2.1 Skewed-gentle algebras

A biquiver is a tuple $(Q_{0},Q_{1},Q_{2})$ , where $Q_{0}$ is the set of vertices, $Q_{1}$ is the set of solid arrows, and $Q_{2}$ is the set of dashed arrows. Let $s,t:Q_{1}\cup Q_{2}\rightarrow Q_{0}$ be the start/terminal functions of arrows. We call an arrow $\unicode[STIX]{x1D6FC}$ in $Q_{1}\cup Q_{2}$ a loop if $s(\unicode[STIX]{x1D6FC})=t(\unicode[STIX]{x1D6FC})$ .

In this paper, we always assume that a biquiver $Q=(Q_{0},Q_{1},Q_{2})$ satisfies:

1. – each arrow in $Q_{2}$ is a dashed loop;

2. – there is at most one loop in $Q_{2}$ at each vertex;

3. – there is no loop in $Q_{1}$ .

Let $Q_{0}^{\operatorname{Sp}}$ be the subset of $Q_{0}$ consisting of vertices where there is a dashed loop in $Q_{2}$ .

Skewed-gentle algebras, modeled on gentle algebras, were introduced in [Reference Geiß and de la PeñaGdlP99] as a certain class of clannish algebras defined in [Reference Crawley-BoeveyCra89].

2.2 Letters

Let $(Q,Z)$ be a skewed-gentle pair. Following [Reference Crawley-BoeveyCra89, Reference GeißGei99], we associate a new biquiver $\widehat{Q}=(\widehat{Q}_{0},\widehat{Q}_{1},\widehat{Q}_{2})$ with $Q=(Q_{0},Q_{1},Q_{2})$ by adding two new vertices $i_{+}$ and $i_{-}$ and two new solid arrows $a_{i_{\pm }}:i\rightarrow i_{\pm }$ for each vertex $i\in Q_{0}$ . That is:

1. – $\widehat{Q}_{0}=Q_{0}\cup \{i_{\pm }\mid i\in Q_{0}\}$ ;

2. – $\widehat{Q}_{1}=Q_{1}\cup \{a_{i_{\pm }}:i\rightarrow i_{\pm }\mid i\in Q_{0}\}$ ;

3. – $\widehat{Q}_{2}=Q_{2}$ .

For example, the biquiver $\widehat{Q}$ associated with the biquiver $Q$ in Example 2.2 is as shown by the following diagram.

For any arrow $\unicode[STIX]{x1D6FC}$ in $\widehat{Q}$ , we define a direct letter $\unicode[STIX]{x1D6FC}$ and an inverse letter $\unicode[STIX]{x1D6FC}^{-1}$ , which are mutually inverse. Let $L$ be the set of all letters. The functions $s,t$ can be extended to $L$ by setting $s(\unicode[STIX]{x1D6FC}^{-1})=t(\unicode[STIX]{x1D6FC})$ and $t(\unicode[STIX]{x1D6FC}^{-1})=s(\unicode[STIX]{x1D6FC})$ . For each $i\in Q_{0}\subset \widehat{Q}_{0}$ , let $L(i):=\{l\in L\mid s(l)=i\}$ . We divide $L(i)$ into two disjoint subsets $L_{+}(i)$ and $L_{-}(i)$ with linear orders such that the subset $L_{\unicode[STIX]{x1D703}}(i)$ has one of the following forms:

1. – $\{a_{i_{\unicode[STIX]{x1D703}}}\}$ ;

2. – $\{a_{i_{\unicode[STIX]{x1D703}}}>\unicode[STIX]{x1D6FC}\}$ ;

3. – $\{\unicode[STIX]{x1D6FD}^{-1}>a_{i_{\unicode[STIX]{x1D703}}}\}$ ;

4. – $\{\unicode[STIX]{x1D6FD}^{-1}>a_{i_{\unicode[STIX]{x1D703}}}>\unicode[STIX]{x1D6FC}\}$ ;

5. – $\{\unicode[STIX]{x1D700}^{-1}>a_{i_{\unicode[STIX]{x1D703}}}>\unicode[STIX]{x1D700}\}$ ,

for $\unicode[STIX]{x1D703}\in \{\pm \}$ , some solid arrows $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$ in $Q_{1}\subset \widehat{Q}_{1}$ and some dashed arrow $\unicode[STIX]{x1D700}$ in $Q_{2}=\widehat{Q}_{2}$ , satisfying that:

1. – for any two solid arrows $\unicode[STIX]{x1D6FE}$ and $\unicode[STIX]{x1D6FF}$ in $Q_{1}\subset \widehat{Q}_{1}$ with $t(\unicode[STIX]{x1D6FF})=s(\unicode[STIX]{x1D6FE})=k$ , we have that $\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6FF}\in Z$ if and only if $\unicode[STIX]{x1D6FE}$ and $\unicode[STIX]{x1D6FF}^{-1}$ are both in $L_{+}(k)$ or $L_{-}(k)$ .

Observe that in the set $L_{\unicode[STIX]{x1D703}}(i)$ , the inverse of an arrow in $Q$ is always greater than the arrow $a_{i_{\unicode[STIX]{x1D703}}}$ , and an arrow in $Q$ is always smaller than the arrow $a_{i_{\unicode[STIX]{x1D703}}}$ . Notice that if $L(i)\neq \{a_{i_{+}},a_{i_{-}}\}$ , then there are exactly two possible choices for the pair $(L_{+}(i),L_{-}(i))$ .

