1pt] (7.3,5) – (8.15,5)
node[pos=0.5,above] $1$;
[->][line width=1pt] (8.7,5) – (9.54,5)
node[pos=0.5,above] $1$;
[->][line width=1pt] (11.45,5) – (12.25,5)
node[pos=0.5,above] $1$;
[->][line width=1pt] (13.95,5) – (14.8,5)
node[pos=0.5,above] $1$;

[->][line width=1pt] (-1.25,-1.25) – (-0.4,-0.25)
node[pos=0.5,below] $\begin{bmatrix}
\end{bmatrix}$;
[->][line width=1pt] (-1.4,1.25) – (-0.4,0.25)
node[pos=0.5,above] $\begin{bmatrix}
\end{bmatrix}$;
[->][line width=1pt] (0.3,0) – (1.25,0)
node[pos=0.5,above] $I_{2}$;
[->][line width=1pt] (3,0) – (4,0)
node[pos=0.5,above] $\begin{bmatrix}
\end{bmatrix}$;
[->][line width=1pt] (4.5,0) – (5.35,0)
node[pos=0.5,above] $I_{3}$;
[->][line width=1pt] (7.05,0) – (8.2,0)
node[pos=0.5,above] $\begin{bmatrix}
\end{bmatrix}$;
[->][line width=1pt] (8.75,0) – (9.54,0)
node[pos=0.5,above] $I_{2}$;
[->][line width=1pt] (11.6,0) – (12.35,0)
node[pos=0.5,above] $\begin{bmatrix}

\end{bmatrix}$;
[->][line width=1pt] (13.95,0) – (14.8,0)
node[pos=0.5,above] $1$;

[->][line width=1pt] (-1.25,-6.25) – (-0.4,-5.25)
node[pos=0.5,below] $\begin{bmatrix}
\end{bmatrix}$;
[->][line width=1pt] (-1.4,-3.75) – (-0.4,-4.75)
node[pos=0.5,above] $\begin{bmatrix}
\end{bmatrix}$;
[->][line width=1pt] (0.4,-5) – (1.25,-5)
node[pos=0.5,above] $I_{2}$;
[->][line width=1pt] (3.25,-5) – (4,-5)
node[pos=0.5,above] $I_{2}$;
[->][line width=1pt] (4.6,-5) – (5.45,-5)
node[pos=0.5,above] $I_{2}$;
[->][line width=1pt] (7.5,-5) – (8.35,-5)
node[pos=0.5,below] $\begin{bmatrix}
\end{bmatrix}$;
[->][line width=1pt] (8.9,-5) – (9.74,-5)
node[pos=0.5,above] $1$;
[->][line width=1pt] (11.45,-5) – (12.25,-5)
node[pos=0.5,above] $0$;
[->][line width=1pt] (13.95,-5) – (14.8,-5)
node[pos=0.5,above] $0$;

[->][line width=1pt] (6.7,4.5) – (6.7,0.5)
node[pos=0.4,right] $f_{k}= \left\{ \begin{array}{lcc}
1 &   if  & 1 \leq k \leq r  \\
\\ \begin{bmatrix}
\frac{1}{2}
\end{bmatrix} &  if &  r+1 \leq k \leq s \\
\\ \begin{bmatrix}
\frac{1}{2}\\
\frac{1}{2}
\end{bmatrix} &  if  & s+1 \leq k \leq i\\
&     &    \\
0 &   if  &  i+1\leq k \leq n
\end{array}
\right.$;

[->>][line width=1pt] (6.7,-0.5) – (6.7,-4.5)
node[pos=0.5,right] $g_{k}= \left\{ \begin{array}{lcc}
0 &   if  & 1 \leq k \leq r  \\
\\ \begin{bmatrix}
\end{bmatrix} &  if &  r+1 \leq k \leq s \\
\\ \begin{bmatrix}
\end{bmatrix} &  if  & s+1 \leq k \leq i\\
\\ \begin{bmatrix}
\end{bmatrix} &  if  & i+1 \leq k \leq n-2\\
&    &      \\
1&  if    & k=n-1    \\
-1&   if  &  k=n
\end{array}
\right.$;

Non-split extension Lemma <ref>

In [7] the author finds a representative for all one dimensional $\Ext$ spaces and two linearly independent representatives when $\Ext_H(N,M)$ is of dimension two, plus another extension that is linearly dependent. While the middle term is already settled when the dimension of $\Ext_H(N,M)$ is one, there are four possible middle terms for a non-split extension when $\Ext_H(N,M)$ is of dimension two. The extension not mentioned in <cit.>, appears in Figure <ref>, Lemma <ref>.

For the next lemma the following notions are important. Let $M,Y$ be indecomposable modules. It is known that $H$ is of finite representation type and the Auslander–Reiten quiver $\Gamma(H)$ is a standard component, so a morphism $M \to Y$ is a composition of paths in $\Gamma (H)$.

Let $M_i$ and $M_\beta$ be indecomposable $H$-modules such that $\dim_K \Ext_H(M_\beta, M_i) =2$, the middle term in a non-split sequence

\[ 0 \to M_i \to Y_j \to M_\beta \to 0 \]
has to be one of the following, where the middle term summands appear in the Auslander–Reiten quiver in Figure <ref>.

s1) $Y_1 = A_1 \oplus B_1 \oplus X_3$ (or $Y_1 = A_1 \oplus B_2 \oplus X_3$)

s2) $Y_2 = A_2 \oplus B_2 \oplus X_3$ (or $Y_2 = A_2 \oplus B_1 \oplus X_3$)

s3) $Y_3 = X_1 \oplus X_2$

s4) $Y_4 = X \oplus X_3$

Consider $\Gamma(H)$ the Auslander–Reiten quiver. Let $M_i$ be the projective $P(i)$, this choice makes the argument easier but $M_i$ can be any indecomposable module such that the two dimensional extension exists as we are using compositions of irreducible morphisms.

The extensions (s1)-(s3) were mentioned in [7], and (s4) appears in Lemma <ref>. We want to show that those four are the only possible non-split extensions. Let $M_\beta$ be an indecomposable such that $\dim_K \Ext_H(M_\beta, M_i) =2$, and let $0 \to M_i \to Y \to M_\beta \to 0 $ be a non-split extension. See the relative positions of $M_i$ and $M_\beta$ in Figure <ref> (a). Observe that depending on the positions of $M_i$ and $M_\beta$, the sectional paths $M_i \to X_3$ and $X_3 \to M_\beta$ may not exist, in that case set $X_3 \simeq 0$.

Possible middle terms for the extension. Lemma <ref>.

We denote the direct sum of indecomposable injective modules $I(a_1) \oplus \cdots \oplus I(a_n)$ by $I(a_1,\ldots,a_m)$, similarly denote $P(a_1, \ldots, a_m)$ the direct sum of indecomposable projective modules. Consider the projective cover and injective envelope of $M_i$ and of $M_\beta$.

While $M_i$ is isomorphic to its projective cover, $P(n-1,n)$ is the projective cover for $M_\beta$. This can be deduced from the position of $M_\beta$ in $\Gamma(H)$. By the properties of projective modules and using snake lemma, we can build a commutative diagram where the vertical lines are epimorphisms.

0 r     P(i) [twoheadrightarrow]d r     P(i,n-1,n) [twoheadrightarrow]dr     P(n-1,n)[twoheadrightarrow]d r

0 r     M_i  r     Y  r     M_βr      0

Similarly, by properties of injective modules and using snake lemma, we can build a commutative diagram where the vertical lines are monomorphisms.

0 r     M_i [hookrightarrow]d  r     Y [hookrightarrow]d r     M_βr [hookrightarrow]d     0

0 r     I(1) r     I(1,r+1,s+1) r     I(r+1,s+1) r

Hence, since there is an epimorphism from $P(i,n-1,n)$ to $Y$ and there is a monomorphism from $Y$ to $I(r+1,s+1)$, the middle term has  to satisfy:

i. $\soc Y $ is a direct summand of $S(1)\oplus S(r+1) \oplus S(s+1)$, and $\ttop Y$ is a direct summand of  $S(i) \oplus S(n-1) \oplus S(n)$,

ii. $\Hom (M_i,Y)\neq 0$ and $\Hom (Y,M_\beta)\neq 0$, and

iii. $\udim Y = \udim M_i + \udim M_\beta $.

If we write the semisimple $\ttop P(M) $ and $\soc I(M)$ for each indecomposable $M$ in $\Gamma (H)$, we obtain a pattern. See in Figure <ref> (b) that the $\ttop$ and $\soc$ conditions (i) are satisfied only by the indecomposables appearing in Figure <ref> (a). It rests to check for which combinations we have $\udim Y$ and $\udim M_i + \udim M_\beta$ and see that the only possibilities are those mentioned in the sequences (s1)-(s4).

§.§ Cluster category

The cluster category of type $D_n$ was defined as  $ \mathcal{C} =\Dd^b ( \modu k Q) / \tau^{-1} [1]$ where $Q$ is an oriented quiver of type $D_n$ in [6]. The cluster category is a particular case of a cluster category from a quiver with potential [12], studied by several authors and introduced in [2]. The hereditary algebra $H$ in the previous section is a particular case of cluster-tilted algebra as in [5], and it is also a Jacobian algebra arising from a surface triangulation [15].

In <cit.> it is proved that every extension in $\mmod H$ can be lifted to a triangle in $\Cc$. Applying the Auslander–Reiten translation to all the triangles induced by the non-split extensions (always considered up to isomorphism) in Section <ref>, we will obtain all the possible middle terms for triangles with indecomposable extremes in $\Cc$.

Let $\Cc$ be a cluster category of Dynkin type. Every non-split triangle with indecomposable extremes is obtained lifting a non-split extension of the form $0\to P_i \to Y \to M \to 0$ on $\mmod H$.

Let $\epsilon$ be a non-split triangle

\[ A \to B \to C \to A[1], \]

where $A$ and $C$ are indecomposable objects in $\Cc$. The category is standard so all morphisms are determined by the Auslander–Reiten quiver and the mesh relations. There is a cluster tilting object $T$ defined as a slice containing $A$ and with the shape of the opposite quiver $Q^{op}$ that gives an equivalence of categories <cit.>:
\[\Hom(T, -)\colon \Cc / (T[1]) \to \mmod H,\]

where $H = KQ$. Then $\Hom_{\Cc}(T, \epsilon)$ is the required non-split extension.

The category $\Cc$ was realized geometrically in <cit.>. The author establishes a bijection between indecomposable objects in $\Cc$ and tagged arcs (the tag being notched or plain) on the punctured disk with $n$ marked points on the boundary, and the irreducible morphisms are expressed as geometric moves and change of tags.
The crossing number $e(\alpha,\beta)$ (see Definition <ref>) between two tagged arcs $\alpha$ and $\beta$ is 0,1 or 2  <cit.>. The crossing number equals the dimension of $\Ext_{\Cc} (X_\alpha ,X_\beta )$ as a $K$-vector space.

Category of tagged arcs on the punctured disk. For each arc $\gamma$, $\tau(\gamma)$ is the arc immediate on the right.

§ CLUSTER ALGEBRAS FROM SURFACES AND SKEIN RELATIONS

In the first part of this section we follow [13]. The reader can find there more details about the following notions, we recall them briefly.

Let $S$ be a connected oriented 2-dimensional Riemann surface with non empty boundary $\partial S$. Fix a non-empty finite set $M$ of marked points. There is at least one marked point on each connected boundary component. The marked points in the interior of $S$ are called punctures. The pair $(S,M)$ is called a bordered marked surface or, in a shortened way, a surface.

* A generalized arc $\alpha$ is a curve in $S$ (up to isotopy) such that: its endpoints are in $M$, except for its endpoints $\alpha$ is disjoint from $\partial S$, $\alpha$ does not cut out an unpunctured monogon or an unpunctured bigon.

* A generalized arc is an arc if moreover it does not cross itself except when its endpoints coincide.

* A closed loop is a closed curve $\zeta$ which is disjoint from $M$ and $\partial S$.

* For any two arcs $\alpha$ and $\beta$, the crossing number $e^\bullet(\alpha, \beta)$ is the minimal number of crossings between curves $\tilde{\alpha}$ and $\tilde{\beta}$ in their respective isotopy classes.

* Two arcs $\alpha$ and $\beta$ are compatible if $e^\bullet(\alpha,\beta)=0$.

Generalized arcs and loops are allowed to self-cross a finite number of times. An ideal triangulation is a maximal collection $\Tt$ of pairwise compatible arcs. It is possible for an ideal triangle to have only two distinct edges, then it is called self-folded triangle. An example is shown in Figure <ref>.

Tagged arcs at a puncture and noose. Self-folded triangle (right).

In <cit.> each ideal triangulation $\Tt$ is associated to a skew-symmetric matrix $B_{\Tt}$ in a puzzle-like game. This matrix can be used as the only piece of information needed to define a cluster algebra.
The cluster algebra associated to the  surface $(S,M)$ is the cluster algebra $\cA (S,M)$ defined from the seed $(\x,B_{\Tt})$, where the initial variables $x_1, \ldots, x_n$ are in  correspondence with the arcs in an ideal triangulation $\Tt$ of $(S,M)$. We will always consider the cluster algebra with  trivial coefficients. We will not give a detailed definition of cluster algebra, we refer the reader to the very nice introduction given in <cit.>.

Ideal triangulations are connected by flips. When $\Tt$ has self-folded triangles the internal edge of that triangle cannot be flipped. To solve this problem Fomin–Shapiro–Thurston defined tagged arcs. When one of the endpoints of the arc is a puncture then we have the option of tagging near this endpoint either notched $\bowtie$ or plain $\bullet$ (in the plain case we may not put a mark in our figures). The tagged arc denoted by $\gamma^{\Join}$ in Figure <ref> replaces the arc $l$ enveloping the puncture in the model [13]. An arc $\alpha$ not at the puncture is equal to its plain version. We call the arc $l$ a noose, as the authors in [19] do, or a loop.

When the surface is the disk with one puncture (see Figure <ref>) there can only be notched tagged arcs at the puncture.
Denote by $\gamma^{\bowtie}$ the notched tagged arc at $p$ and by $\gamma^{\bullet}$ the plain version, see Figure <ref>.

While one can define the crossing number for tagged arcs in general, we will give a definition only for the punctured disk.

Let $\alpha$ and $\beta$ be two tagged arcs on the punctured disk. Then the crossing number $e(\alpha, \beta)$ is computed as follows

* if one of them is not an arc at the puncture then $e(\alpha, \beta) = e^\bullet(\alpha^\bullet,\beta^\bullet)$

* if both are arcs at the puncture and they have the same tag then $e(\alpha, \beta) = 0$, if they have different tag then the notched arc, say $\alpha$, should be replaced with the corresponding noose $l$: $e(\alpha,\beta) = e^\bullet (l,\beta )$.

A maximal set of non-crossing tagged arcs is a triangulation. Every ideal triangulation not containing self-folded triangles can be associated to a tagged triangulation in a trivial way, where all arcs at the puncture are tagged plain.