2.3 Words

A word $\mathfrak{m}$ is a sequence $\unicode[STIX]{x1D714}_{m}\cdots \unicode[STIX]{x1D714}_{2}\unicode[STIX]{x1D714}_{1}$ of letters in $L$ satisfying that for any $1\leqslant j\leqslant m-1$ , $\unicode[STIX]{x1D714}_{j}^{-1}\in L_{\unicode[STIX]{x1D703}}(i)$ and $\unicode[STIX]{x1D714}_{j+1}\in L_{\unicode[STIX]{x1D703}^{\prime }}(i)$ for different $\unicode[STIX]{x1D703},\unicode[STIX]{x1D703}^{\prime }\in \{\pm \}$ and some $i\in Q_{0}\subset \widehat{Q}_{0}$ . We call $\unicode[STIX]{x1D714}_{1}$ the first letter of $\mathfrak{m}$ and $\unicode[STIX]{x1D714}_{m}$ the last letter of $\mathfrak{m}$ . Since both $L_{\unicode[STIX]{x1D703}}(i)$ and $L_{\unicode[STIX]{x1D703}^{\prime }}(i)$ are subsets of  $L(i)$ , we have  $t(\unicode[STIX]{x1D714}_{j})=s(\unicode[STIX]{x1D714}_{j+1})=i$ . The functions $s,t$ can be generalized to the set of words by $s(\mathfrak{m}):=s(\unicode[STIX]{x1D714}_{1})$ and $t(\mathfrak{m}):=t(\unicode[STIX]{x1D714}_{m})$ . The inverse of a word $\mathfrak{m}=\unicode[STIX]{x1D714}_{m}\cdots \unicode[STIX]{x1D714}_{2}\unicode[STIX]{x1D714}_{1}$ is defined as $\mathfrak{m}^{-1}:=\unicode[STIX]{x1D714}_{1}^{-1}\unicode[STIX]{x1D714}_{2}^{-1}\cdots \unicode[STIX]{x1D714}_{m}^{-1}$ . The product $\mathfrak{n}\mathfrak{m}$ of two words $\mathfrak{m}=\unicode[STIX]{x1D714}_{m}\cdots \unicode[STIX]{x1D714}_{2}\unicode[STIX]{x1D714}_{1}$ and $\mathfrak{n}=\unicode[STIX]{x1D714}_{m+r}\cdots \unicode[STIX]{x1D714}_{m+2}\unicode[STIX]{x1D714}_{m+1}$ is defined to be $\unicode[STIX]{x1D714}_{m+r}\cdots \unicode[STIX]{x1D714}_{m+2}\unicode[STIX]{x1D714}_{m+1}\unicode[STIX]{x1D714}_{m}\cdots \unicode[STIX]{x1D714}_{2}\unicode[STIX]{x1D714}_{1}$ if this is again a word.

A letter is called punctured if it is of the from $a_{i_{\unicode[STIX]{x1D703}}}$ or $a_{i_{\unicode[STIX]{x1D703}}}^{-1}$ such that $L_{\unicode[STIX]{x1D703}}(i)=\{\unicode[STIX]{x1D700}^{-1}>a_{i_{\unicode[STIX]{x1D703}}}>\unicode[STIX]{x1D700}\}$ for some dashed arrow $\unicode[STIX]{x1D700}$ . A word $\mathfrak{m}$ is called left inextensible (respectively right inextensible) if there is no letter $l$ such that $l\mathfrak{m}$ (respectively $\mathfrak{m}l$ ) is again a word. A word is called maximal if it is both left and right inextensible. It is obvious that a word $\mathfrak{m}=\unicode[STIX]{x1D714}_{m}\cdots \unicode[STIX]{x1D714}_{1}$ is right (respectively left) inextensible if and only if $\unicode[STIX]{x1D714}_{1}$ (respectively $\unicode[STIX]{x1D714}_{m}$ ) is of the form $a_{i_{\unicode[STIX]{x1D703}}}^{-1}$ (respectively $a_{i_{\unicode[STIX]{x1D703}}}$ ). In a word, except for its first letter (respectively last letter), there are no letters of the form $a_{i_{\unicode[STIX]{x1D703}}}^{-1}$ (respectively $a_{i_{\unicode[STIX]{x1D703}}}$ ). In particular, each word contains at most two punctured letters.

2.4 Orders

The linear orders in the sets $L_{\pm }(i),$ $i\in Q_{0}\subset \widehat{Q}_{0}$ , induce a partial order ${\geqslant}$ on the set of words, such that $\mathfrak{m}>\mathfrak{r}$ if and only if $\mathfrak{m}=\unicode[STIX]{x1D714}_{m}\cdots \unicode[STIX]{x1D714}_{2}\unicode[STIX]{x1D714}_{1}$ and $\mathfrak{r}=\unicode[STIX]{x1D708}_{r}\cdots \unicode[STIX]{x1D708}_{2}\unicode[STIX]{x1D708}_{1}$ satisfy $\unicode[STIX]{x1D714}_{j}\cdots \unicode[STIX]{x1D714}_{1}=\unicode[STIX]{x1D708}_{j}\cdots \unicode[STIX]{x1D708}_{1}$ and $\unicode[STIX]{x1D714}_{j+1}>\unicode[STIX]{x1D708}_{j+1}$ for some $j\geqslant 0$ .

For each vertex $i\in Q_{0}$ , let $W_{\pm }(i)$ be the set of left inextensible words whose first letter is in $L_{\pm }(i)$ . By construction, $W_{\pm }(i)$ are linearly ordered. For each word $\mathfrak{m}$ in $W_{\unicode[STIX]{x1D703}}(i)$ , we use $\operatorname{[+]}\mathfrak{m}$ to denote its successor (if it exists) and use $\operatorname{[-]}\mathfrak{m}$ to denote its predecessor (if it exists). So we have $\operatorname{[-]}\mathfrak{m}>\mathfrak{m}>\operatorname{[+]}\mathfrak{m}$ . In case $\mathfrak{m}$ is right inextensible, we set $\mathfrak{m}\operatorname{[+]}:=(\operatorname{[+]}\mathfrak{m}^{-1})^{-1}$ and $\mathfrak{m}\operatorname{[-]}:=(\operatorname{[-]}\mathfrak{m}^{-1})^{-1}$ .

Let $\mathfrak{m}$ be a maximal word. By [Reference GeißGei99], $\operatorname{[+]}(\mathfrak{m}\operatorname{[+]})=(\operatorname{[+]}\mathfrak{m})\operatorname{[+]}$ provided both of them exist; denote by $\operatorname{[+]}\mathfrak{m}\operatorname{[+]}$ one (or both) of them.

2.5 Admissible words

For technical reasons, we introduce a special letter $\unicode[STIX]{x1D700}^{\ast }$ for each dashed loop $\unicode[STIX]{x1D700}$ and a map $F$ on letters which sends the elements in $\{\unicode[STIX]{x1D700}^{-1}>a_{i_{\unicode[STIX]{x1D703}}}>\unicode[STIX]{x1D700}\}$ to $\unicode[STIX]{x1D700}^{\ast }$ and preserves the other letters.

A maximal word $\mathfrak{m}=\unicode[STIX]{x1D714}_{m}\cdots \unicode[STIX]{x1D714}_{1}$ is called admissible if the following conditions hold.

1. (A1) For each $\unicode[STIX]{x1D714}_{i}=\unicode[STIX]{x1D700}$ with $\unicode[STIX]{x1D700}$ a dashed loop, we have that $\unicode[STIX]{x1D714}_{1}^{-1}\cdots \unicode[STIX]{x1D714}_{i-1}^{-1}>\unicode[STIX]{x1D714}_{m}\cdots \unicode[STIX]{x1D714}_{i+1}$ , and for each $\unicode[STIX]{x1D714}_{i}=\unicode[STIX]{x1D700}^{-1}$ with $\unicode[STIX]{x1D700}$ a dashed loop, we have that $\unicode[STIX]{x1D714}_{1}^{-1}\cdots \unicode[STIX]{x1D714}_{i-1}^{-1}<\unicode[STIX]{x1D714}_{m}\cdots \unicode[STIX]{x1D714}_{i+1}$ .

2. (A2) If $\mathfrak{m}$ contains two punctured letters, then $F(\mathfrak{m})$ is not a proper power of $F(\mathfrak{m}^{\prime })$ for any maximal word $\mathfrak{m}^{\prime }$ containing two punctured letters, where

   $$\begin{eqnarray}F(\mathfrak{m}):=F(\unicode[STIX]{x1D714}_{m})\cdots F(\unicode[STIX]{x1D714}_{1})F(\unicode[STIX]{x1D714}_{2}^{-1})\cdots F(\unicode[STIX]{x1D714}_{m-1}^{-1}).\end{eqnarray}$$
   Let $\mathfrak{X}$ be the set of admissible words. Note that if $\mathfrak{m}$ is in $\mathfrak{X}$ then so is its inverse $\mathfrak{m}^{-1}$ . Let $\overline{\mathfrak{X}}$ be the set of all equivalence classes in $\mathfrak{X}$ with respect to $\mathfrak{m}\simeq \mathfrak{m}^{-1}$ . But when we say an element $\mathfrak{m}$ in $\overline{\mathfrak{X}}$ , we always mean that $\mathfrak{m}$ is a representative in an equivalence class.

2.6 Known results

In this subsection, we collect some results on indecomposable modules of skewed-gentle algebras and homomorphism spaces between them.

Let $\mathfrak{m}\in \overline{\mathfrak{X}}$ . Associate an indeterminate with each punctured letter in $\mathfrak{m}$ and let $A_{\mathfrak{m}}$ be the $\mathbf{k}$ -algebra generated by these indeterminates $x$ with relations $x^{2}=x$ . A one-dimensional module $N$ of $A_{\mathfrak{m}}$ is an algebra homomorphism $N:A_{\mathfrak{m}}\rightarrow \mathbf{k}$ . It is determined (up to isomorphism) by the values $N(x)\in \{0,1\}$ . More precisely:

1. – if $\mathfrak{m}$ contains no punctured letters, then $A_{\mathfrak{m}}=\mathbf{k}$ and there is one one-dimensional module $N=\mathbf{k}$ ;

2. – if $\mathfrak{m}$ contains one punctured letter, then $A_{\mathfrak{m}}=\mathbf{k}[x]/(x^{2}-x)$ and there are two one-dimensional modules: $N=\mathbf{k}_{a}$ with $\mathbf{k}_{a}(x)=a$ , for $a\in \{0,1\}$ . Moreover, we have

   (2.1) $$\begin{eqnarray}\displaystyle \dim _{\mathbf{k}}\operatorname{Hom}_{A_{\mathfrak{m}}}(\mathbf{k}_{u},\mathbf{k}_{v})=\unicode[STIX]{x1D6FF}_{u,v}\quad \forall u,v\in \{0,1\}; & & \displaystyle\end{eqnarray}$$

3. – if $\mathfrak{m}$ contains two punctured letters, then $A_{\mathfrak{m}}=\mathbf{k}\langle x,y\rangle /(x^{2}-x,y^{2}-y)$ and there are four one-dimensional modules: $N=\mathbf{k}_{a,b}$ with $\mathbf{k}_{a,b}(x)=a$ and $\mathbf{k}_{a,b}(y)=b$ , for $a,b\in \{0,1\}$ . Moreover, we have

   (2.2) $$\begin{eqnarray}\displaystyle \dim _{\mathbf{k}}\operatorname{Hom}_{A_{\mathfrak{m}}}(\mathbf{k}_{u,u^{\prime }},\mathbf{k}_{v,v^{\prime }})=\unicode[STIX]{x1D6FF}_{u,v}\unicode[STIX]{x1D6FF}_{u^{\prime },v^{\prime }}\quad \forall u,v,u^{\prime },v^{\prime }\in \{0,1\}. & & \displaystyle\end{eqnarray}$$

The Auslander–Reiten translation $\unicode[STIX]{x1D70F}$ can be interpreted by the order of words.

For technical reasons, we also consider a trivial word $1_{i}$ corresponding to each vertex $i\in Q_{0}\subset \widehat{Q}_{0}$ . Let $\mathfrak{m}=\unicode[STIX]{x1D714}_{m}\cdots \unicode[STIX]{x1D714}_{1}$ be a word in $\overline{\mathfrak{X}}$ . For any integers $i,j$ with $0\leqslant i<j\leqslant m+1$ , we consider the subword $\mathfrak{m}_{(i,j)}$ of $\mathfrak{m}$ between $i$ and $j$ defined as

where $1_{t(\unicode[STIX]{x1D714}_{0})}:=1_{s(\unicode[STIX]{x1D714}_{1})}$ .

Let $\mathfrak{m}=\unicode[STIX]{x1D714}_{m}\cdots \unicode[STIX]{x1D714}_{1}$ and $\mathfrak{r}=\unicode[STIX]{x1D708}_{r}\cdots \unicode[STIX]{x1D708}_{1}$ be two words in $\bar{\mathfrak{X}}$ . A pair $((i,j),(h,l))$ of pairs of integers $i,j,h,l$ with $0\leqslant i<j\leqslant m+1$ and $0\leqslant h<l\leqslant r+1$ is called an int-pair from $\mathfrak{m}$ to $\mathfrak{r}$ if one of the following conditions holds:

1. – $\mathfrak{m}_{(i,j)}=\mathfrak{r}_{(h,l)}$ , $\unicode[STIX]{x1D714}_{i}^{-1}<\unicode[STIX]{x1D708}_{h}^{-1}$ and $\unicode[STIX]{x1D714}_{j}<\unicode[STIX]{x1D708}_{l}$ ;

2. – $\mathfrak{m}_{(i,j)}=(\mathfrak{r}_{(h,l)})^{-1}$ , $\unicode[STIX]{x1D714}_{i}^{-1}<\unicode[STIX]{x1D708}_{l}$ and $\unicode[STIX]{x1D714}_{j}<\unicode[STIX]{x1D708}_{h}^{-1}$ ;

where if an inequality contains at least one of $\unicode[STIX]{x1D714}_{0}$ , $\unicode[STIX]{x1D714}_{m+1}$ , $\unicode[STIX]{x1D708}_{0}$ and $\unicode[STIX]{x1D708}_{r+1}$ then we assume that it holds automatically. Let $H^{\mathfrak{m},\mathfrak{r}}$ be the set of int-pairs from $\mathfrak{m}$ and $\mathfrak{r}$ .