§.§ Skein relations

Given an ideal triangulation $\Tt$ without self-folded triangles, generalized arcs and loops define cluster algebra elements <cit.>. Each finite set $C$ of generalized arcs or closed loops is called a multicurve, such set is associated to a cluster algebra element defined by the product of all elements $x_\alpha$ with $\alpha\in C$, see <cit.>.  A multicurve $C$ is represented drawing each curve $\alpha \in C$ over $(S,M)$.

Let $\gamma^{\bowtie} $ and $\gamma^{\bullet}$ be as in Figure <ref>, the elements in $\cA (S,M)$ satisfy $x_{\gamma^{\bowtie}} x_{\gamma^{\bullet}} = x_{l}$. This definition is compatible with Ptolemy relations for $\lambda$-lengths and exchange relations of cluster variables, see <cit.>, and agrees with the combinatorial definition in <cit.>. When $\alpha$ is a tagged arc, $x_\alpha$ is a cluster variable. If $\epsilon$ is a curve isotopic to a boundary segment, we set  $x_{\epsilon} = 1$.

We need to recall some identities in $\cA(S,M)$. Such identities arise from smoothing arcs crossing in the interior of $S$. The next definitions follow the work developed in [14, 19, 18, 4]. Let $\alpha$, $\beta$ and $\delta$ be generalized arcs, tagged plain (this includes the noose), or closed loops. Let $\alpha$ be a generalized arc crossing $\beta$ at a point $x$, and let $\delta$ be such that it has a self-intersection at a point $x$.

For the crossings as those in Figure <ref> (1) and (2),

* the smoothing of $\{ \alpha, \beta\}$ is the pair of multicurves $\alpha^+ \beta$, obtained walking along $\alpha$, stopping just before the point $x$ and turning left following $\beta$, and $\alpha^- \beta$, obtained in a similar way but turning right at $x$.

* the smoothing of $\{ \delta \}$ is the pair of multicurves $\{ \zeta, \gamma \}$,  obtained walking along $\delta$ and avoiding the point $x$ by turning left, and $\{ \lambda \}$, obtained walking along $\delta$ and avoiding the point $x$ by turning right.

Smoothing arcs.

The smoothing is related to the identities over $\cA (S,M)$, called skein relations:

\[ (1) \hspace{5pt} x_\alpha  x_\beta = x_{\alpha^+ \beta} + x_{\alpha^- \beta} \hspace{15pt} ; \hspace{15pt} (2) \hspace{5pt}
x_\delta = x_{\gamma, \zeta} + x_\lambda.\]

We will need two identities more. The first one says that a closed loop $\sigma $ contractible to a puncture gives us the element $x_\sigma = +2 \in \cA (S,M)$. For the second identity, if $\alpha$ and $\beta$ are tagged arcs crossing at a point $x$ and one of them, say $\alpha$, has a puncture as an endpoint and it is tagged notched there, we can proceed as in Figure <ref> (1), preserving the notched end in the multicurves $\alpha^+ \beta$ and $\alpha^- \beta$, as the reader can see in the example in Figure <ref> (4). The equation $(1)$ still holds and it is obtained from the plain version applying the automorphism change of tag from <cit.>.

Let $\mathcal{A}(S,M)$ be the cluster algebra arising from the punctured disk. Let the arcs in the next figure represent the corresponding elements in $\cA (S,M)$. The following identities hold.

Part (a) is easily obtained after multiplying the expression by $x_{\gamma^\bullet}$, where $\gamma^{\Join} $ is the notched arc in the figure. Then the skein relation $(1)$ can be applied and $x_{\gamma^\bullet}$ can be extracted as a common factor. This gives formula (a). Observe that we can consider that the puncture is the intersection point $x$ and the intersection is determined by the curves having different tag. The smoothing is obtained avoiding the puncture and by turning right and left. In this case we will use the notation $\alpha^+ \beta , \alpha^- \beta$.

We will obtain the first half of (b), as the rest is symmetric. First observe the following identities in Figure <ref>, where we use (a), and (2) and (3) from Figure <ref>

Proof of Proposition <ref>.

The formulas in Figure <ref> imply the identity in Figure <ref>. Now we can apply skein relations for a double intersection. We have the identity in Figure <ref>, and a symmetrical formula. Thus we obtain part (b).

Proof of Proposition <ref>.

Proof of Proposition <ref>.

In the next section we will interpret Proposition <ref> (b) as a generalization of

\[ (1) \ \ \   x_\alpha  x_\beta = x_{\alpha^+ \beta} + x_{\alpha^- \beta}\]

which is in time the cluster multiplication formula (see [9, 20]) for string objects $X_\alpha$ and $X_\beta$ with $\dim_K (X_\alpha, X_\beta) = 1$. The reader can find the definition of string objects and cluster categories arising from unpunctured surfaces in [8]. The geometric relation between the one dimensional extension space and the equation (1) was proven in [10]. This work shows how the middle terms arise from (unpunctured) skein relations.

In the next section we are going to show how middle terms of non-split triangles arise from skein and punctured skein relations.

§ EXTENSIONS FROM PUNCTURED SKEIN RELATIONS

We know from Section <ref> that non-split triangles in the cluster category $\Cc$ of type $D$ can be found using non-split extensions in $\mmod H$. We will use the non-split extensions mentioned in Section <ref>.

Take the algebra $KQ=H$ defined by the triangulation in Figure <ref> (that is again the one defined by the oriented quiver in Section <ref>). Said algebra is defined as $\End^{op}_{\mathcal{C}}(T)$. Denote by $T_i[1] $ the shift of each cluster tilting object summand and use the short notation $i$ for the arc in $ \mathcal{T}$. The projective module $P(i)$ corresponds to the arc $\tau^{-1}(i)$ in the geometric realization. Recall that $\tau$ acts on arcs by rotating the endpoints and changing tag when the arc is at the puncture, see Figure <ref>.

Triangulation $\mathcal{T}$ that defines the Jacobian algebra $H=KQ$.

Denote an arc of the form $\tau^{-1}(i)$ by $\alpha$, and the module $P(i)$ by $M_{\alpha}$. By abuse of notation we will also denote the endpoints of each arc $i \in [1,n-2]$ by $i$, and we complete the list naming clockwise the remaining marked points $n-1$ and $n$ (Figure <ref>). Each arc $\alpha$ not at the puncture has endpoints $p,n$, where $p=i+1$. If one of the endpoints of $\alpha$ is the puncture, that is for $i \in \{n-1,n \}$, then the other endpoint is $p=n$.

For each $\alpha$, let $\beta \notin \mathcal{T}$ be an arc such that $e(\alpha, \beta) > 0$, and denote by $M_{\beta}$ the corresponding $H$-module. There is a correspondence between arcs not in $\mathcal{T}$ and $H$ modules, as the reader can find in <cit.>. The triangulation $\mathcal{T}$ (i.e. the position of the chosen cluster tilting object) is such that for $\alpha$ and $\beta$ we have $\dim_K \Ext^1 (M_{\beta},M_{\alpha}) = e(\alpha, \beta) $ and  the morphisms involved in the sequence do not factor through the ideal of morphisms $ \add (\oplus_{i=1}^n T_i [1]) = (T[1])$ in $\Cc$.

§.§ Cases where $e(\alpha, \beta) =1$

In this section we obtain each one of the non-split extensions in which $ \dim_K \Ext^1(M_{\beta},M_{\alpha}) = 1$. They are exactly those described in <cit.> and the reader can compare the Figure <ref> and the Auslander–Reiten quiver in the after mentioned article. Now we can interpret each middle term as a multicurve $\alpha^+ \beta$ obtained by smoothing the crossing between the extremes $\alpha$ and $\beta$. We find $\alpha^+ \beta$ walking along $\alpha$ and turning right at the intersection continuing on $\beta$.  Note that a multicurve defines a module as follows: if $\alpha^+ \beta = \{ \gamma_1, \ldots, \gamma_m \}$, then $M_{\alpha^+ \beta} = \oplus_{j=1}^m M_{\gamma_j}$, and when $\gamma_j$ is isotopic to a boundary segment $M_{\gamma_j} = 0$.

Non-split extensions in $ \Ext^1(M_{\beta},M_{\alpha})$. The middle term  is obtained as $\alpha^+ \beta$.

Observe the middle terms in the non-split sequences (1), (2) and (3) in Figure <ref>. For the case (1) we need the identity $x_l = x_{\gamma^{\bowtie}} x_{\gamma^\bullet}$. In case (2) the intersection is at the puncture, so the smoothing $\alpha^+ \beta$ is obtained walking on $\alpha$, avoiding the puncture (by turning right) and continuing on $\beta$. This is one of the terms in Proposition <ref> (a).

See the middle terms in the non-split sequences (4) and (5) in Figure <ref>.
In the case of sequence (5), the smoothing carries a tag as explained in Figure <ref> (4).

Non-split extensions in $ \Ext^1(M_{\beta},M_{\alpha})$. The middle term is obtained as $\alpha^+ \beta$.

§.§ Cases where $e(\alpha, \beta) =2$

We obtain now the non-split extensions appearing in Lemma <ref>. The reader can see the summands of $x_{\alpha^+ \beta}$ in Figure <ref>. They correspond to the four rightmost terms in <ref> (b).

\[ 2 x_{\alpha} x_{\beta} = (x_{\alpha^- \beta}) + (x_{\alpha^+ \beta}) = (x_{\alpha^- \beta,x} + x_{\alpha^- \beta,y}) + (x_{\alpha^+ \beta,x} + x_{\alpha^+ \beta,y}),\]

We define the smoothing $\alpha^* \beta, z$ as: walk along $\alpha$ until you find the point $z$ as the first intersection between $\alpha$ and $\beta$, then turn right if $*=+$ or left if $*=-$.

From now on we denote by $\alpha^+ \beta$ the set of multicurves obtained by punctured skein relations turning right. When $e(\alpha,\beta) = 1$, then $\alpha^+ \beta$ is a single multicurve. When $e(\alpha, \beta) = 2$ the set $\alpha^+ \beta $ contains four multicurves that appear in Figure <ref>.

The smoothing $\alpha^+ \beta$ at points $x$ and $y$ in the interior of the disk.

Two non-split extensions that define a basis of $ \Ext^1(M_\beta,M_\alpha)$.

Non-split extension in $ \Ext^1(M_\beta,M_\alpha)$.

§ NON-SPLIT EXTENSIONS OF TYPE $D$

During this section we will need the cluster tilting equivalence <cit.>. A cluster tilting object $T$ is a maximal basic object such that $\Hom(T,T[1]) = \Ext (T,T)=0$. In the geometric models, said object $T$ is represented by a triangulation $\mathcal{T}$, as $e(\alpha, \beta)$ represents the dimension of the corresponding $\Ext(X_\alpha, X_\beta)$ as $K$ vector space. Given a cluster tilting object $T$, the $\Hom$ functor produces an equivalence $\Hom(T, -)\colon \Cc / (T[1]) \to \mmod B$ for $B= \End_{\mathcal{C}} (T)^{op}$.

We have the next result.

All non-trivial triangles with indecomposable extreme terms in the cluster category of type $D$ can be obtained via (punctured) skein relations.
The triangles have the form

\[ X_\alpha \to X_C \to X_\beta \to X_{\alpha} [1], \]

where $C$ is a multicurve in $\alpha^+ \beta$.

From Lemma <ref> the extensions (1)-(9) lift to triangles in $\Cc$.  When we consider the category $\Cc$ the marked point $n$ on the boundary is not distinguished anymore  and the geometric figures (1)-(9) represent triangles independently of the arcs endpoints. The arc crossings in (1)-(9) are all the possible crossings of two arcs over the punctured disk, hence we find all the possible non-split extensions with indecomposable extremes.

We found the possible middle terms of
\[ X_\alpha \to X_C \to X_\beta \to X_{\alpha} [1] \]
performing a smoothing $\alpha^+ \beta$. On the other hand, we can do $\alpha^- \beta = \beta^+ \alpha$, in that case we obtain the middle terms for
\[ X_\beta \to X_E \to X_\alpha \to X_{\beta} [1]. \]

Now that we have identified all the triangles in $\mathcal{C}$ we ask, given a cluster tilting object $T$, when a triangle gives rise to a short exact sequence in $\mmod B$. We will use the next lemma.

Let $T$ be a cluster tilting object and $B = \End (T)^{op}$.
Let $A \to E \to C \to A [1] $ be a triangle in a cluster category such that the identity

\begin{equation*}
\dim_K \Hom(T,A) + \dim_K \Hom(T,C)  = \dim_K \Hom(T,E),
\end{equation*}

holds. Then

\begin{equation*}
0 \to  \Hom(T,A) \to  \Hom(T,E)  \to  \Hom(T,C) \to 0
\end{equation*}

is a short exact sequence in $\mmod B$.

We start with the triangle $A \ra E \ra C \ra A[1]$ and apply $\Hom (T,-)$. Since it is a cohomological functor, this yields a long exact sequence in $\mmod B$:

\[
\cdots \ra \Hom(T,C[-1]) \ra \Hom(T,A) \xrightarrow{f}  \Hom(T,E)  \xrightarrow{g}  \Hom(T,C) \ra \Hom(T,A[1]) \ra \cdots
\]

Now we can truncate this long exact sequence and obtain the (shorter) exact sequence

\[
0 \ra ker f \ra \Hom(T,A) \xrightarrow{f}  \Hom(T,E)  \xrightarrow{g}  \Hom(T,C) \ra coker g \ra 0
\]

Basic linear algebra shows that

\[
\dim_K ker f - \dim_K \Hom(T,A) + \dim_K  \Hom(T,E)  - \dim_K \Hom(T,C) + \dim coker g = 0.
\]

From the hypothesis, we get $\dim_K ker f + \dim_K coker g = 0$. Hence $\dim_K ker f = \dim_K coker g = 0$ and the sequence

\[
0 \to  \Hom(T,A) \ra  \Hom(T,E)  \ra  \Hom(T,C) \to 0
\]

is exact.

Let $M_\gamma$ be a $B$ module for $B=KQ/I = \End_\Cc^{op}(T)$ a cluster tilted algebra of type $D$ (or more generally for a $2$-Calabi–Yau tilted algebra from a cluster category arising from a surface). Then $M_\gamma$ is such that as a $(Q,I) $ representation $\dim_K (M_\gamma)_i = e(i, \gamma)$, where we denote by $i$ the arc corresponding to $T_i [1]$ and also we may denote by $i$ the associated vertex of the quiver $Q$. This is because

\[ (M_\gamma)_i = \Hom_B(P_i,M_\gamma) \simeq \dfrac{\Hom_{\mathcal{C}}(T_i, X_\gamma)} {(T[1])} \simeq \Hom_{\mathcal{C}} (T_i, X_\gamma)\simeq \Hom_{\mathcal{C}} (T_i[1], X_\gamma[1])=\Ext_{\mathcal{C}} (T_i [1], X_\gamma)\]

where the third $K$-isomorphism holds as $\Hom_{\mathcal{C}}(T_i,T[1]) = 0$.

Let $\mathcal{T}$ be a triangulation of the punctured disk.
Let $C = \{ \gamma_i , \ldots , \gamma_t \}$ be a multicurve. We call the total dimension of $C$, and denote by $d(C)$, the total number of crossings between the multicurve and $\mathcal{T}$.

Let $T$ be a cluster tilting object in $\mathcal{C}$ and $\mathcal{T}$ the corresponding triangulation on the punctured disk. Let $\alpha$ and $\beta$ be two arcs such that $\alpha, \beta \notin \mathcal{T}$ and $e(\alpha, \beta) \geq 1$. For the multicurve $\{ \alpha, \beta \}$ we have $d(\{\alpha, \beta \})= \sum_j e(j, \alpha) +  e(j, \beta)$ where $j$ runs over all the arcs in $\mathcal{T}$. That is, from Remark <ref>, $d(\{\alpha, \beta \})$ is the dimension of $M_\alpha \oplus M_\beta$ as a $K$ vector space.

Let $\alpha$ and $\beta$ be arcs in the disk such that $e(\alpha, \beta) \geq 1$. Let $ X_\alpha \to X_C \to X_\beta \to X_{\alpha} [1] $ and $ X_\beta \to X_E \to X_\alpha \to X_{\beta} [1] $ be non-split triangles with indecomposable extremes in the cluster category of type $D$, where $C \in \alpha^+ \beta$ and $E \in \alpha^- \beta$, and let $\mathcal{T}$ be a triangulation.
Then there is a non-split extension

\[ 0 \to M_{\alpha} \to M_{C} \to M_{\beta} \to 0 \]

if and only if $d(C) = d(\{ \alpha, \beta\})$, and there is a non-split extension

\[ 0 \to M_{\beta} \to M_{E} \to M_{\alpha} \to 0 \]

if and only if $d(E) = d(\{ \alpha, \beta\})$

This is a direct consequence or Theorem <ref>, Lemma <ref> and Remark <ref>.

Observe that this result is related to <cit.> since a pair of string modules $M_\alpha$ and $M_\beta$ in the category $\mmod B$, where $B$ is a Jacobian algebra arising from a surface without punctures, that cross in a 3-cycle are such that $M_C$ does not preserve total dimension. And this type of crossing is the only one where the dimension is not preserved.

§ EXAMPLES

In this section, we will compute some examples of extensions given by punctured skein relations as shown in Figures <ref>, <ref>, <ref> and <ref>. The examples are not exclusively in type $D$, as the results in the previous section apply to  geometric configurations that resemble the punctured disk. The notation and results used in this section follow [11, 12, 15], but we briefly recall them now.

Let ($Q(\tau)$, $P(\tau)$) be a quiver with potential for a  triangulation $\tau$ of the surface and $P(\tau)=\displaystyle{ \sum_{\triangle}S^{\triangle}} +S^{p}$  the potential defined by Labardini-Fragoso [15], where $\triangle$ runs over the set of the interior triangles in $\tau$ oriented in clockwise and $S^{p}$ is the cycle around the punctured $p$ oriented in counterclockwise.

A representation of a quiver with potential ($Q$,$P$) is a pair $(M,P)$, where $P$ is a potential and $M$ consists of the two following families.

1) A family $(M_{i})_{i\in Q_{0}}$ of  finite dimensional $K$ vector spaces.

2) A family $(\varphi_{\alpha}:M_{s(\alpha)}\longrightarrow {}{M_{t(\alpha)}})_{\alpha \in Q_1}$ of $K$-linear transformations such that  $\partial_{\alpha}(P)=0$,   for all $\alpha \in Q_{1} $. Where $\partial_{a}$ is the cyclic derivative defined for all arrows $a\in Q_{1}$ and each cycle $a_{1}...a_{d}$ in $Q$:

\[\partial_{a}(a_{1}...a_{d})=\displaystyle{\sum_{i=1}^{d}}\delta_{a,a_{i}}a_{i+1}...a_{d}a_{1}...a_{i-1}\]

where $\delta_{a,a_{i}}$ is the Kronecker delta and extending linearly and continuously to obtain a morphism $\partial_{a}:\langle \langle R \rangle \rangle_{cyc} \rightarrow \langle \langle R \rangle \rangle$, for more details see <cit.>.

For an arc $j$  which does not belong to $\tau$, let $\gamma_{q_{0},q_{1}}=[q_{0},q_{1}]_{j}$ be the segment of  $j$ walking from $q_{0}$ to $q_{1}$. The segment $\gamma_{q_{0},q_{1}}$ surrounds the punture $p$ counterclockwise  if locally $\gamma_{q_{0},q_{1}}$ is the segment shown in  Figure  <ref>.

[black] (-5,0)  circle (2pt)
(-5,-4)  circle (2pt)
(-8.5,-2)  circle (2pt)
(-9.5,-2)  circle (1pt)
(-9.5,-2.5)  circle (1pt)
(-9.5,-1.5)  circle (1pt);

[line width=1pt] (-5, 0) – (-5, -4)
node[pos=0.5,left] $\overline{j}$;
[line width=1pt] (-5, 0) – (-8.5, -2);
[line width=1pt] (-5, -4) – (-8.5, -2);
[line width=1pt] (-8.5, -2) – (-8.5,0);
[line width=1pt] (-8.5, -2) – (-10.5,-0.5);
[line width=1pt] (-8.5, -2) – (-10.5,-3.5);
[line width=1pt] (-8.5, -2) – (-8.5,-4)
node[pos=0.05,left] $p$;

[red] (-5,-1)  circle (2pt)
(-5,-3)  circle (2pt)
(-6.35,-3.2)  circle (2pt)
(-8.5,-3.3)  circle (2pt)
(-9.8,-3)  circle (2pt)
(-9.8,-1)  circle (2pt)
(-8.5,-0.45)  circle (2pt)
(-6.05,-0.6)  circle (2pt);

[color=red][line width=1.5pt] (-5,-1)  .. controls(-8, 0.5) and (-10.5,-1) ..  (-10.5,-2)
node[pos=0,right] $q_{0}$
node[pos=0.11,above] $r_{1}$
node[pos=0.4,above] $r_{2}$;

[color=red][line width=1.5pt] (-5,-3)  .. controls(-8, -3.5) and (-10.5,-3.5) ..  (-10.5,-2)
node[pos=0,right] $q_{1}$
node[pos=0.15,below] $r_{l}$;

$\gamma_{q_{0},q_{1}}$ surrounds the puncture $p$.

A detour curve $d_{q_{0},q_{1}}^{\Delta }$ is drawn inside $\Delta$  if  $\gamma_{q_{0},q_{1}}=[q_{0},q_{1}]_{j}$ surrounds the puncture $p$,  as is shown in Figure <ref>.

[black] (-5,0)  circle (2pt)
(-5,-4)  circle (2pt)
(-8.5,-2)  circle (2pt)
(-9.5,-2)  circle (1pt)
(-9.5,-2.5)  circle (1pt)
(-9.5,-1.5)  circle (1pt);

[line width=1pt] (-5, 0) – (-5, -4)
node[pos=0.4,left] $\overline{j}$;
[line width=1pt] (-5, 0) – (-8.5, -2);
[line width=1pt] (-5, -4) – (-8.5, -2);
[line width=1pt] (-8.5, -2) – (-8.5,0);
[line width=1pt] (-8.5, -2) – (-10.5,-0.5);
[line width=1pt] (-8.5, -2) – (-10.5,-3.5);
[line width=1pt] (-8.5, -2) – (-8.5,-4)
node[pos=0.05,left] $p$;

[red] (-5,-1)  circle (2pt)
(-5,-3)  circle (2pt)
(-6.35,-3.2)  circle (2pt)
(-8.5,-3.3)  circle (2pt)
(-9.8,-3)  circle (2pt)
(-9.8,-1)  circle (2pt)
(-8.5,-0.45)  circle (2pt)
(-6.05,-0.6)  circle (2pt);

[color=red][line width=1.5pt] (-5,-1)  .. controls(-8, 0.5) and (-10.5,-1) ..  (-10.5,-2)
node[pos=0,right] $q_{0}$
node[pos=0.11,above] $r_{1}$
node[pos=0.4,above] $r_{2}$;

[color=red][line width=1.5pt] (-5,-3)  .. controls(-8, -3.5) and (-10.5,-3.5) ..  (-10.5,-2)
node[pos=0,right] $q_{1}$
node[pos=0.15,below] $r_{l}$;

[->][color=brown][line width=1pt] (-6.05,-0.7)  –   (-5.1,-2.9)
node[pos=0.3,left] $d_{q_{0},q_{1}}^{\Delta }$;

Detour curve.

Let $j$ be an arc that does not belong to $\tau$, an arc representation is defined. In case of a punctured disk the arc representation $M_j =(M,P)$ is defined easily following <cit.>.

For every vertex $i\in Q_{0}$ the $K$ vector space $M_i$ is defined as:

1) If the arc $j$ is not the arc incident at $p$ labeled $\bowtie$ then $M_{i}=K^{{\mathds A}}$, where ${\mathds A}$ is the number of intersection points of $j$ with $i$.

2) Let $j$ be the arc incident at $p$ labeled $\bowtie$ and $q$ the other endpoint of $j$,  the arc $j$  is replaced by the loop $j^{o}$  based at $q$ which surrounds the puncture $p$.

$\bullet$ If the arc $j'$ with endpoints $q$ and $p$ labeled plain belongs to $\tau$ then $M_{i}=K^{{\mathds A}}$, where ${{\mathds A}}$ is the number of  intersection points of $j^{o}$ with $i$.

$\bullet$  If the arc $j'$ with endpoints $q$ and $p$ labeled plain  does not belong to $\tau$ then there is an arc $\overline{j}\in \tau$ not incident at $p$, such that $j^{o}=[q,q_{0}]_{j^{o}}\cup \gamma_{q_{0},q_{1}}\cup[q_{1},q]_{j^{o}}$ and $\gamma_{q_{0},q_{1}}$ surrounds the puncture $p$ counterclockwise and $q_{o},q_{1}\in \overline{j}$. The vector space is $M_{i}=K^{\mathds A}$ if $i\neq \overline{j}$, on the other hand,  $M_{i}=\frac{K^{A}}{K_{q_{0}}}$ for $i=\overline{j}$, where ${\mathds A}$ is the number of intersection points of $\gamma_{q_{0},q_{1}}\cup[q_{1},q]_{j^{o}}$  with the arc $i$ and $K_{q_{0}}$ is the copy of the field corresponding to the intersection point of $j^{o}$ with $\overline{j}$.

For every arrow $\alpha: i\longrightarrow k$ in $Q_{1}$ the linear transformation $\varphi_{\alpha}$ is defined in two steps. First we define a linear transformation $\overline{\varphi}_{\alpha}$ and then $\varphi_{\alpha}$ it is defined modifying $\overline{\varphi}_{\alpha}$.

If $j$ is an arc with labeled plain and $b_{1},...,b_{m}$ and $c_{1},...,c_{t}$ is an enumeration of the intersection points of $j$ with $i$ and $k$ respectively. The entry $m_{c_{h},b_{l}}$ of the matrix $\overline{\varphi}_{\alpha}$ is defined as follows:

$$m_{c_{h},b_{l}}= \left\{ \begin{array}{lcc}
1 &     \mbox{if the interior of $[b_{l},c_{h}]_{j}$ does not intersect any arc of $\tau$ }.\\
1  &  \mbox{if there is a segment $\gamma_{c_{h'},c_{h}}$ of $j$ which sourrounds the puncture $p$}\\
&  \mbox{counterclockwise and $r_{1}=b_{l}$}.\\
0   & \mbox{else}

\end{array}
\right.$$

If $j$ is the arc incident at $p$ labeled  $\bowtie$ and $q$ is the other end of $j$ then;

$\bullet$ The arc $j$ is replaced by the loop $j^{o}$ based on $q$ in the definition of $m_{c_{h},b_{l}}$ if the arc $j'$ with endpoints $q$ and $p$ labeled plain belongs to $\tau$.

$\bullet$ The arc $j$ is replaced by $\gamma_{q_{0},q_{1}}\cup[q_{1},q]_{j^{o}}$  in the definition of $m_{c_{h},b_{l}}$ if the arc $j'$ with endpoints $q$ and $p$ labeled plain does not belong to $\tau$.

Now, we will define the linear transformation $\varphi_{\alpha}$. If  $j$ is the arc incident at $p$  labeled  $\bowtie$ and $q$ the other endpoint of $j$  then;

$\bullet$$\varphi_{\alpha}=\overline{\varphi_{\alpha}}$ if the arc $j'$ with endpoints $q$ and $p$ labeled plain belongs to $\tau$.

$$\varphi_{\alpha}= \left\{ \begin{array}{lcc}
\overline{\varphi}_{\alpha} &     \mbox{if $i\neq \overline{j}\neq {k}$}\\
\overline{\varphi}^{q_{0}}_{\alpha}  &  \mbox{if $i=\overline{j}$}\\
^{q_{0}}\overline{\varphi}_{\alpha}  &  \mbox{if $k=\overline{j}$}
\end{array}
\right.$$

if the arc $j'$ with endpoints $q$ and $p$  labeled plain does not belong to $\tau$. Where, $\overline{\varphi}^{q_{0}}_{\alpha}$ is the matrix obtained from  $\overline{\varphi}_{\alpha}$ by removing the column that corresponds to the intersection point $q_{0}$ of $\gamma_{q_{0},q_{1}}\cup[q_{1},q]_{j^{o}}$ with $i$, and $^{q_{0}}\overline{\varphi}_{\alpha}$ is the matrix obtained from  $\overline{\varphi}_{\alpha}$ by removing the arrow  which corresponds to the intersection point $q_{0}$ of $\gamma_{q_{0},q_{1}}\cup[q_{1},q]_{j^{o}}$ with $k$.

On the other hand, if $j$ is an arc labeled plain in both endpoints then $\varphi_{\alpha}=\overline{\varphi}_{\alpha}$.

§.§ Examples on the disk

See the next arcs crossing over a triangulated punctured disk.