3.1 Jacobian algebras and Ginzburg dg algebras

Let $Q$ be a finite quiver and $W$ a potential on $Q$ , that is, a sum of cycles in $Q$ . The Jacobian algebra of the quiver with potential $(Q,W)$ is the quotient

where $\widehat{\mathbf{k}Q}$ is the complete path algebra of $Q$ , $\unicode[STIX]{x2202}W=\langle \unicode[STIX]{x2202}_{a}W:a\in Q_{1}\rangle$ and $\overline{\unicode[STIX]{x2202}W}$ is the closure of $\unicode[STIX]{x2202}W$ in $\widehat{\mathbf{k}Q}$ (cf. [Reference Derksen, Weyman and ZelevinskyDWZ08]).

The Jacobian algebra is the 0th cohomology of its refinement, the Ginzburg dg algebra $\unicode[STIX]{x1D6E4}=\unicode[STIX]{x1D6E4}(Q,W)$ of $(Q,W)$ (see the construction in [Reference KellerKel12, § 7.2]). There are three categories associated with $\unicode[STIX]{x1D6E4}$ , namely:

1. – the finite dimensional derived category ${\mathcal{D}}_{fd}(\unicode[STIX]{x1D6E4})$ of $\unicode[STIX]{x1D6E4}$ , which is a 3-Calabi–Yau category;

2. – the perfect derived category $\operatorname{per}(\unicode[STIX]{x1D6E4})$ of $\unicode[STIX]{x1D6E4}$ , which contains ${\mathcal{D}}_{fd}(\unicode[STIX]{x1D6E4})$ ;

3. – the cluster category ${\mathcal{C}}(\unicode[STIX]{x1D6E4})$ of $\unicode[STIX]{x1D6E4}$ , which is the (triangulated) 2-Calabi–Yau quotient category

   (3.1) $$\begin{eqnarray}\displaystyle {\mathcal{C}}(\unicode[STIX]{x1D6E4}):=\operatorname{per}(\unicode[STIX]{x1D6E4})/{\mathcal{D}}_{fd}(\unicode[STIX]{x1D6E4}). & & \displaystyle\end{eqnarray}$$

Furthermore there is a canonical cluster tilting object $T_{\unicode[STIX]{x1D6E4}}$ in ${\mathcal{C}}(\unicode[STIX]{x1D6E4})$ induced by the silting object $\unicode[STIX]{x1D6E4}$ in $\operatorname{per}\unicode[STIX]{x1D6E4}$ such that

See [Reference AmiotAmi09, Theorem 3.5] and [Reference Keller and ReitenKR07, § 2.1, Proposition (c)].

For a vertex $i$ of $Q$ , let $\unicode[STIX]{x1D707}_{i}(Q,W)$ be the mutation of $(Q,W)$ at $i$ in the sense of [Reference Derksen, Weyman and ZelevinskyDWZ08], see also Appendix B. By [Reference Keller and YangKY11], there exists a canonical triangulated equivalence

3.2 Quivers with potential from marked surfaces

Throughout the article, $\mathbf{S}$ denotes a marked surface with non-empty boundary in the sense of [Reference Fomin, Shapiro and ThurstonFST08], that is, a compact connected oriented surface $\mathbf{S}$ with a finite set $\mathbf{M}$ of marked points on its boundary $\unicode[STIX]{x2202}\mathbf{S}$ and a finite set $\mathbf{P}$ of punctures in its interior $\mathbf{S}\setminus \unicode[STIX]{x2202}\mathbf{S}$ such that the following conditions hold:

1. – each connected component of $\unicode[STIX]{x2202}\mathbf{S}$ contains at least one marked point;

2. – $\mathbf{S}$ is not closed, i.e.  $\unicode[STIX]{x2202}\mathbf{S}\neq \emptyset$ ;

3. – the rank

   (3.4) $$\begin{eqnarray}n=6g+3p+3b+m-6\end{eqnarray}$$
   of the surface is positive, where $g$ is the genus of $\mathbf{S}$ , $b$ the number of boundary components, $m=|\mathbf{M}|$ the number of marked points and $p=|\mathbf{P}|$ the number of punctures.

Figure 1. The completions of curves.

Figure 2. Once-punctured monogons.

Figure 3. The tagged rotations of three tagged curves in $\mathbf{S}$ .

We explain the intersection number of two tagged curves in some special cases.

Figure 4. The punctured intersections.

By Remark 3.5, the following definitions of ideal triangulations and tagged triangulations are equivalent to the original ones in [Reference Fomin, Shapiro and ThurstonFST08].

Any ideal/tagged triangulation $\mathbf{T}$ of $\mathbf{S}$ consists of $n$ ordinary/tagged curves (see [Reference Fomin, Shapiro and ThurstonFST08, Proposition 2.10, Theorem 7.9]), where $n$ is the rank of $\mathbf{S}$ (cf. (3.4)). We require $n>0$ and exclude the case of a once-punctured monogon (where $n=1$ ) in the proofs. However, all the results hold in this case by direct checking and thus we will not exclude this case in the statements.

A triangle in $\mathbf{T}$ has three distinct sides unless it is a self-folded triangle as in the left picture of Figure 5, where we call $\unicode[STIX]{x1D6FC}$ the folded side and $\unicode[STIX]{x1D6FD}$ the remaining side.

Figure 5. The self-folded triangle and the corresponding tagged version.

The flip of an ideal triangulation $\mathbf{T}$ , with respect to a curve $\unicode[STIX]{x1D6FC}$ in $\mathbf{T}$ , is the unique ideal triangulation $\mathbf{T}^{\prime }$ (if it exists) that shares all curves in $\mathbf{T}$ but $\unicode[STIX]{x1D6FC}$ . One can always flip an ideal triangulation with respect to a curve unless it is the folded side of a self-folded triangle. To overcome this shortcoming, Fomin et al. [Reference Fomin, Shapiro and ThurstonFST08] introduced tagged triangulations, with tagged flips, so that every tagged triangulation can be flipped with respect to any tagged curve in it. The exchange graph of tagged triangulations with tagged flips is denoted by $\operatorname{EG}^{\times }(\mathbf{S})$ , that is, the graph whose vertices are tagged triangulations and whose edges are tagged flips.