(0,0) circle (2cm);

[black] 	(0,0)  circle (2pt)
(1.7,1.05)  circle (2pt)
(-1.5,1.35)  circle (2pt)
(-2,0)  circle (2pt)
(1.5,-1.35)  circle (2pt)
(-1.5,-1.35)  circle (2pt)
(0,-2)  circle (2pt)
(0,2)  circle (2pt);

[color=blue][line width=1pt] (-1.5,1.35) – (-1.5,-1.35)
node[pos=0.25,left] 1;
[color=blue][line width=1pt] (0,2) – (-1.5,-1.35)
node[pos=0.75,left] 2;
[color=blue][line width=1pt] (0,2) – (0,0)
node[pos=0.5,left] 4;
[color=blue][line width=1pt] (0,2) – (1.5,-1.35)
node[pos=0.75,right] 6;
[color=blue][line width=1pt] (0,0) – (-1.5,-1.35)
node[pos=0.5,right] 3;
[color=blue][line width=1pt] (0,0) – (1.5,-1.35)
node[pos=0.6,right] 5;
[color=blue][line width=1pt] (-1.5,-1.35) – (1.5,-1.35)
node[pos=0.35,below] 7;

[color=red][line width=1pt] (-2,0) .. controls(0, 1) and (1.5,0.5) ..  (0,-2)
node[pos=0.5,above] $\alpha$;

[color=green][line width=1pt] (-1.5,1.35) .. controls(-1, -1.2) and (1.25,-0.65) ..  (1.7,1.05)
node[pos=0.5,below] $\beta$;

[->][line width=0.5pt] (-2.65,0) – (-3.85,0)
node[pos=0.5,above] $\alpha_{1}$;

[->][line width=0.5pt] (-2.35,0.1) – (-1.15,1)
node[pos=0.5,above] $\alpha_{2}$;

[->][line width=0.5pt] (-1.15,-1.1) – (-2.35,-0.1)
node[pos=0.5,below] $\alpha_{3}$;

[->][line width=0.5pt] (-1,0.95) – (-1,-0.95)
node[pos=0.5,left] $\alpha_{4}$;

[->][line width=0.5pt] (0.35,0.1) – (-0.85,1)
node[pos=0.5,below] $\alpha_{5}$;

[->][line width=0.5pt] (-0.855,-1.05) – (0.35,-0.1)
node[pos=0.5,above] $\alpha_{6}$;

[->][line width=0.5pt] (-0.85,1.15) – (1.85,1.15)
node[pos=0.5,above] $\alpha_{7}$;

[<-][line width=0.5pt] (-0.85,-1.15) – (1.85,-1.15)
node[pos=0.5,below] $\alpha_{8}$;

[->][line width=0.5pt] (1.85,1) – (0.65,0.1)
node[pos=0.5,above] $\alpha_{9}$;

[->][line width=0.5pt]  (0.65,-0.1) – (1.85,-1)
node[pos=0.5,above] $\alpha_{10}$;

Left: triangulation $\tau$ of the disk and arcs $\alpha$ and $\beta$. Right: quiver $Q(\tau)$

See Figure <ref>. The arc representations  $M_\alpha$ and $M_\beta$ are the following,

[scale=0.9 ]

[->][line width=0.5pt] (-2.65,0) – (-3.85,0)
node[pos=0.5,above] $1$;

[->][line width=0.5pt] (-2.35,0.1) – (-1.15,1)
node[pos=0.5,above] $1$;

[->][line width=0.5pt] (-1.15,-1.1) – (-2.35,-0.1)
node[pos=0.5,below] $0$;

[->][line width=0.5pt] (-1,0.95) – (-1,-0.95)
node[pos=0.5,left] $0$;

[->][line width=0.5pt] (0.35,0.1) – (-0.85,1)
node[pos=0.5,below] $1$;

[->][line width=0.5pt](-0.855,-1.05) – (0.3,-0.15)
node[pos=0.5,above] $0$;

[->][line width=0.5pt] (-0.85,1.15) – (1.85,1.15)
node[pos=0.5,above] $0$;

[<-][line width=0.5pt] (-0.85,-1.15) – (1.85,-1.15)
node[pos=0.5,below] $0$;

[->][line width=0.5pt] (1.85,1) – (0.7,0.15)
node[pos=0.5,above] $0$;

[->][line width=0.5pt]  (0.65,-0.1) – (1.85,-1)
node[pos=0.5,above] $1$;

[->][line width=0.5pt] (-2.65,0) – (-3.85,0)
node[pos=0.5,above] $0$;

[->][line width=0.5pt] (-2.35,0.1) – (-1.15,1)
node[pos=0.5,above] $0$;

[->][line width=0.5pt] (-1.15,-1.1) – (-2.35,-0.1)
node[pos=0.5,below] $1$;

[->][line width=0.5pt] (-1,0.95) – (-1,-0.95)
node[pos=0.5,left] $0$;

[->][line width=0.5pt] (0.35,0.1) – (-0.85,1)
node[pos=0.5,below] $0$;

[->][line width=0.5pt] (-0.855,-1.05) – (0.3,-0.15)
node[pos=0.5,above] $1$;

[->][line width=0.5pt] (-0.85,1.15) – (1.85,1.15)
node[pos=0.5,above] $0$;

[<-][line width=0.5pt] (-0.85,-1.15) – (1.85,-1.15)
node[pos=0.5,below] $0$;

[->][line width=0.5pt] (1.85,1) – (0.7,0.15)
node[pos=0.5,above] $1$;

[->][line width=0.5pt]  (0.65,-0.1) – (1.85,-1)
node[pos=0.5,above] $0$;

Arc representations  $M_{\alpha}$ (left) and $M_{\beta}$ (right).

The possible four middle terms are defined by four multicurves (a)-(d). By Proposition <ref>, we see that in the four cases the number of crossings of each multicurve with the triangulation $\tau$ is preserved so each one of the multicurves in Figure <ref> (a)-(d) defines a middle term in an extension.

(0,0) circle (2cm);
(0,0) circle (2cm);

[black] 	(0,0)  circle (2pt)
(1.7,1.05)  circle (2pt)
(-1.5,1.35)  circle (2pt)
(-2,0)  circle (2pt)
(1.5,-1.35)  circle (2pt)
(-1.5,-1.35)  circle (2pt)
(0,-2)  circle (2pt)
(0,2)  circle (2pt);

[color=blue][line width=1pt] (-1.5,1.35) – (-1.5,-1.35)
node[pos=0.25,left] 1;
[color=blue][line width=1pt] (0,2) – (-1.5,-1.35)
node[pos=0.75,left] 2;
[color=blue][line width=1pt] (0,2) – (0,0)
node[pos=0.5,left] 4;
[color=blue][line width=1pt] (0,2) – (1.5,-1.35)
node[pos=0.75,right] 6;
[color=blue][line width=1pt] (0,0) – (-1.5,-1.35)
node[pos=0.5,right] 3;
[color=blue][line width=1pt] (0,0) – (1.5,-1.35)
node[pos=0.6,right] 5;
[color=blue][line width=1pt] (-1.5,-1.35) – (1.5,-1.35)
node[pos=0.35,below] 7;

(-1.55,1.6) node$r$;

[color=red][line width=1pt] (-2,0) –  (0,0)
node[pos=0.4,above] $\gamma_{1}$;

[color=red][line width=1pt] (-1.5,1.35) –  (0,0)
node[pos=0.4,above] $\gamma_{2}$;

[color=red][line width=1pt] (1.7,1.05) – (0,-2)
node[pos=0.2,left] $\gamma_{3}$;

(0,0) circle (2cm);
(0,0) circle (2cm);

[black] 	(0,0)  circle (2pt)
(1.7,1.05)  circle (2pt)
(-1.5,1.35)  circle (2pt)
(-2,0)  circle (2pt)
(1.5,-1.35)  circle (2pt)
(-1.5,-1.35)  circle (2pt)
(0,-2)  circle (2pt)
(0,2)  circle (2pt);

[color=blue][line width=1pt] (-1.5,1.35) – (-1.5,-1.35)
node[pos=0.25,left] 1;
[color=blue][line width=1pt] (0,2) – (-1.5,-1.35)
node[pos=0.75,left] 2;
[color=blue][line width=1pt] (0,2) – (0,0)
node[pos=0.5,left] 4;
[color=blue][line width=1pt] (0,2) – (1.5,-1.35)
node[pos=0.75,right] 6;
[color=blue][line width=1pt] (0,0) – (-1.5,-1.35)
node[pos=0.5,right] 3;
[color=blue][line width=1pt] (0,0) – (1.5,-1.35)
node[pos=0.6,right] 5;
[color=blue][line width=1pt] (-1.5,-1.35) – (1.5,-1.35)
node[pos=0.35,below] 7;

(-1.55,1.6) node$r$;

[color=red][line width=1pt] (-2,0) –  (0,0)
node[pos=0.4,above] $\gamma_{1}$;

[color=red][line width=1pt] (-1.5,1.35) –  (0,0)
node[pos=0.4,above] $\gamma_{2}$;

[color=red][line width=1pt] (1.7,1.05) – (0,-2)
node[pos=0.2,left] $\gamma_{3}$;

(0,0) circle (2cm);

[black] 	(0,0)  circle (2pt)
(1.7,1.05)  circle (2pt)
(-1.5,1.35)  circle (2pt)
(-2,0)  circle (2pt)
(1.5,-1.35)  circle (2pt)
(-1.5,-1.35)  circle (2pt)
(0,-2)  circle (2pt)
(0,2)  circle (2pt);

[color=blue][line width=1pt] (-1.5,1.35) – (-1.5,-1.35)
node[pos=0.25,left] 1;
[color=blue][line width=1pt] (0,2) – (-1.5,-1.35)
node[pos=0.75,left] 2;
[color=blue][line width=1pt] (0,2) – (0,0)
node[pos=0.5,left] 4;
[color=blue][line width=1pt] (0,2) – (1.5,-1.35)
node[pos=0.75,right] 6;
[color=blue][line width=1pt] (0,0) – (-1.5,-1.35)
node[pos=0.5,left] 3;
[color=blue][line width=1pt] (0,0) – (1.5,-1.35)
node[pos=0.6,right] 5;
[color=blue][line width=1pt] (-1.5,-1.35) – (1.5,-1.35)
node[pos=0.35,below] 7;

(-1.55,1.6) node$r$;

[color=red][line width=1pt] (-2,0) .. controls(-0.5, -1) and (1,-0.5) ..  (1.7,1.05)
node[pos=0.4,below] $\gamma_{1}$;

[color=red][line width=1pt] (-1.5,1.35) .. controls(-0.5, 1) and (1.5,0.5) ..  (0,-2)
node[pos=0.5,right] $\gamma_{2}$;

(0,0) circle (2cm);
(0,0) circle (2cm);

[black] 	(0,0)  circle (2pt)
(1.7,1.05)  circle (2pt)
(-1.5,1.35)  circle (2pt)
(-2,0)  circle (2pt)
(1.5,-1.35)  circle (2pt)
(-1.5,-1.35)  circle (2pt)
(0,-2)  circle (2pt)
(0,2)  circle (2pt);

[color=blue][line width=1pt] (-1.5,1.35) – (-1.5,-1.35)
node[pos=0.25,left] 1;
[color=blue][line width=1pt] (0,2) – (-1.5,-1.35)
node[pos=0.75,left] 2;
[color=blue][line width=1pt] (0,2) – (0,0)
node[pos=0.5,left] 4;
[color=blue][line width=1pt] (0,2) – (1.5,-1.35)
node[pos=0.75,right] 6;
[color=blue][line width=1pt] (0,0) – (-1.5,-1.35)
node[pos=0.5,right] 3;
[color=blue][line width=1pt] (0,0) – (1.5,-1.35)
node[pos=0.6,right] 5;
[color=blue][line width=1pt] (-1.5,-1.35) – (1.5,-1.35)
node[pos=0.35,below] 7;

(-1.55,1.6) node$r$;

[color=red][line width=1pt] (-2,0) .. controls(1.5, -1.25) and (1,0.75) ..  (-1.5,1.35)
node[pos=0.75,right] $\gamma_{1}$;

[color=red][line width=1pt] (0,-2) –  (1.7,1.05)
node[pos=0.75,right] $\gamma_{2}$;

Multicurves (a), (b), (c) and (d) -in lexicographic order- corresponding to items (6), (7), (8) and (9) of Figures <ref> and <ref>.

To compute $M_{\gamma_{1}}\oplus M_{\gamma_{2}}\oplus M_{\gamma_{3}}$ in (a) Figure <ref>, the arc representations $M_{\gamma_{2}}$ and $M_{\gamma_{3}}$ are computed easily, while to compute $M_{\gamma_{1}}$, the arc $\gamma_{1}$  is replaced by the curve $\gamma_{q_{0},q_{1}}\cup[q_{1},n]_{j^{o}}$ and the detour curve is drawn. The arc representation computation for (b) is very similar to (a) and it is omitted.
See the arc representations in Figure <ref> and the extensions in Figure <ref>.

(0,0) circle (2cm);
(0,0) circle (2cm);

[black] 	(0,0)  circle (2pt)
(1.7,1.05)  circle (2pt)
(-1.5,1.35)  circle (2pt)
(-2,0)  circle (2pt)
(1.5,-1.35)  circle (2pt)
(-1.5,-1.35)  circle (2pt)
(0,-2)  circle (2pt)
(0,2)  circle (2pt);

[color=blue][line width=1pt] (-1.5,1.35) – (-1.5,-1.35)
node[pos=0.25,left] 1;
[color=blue][line width=1pt] (0,2) – (-1.5,-1.35)
node[pos=0.25,left] 2;
[color=blue][line width=1pt] (0,2) – (0,0)
node[pos=0.5,left] 4;
[color=blue][line width=1pt] (0,2) – (1.5,-1.35)
node[pos=0.75,right] 6;
[color=blue][line width=1pt] (0,0) – (-1.5,-1.35)
node[pos=0.35,right] 3;
[color=blue][line width=1pt] (0,0) – (1.5,-1.35)
node[pos=0.6,right] 5;
[color=blue][line width=1pt] (-1.5,-1.35) – (1.5,-1.35)
node[pos=0.35,below] 7;

[color=red][line width=0.5pt] (-2,0) –  (0,0);
[->][color=brown][line width=0.75pt] (-0.85,-0.7) –  (-0.6,0.4)
node[pos=0.4,right] $d$;

[color=magenta][line width=1pt] (-2,0) .. controls(1, 1.5) and (1,-1) .. (-1.25,-0.75);

Curve $\gamma_{q_{0},q_{1}}\cup[q_{1},n]_{j^{o}}$.