For each curve $\unicode[STIX]{x1D6FE}$ in $\mathbf{C}(\mathbf{S})$ with $\operatorname{Int}(\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D6FE})=0$ , we define its tagged version $\unicode[STIX]{x1D6FE}^{\times }$ to be $(\unicode[STIX]{x1D6FE},\emptyset )$ unless $\unicode[STIX]{x1D6FE}$ is a loop enclosing a puncture, as $\unicode[STIX]{x1D6FD}$ in the left picture of Figure 5. In that case $\unicode[STIX]{x1D6FD}^{\times }$ , as in the right picture of Figure 5, is defined to be $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D705})$ , where $\unicode[STIX]{x1D6FC}$ is the unique curve without self-intersections enclosed by $\unicode[STIX]{x1D6FD}$ and $\unicode[STIX]{x1D705}(t)=1$ for $t$ with $\unicode[STIX]{x1D6FC}(t)\in \mathbf{P}$ . In this way, each ideal triangulation $\mathbf{T}$ induces a tagged triangulation $\mathbf{T}^{\times }$ consisting of the tagged versions of all curves in $\mathbf{T}$ .

For each ideal triangulation $\mathbf{T}$ , there is an associated quiver with potential $(Q_{\mathbf{T}},W_{\mathbf{T}})$ (cf. [Reference Fomin, Shapiro and ThurstonFST08, Reference Irelli and Labardini-FragosoIL12]). In the paper, we only study $(Q_{\mathbf{T}},W_{\mathbf{T}})$ in the case when $\mathbf{T}$ is an admissible triangulation in the following sense (see Figure 14 for example).

In particular, in such a triangulation, the folded side of each self-folded triangle connects a marked point in $\mathbf{M}$ and a puncture in $\mathbf{P}$ .

In an admissible triangulation $\mathbf{T}$ , for a curve $\unicode[STIX]{x1D6FC}\in \mathbf{T}$ , let $\unicode[STIX]{x1D70B}_{\mathbf{T}}(\unicode[STIX]{x1D6FC})$ be the curve defined as follows: if $\unicode[STIX]{x1D6FC}$ is the folded side of a self-folded triangle in $\mathbf{T}$ (see the left picture of Figure 5), then $\unicode[STIX]{x1D70B}_{\mathbf{T}}(\unicode[STIX]{x1D6FC})$ is the corresponding remaining side (i.e.  $\unicode[STIX]{x1D6FD}$ in the left picture of Figure 5); if there is no such triangle, set $\unicode[STIX]{x1D70B}_{\mathbf{T}}(\unicode[STIX]{x1D6FC})=\unicode[STIX]{x1D6FC}$ . The associated quiver with potential $(Q_{\mathbf{T}},W_{\mathbf{T}})$ is given by the following data (see Figure 6):

1. – the vertices of $Q_{\mathbf{T}}$ are labeled by the curves in $\mathbf{T}$ ;

2. – there is an arrow from $i$ to $j$ whenever there is a non-self-folded triangle in $\mathbf{T}$ having $\unicode[STIX]{x1D70B}_{\mathbf{T}}(i)$ and $\unicode[STIX]{x1D70B}_{\mathbf{T}}(j)$ as edges with $\unicode[STIX]{x1D70B}_{\mathbf{T}}(j)$ following $\unicode[STIX]{x1D70B}_{\mathbf{T}}(i)$ in the clockwise orientation (which is induced by the orientation of $\mathbf{S}$ ). For instance, the quiver for a non-self-folded triangle is shown in Figure 6;

3. – each subset $\{i,j,k\}$ of $\mathbf{T}$ with $\unicode[STIX]{x1D70B}_{\mathbf{T}}(i)$ , $\unicode[STIX]{x1D70B}_{\mathbf{T}}(j)$ , $\unicode[STIX]{x1D70B}_{\mathbf{T}}(k)$ forming an interior non-self-folded triangle in $\mathbf{T}$ yields a unique three-cycle up to cyclic permutation. The potential $W_{\mathbf{T}}$ is the sum of all such three-cycles.

Figure 6. The quiver associated with a non-self-folded triangle.

Then by § 3.1, there is an associated cluster category, denoted by ${\mathcal{C}}(\mathbf{T})$ .

3.3 Correspondence

The objects and morphisms in ${\mathcal{C}}(\mathbf{T})$ are expected to correspond to curves and intersection numbers, respectively. A cluster tilting object $T=\bigoplus _{j=1}^{n}T_{j}$ in a cluster category ${\mathcal{C}}$ is an object satisfying $\operatorname{Ext}_{{\mathcal{C}}}^{1}(T,X)=0$ if and only if $X\in \mathsf{add}\,T$ . The mutation $\unicode[STIX]{x1D707}_{i}$ at the $i$ th indecomposable direct summand acts on a cluster tilting object $T=\bigoplus _{j=1}^{n}T_{j}$ , by replacing $T_{i}$ with the unique indecomposable object $T_{i}^{\prime }\ncong T_{i}$ satisfying that $(T\setminus T_{i})\oplus T_{i}^{\prime }$ is a cluster tilting object.

In the unpunctured case, we have the following known results.

In the punctured case, we also know the following. Recall that a rigid indecomposable object in ${\mathcal{C}}(\mathbf{T})$ is reachable if it is a summand of some cluster tilting object, which is obtained from the canonical cluster tilting object by a sequence of mutations.

3.4 Cluster(-tilting) exchange graphs

The cluster(-tilting) exchange graph $\operatorname{CEG}({\mathcal{C}})$ of a cluster category ${\mathcal{C}}$ is the graph whose vertices are cluster tilting objects and whose edges are mutations. There are the following known results about connectedness of cluster exchange graphs.

4.1 Skewed-gentle algebras from admissible triangulations

Let $\mathbf{T}$ be an admissible triangulation of $\mathbf{S}$ , i.e. every puncture is in a self-folded triangle (see Lemma A.1 for the existence of $\mathbf{T}$ ), with the associated quiver with potential $(Q_{\mathbf{T}},W_{\mathbf{T}})$ and the cluster category ${\mathcal{C}}(\mathbf{T})$ .

Let $\mathbf{T}^{o}$ be the subset of $\mathbf{T}$ consisting of curves whose endpoints are in $\mathbf{M}$ . Now we associate a biquiver $Q^{\mathbf{T}}=(Q_{0}^{\mathbf{T}},Q_{1}^{\mathbf{T}},Q_{2}^{\mathbf{T}})$ with potential $W^{\mathbf{T}}$ as follows:

1. – $Q_{0}^{\mathbf{T}}=\mathbf{T}^{o}$ , i.e. curves in $\mathbf{T}$ which are sides of non-self-folded triangles correspond to vertices in $Q_{0}^{\mathbf{T}}$ ;

2. – there is a solid arrow from $i$ to $j$ in $Q_{1}^{\mathbf{T}}$ whenever there is a non-self-folded triangle $\unicode[STIX]{x1D6E5}$ in $\mathbf{T}$ such that $\unicode[STIX]{x1D6E5}$ has sides $i$ and $j$ with $j$ following $i$ in the clockwise orientation;

3. – there is a dashed loop at $i$ in $Q_{2}^{\mathbf{T}}$ , denoted by $\unicode[STIX]{x1D700}_{i}$ , whenever $i$ is the remaining side of a self-folded triangle;

4. – each non-self-folded triangle in $\mathbf{T}$ induces a unique three-cycle up to cyclic permutation. The potential $W^{\mathbf{T}}$ is the sum of all such three-cycles.

See the example in § 6. By construction, there are no loops in $Q_{1}^{\mathbf{T}}$ , any arrow in $Q_{2}^{\mathbf{T}}$ is a loop and there is at most one loop in $Q_{2}^{\mathbf{T}}$ at each vertex. Hence the biquiver $Q^{\mathbf{T}}$ satisfies the conditions on biquivers in § 2.1. Let $Z=\unicode[STIX]{x2202}W^{\mathbf{T}}=\{\unicode[STIX]{x2202}_{a}W^{\mathbf{T}}:a\in Q_{1}^{\mathbf{T}}\}$ . We have the following straightforward observation.

Figure 7. Non-self-folded triangles with the corresponding quivers.

Figure 8. Self-folded triangles with the corresponding quivers.

Let $Q_{0}^{\operatorname{Sp}}$ be the subset of $Q_{0}^{\mathbf{T}}$ consisting of vertices where there are dashed loops.

Now we prove that the Jacobian algebra ${\mathcal{P}}(Q_{\mathbf{T}},W_{\mathbf{T}})$ is isomorphic to the skewed-gentle algebra $\unicode[STIX]{x1D6EC}^{\mathbf{T}}$ .

4.2 Correspondence

Denote by $\mathsf{rep}(Q^{\mathbf{T}},W^{\mathbf{T}})$ the category of finite dimensional $\mathbf{k}$ -linear representations of $Q$ bounded by $R=\unicode[STIX]{x2202}W^{\mathbf{T}}\cup \{\unicode[STIX]{x1D700}^{2}=\unicode[STIX]{x1D700}\mid \unicode[STIX]{x1D700}\in Q_{2}^{\mathbf{T}}\}$ . By the equivalence (3.2) and Proposition 4.4, we have an equivalence

where $T_{\mathbf{T}}$ denotes the canonical cluster tilting object in ${\mathcal{C}}(\mathbf{T})$ (cf. § 3.1). So one can regard the set of indecomposable objects in ${\mathcal{C}}(\mathbf{T})$ as the union of the set of indecomposable representations in $\mathsf{rep}(Q^{\mathbf{T}},W^{\mathbf{T}})$ and the set of indecomposable direct summands of $T_{\mathbf{T}}$ . Note that indecomposable direct summands of $T_{\mathbf{T}}$ are indexed by curves in $\mathbf{T}$ and hence also by tagged curves in $\mathbf{T}^{\times }$ (which is the tagged version of $\mathbf{T}$ ). Thus, we can write

To describe indecomposable representations in $\mathsf{rep}(Q^{\mathbf{T}},W^{\mathbf{T}})$ , we shall use notions and notation stated in § 2. Recall that from the biquiver $Q^{\mathbf{T}}$ , in § 2.2, we constructed a new biquiver $\widehat{Q}=(\widehat{Q}_{0},\widehat{Q}_{1},\widehat{Q}_{2})$ by adding two new solid arrows $a_{i_{\pm }}$ (whose terminal vertices are also new vertices) for each vertex $i\in Q_{0}^{\mathbf{T}}$ . Recall that a letter is an arrow in $\widehat{Q}$ or its inverse. For each $i\in Q_{0}^{\mathbf{T}}\subset \widehat{Q}_{0}$ the set $L(i)=\{l\in L\mid s(l)=i\}$ is divided into two disjoint subsets $L_{\unicode[STIX]{x1D703}}(i)$ , $\unicode[STIX]{x1D703}\in \{\pm \}$ , with linear orders satisfying certain conditions.

See Figure 9 (cf. Figures 7 and 8), where the disjoint subsets $L_{\unicode[STIX]{x1D703}}(i)$ are given by $L_{+}(i)=\{\unicode[STIX]{x1D6FC}_{1}^{-1}>a_{i_{+}}>\unicode[STIX]{x1D6FD}_{1}\}$ and $L_{-}(i)=\{\unicode[STIX]{x1D6FC}_{2}^{-1}>a_{i_{-}}>\unicode[STIX]{x1D6FD}_{2}\}$ for the left picture and by $L_{+}(i)=\{\unicode[STIX]{x1D6FC}^{-1}>a_{i_{+}}>\unicode[STIX]{x1D6FD}\}$ and $L_{-}(i)=\{\unicode[STIX]{x1D700}^{-1}>a_{i_{-}}>\unicode[STIX]{x1D700}\}$ for the right picture. Note that $\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},\unicode[STIX]{x1D6FC}_{1},\unicode[STIX]{x1D6FD}_{1},\unicode[STIX]{x1D6FC}_{2},\unicode[STIX]{x1D6FD}_{2}$ might not exist.

Figure 9. The arc segments associated with letters.

By Construction 4.7, any arc segment associated with a letter in $L(i)$ starts at $i$ . The following lemma gives some basic topological interpretation of notions about letters.

Figure 10. The orders.