Then, the arc representations of the middle terms (a), (c) and (d) -in lexicographic order- from Figure <ref> are the following

[->][line width=0.5pt] (-2.72,0) – (-3.85,0)
node[pos=0.5,above] $ \begin{bmatrix}
\end{bmatrix} $;

[->][line width=0.5pt] (-2.27,0.18) – (-1.15,1)
node[pos=0.5,left] $ \begin{bmatrix}
\end{bmatrix} $;

[->][line width=0.5pt] (-1.15,-1.1) – (-2.35,-0.1)
node[pos=0.5,below] $ \begin{bmatrix}
\end{bmatrix} $;

[->][line width=0.5pt] (-1,0.95) – (-1,-0.95)
node[pos=0.5,left] $0$;

[->][line width=0.5pt] (0.3,0.15) – (-0.85,1)
node[pos=0.5,right] $ \begin{bmatrix}
\end{bmatrix} $;

[->][line width=0.5pt] (-0.855,-1.05) – (0.27,-0.15)
node[pos=0.5,above] $ \begin{bmatrix}
\end{bmatrix} $;

[->][line width=0.5pt] (-0.85,1.15) – (1.85,1.15)
node[pos=0.5,above] $0$;

[<-][line width=0.5pt] (-0.85,-1.15) – (1.85,-1.15)
node[pos=0.5,below] $0$;

[->][line width=0.5pt] (1.85,1) – (0.72,0.17)
node[pos=0.35,below]  $ \begin{bmatrix}
\end{bmatrix} $;

[->][line width=0.5pt]  (0.65,-0.1) – (1.85,-1)
node[pos=0.65,right]$ \begin{bmatrix}
\end{bmatrix} $;

[->][line width=0.5pt] (-2.72,0) – (-3.85,0)
node[pos=0.5,above] $ \begin{bmatrix}
\end{bmatrix} $;

[->][line width=0.5pt] (-2.27,0.18) – (-1.15,1)
node[pos=0.5,left] $ \begin{bmatrix}
\end{bmatrix} $;

[->][line width=0.5pt] (-1.15,-1.1) – (-2.35,-0.1)
node[pos=0.5,below] $ \begin{bmatrix}
\end{bmatrix} $;

[->][line width=0.5pt] (-1,0.95) – (-1,-0.95)
node[pos=0.5,left] $0$;

[->][line width=0.5pt] (0.3,0.15) – (-0.85,1)
node[pos=0.5,right] $ \begin{bmatrix}
\end{bmatrix} $;

[->][line width=0.5pt] (-0.855,-1.05) – (0.27,-0.15)
node[pos=0.5,above] $ \begin{bmatrix}
\end{bmatrix} $;

[->][line width=0.5pt] (-0.85,1.15) – (1.85,1.15)
node[pos=0.5,above] $0$;

[<-][line width=0.5pt] (-0.85,-1.15) – (1.85,-1.15)
node[pos=0.5,below] $0$;

[->][line width=0.5pt] (1.85,1) – (0.72,0.17)
node[pos=0.35,below]  $ \begin{bmatrix}
\end{bmatrix} $;

[->][line width=0.5pt]  (0.65,-0.1) – (1.85,-1)
node[pos=0.65,right]$ \begin{bmatrix}
\end{bmatrix} $;

[->][line width=0.5pt] (-2.72,0) – (-3.85,0)
node[pos=0.5,above] $ \begin{bmatrix}
\end{bmatrix} $;

[->][line width=0.5pt] (-2.27,0.18) – (-1.15,1)
node[pos=0.5,left] $ \begin{bmatrix}
\end{bmatrix} $;

[->][line width=0.5pt] (-1.15,-1.1) – (-2.35,-0.1)
node[pos=0.5,below] $ \begin{bmatrix}
\end{bmatrix} $;

[->][line width=0.5pt] (-1,0.95) – (-1,-0.95)
node[pos=0.5,left] $0$;

[->][line width=0.5pt] (0.3,0.15) – (-0.85,1)
node[pos=0.5,right] $ \begin{bmatrix}
\end{bmatrix} $;

[->][line width=0.5pt] (-0.855,-1.05) – (0.27,-0.15)
node[pos=0.5,above] $ \begin{bmatrix}
\end{bmatrix} $;

[->][line width=0.5pt] (-0.85,1.15) – (1.85,1.15)
node[pos=0.5,above] $0$;

[<-][line width=0.5pt] (-0.85,-1.15) – (1.85,-1.15)
node[pos=0.5,below] $0$;

[->][line width=0.5pt] (1.85,1) – (0.72,0.17)
node[pos=0.35,below]  $ \begin{bmatrix}
\end{bmatrix} $;

[->][line width=0.5pt]  (0.65,-0.1) – (1.85,-1)
node[pos=0.65,right]$ \begin{bmatrix}
\end{bmatrix} $;

Arc representations corresponding to middle terms (a), (c) and (d).

The extensions (a), (c) and (d) for our example are the following;

(-0.5,0)node$M_{\gamma_{1}}\oplus M_{\gamma_{2}}\oplus M_{\gamma_{3}}$;

[->][line width=0.5pt] (-3.25,0) – (-2.85,0);
[->][line width=0.5pt] (-2.2,0) – (-1.75,0);
[->][line width=0.5pt] (0.6,0) – (1.45,0);
[->][line width=0.5pt] (2.05,0) – (2.6,0);

(-1.75,2)node$ \left\{ \begin{array}{lc}
1 &     \mbox{if $i=1,4,7$}\\
&     \\
\begin{bmatrix}
1 \\
\end{bmatrix}  &  \mbox{if $i=2,5$}\\
&     \\
0  &  \mbox{if $i=3,6$}
\end{array}
\right.$;

(1.5,2)node$ \left\{ \begin{array}{lc}
1 &     \mbox{if $i=6$}\\
-1 &     \mbox{if $i=3$}\\
&     \\
\begin{bmatrix}
\end{bmatrix}  &  \mbox{if $i=2,5$}\\
&     \\
0  &  \mbox{if $i=1,4,7$}
\end{array}
\right.$;

(-0.5,0)node $M_{\gamma_{1}}\oplus M_{\gamma_{2}}$;

[->][line width=0.5pt] (-3.25,0) – (-2.85,0);
[->][line width=0.5pt] (-2.2,0) – (-1.5,0);
[->][line width=0.5pt] (0.4,0) – (1.45,0);
[->][line width=0.5pt] (2.05,0) – (2.6,0);

(-1.75,2.5)node$ \left\{ \begin{array}{lc}
1 &     \mbox{if $i=1,4,7$}\\
&     \\
\begin{bmatrix}
1 \\
\end{bmatrix}  &  \mbox{if $i=2$}\\
&     \\

\begin{bmatrix}
0 \\
\end{bmatrix}  &  \mbox{if $i=5$}\\
&     \\
0  &  \mbox{if $i=3,6$}
\end{array}
\right.$;

(1.5,2.5)node$ \left\{ \begin{array}{lc}
1 &     \mbox{if $i=6,3$}\\
\begin{bmatrix}
\end{bmatrix} &     \mbox{if $i=5$}\\
&     \\
\begin{bmatrix}
\end{bmatrix}  &  \mbox{if $i=2$}\\
&     \\
0  &  \mbox{if $i=1,4,7$}
\end{array}
\right.$;

(-0.5,0)node$M_{\gamma_{1}}\oplus M_{\gamma_{2}}$;

[->][line width=0.5pt] (-3.25,0) – (-2.85,0);
[->][line width=0.5pt] (-2.2,0) – (-1.5,0);
[->][line width=0.5pt] (0.4,0) – (1.45,0);
[->][line width=0.5pt] (2.05,0) – (2.6,0);

(-1.75,2.5)node$ \left\{ \begin{array}{lc}
1 &     \mbox{if $i=4,7$}\\
2 &     \mbox{if $i=1$}\\
&     \\
\begin{bmatrix}
1 \\
\end{bmatrix}  &  \mbox{if $i=5$}\\
&     \\
\begin{bmatrix}
2 \\
\end{bmatrix}  &  \mbox{if $i=2$}\\
&     \\
0  &  \mbox{if $i=3,6$}
\end{array}
\right.$;

(1.5,2.5)node$ \left\{ \begin{array}{lc}
1 &     \mbox{if $i=6$}\\
-1 &     \mbox{if $i=3$}\\
&     \\
\begin{bmatrix}
\end{bmatrix}  &  \mbox{if $i=5$}\\
&     \\
\begin{bmatrix}
\end{bmatrix}  &  \mbox{if $i=2$}\\

&     \\
0  &  \mbox{if $i=1,4,7$}
\end{array}
\right.$;

Extensions (a), (c) and (d).

See the arcs crossing on the triangulated surface.

(0,0) circle (2cm);

[black] 	(0,0)  circle (2pt)
(1.9,0.55)  circle (2pt)
(-1.5,1.35)  circle (2pt)
(-2,0)  circle (2pt)
(0.85,-1.8)  circle (2pt)
(-1.5,-1.35)  circle (2pt)

(0,2)  circle (2pt)

(1.3,1.5)  circle (2pt)
(1.9,-0.65)  circle (2pt);

[color=blue][line width=1pt] (-1.5,1.35) – (-1.5,-1.35)
node[pos=0.25,left] 1;
[color=blue][line width=1pt] (0,2) .. controls(-1.15, 1) and (-1.25,-0.75) .. (-1.5,-1.35)
node[pos=0.25,left] 2;
[color=blue][line width=1pt] (-1.5,-1.35)  .. controls(-0.35, 1.25) and (0.5,1.25) .. (0.85,-1.8)
node[pos=0.5,above] 3;
[color=blue][line width=1pt] (0,2) .. controls(1.15, 1) and (0.85,-0.75) .. (0.85,-1.8)
node[pos=0.25,right] 6;
[color=blue][line width=1pt] (0,0) – (0.85,-1.8)
node[pos=0.5,left] 5;
[color=blue][line width=1pt] (0,0) – (-1.5,-1.35)
node[pos=0.5,right] 4;

[color=blue][line width=1pt] (0.85,-1.8) – (1.3,1.5)
node[pos=0.25,right] 7;

[color=blue][line width=1pt]  (1.3,1.5)  – (1.9,-0.65)
node[pos=0.65,right] 8;

[color=red][line width=1.5pt] (-2,0) –  (0,0)
node[pos=0.35,above] $\alpha$;

[color=green][line width=1.5pt] (0.85,-1.8) .. controls(-1, -1.2) and (-1,1.2) ..  (1.3,1.5)
node[pos=0.25,left] $\beta$;

[scale=1 ]

[->][line width=0.5pt] (-1.65,1.1) – (-2.85,0.1)
node[pos=0.5,above] $\alpha_{1}$;

[->][line width=0.5pt] (-1.35,1.15) – (1.35,1.15)
node[pos=0.5,above] $\alpha_{2}$;

[->][line width=0.5pt] (1.35,1.05) – (0.15,0.1)
node[pos=0.5,right] $\alpha_{3}$;

[->][line width=0.5pt] (-0.15,0.05) – (-1.35,1)
node[pos=0.5,left] $\alpha_{4}$;

[->][line width=0.5pt] (-1.35,-1.05) – (-0.15,-0.15)
node[pos=0.5,left] $\alpha_{5}$;

[->][line width=0.5pt] (0.15,-0.1) – (1.35,-1.05)
node[pos=0.5,right] $\alpha_{6}$;

[->][line width=0.5pt] (2.85,1.15) – (1.65,1.15)
node[pos=0.5,above] $\alpha_{7}$;

[->][line width=0.5pt] (3,1) – (3,-0.3)
node[pos=0.5,right] $\alpha_{8}$;

Left: triangulation $\tau$ and arcs $\alpha$ and $\beta$. Right: quiver $Q(\tau)$

See the triangulation and the arcs $\alpha$ and $\beta$ in Figure <ref>. The arc representations $M_\alpha$ and $M_\beta$ are:

[scale=1 ]

[->][line width=0.5pt] (-1.65,1.1) – (-2.85,0.1)
node[pos=0.5,above] $1$;

[->][line width=0.5pt] (-1.35,1.15) – (1.35,1.15);

[->][line width=0.5pt] (1.35,1.05) – (0.15,0.1);

[->][line width=0.5pt] (-0.15,0.05) – (-1.35,1)
node[pos=0.5,left] $1$;

[->][line width=0.5pt] (-1.35,-1.05) – (-0.15,-0.15)
node[pos=0.5,left] $1$;

[->][line width=0.5pt] (0.15,-0.1) – (1.35,-1.05)
node[pos=0.5,right] $1$;

[->][line width=0.5pt] (2.85,1.15) – (1.65,1.15);

[->][line width=0.5pt] (3,1) – (3,-0.3);

[->][line width=0.5pt] (-1.65,1.1) – (-2.85,0.1);

[->][line width=0.5pt] (-1.35,1.15) – (1.35,1.15);

[->][line width=0.5pt] (1.35,1.05) – (0.15,0.1)
node[pos=0.5,right] $1$;

[->][line width=0.5pt] (-0.15,0.05) – (-1.35,1);

[->][line width=0.5pt] (-1.35,-1.05) – (-0.15,-0.15)
node[pos=0.5,left] $1$;

[->][line width=0.5pt] (0.15,-0.1) – (1.35,-1.05);

[->][line width=0.5pt] (2.85,1.15) – (1.65,1.15);

[->][line width=0.5pt] (3,1) – (3,-0.3);

Arc representations  $M_{\alpha}$ and $M_{\beta}$ form Figure <ref>

In this example there is one possible middle term as it is shown in Figure <ref>. And from Proposition <ref> we know that it will be a middle term in a non-split extension.

(0,0) circle (2cm);

[black] 	(0,0)  circle (2pt)
(1.9,0.55)  circle (2pt)
(-1.5,1.35)  circle (2pt)
(-2,0)  circle (2pt)
(0.85,-1.8)  circle (2pt)
(-1.5,-1.35)  circle (2pt)

(0,2)  circle (2pt)

(1.3,1.5)  circle (2pt)
(1.9,-0.65)  circle (2pt);

[color=blue][line width=1pt] (-1.5,1.35) – (-1.5,-1.35)
node[pos=0.25,left] 1;
[color=blue][line width=1pt] (0,2) .. controls(-1.15, 1) and (-1.25,-0.75) .. (-1.5,-1.35)
node[pos=0.25,left] 2;
[color=blue][line width=1pt] (-1.5,-1.35)  .. controls(-0.35, 1.25) and (0.5,1.25) .. (0.85,-1.8)
node[pos=0.5,above] 3;
[color=blue][line width=1pt] (0,2) .. controls(1.15, 1) and (0.85,-0.75) .. (0.85,-1.8)
node[pos=0.25,right] 6;
[color=blue][line width=1pt] (0,0) – (0.85,-1.8)
node[pos=0.5,left] 5;
[color=blue][line width=1pt] (0,0) – (-1.5,-1.35)
node[pos=0.5,right] 4;

[color=blue][line width=1pt] (0.85,-1.8) – (1.3,1.5)
node[pos=0.25,right] 7;

[color=blue][line width=1pt]  (1.3,1.5)  – (1.9,-0.65)
node[pos=0.65,right] 8;

[color=red][line width=1pt] (-2,0) –  (0.85,-1.8)
node[pos=0.65,left] $\gamma_{1}$;

[color=red][line width=1pt] (0,0) –  (1.3,1.5)
node[pos=0.5,left] $\gamma_{2}$;

Arcs that define a middle term $M_{\gamma_1} \oplus M_{\gamma_2}$ in a non-split extension.

We compute the arc representation $M_{\gamma_{1}}\oplus M_{\gamma_{2}}$ from Figure <ref>. The arc representation $M_{\gamma_{1}}$ is computed easily, while to compute $M_{\gamma_{2}}$, the arc $\gamma_{2}$ is replaced by the curve $\gamma_{q_{0},q_{1}}\cup[q_{1},q]_{j^{o}}$ and the detour curve is drawn. Then the arc representation is the following;

[->][line width=0.5pt] (-1.65,1.1) – (-2.85,0.1)
node[pos=0.5,above] $1$;

[->][line width=0.5pt] (-1.35,1.15) – (1.35,1.15)
node[pos=0.5,above] $0$;

[->][line width=0.5pt] (1.35,1.05) – (0.23,0.18)
node[pos=0.25,below] $ \begin{bmatrix}
\end{bmatrix} $;

[->][line width=0.5pt] (-0.15,0.05) – (-1.35,1)
node[pos=0.5,left] $ \begin{bmatrix}
\end{bmatrix} $;

[->][line width=0.5pt] (-1.25,-1) – (-0.2,-0.2)
node[pos=0.5,left] $I_{2}$;

[->][line width=0.5pt] (0.15,-0.1) – (1.35,-1.05)
node[pos=0.6,right] $ \begin{bmatrix}
\end{bmatrix} $;

[->][line width=0.5pt] (2.85,1.15) – (1.65,1.15);

[->][line width=0.5pt] (3,1) – (3,-0.3);

Arc representation $M_{\gamma_{1}}\oplus M_{\gamma_{2}}$

The extension is shown next.