Recall that a word is a sequence $\mathfrak{m}=\unicode[STIX]{x1D714}_{m}\cdots \unicode[STIX]{x1D714}_{1}$ of letters in $L$ such that for any $1\leqslant j\leqslant m-1$ , $\unicode[STIX]{x1D714}_{j}^{-1}\in L_{\unicode[STIX]{x1D703}}(i)$ and $\unicode[STIX]{x1D714}_{j+1}\in L_{\unicode[STIX]{x1D703}^{\prime }}(i)$ for different $\unicode[STIX]{x1D703},\unicode[STIX]{x1D703}^{\prime }\in \{\pm \}$ and some $i\in Q_{0}^{\mathbf{T}}$ . By Lemma 4.8(1), this condition is equivalent to $\mathfrak{a}(\unicode[STIX]{x1D714}_{j}^{-1})$ and $\mathfrak{a}(\unicode[STIX]{x1D714}_{j+1})$ starting at the same curve in $\mathbf{T}$ , but being in the two triangles adjacent to this curve, respectively. Hence we can glue the two arc segments $\mathfrak{a}(\unicode[STIX]{x1D714}_{j})$ and $\mathfrak{a}(\unicode[STIX]{x1D714}_{j+1})$ together to get a curve segment.

Recall that for two words $\mathfrak{m}=\unicode[STIX]{x1D714}_{m}\cdots \unicode[STIX]{x1D714}_{2}\unicode[STIX]{x1D714}_{1}$ and $\mathfrak{r}=\unicode[STIX]{x1D708}_{r}\cdots \unicode[STIX]{x1D708}_{2}\unicode[STIX]{x1D708}_{1}$ , $\mathfrak{m}>\mathfrak{r}$ if and only if there is $j$ such that $\unicode[STIX]{x1D714}_{j}\cdots \unicode[STIX]{x1D714}_{1}=\unicode[STIX]{x1D708}_{j}\cdots \unicode[STIX]{x1D708}_{1}$ and $\unicode[STIX]{x1D714}_{j+1}>\unicode[STIX]{x1D708}_{j+1}$ .

Recall that a word $\mathfrak{m}=\unicode[STIX]{x1D714}_{m}\cdots \unicode[STIX]{x1D714}_{1}$ is maximal if and only if $\unicode[STIX]{x1D714}_{1}=a_{i_{\unicode[STIX]{x1D703}}}^{-1}$ and $\unicode[STIX]{x1D714}_{m}=a_{j_{\unicode[STIX]{x1D703}^{\prime }}}$ for some $i,j\in Q_{0}^{\mathbf{T}}$ and some $\unicode[STIX]{x1D703},\unicode[STIX]{x1D703}^{\prime }\in \{\pm \}$ .

Recall from § 2.5 that a maximal word $\mathfrak{m}=\unicode[STIX]{x1D714}_{m}\cdots \unicode[STIX]{x1D714}_{1}$ is called admissible if the following hold.

1. (A1) For each $\unicode[STIX]{x1D714}_{i}=\unicode[STIX]{x1D700}$ with $\unicode[STIX]{x1D700}$ a dashed loop, we have that $\unicode[STIX]{x1D714}_{1}^{-1}\cdots \unicode[STIX]{x1D714}_{i-1}^{-1}>\unicode[STIX]{x1D714}_{m}\cdots \unicode[STIX]{x1D714}_{i+1}$ , and for each $\unicode[STIX]{x1D714}_{i}=\unicode[STIX]{x1D700}^{-1}$ with $\unicode[STIX]{x1D700}$ a dashed loop, we have that $\unicode[STIX]{x1D714}_{1}^{-1}\cdots \unicode[STIX]{x1D714}_{i-1}^{-1}<\unicode[STIX]{x1D714}_{m}\cdots \unicode[STIX]{x1D714}_{i+1}$ .

2. (A2) If $\mathfrak{m}$ contains two punctured letters then $F(\mathfrak{m})$ is not a proper power of $F(\mathfrak{m}^{\prime })$ for any maximal word $\mathfrak{m}^{\prime }$ containing two punctured letters, where

   $$\begin{eqnarray}F(\mathfrak{m}):=F(\unicode[STIX]{x1D714}_{m})\cdots F(\unicode[STIX]{x1D714}_{1})F(\unicode[STIX]{x1D714}_{2}^{-1})\cdots F(\unicode[STIX]{x1D714}_{m-1}^{-1}).\end{eqnarray}$$

On the other hand, recall from Definition 3.1 that a pair $(\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D705})$ is called a tagged curve if the following hold.

1. (T1) The curve $\unicode[STIX]{x1D6FE}$ does not cut out a once-punctured monogon by a self-intersection (including endpoints), cf. Figure 2.

2. (T2) If $\unicode[STIX]{x1D6FE}(0),\unicode[STIX]{x1D6FE}(1)\in \mathbf{P}$ , then the completion $\bar{\unicode[STIX]{x1D6FE}}$ is not a proper power of a closed curve in the sense of the multiplication in the fundamental group of $\mathbf{S}$ .

Let $\mathbf{C}_{0}(\mathbf{S})$ be the subset of $\mathbf{C}(\mathbf{S})\setminus \mathbf{T}$ consisting of curves satisfying the conditions (T1) and (T2). That is, $\mathbf{C}_{0}(\mathbf{S})$ consists of curves $\unicode[STIX]{x1D6FE}$ in $\mathbf{C}(\mathbf{S})\setminus \mathbf{T}$ such that there exists a tagged curve $(\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D705})$ for some $\unicode[STIX]{x1D705}$ .

By Lemmas 4.11 and 4.12, we have the following.

For each $\unicode[STIX]{x1D6FE}\in \mathbf{C}_{0}(\mathbf{S})$ , we denote by $\mathfrak{m}_{\unicode[STIX]{x1D6FE}}$ the preimage of $\unicode[STIX]{x1D6FE}$ under the bijection (4.3). Recall from § 2.6 that the algebra $A_{\mathfrak{m}_{\unicode[STIX]{x1D6FE}}}$ is generated by the indeterminates $x$ associated with punctured letters in $\mathfrak{m}_{\unicode[STIX]{x1D6FE}}$ with $x^{2}=x$ . So by Lemma 4.8(3) indeterminates of $A_{\mathfrak{m}_{\unicode[STIX]{x1D6FE}}}$ are indexed by endpoints of $\unicode[STIX]{x1D6FE}$ that are punctures.

4.3 Flips and mutations

We study $\diamondsuit$ -flips of an admissible triangulation $\mathbf{T}$ in this subsection. Recall that the function $\unicode[STIX]{x1D70B}_{\mathbf{T}}$ on $\mathbf{T}$ is defined as follows: if $\unicode[STIX]{x1D6FE}$ is the folded side of a self-folded triangle in $\mathbf{T}$ , then $\unicode[STIX]{x1D70B}_{\mathbf{T}}(\unicode[STIX]{x1D6FE})$ is the corresponding remaining side; otherwise, $\unicode[STIX]{x1D70B}_{\mathbf{T}}(\unicode[STIX]{x1D6FE})=\unicode[STIX]{x1D6FE}$ .