(-0.5,0)node$M_{\gamma_{1}}\oplus M_{\gamma_{2}}$;

[->][line width=0.5pt] (-3.25,0) – (-2.85,0);
[->][line width=0.5pt] (-2.2,0) – (-1.5,0);
[->][line width=0.5pt] (0.4,0) – (1.45,0);
[->][line width=0.5pt] (2.05,0) – (2.6,0);

(-1.75,2)node$ \left\{ \begin{array}{lc}
1 &     \mbox{if $i=1,2,5$}\\
&     \\
\begin{bmatrix}
1 \\
\end{bmatrix}  &  \mbox{if $i=3,4$}\\
&     \\
0  &  \mbox{if $i=6,7,8$}
\end{array}
\right.$;

(1.5,2)node$ \left\{ \begin{array}{lc}
1 &     \mbox{if $i=6$}\\
&     \\
\begin{bmatrix}
\end{bmatrix}  &  \mbox{if $i=4,3$}\\
&     \\
0  &  \mbox{if $i=1,2,5,7,8$}
\end{array}
\right.$;

Non-split extension from Figure <ref>

§.§ Extensions in other surfaces

In this section we apply the results in Section <ref> to other surfaces.

The cluster category of a punctured surface $\mathcal{C}(S,M)$ was defined in [22]. The authors establish a bijection between tagged curves and string objects in  $\mathcal{C}(S,M)$. Also in <cit.> they show how Iyama-Yoshino reduction (see <cit.>) can be used in these categories. If we define a quiver with potential from a triangulation $\tau$ of $(S,M)$, such that $\gamma_0 \in \tau$, as we see in the Figure <ref> then the category $\mathcal{C}(S,M)/\gamma_0$ (see <cit.>) is obtained as a cluster category from a quiver with potential $(Q(\tau),P(\tau))$ by removing the vertex corresponding to $\gamma_0$ in $\tau$ and also removing the terms containing arrows having that vertex as a source or target in $P(\tau)$.

Surface $(S,M)$ and arc $\gamma_0$ that splits the surface into two surfaces.

Cutting along $\gamma_0$ splits the surface in Figure <ref>. On one side we have an annulus of type $A_{2,1}$, and on the other side we have a punctured disk, where we see $\alpha$ and $\beta$ crossing. So the arcs $\alpha$ and $\beta$ lie on a disk and are such that $e(\alpha, \beta)  =2$. We can find a triangulation $\tau$ of $(S,M)$ containing $\gamma_0$ such that the portion of the triangulation on the disk is as in Figure <ref>. Then we can define the associated quiver $Q(\tau)$ and potential $P(\tau)$ such that each one of the four multicurves in $\alpha^+ \beta$ correspond to middle terms of non-split extensions. These extensions lift to non-split triangles in $\mathcal{C}(S,M)$.

(0,0) circle (3.5cm);
[blue,fill=blue!25] (-1.5,0.8) circle (0.5cm);

[black] 	(0,0)  circle (2pt)
(0,3.5)  circle (2pt)
(3.5,0)  circle (2pt)
(-2.95,1.9)  circle (2pt)
(2.45,2.5)  circle (2pt)
(2.45,-2.5)  circle (2pt)
(-2.85,-2)  circle (2pt)
(-1.9,1.1)  circle (2pt)
(0,-3.5)  circle (2pt);

[color=blue][line width=1pt](-1.9,1.1) .. controls(-2.65, 0.35) and (-1.5,0) .. (0,0)
node[pos=0.25,below] 1;
[color=blue][line width=1pt](-1.9,1.1) .. controls(-1.7, 2.05) and (-0.85,1) .. (0,0)
node[pos=0.25,above] 2;
[color=blue][line width=1pt] (0,0)  .. controls(-4.5, -1.75) and (-1.75,5) .. (0,0)
node[pos=0.5,left] 10;
[color=blue][line width=1pt] (-2.95,1.9)  .. controls(-3.5, -1.5) and (-1,-0.5) .. (0,0)
node[pos=0.25,below] 8;
[color=blue][line width=1pt] (-2.95,1.9)  .. controls(-1.5, 3) and (-0.25,2.5) .. (0,0)
node[pos=0.45,above] 4;
[color=blue][line width=1pt] (-2.95,1.9)  .. controls(-4.5, -2) and (0,-3.5) .. (0,-3.5)
node[pos=0.75,above] 9;
[color=blue][line width=1pt] (-2.95,1.9)  .. controls(-1.7,3.5) and (1.7,3) .. (2.45,2.5)
node[pos=0.25,above] 3;
[color=blue][line width=1pt] (0,0) – (2.45,2.5)
node[pos=0.75,left] 6;
[color=blue][line width=1pt] (0,0) – (0,-3.5)
node[pos=0.75,left] 7;
[color=blue][line width=1pt] (0,-3.5) – (2.45,2.5)
node[pos=0.5,left] 5;
[color=blue][line width=1pt] (0,-3.5) –(3.5,0)
node[pos=0.5,left] 11;

[color=red][line width=1pt] (0,3.5) .. controls(-2,-2) and (0.75,-1) .. (3.5,0)
node[pos=0.5,left] $\alpha$;

[color=green][line width=1pt] (-2.85,-2) .. controls(0.5,2.5) and (1,1.25) ..(2.45,-2.5)
node[pos=0.5,above] $\beta$;

[scale=0.8 ]

[->][line width=0.5pt] (-3.3,3.35) – (-3.3,2.05)
node[pos=0.5,left] $\alpha_{1}$;
[->][line width=0.5pt] (-2.9,3.35) – (-2.9,2.05)
node[pos=0.5,right] $\alpha_{17}$;
[->][line width=0.5pt] (-3.1,1.55) – (-3.1,0.25)
node[pos=0.5,left] $\alpha_{2}$;
[->][line width=0.5pt] (-2.85,-0.15) – (-1.7,-1.1)
node[pos=0.5,left] $\alpha_{3}$;
[->][line width=0.5pt] (-1.25,-1.15) – (0.25,-1.15)
node[pos=0.5,below] $\alpha_{4}$;
[->][line width=0.5pt] (0.75,-1.1)– (1.8,-0.2)
node[pos=0.5,below] $\alpha_{5}$;
[<-][line width=0.5pt] (1.95,0.25)– (1.55,1.25)
node[pos=0.5,right] $\alpha_{14}$;
[<-][line width=0.5pt]  (1.15,1.45)– (-0.25,1.15)
node[pos=0.5,above] $\alpha_{15}$;
[->][line width=0.5pt]   (-0.75,1.05)–  (-2.8,0)
node[pos=0.5,above] $\alpha_{7}$;
[<-][line width=0.5pt]   (-1.4,-1.3)–  (-0.6,-2.75)
node[pos=0.5,left] $\alpha_{9}$;
[->][line width=0.5pt]  (0.4,-1.35)–   (-0.3,-2.75)
node[pos=0.5,right] $\alpha_{10}$;
[->][line width=0.5pt]  (2.05,-0.2)–   (2.45,-1.4)
node[pos=0.5,right] $\alpha_{12}$;
[->][line width=0.5pt](2.25,-1.6)–(0.75,-1.25)
node[pos=0.5,below] $\alpha_{11}$;
[->][line width=0.5pt] (2.5,-2.8)–(2.5,-1.85)
node[pos=0.5,right] $\alpha_{13}$;
[->][line width=0.5pt] (-1.45,-0.95)–(-0.55,0.95)
node[pos=0.5,right] $\alpha_{8}$;
[->][line width=0.5pt] (1.8,0.05)–(-0.4,1)
node[pos=0.5,right] $\alpha_{6}$;
[->][line width=0.5pt] (-2.9,0.25) .. controls(-1.5,1) and (-1.5,2.5) ..   (-2.9,3.6)
node[pos=0.5,right] $\alpha_{16}$;

Left: Surface with punctures with a triangulation $\widetilde{\tau}$. Right: quiver $Q(\widetilde{\tau})$.

Let $\widetilde{\tau}$ be a triangulation of $(S,M)$ in Figure <ref>. Let $(Q(\widetilde{\tau}), P(\widetilde{\tau}))$ the quiver with potential and $B = K Q(\widetilde{\tau}) / I$ the Jacobian algebra defined by $(Q(\widetilde{\tau}), P(\widetilde{\tau}))$. By Proposition <ref>, for each multicurve $C \in \alpha^+ \beta$ such that the total dimension of $C$ and $\{\alpha, \beta \}$ coincide we can find a non-split sequence in $\mmod B$.

[scale=1 ]

[->][line width=0.5pt] (-3.3,3.35) – (-3.3,2.05)
node[pos=0.5,left] $1$;
[->][line width=0.5pt] (-2.9,3.35) – (-2.9,2.05)
node[pos=0.5,right] $0$;
[->][line width=0.5pt] (-3.1,1.55) – (-3.1,0.25)
node[pos=0.5,left] $ \begin{bmatrix}
\end{bmatrix} $;
[->][line width=0.5pt] (-2.85,-0.15) – (-1.7,-1.1)
node[pos=0.5,left] $ \begin{bmatrix}
\end{bmatrix} $;
[->][line width=0.5pt] (-1.25,-1.15) – (0.25,-1.15)
node[pos=0.5,below] $1$;
[->][line width=0.5pt] (0.75,-1.1)– (1.8,-0.2);
[<-][line width=0.5pt] (1.95,0.25)– (1.55,1.25);
[<-][line width=0.5pt]  (1.15,1.45)– (-0.25,1.15)
node[pos=0.5,above] $1$;
[->][line width=0.5pt]   (-0.75,1.05)–  (-2.8,0)
node[pos=0.25,above]  $ \begin{bmatrix}
\end{bmatrix} $;
[<-][line width=0.5pt]   (-1.4,-1.3)–  (-0.6,-2.75);
[->][line width=0.5pt]  (0.4,-1.35)–   (-0.3,-2.75);
[->][line width=0.5pt]  (2.05,-0.2)–   (2.45,-1.4);
[->][line width=0.5pt](2.25,-1.6)–(0.75,-1.25)
node[pos=0.5,below] $1$;
[->][line width=0.5pt] (2.5,-2.8)–(2.5,-1.85);
[->][line width=0.5pt] (-1.45,-0.95)–(-0.55,0.95)
node[pos=0.5,right] $0$;
[->][line width=0.5pt] (1.8,0.05)–(-0.4,1);
[->][line width=0.5pt] (-2.9,0.25) .. controls(-1.5,1) and (-1.5,2.5) ..   (-2.9,3.6)
node[pos=0.75,right] $ \begin{bmatrix}
\end{bmatrix} $;

[->][line width=0.5pt] (-3.3,3.35) – (-3.3,2.05)
node[pos=0.5,left] $1$;
[->][line width=0.5pt] (-2.9,3.35) – (-2.9,2.05)
node[pos=0.5,right] $0$;
[->][line width=0.5pt] (-3.1,1.55) – (-3.1,0.25)
node[pos=0.5,left] $ \begin{bmatrix}
\end{bmatrix} $;
[->][line width=0.5pt] (-2.85,-0.15) – (-1.7,-1.1)
node[pos=0.5,left] $ \begin{bmatrix}
\end{bmatrix} $;
[->][line width=0.5pt] (-1.25,-1.15) – (0.25,-1.15);
[->][line width=0.5pt] (0.75,-1.1)– (1.8,-0.2);
[<-][line width=0.5pt] (1.95,0.25)– (1.55,1.25);
[<-][line width=0.5pt]  (1.15,1.45)– (-0.25,1.15);
[->][line width=0.5pt]   (-0.75,1.05)–  (-2.8,0)
node[pos=0.25,above] $ \begin{bmatrix}
\end{bmatrix} $;
[<-][line width=0.5pt]   (-1.4,-1.3)–  (-0.6,-2.75)
node[pos=0.5,left] $1$;
[->][line width=0.5pt]  (0.4,-1.35)–   (-0.3,-2.75);
[->][line width=0.5pt]  (2.05,-0.2)–   (2.45,-1.4)
node[pos=0.5,right] $1$;
[->][line width=0.5pt](2.25,-1.6)–(0.75,-1.25);
[->][line width=0.5pt] (2.5,-2.8)–(2.5,-1.85)
node[pos=0.5,right] $1$;
[->][line width=0.5pt] (-1.45,-0.95)–(-0.55,0.95)
node[pos=0.5,right] $0$;
[->][line width=0.5pt] (1.8,0.05)–(-0.4,1)
node[pos=0.5,right] $1$;
[->][line width=0.75pt] (-2.9,0.25) .. controls(-1.5,1) and (-1.5,2.5) ..   (-2.9,3.6)
node[pos=0.75,right] $ \begin{bmatrix}
\end{bmatrix} $;

Arc representations  $M_{\alpha}$ (left) and $M_{\beta}$ (right).

The possible middle terms are given by the next multicurves.

(0,0) circle (3.5cm);
[blue,fill=blue!25] (-1.5,0.8) circle (0.5cm);

[black] 	(0,0)  circle (2pt)
(0,3.5)  circle (2pt)
(3.5,0)  circle (2pt)
(-2.95,1.9)  circle (2pt)
(2.45,2.5)  circle (2pt)
(2.45,-2.5)  circle (2pt)
(-2.85,-2)  circle (2pt)
(-1.9,1.1)  circle (2pt)
(0,-3.5)  circle (2pt);

[color=blue][line width=1pt](-1.9,1.1) .. controls(-2.65, 0.35) and (-1.5,0) .. (0,0)
node[pos=0.25,below] 1;
[color=blue][line width=1pt](-1.9,1.1) .. controls(-1.7, 2.05) and (-0.85,1) .. (0,0)
node[pos=0.25,above] 2;
[color=blue][line width=1pt] (0,0)  .. controls(-4.5, -1.75) and (-1.75,5) .. (0,0)
node[pos=0.5,left] 10;
[color=blue][line width=1pt] (-2.95,1.9)  .. controls(-3.5, -1.5) and (-1,-0.5) .. (0,0)
node[pos=0.25,below] 8;
[color=blue][line width=1pt] (-2.95,1.9)  .. controls(-1.5, 3) and (-0.25,2.5) .. (0,0)
node[pos=0.45,above] 4;
[color=blue][line width=1pt] (-2.95,1.9)  .. controls(-4.5, -2) and (0,-3.5) .. (0,-3.5)
node[pos=0.75,above] 9;
[color=blue][line width=1pt] (-2.95,1.9)  .. controls(-1.7,3.5) and (1.7,3) .. (2.45,2.5)
node[pos=0.25,above] 3;
[color=blue][line width=1pt] (0,0) – (2.45,2.5)
node[pos=0.75,left] 6;
[color=blue][line width=1pt] (0,0) – (0,-3.5)
node[pos=0.75,left] 7;
[color=blue][line width=1pt] (0,-3.5) – (2.45,2.5)
node[pos=0.5,left] 5;
[color=blue][line width=1pt] (0,-3.5) –(3.5,0)
node[pos=0.75,left] 11;

[color=red][line width=1pt] (-2.85,-2) .. controls(-1.5, -1.5) and (0.5,1) ..  (0,3.5)
node[pos=0.25,right] $\gamma_{1}$;

[color=red][line width=1pt] (3.5,0) –  (0,0)
node[pos=0.4,above] $\gamma_{2}$;

[color=red][line width=1pt] (2.45,-2.5) – (0,0)
node[pos=0.2,left] $\gamma_{3}$;

(0,0) circle (3.5cm);
[blue,fill=blue!25] (-1.5,0.8) circle (0.5cm);

[black] 	(0,0)  circle (2pt)
(0,3.5)  circle (2pt)
(3.5,0)  circle (2pt)
(-2.95,1.9)  circle (2pt)
(2.45,2.5)  circle (2pt)
(2.45,-2.5)  circle (2pt)
(-2.85,-2)  circle (2pt)
(-1.9,1.1)  circle (2pt)
(0,-3.5)  circle (2pt);

[color=blue][line width=1pt](-1.9,1.1) .. controls(-2.65, 0.35) and (-1.5,0) .. (0,0)
node[pos=0.25,below] 1;
[color=blue][line width=1pt](-1.9,1.1) .. controls(-1.7, 2.05) and (-0.85,1) .. (0,0)
node[pos=0.25,above] 2;
[color=blue][line width=1pt] (0,0)  .. controls(-4.5, -1.75) and (-1.75,5) .. (0,0)
node[pos=0.5,left] 10;
[color=blue][line width=1pt] (-2.95,1.9)  .. controls(-3.5, -1.5) and (-1,-0.5) .. (0,0)
node[pos=0.25,below] 8;
[color=blue][line width=1pt] (-2.95,1.9)  .. controls(-1.5, 3) and (-0.25,2.5) .. (0,0)
node[pos=0.45,above] 4;
[color=blue][line width=1pt] (-2.95,1.9)  .. controls(-4.5, -2) and (0,-3.5) .. (0,-3.5)
node[pos=0.75,above] 9;
[color=blue][line width=1pt] (-2.95,1.9)  .. controls(-1.7,3.5) and (1.7,3) .. (2.45,2.5)
node[pos=0.25,above] 3;
[color=blue][line width=1pt] (0,0) – (2.45,2.5)
node[pos=0.75,left] 6;
[color=blue][line width=1pt] (0,0) – (0,-3.5)
node[pos=0.75,left] 7;
[color=blue][line width=1pt] (0,-3.5) – (2.45,2.5)
node[pos=0.5,left] 5;
[color=blue][line width=1pt] (0,-3.5) –(3.5,0)
node[pos=0.5,left] 11;

[color=red][line width=1pt] (-2.85,-2) .. controls(-1.5, -1.5) and (0.5,1) ..  (0,3.5)
node[pos=0.25,right] $\gamma_{1}$;

[color=red][line width=1pt] (3.5,0) –  (0,0)
node[pos=0.4,above] $\gamma_{2}$;

[color=red][line width=1pt] (2.45,-2.5) – (0,0)
node[pos=0.2,left] $\gamma_{3}$;

(0,0) circle (3.5cm);
[blue,fill=blue!25] (-1.5,0.8) circle (0.5cm);

[black] 	(0,0)  circle (2pt)
(0,3.5)  circle (2pt)
(3.5,0)  circle (2pt)
(-2.95,1.9)  circle (2pt)
(2.45,2.5)  circle (2pt)
(2.45,-2.5)  circle (2pt)
(-2.85,-2)  circle (2pt)
(-1.9,1.1)  circle (2pt)
(0,-3.5)  circle (2pt);

[color=blue][line width=1pt](-1.9,1.1) .. controls(-2.65, 0.35) and (-1.5,0) .. (0,0)
node[pos=0.25,below] 1;
[color=blue][line width=1pt](-1.9,1.1) .. controls(-1.7, 2.05) and (-0.85,1) .. (0,0)
node[pos=0.25,above] 2;
[color=blue][line width=1pt] (0,0)  .. controls(-4.5, -1.75) and (-1.75,5) .. (0,0)
node[pos=0.5,left] 10;
[color=blue][line width=1pt] (-2.95,1.9)  .. controls(-3.5, -1.5) and (-1,-0.5) .. (0,0)
node[pos=0.25,below] 8;
[color=blue][line width=1pt] (-2.95,1.9)  .. controls(-1.5, 3) and (-0.25,2.5) .. (0,0)
node[pos=0.45,above] 4;
[color=blue][line width=1pt] (-2.95,1.9)  .. controls(-4.5, -2) and (0,-3.5) .. (0,-3.5)
node[pos=0.75,above] 9;
[color=blue][line width=1pt] (-2.95,1.9)  .. controls(-1.7,3.5) and (1.7,3) .. (2.45,2.5)
node[pos=0.25,above] 3;
[color=blue][line width=1pt] (0,0) – (2.45,2.5)
node[pos=0.75,left] 6;
[color=blue][line width=1pt] (0,0) – (0,-3.5)
node[pos=0.75,left] 7;
[color=blue][line width=1pt] (0,-3.5) – (2.45,2.5)
node[pos=0.5,left] 5;
[color=blue][line width=1pt] (0,-3.5) –(3.5,0)
node[pos=0.5,left] 11;

[color=red][line width=1pt] (-2.85,-2) .. controls(-1.5, 1) and (0.5,1.5) ..  (3.5,0)
node[pos=0.6,above] $\gamma_{1}$;

[color=red][line width=1pt] (0,3.5) .. controls(-1.5, 1) and (-0.5,-1.5) ..  (2.45,-2.5)
node[pos=0.6,left] $\gamma_{2}$;

(0,0) circle (3.5cm);
[blue,fill=blue!25] (-1.5,0.8) circle (0.5cm);

[black] 	(0,0)  circle (2pt)
(0,3.5)  circle (2pt)
(3.5,0)  circle (2pt)
(-2.95,1.9)  circle (2pt)
(2.45,2.5)  circle (2pt)
(2.45,-2.5)  circle (2pt)
(-2.85,-2)  circle (2pt)
(-1.9,1.1)  circle (2pt)
(0,-3.5)  circle (2pt);

[color=blue][line width=1pt](-1.9,1.1) .. controls(-2.65, 0.35) and (-1.5,0) .. (0,0)
node[pos=0.25,below] 1;
[color=blue][line width=1pt](-1.9,1.1) .. controls(-1.7, 2.05) and (-0.85,1) .. (0,0)
node[pos=0.25,above] 2;
[color=blue][line width=1pt] (0,0)  .. controls(-4.5, -1.75) and (-1.75,5) .. (0,0)
node[pos=0.5,left] 10;
[color=blue][line width=1pt] (-2.95,1.9)  .. controls(-3.5, -1.5) and (-1,-0.5) .. (0,0)
node[pos=0.25,below] 8;
[color=blue][line width=1pt] (-2.95,1.9)  .. controls(-1.5, 3) and (-0.25,2.5) .. (0,0)
node[pos=0.45,above] 4;
[color=blue][line width=1pt] (-2.95,1.9)  .. controls(-4.5, -2) and (0,-3.5) .. (0,-3.5)
node[pos=0.75,above] 9;
[color=blue][line width=1pt] (-2.95,1.9)  .. controls(-1.7,3.5) and (1.7,3) .. (2.45,2.5)
node[pos=0.25,above] 3;
[color=blue][line width=1pt] (0,0) – (2.45,2.5)
node[pos=0.75,left] 6;
[color=blue][line width=1pt] (0,0) – (0,-3.5)
node[pos=0.75,left] 7;
[color=blue][line width=1pt] (0,-3.5) – (2.45,2.5)
node[pos=0.5,left] 5;
[color=blue][line width=1pt] (0,-3.5) –(3.5,0)
node[pos=0.5,left] 11;

[color=red][line width=1pt] (-2.85,-2) .. controls(-1.75, -1.5) and (0,1.5) ..  (0,3.5)
node[pos=0.25,right] $\gamma_{1}$;

[color=red][line width=1pt](2.45,-2.5) .. controls(-2.5, -1) and (-1.5,2.5) .. (3.5,0)
node[pos=0.7,above] $\gamma_{2}$;

Multicurves in $\alpha^+ \beta$. Label the multicurves (a) - (d) in lexicographic order.

Observe that each multicurve $C$ satisfies $d(C) = d(\{\alpha, \beta \})$. To compute $M_{\gamma_{1}}\oplus M_{\gamma_{2}}\oplus M_{\gamma_{3}}$ in  item (a) of Figure <ref>, the arc representations $M_{\gamma_{1}}$ and $M_{\gamma_{3}}$ are computed easily, while to compute $M_{\gamma_{2}}$, the arc $\gamma_{2}$ is replaced by the curve $\gamma_{q_{0},q_{1}}\cup[q_{1},n]_{j^{o}}$ and the detour curve is drawn. The arc representation in item (b) can be found in a similar way.

[$M_{\gamma_{1}}\oplus M_{\gamma_{2}}\oplus M_{\gamma_{3}}$ Figure <ref>. ]

[->][line width=0.5pt] (-3.3,3.35) – (-3.3,2.05)
node[pos=0.5,left] $I_{2}$;
[->][line width=0.5pt] (-2.9,3.35) – (-2.9,2.05)
node[pos=0.5,right] $0$;
[->][line width=0.5pt] (-3.1,1.55) – (-3.1,0.25)
node[pos=0.5,left] $ \begin{bmatrix}
\end{bmatrix} $;
[->][line width=0.5pt] (-2.85,-0.15) – (-1.7,-1.1)
node[pos=0.35,left] $ \begin{bmatrix}
\end{bmatrix} $;
[->][line width=0.5pt] (-1.25,-1.15) – (0.25,-1.15)
node[pos=0.5,below] $ \begin{bmatrix}
\end{bmatrix} $;
[->][line width=0.5pt] (0.75,-1.1)– (1.8,-0.2)
node[pos=0.5,below] $0$;
[<-][line width=0.5pt] (1.95,0.25)– (1.55,1.25)
node[pos=0.5,right] $0$;
[<-][line width=0.5pt]  (1.15,1.45)– (-0.25,1.15)
node[pos=0.5,above] $ \begin{bmatrix}
\end{bmatrix} $;
[->][line width=0.5pt]   (-0.75,1.05)–  (-2.8,0)
node[pos=0.25,above]$ \begin{bmatrix}
\end{bmatrix} $;
[<-][line width=0.5pt]   (-1.4,-1.3)–  (-0.6,-2.75)
node[pos=0.5,left] $ \begin{bmatrix}
\end{bmatrix} $;
[->][line width=0.5pt]  (0.4,-1.35)–   (-0.3,-2.75)
node[pos=0.5,right] $0$;
[->][line width=0.5pt]  (2.05,-0.2)–   (2.45,-1.4)
node[pos=0.5,right] $ \begin{bmatrix}
\end{bmatrix} $;
[->][line width=0.5pt](2.25,-1.6)–(0.75,-1.25)
node[pos=0.5,below] $ \begin{bmatrix}
\end{bmatrix} $;
[->][line width=0.5pt] (2.5,-2.8)–(2.5,-1.85)
node[pos=0.5,right] $ \begin{bmatrix}
\end{bmatrix} $;
[->][line width=0.5pt] (-1.45,-0.95)–(-0.55,0.95)
node[pos=0.5,right] $0$;
[->][line width=0.5pt] (1.8,0.05)–(-0.4,1)
node[pos=0.5,below] $ \begin{bmatrix}
\end{bmatrix} $;
[->][line width=0.5pt] (-2.9,0.25) .. controls(-1.5,1) and (-1.5,2.5) ..   (-2.8,3.6)
node[pos=0.85,right] $ \begin{bmatrix}
\end{bmatrix} $;

[->][line width=0.5pt] (-3.3,3.35) – (-3.3,2.05)
node[pos=0.5,left] $I_{2}$;
[->][line width=0.5pt] (-2.9,3.35) – (-2.9,2.05)
node[pos=0.5,right] $0$;
[->][line width=0.5pt] (-3.1,1.55) – (-3.1,0.25)
node[pos=0.5,left] $ \begin{bmatrix}
\end{bmatrix} $;
[->][line width=0.5pt] (-2.85,-0.15) – (-1.7,-1.1)
node[pos=0.33,left] $ \begin{bmatrix}
\end{bmatrix} $;
[->][line width=0.5pt] (-1.25,-1.15) – (0.25,-1.15)
node[pos=0.5,below] $ \begin{bmatrix}
\end{bmatrix} $;
[->][line width=0.5pt] (0.75,-1.1)– (1.8,-0.2)
node[pos=0.5,below] $0$;
[<-][line width=0.5pt] (1.95,0.25)– (1.55,1.25)
node[pos=0.5,right] $0$;
[<-][line width=0.5pt]  (1.15,1.45)– (-0.25,1.15)
node[pos=0.5,above] $ \begin{bmatrix}
\end{bmatrix} $;
[->][line width=0.5pt]   (-0.75,1.05)–  (-2.8,0)
node[pos=0.25,above] $ \begin{bmatrix}
\end{bmatrix} $;
[<-][line width=0.5pt]   (-1.4,-1.3)–  (-0.6,-2.75);
[->][line width=0.5pt]  (0.4,-1.35)–   (-0.3,-2.75);
[->][line width=0.5pt]  (2.05,-0.2)–   (2.45,-1.4)
node[pos=0.5,right] $ \begin{bmatrix}
\end{bmatrix} $;
[->][line width=0.5pt](2.25,-1.6)–(0.75,-1.25)
node[pos=0.5,below] $ \begin{bmatrix}
\end{bmatrix} $;
[->][line width=0.5pt] (2.5,-2.8)–(2.5,-1.85)
node[pos=0.5,right] $ \begin{bmatrix}
\end{bmatrix} $;
[->][line width=0.5pt] (-1.45,-0.95)–(-0.55,0.95)
node[pos=0.5,right] $0$;
[->][line width=0.5pt] (1.8,0.05)–(-0.4,1)
node[pos=0.5,below] $ \begin{bmatrix}
\end{bmatrix} $;
[->][line width=0.5pt] (-2.9,0.25) .. controls(-1.5,1) and (-1.5,2.5) ..   (-2.8,3.6)
node[pos=0.85,right]$ \begin{bmatrix}
\end{bmatrix} $;

[->][line width=0.5pt] (-3.3,3.35) – (-3.3,2.05)
node[pos=0.5,left] $I_{2}$;
[->][line width=0.5pt] (-2.9,3.35) – (-2.9,2.05)
node[pos=0.5,right] $0$;
[->][line width=0.5pt] (-3.1,1.55) – (-3.1,0.25)
node[pos=0.5,left] $ \begin{bmatrix}
\end{bmatrix} $;
[->][line width=0.5pt] (-2.85,-0.15) – (-1.7,-1.1)
node[pos=0.33,left] $ \begin{bmatrix}
\end{bmatrix} $;
[->][line width=0.5pt] (-1.25,-1.15) – (0.25,-1.15)
node[pos=0.5,below] $ \begin{bmatrix}
\end{bmatrix} $;
[->][line width=0.5pt] (0.75,-1.1)– (1.8,-0.2)
node[pos=0.5,below] $0$;
[<-][line width=0.5pt] (1.95,0.25)– (1.55,1.25)
node[pos=0.5,right] $0$;
[<-][line width=0.5pt]  (1.15,1.45)– (-0.25,1.15)
node[pos=0.5,above] $ \begin{bmatrix}
\end{bmatrix} $;
[->][line width=0.5pt]   (-0.75,1.05)–  (-2.8,0)
node[pos=0.25,above] $ \begin{bmatrix}
\end{bmatrix} $;
[<-][line width=0.5pt]   (-1.4,-1.3)–  (-0.6,-2.75)
node[pos=0.5,left]  $ \begin{bmatrix}
\end{bmatrix} $;
[->][line width=0.5pt]  (0.4,-1.35)–   (-0.3,-2.75)
node[pos=0.5,right] $0$;
[->][line width=0.5pt]  (2.05,-0.2)–   (2.45,-1.4)
node[pos=0.5,right] $ \begin{bmatrix}
\end{bmatrix} $;
[->][line width=0.5pt](2.25,-1.6)–(0.75,-1.25)
node[pos=0.5,below] $ \begin{bmatrix}
\end{bmatrix} $;
[->][line width=0.5pt] (2.5,-2.8)–(2.5,-1.85)
node[pos=0.5,right] $ \begin{bmatrix}
\end{bmatrix} $;
[->][line width=0.5pt] (-1.45,-0.95)–(-0.55,0.95)
node[pos=0.5,right] $0$;
[->][line width=0.5pt] (1.8,0.05)–(-0.4,1)
node[pos=0.5,below] $ \begin{bmatrix}
\end{bmatrix} $;
[->][line width=0.5pt] (-2.9,0.25) .. controls(-1.5,1) and (-1.5,2.5) ..   (-2.8,3.6)
node[pos=0.85,right]$ \begin{bmatrix}
\end{bmatrix} $;

(c) $M_{\gamma_{1}}\oplus M_{\gamma_{2}}$ Figure <ref>.                       (d) $M_{\gamma_{1}}\oplus M_{\gamma_{2}}$ Figure <ref>.