Note that there are two types of $\diamondsuit$ -flips: when $i$ is not a side of a self-folded triangle, the corresponding $\diamondsuit$ -flip is an ordinary flip, and when $i$ is the remaining side of a self-folded triangle, the corresponding $\diamondsuit$ -flip is a combination of two ordinary flips occurring inside a once-punctured digon (see Figure A.1). Recall that an ordinary flip of a triangulation $\mathbf{T}^{\prime }$ is a new triangulation $\mathbf{T}^{\prime \prime }$ which shares all curves with $\mathbf{T}^{\prime }$ except for one.

The $\diamondsuit$ -flips of an admissible triangulation $\mathbf{T}$ are indexed by the vertices of the quiver $Q^{\mathbf{T}}$ . Define the mutation $(Q^{\prime },W^{\prime })=\unicode[STIX]{x1D707}_{i}(Q^{\mathbf{T}},W^{\mathbf{T}})$ at a vertex $i\in Q_{0}^{\mathbf{T}}$ to be $(Q_{0}^{\prime },Q_{1}^{\prime },W^{\prime })=\unicode[STIX]{x1D707}_{i}(Q_{0},Q_{1},W)$ in the sense of [Reference Derksen, Weyman and ZelevinskyDWZ08] with $Q_{2}^{\prime }=Q_{2}$ . By [Reference Fomin, Shapiro and ThurstonFST08, Reference Labardini-FragosoLab09a],

for each $i\in \mathbf{T}^{o}=Q_{0}^{\mathbf{T}}$ , where $f_{i}$ is the $\diamondsuit$ -flip associated with $i$ . Then by [Reference Keller and YangKY11], there is an equivalence $\widetilde{\unicode[STIX]{x1D707}}_{i}:{\mathcal{C}}(\mathbf{T})\simeq {\mathcal{C}}(\mathbf{T}^{\prime })$ .

For each tagged curve $(\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D705})$ , the corresponding object $X_{(\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D705})}^{\mathbf{T}}$ in ${\mathcal{C}}(\mathbf{T})$ is given by the associated representation $M_{(\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D705})}^{\mathbf{T}}$ in $\mathsf{rep}(Q^{\mathbf{T}},W^{\mathbf{T}})$ through the equivalence $F_{\mathbf{T}}:{\mathcal{C}}(\mathbf{T})/(T_{\mathbf{T}})\rightarrow \mathsf{rep}(Q^{\mathbf{T}},W^{\mathbf{T}})$ . Then the equivalence $\widetilde{\unicode[STIX]{x1D707}}_{i}$ should be compatible with some mutation of representations. However, there is not a well-defined mutation on $\mathsf{rep}(Q^{\mathbf{T}},W^{\mathbf{T}})$ . Instead, we shall use decorated representations and their mutations introduced in [Reference Derksen, Weyman and ZelevinskyDWZ08]. A decorated representation of $(Q^{\mathbf{T}},W^{\mathbf{T}})$ is a pair $(M,V)$ , where $M\in \mathsf{rep}(Q^{\mathbf{T}},W^{\mathbf{T}})$ is a usual representation of $(Q^{\mathbf{T}},W^{\mathbf{T}})$ and $V$ is a representation of $(Q_{0}^{\mathbf{T}},Q_{2}^{\mathbf{T}})$ bounded by $\{\unicode[STIX]{x1D700}^{2}-\unicode[STIX]{x1D700}\mid \unicode[STIX]{x1D700}\in Q_{2}^{\mathbf{T}}\}$ .

Let $(\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D705})$ be a tagged curve in $\mathbf{C}^{\times }(\mathbf{S})$ and $(M_{(\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D705})}^{\mathbf{T}},V_{(\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D705})}^{\mathbf{T}})$ the corresponding decorated representation. Construct $\unicode[STIX]{x1D707}_{i}(M_{(\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D705})}^{\mathbf{T}},V_{(\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D705})}^{\mathbf{T}})$ as in Appendix B. Now we prove that the map $X^{\mathbf{T}}$ from tagged curves to string objects is independent of the choice of the admissible triangulation  $\mathbf{T}$ .

5.1 AR-translation and AR-triangles

Note that we have chosen an admissible triangulation $\mathbf{T}$ and have a bijection $X^{\mathbf{T}}:\mathbf{C}^{\times }(\mathbf{S})\rightarrow \mathfrak{S}(\mathbf{T})$ from the set of tagged curves to the set of string objects in the cluster category ${\mathcal{C}}(\mathbf{T})$ . The tagged rotation (cf. Definition 3.2 and Figure 3) $\unicode[STIX]{x1D70C}$ is a permutation on $\mathbf{C}^{\times }(\mathbf{S})$ while the shift functor $[1]$ in ${\mathcal{C}}(\mathbf{T})$ gives a permutation on the set $\mathfrak{S}(\mathbf{T})$ . We will give a straightforward proof of Theorem 3.10, with a slight generalization to tagged curves.

For a curve $\unicode[STIX]{x1D6FE}$ in $\mathbf{C}(\mathbf{S})$ with $\unicode[STIX]{x1D6FE}(1)\in \mathbf{M}$ , denote by $\operatorname{[+]}\unicode[STIX]{x1D6FE}$ the curve obtained from $\unicode[STIX]{x1D6FE}$ by moving $\unicode[STIX]{x1D6FE}(1)$ along the boundary anticlockwise to the next marked point; dually, for a curve $\unicode[STIX]{x1D6FE}$ in $\mathbf{C}(\mathbf{S})$ with $\unicode[STIX]{x1D6FE}(0)\in \mathbf{M}$ , denote by $\unicode[STIX]{x1D6FE}\operatorname{[+]}$ the curve obtained from $\unicode[STIX]{x1D6FE}$ by moving $\unicode[STIX]{x1D6FE}(0)$ along the boundary anticlockwise to the next marked point. We first show the following lemma, where $\mathfrak{m}_{\unicode[STIX]{x1D6FE}}$ is the word associated with $\unicode[STIX]{x1D6FE}$ defined by the bijection (4.3) and $\operatorname{[+]}\mathfrak{m}$ and $\mathfrak{m}\operatorname{[+]}$ are defined in § 2.4. Recall that the set $\mathbf{C}_{0}(\mathbf{S})$ consists of curves $\unicode[STIX]{x1D6FE}$ in $\mathbf{C}(\mathbf{S})\setminus \mathbf{T}$ such that there exists a tagged curve $(\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D705})$ for some $\unicode[STIX]{x1D705}$ .

Figure 12. Cutting: five cases.