Middle terms of non-split extensions.

The non-split extensions are the following;

(-0.5,0)node$M_{\gamma_{1}}\oplus M_{\gamma_{2}}\oplus M_{\gamma_{3}}$;

[->][line width=0.5pt] (-3.25,0) – (-2.85,0);
[->][line width=0.5pt] (-2.2,0) – (-1.75,0);
[->][line width=0.5pt] (0.6,0) – (1.45,0);
[->][line width=0.5pt] (2.05,0) – (2.6,0);

(-2.5,2.5)node$ \left\{ \begin{array}{lc}
1 &     \mbox{if $i=3,7$}\\
0  &  \mbox{if $i=6,9,11$}\\
&     \\
\begin{bmatrix}
1 \\
\end{bmatrix}  &  \mbox{if $i=1,2,4,5,8$}\\
&     \\
\begin{bmatrix}
1 &0\\
\end{bmatrix}  &  \mbox{if $i=10$}
\end{array}
\right.$;

(1.75,2.5)node$ \left\{ \begin{array}{lc}
1 &     \mbox{if $i=9,11$}\\
-1 &     \mbox{if $i=6$}\\
&     \\
0  &  \mbox{if $i=7,3$}\\
&     \\
\begin{bmatrix}
\end{bmatrix}  &  \mbox{if $i=5$}\\
&     \\
\begin{bmatrix}
\end{bmatrix}  &  \mbox{if $i=1,2,4,8$}\\
&     \\
\begin{bmatrix}
\end{bmatrix}  &  \mbox{if $i=10$}
\end{array}
\right.$;

(-0.5,0)node $M_{\gamma_{1}}\oplus M_{\gamma_{2}}$;

[->][line width=0.5pt] (-3.25,0) – (-2.85,0);
[->][line width=0.5pt] (-2.2,0) – (-1.5,0);
[->][line width=0.5pt] (0.4,0) – (1.45,0);
[->][line width=0.5pt] (2.05,0) – (2.6,0);

(-2.5,2.5)node$ \left\{ \begin{array}{lc}
1 &     \mbox{if $i=3,7$}\\
0  &  \mbox{if $i=6,9,11$}\\
&     \\
\begin{bmatrix}
1 \\
\end{bmatrix}  &  \mbox{if $i=1,2,4,5,8$}\\
&     \\
\begin{bmatrix}
1 &0\\
\end{bmatrix}  &  \mbox{if $i=10$}
\end{array}
\right.$;

(1.75,2.5)node$ \left\{ \begin{array}{lc}
1 &     \mbox{if $i=6$}\\
-1 &     \mbox{if $i=11$}\\
&     \\
0  &  \mbox{if $i=3,7,9$}\\
&     \\
\begin{bmatrix}
\end{bmatrix}  &  \mbox{if $i=1,2,4,5,8$}\\
&     \\
\begin{bmatrix}
\end{bmatrix}  &  \mbox{if $i=10$}
\end{array}
\right.$;

(-0.5,0)node$M_{\gamma_{1}}\oplus M_{\gamma_{2}}$;

[->][line width=0.5pt] (-3.25,0) – (-2.85,0);
[->][line width=0.5pt] (-2.2,0) – (-1.5,0);
[->][line width=0.5pt] (0.4,0) – (1.45,0);
[->][line width=0.5pt] (2.05,0) – (2.6,0);

(-2.5,3)node$ \left\{ \begin{array}{lc}
1 &     \mbox{if $i=3,7$}\\
0  &  \mbox{if $i=6,9,11$}\\
&     \\
\begin{bmatrix}
1 \\
\end{bmatrix}  &  \mbox{if $i=1,2,4,8$}\\
&     \\
\begin{bmatrix}
2 \\
\end{bmatrix}  &  \mbox{if $i=5$}\\
&     \\
\begin{bmatrix}
1 &0\\
\end{bmatrix}  &  \mbox{if $i=10$}
\end{array}
\right.$;

(1.75,2.75)node$ \left\{ \begin{array}{lc}
1 &     \mbox{if $i=9$}\\
-1 &     \mbox{if $i=6$}\\
-2&     \mbox{if $i=11$}\\
&     \\
0  &  \mbox{if $i=3,7$}\\
&     \\
\begin{bmatrix}
\end{bmatrix}  &  \mbox{if $i=1,2,4,8$}\\
&     \\
\begin{bmatrix}
\end{bmatrix}  &  \mbox{if $i=5$}\\
&     \\
\begin{bmatrix}
\end{bmatrix}  &  \mbox{if $i=10$}
\end{array}
\right.$;

Non-split extensions obtained via punctured skein relations.

[1]
Ibrahim Assem, Thomas Brüstle, Gabrielle Charbonneau-Jodoin, and Pierre-Guy
Gentle algebras arising from surface triangulations.
Algebra Number Theory, 4(2):201–229, 2010.

[2]
Claire Amiot.
Cluster categories for algebras of global dimension 2 and quivers
with potential.
In Annales de l'institut Fourier, volume 59, pages 2525–2590,

[3]
Ibrahim Assem, Daniel Simson, and Andrzej Skowroński.
Elements of the representation theory of associative algebras.
Vol. 1, volume 65 of London Mathematical Society Student Texts.
Cambridge University Press, Cambridge, 2006.
Techniques of representation theory.

[4]
Ibrahim Assem, Vasilisa Shramchenko, and Ralf Schiffler.
Cluster automorphisms and compatibility of cluster variables.
Glasg. Math. J., 56(3):705–720, 2014.

[5]
Aslak Bakke Buan, B. Marsh, and Idun Reiten.
Cluster-tilted algebras.
Transactions of the American Mathematical Society,
359(1):323–332, 2007.

[6]
Aslak Bakke Buan, B. Marsh, Markus Reineke, Idun Reiten, and Gordana Todorov.
Tilting theory and cluster combinatorics.
Adv. Math., 204(2):572–618, 2006.

[7]
Peter Brown.
Non-split extensions over representation-finite hereditary algebras.
Comm. Algebra, 26(4):1005–1015, 1998.

[8]
Thomas Brüstle and Jie Zhang.
On the cluster category of a marked surface without punctures.
Algebra Number Theory, 5(4):529–566, 2011.

[9]
Philippe Caldero and Bernhard Keller.
From triangulated categories to cluster algebras.
Invent. Math., 172(1):169–211, 2008.

[10]
Ilke Canakci and Sibylle Schroll.
Extensions in Jacobian algebras and cluster categories of marked
Adv. Math., 313:1–49, 2017.
With an appendix by Claire Amiot.

[11]
Salomón Domínguez.
Arc representations.
arXiv preprint arXiv:1709.09521v1, 2017.

[12]
Harm Derksen, Jerzy Weyman, and Andrei Zelevinsky.
Quivers with potentials and their representations. I. Mutations.
Selecta Math. (N.S.), 14(1):59–119, 2008.

[13]
Sergey Fomin, Michael Shapiro, and Dylan Thurston.
Cluster algebras and triangulated surfaces. I. Cluster complexes.
Acta Math., 201(1):83–146, 2008.

[14]
Sergey Fomin and Dylan Thurston.
Cluster algebras and triangulated surfaces. part ii: Lambda lengths.
arXiv preprint arXiv:1210.5569, 2012.

[15]
Daniel Labardini-Fragoso.
Quivers with potentials associated to triangulated surfaces.
Proc. Lond. Math. Soc. (3), 98(3):797–839, 2009.

[16]
Daniel Labardini-Fragoso.
On triangulations, quivers with potentials and mutations.
Mexican Mathematicians Abroad, 657:103, 2016.

[17]
B. R. Marsh and Y. Palu.
Coloured quivers for rigid objects and partial triangulations: the
unpunctured case.
Proceedings of the London Mathematical Society,
108(2):411–440, 2014.

[18]
Gregg Musiker, Ralf Schiffler, and Lauren Williams.
Positivity for cluster algebras from surfaces.
Adv. Math., 227(6):2241–2308, 2011.

[19]
Gregg Musiker and Lauren Williams.
Matrix formulae and skein relations for cluster algebras from
Int. Math. Res. Not. IMRN, (13):2891–2944, 2013.

[20]
Yann Palu.
Cluster characters II: a multiplication formula.
Proc. Lond. Math. Soc. (3), 104(1):57–78, 2012.

[21]
Pierre-Guy Plamondon.
Cluster characters for cluster categories with infinite-dimensional
morphism spaces.
Advances in Mathematics, 227(1):1–39, 2011.

[22]
Yu Qiu and Yu Zhou.
Cluster categories for marked surfaces: punctured case.
Compos. Math., 153(9):1779–1819, 2017.

[23]
Ralf Schiffler.
A geometric model for cluster categories of type $d_n$.
Journal of Algebraic Combinatorics, 27(1):1–21, 2008.

[24]
Ralf Schiffler.
Quiver representations.
CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC.
Springer, Cham, 2014.

[1]
Ibrahim Assem, Thomas Brüstle, Gabrielle Charbonneau-Jodoin, and Pierre-Guy
Gentle algebras arising from surface triangulations.
Algebra Number Theory, 4(2):201–229, 2010.

[2]
Claire Amiot.
Cluster categories for algebras of global dimension 2 and quivers
with potential.
In Annales de l'institut Fourier, volume 59, pages 2525–2590,

[3]
Ibrahim Assem, Daniel Simson, and Andrzej Skowroński.
Elements of the representation theory of associative algebras.
Vol. 1, volume 65 of London Mathematical Society Student Texts.
Cambridge University Press, Cambridge, 2006.
Techniques of representation theory.

[4]
Ibrahim Assem, Vasilisa Shramchenko, and Ralf Schiffler.
Cluster automorphisms and compatibility of cluster variables.
Glasg. Math. J., 56(3):705–720, 2014.

[5]
Aslak Bakke Buan, B. Marsh, and Idun Reiten.
Cluster-tilted algebras.
Transactions of the American Mathematical Society,
359(1):323–332, 2007.

[6]
Aslak Bakke Buan, B. Marsh, Markus Reineke, Idun Reiten, and Gordana Todorov.
Tilting theory and cluster combinatorics.
Adv. Math., 204(2):572–618, 2006.

[7]
Peter Brown.
Non-split extensions over representation-finite hereditary algebras.
Comm. Algebra, 26(4):1005–1015, 1998.

[8]
Thomas Brüstle and Jie Zhang.
On the cluster category of a marked surface without punctures.
Algebra Number Theory, 5(4):529–566, 2011.

[9]
Philippe Caldero and Bernhard Keller.
From triangulated categories to cluster algebras.
Invent. Math., 172(1):169–211, 2008.

[10]
Ilke Canakci and Sibylle Schroll.
Extensions in Jacobian algebras and cluster categories of marked
Adv. Math., 313:1–49, 2017.
With an appendix by Claire Amiot.

[11]
Salomón Domínguez.
Arc representations.
arXiv preprint arXiv:1709.09521v1, 2017.

[12]
Harm Derksen, Jerzy Weyman, and Andrei Zelevinsky.
Quivers with potentials and their representations. I. Mutations.
Selecta Math. (N.S.), 14(1):59–119, 2008.

[13]
Sergey Fomin, Michael Shapiro, and Dylan Thurston.
Cluster algebras and triangulated surfaces. I. Cluster complexes.
Acta Math., 201(1):83–146, 2008.

[14]
Sergey Fomin and Dylan Thurston.
Cluster algebras and triangulated surfaces. part ii: Lambda lengths.
arXiv preprint arXiv:1210.5569, 2012.

[15]
Daniel Labardini-Fragoso.
Quivers with potentials associated to triangulated surfaces.
Proc. Lond. Math. Soc. (3), 98(3):797–839, 2009.

[16]
Daniel Labardini-Fragoso.
On triangulations, quivers with potentials and mutations.
Mexican Mathematicians Abroad, 657:103, 2016.

[17]
B. R. Marsh and Y. Palu.
Coloured quivers for rigid objects and partial triangulations: the
unpunctured case.
Proceedings of the London Mathematical Society,
108(2):411–440, 2014.

[18]
Gregg Musiker, Ralf Schiffler, and Lauren Williams.
Positivity for cluster algebras from surfaces.
Adv. Math., 227(6):2241–2308, 2011.

[19]
Gregg Musiker and Lauren Williams.
Matrix formulae and skein relations for cluster algebras from
Int. Math. Res. Not. IMRN, (13):2891–2944, 2013.

[20]
Yann Palu.
Cluster characters II: a multiplication formula.
Proc. Lond. Math. Soc. (3), 104(1):57–78, 2012.

[21]
Pierre-Guy Plamondon.
Cluster characters for cluster categories with infinite-dimensional
morphism spaces.
Advances in Mathematics, 227(1):1–39, 2011.

[22]
Yu Qiu and Yu Zhou.
Cluster categories for marked surfaces: punctured case.
Compos. Math., 153(9):1779–1819, 2017.

[23]
Ralf Schiffler.
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