\documentclass[submission]{FPSAC2017}

\articlenumber{16}
\addbibresource{16_Garver_McConville_Serhiyenko.bib}


\newtheorem{thm}{Theorem}
\newtheorem{lem}{Lemma}
\newtheorem{definition}{Definition}
\newtheorem{corollary}{Corollary}
\newtheorem{proposition}{Proposition}



\usepackage{lipsum}

\usepackage[all]{xy}
\usepackage{mathabx}
\usepackage{tikz}






\title[Minimal Length Maximal Green Sequences]{Minimal Length Maximal Green Sequences}

\author[Garver, McConville and Serhiyenko]{Alexander Garver\thanks{\href{mailto:alexander.garver@lacim.ca}{alexander.garver@lacim.ca}}\addressmark{1},  Thomas McConville\thanks{\href{mailto:thomasmc@mit.edu}{thomasmc@mit.edu}}\addressmark{2} \and Khrystyna Serhiyenko\thanks{\href{mailto:khrystyna.serhiyenko@berkeley.edu}{khrystyna.serhiyenko@berkeley.edu}}\addressmark{3}}

\address{\addressmark{1}Laboratoire de Combinatoire et d'Informatique Math\'ematique, \\ Universit\'e du Qu\'ebec \`a Montr\'eal \\ \addressmark{2}Department of Mathematics, Massachusetts Institute of Technology\\ \addressmark{3}Department of Mathematics, University of California, Berkeley}

\received{\today}

\abstract{Maximal green sequences are important objects in representation theory, cluster algebras, and string theory. It is an open problem to determine what lengths are achieved by maximal green sequences of a quiver. We use the combinatorics of surface triangulations to address this problem. Our main result is a formula for the length of minimal length maximal green sequences of quivers defined by triangulations of an annulus or a punctured disk.} %\textcolor{red}{STUFF}}%\lipsum[1]

\resume{Les suites vertes maximales sont des objets importants dans la th\'eorie des repr\'esentation, les alg\`ebres amass\'ees et la th\'eorie des cordes. C'est un probl\`eme ouvert que de d\'eterminer les longueurs que peuvent prendre les suites vertes maximales d'un {carquois}. Nous utilisons la combinatoire des triangulations de surface pour \'etudier ce probl\`eme. Notre r\'esultat principal est une formule pour la longueur minimale des suites vertes maximales de carquois d\'efinis par des triangulations d'un anneau ou d'un disque perfor\'e.}%\lipsum[2]

\keywords{quiver mutation, maximal green sequence, triangulated surface}

\begin{document}

\maketitle

\section{Introduction}

A maximal green sequence is a distinguished sequence of local transformations, known as \textbf{mutations}, of a given \textbf{quiver} (i.e., directed graph). Maximal green sequences were introduced by Keller in \cite{keller2011cluster} in order to obtain combinatorial formulas for the refined Donaldson-Thomas invariants of Kontsevich and Soibelman \cite{KS}. They are also important in string theory \cite{alim2013bps}, representation theory \cite{bdp, by14}, and cluster algebras \cite{fomin2002cluster}. 

Recently, there have been many developments on the combinatorics of maximal green sequences (for example, see \cite{garver2014maximal, muller2016existence, mills2016maximal} and references therein). Our goal is to add to the known combinatorics by developing a numerical invariant of the set of maximal green sequences of $Q$: the length of minimal length maximal green sequences.


This invariant is natural from the perspective of cluster algebras. Fixing a quiver $Q$ induces an orientation of the edges of the corresponding exchange graph, and the maximal green sequences of $Q$ are in natural bijection with finite length maximal directed paths in the resulting oriented exchange graph (see \cite{bdp}). Examples of oriented exchange graphs include the Hasse diagrams of Tamari lattices and Cambrian lattices of type $\mathbb{A}$, $\mathbb{D}$, and $\mathbb{E}$ \cite{ReadingCamb}. In these examples the minimal length maximal green sequences always have length equal to the number of vertices of $Q$, but in general the minimal length of a maximal green sequence is not known. Thus understanding the length of minimal length maximal green sequences provides new information about oriented exchange graphs.




Our main result (\cref{Thm:main}) is a formula for this minimal length when $Q$ is of \textbf{cluster type} $\mathbb{D}_n$ and \textbf{cluster type} $\widetilde{\mathbb{A}}_{n-1}$ (i.e., $Q$ is in the mutation class of a type $\mathbb{D}_n$ or of an affine type $\mathbb{A}_{n-1}$ Dynkin quiver). This number was calculated in cluster type $\mathbb{A}$ in \cite{cormier2015minimal}.

\vspace{.5em}

In \cref{Sec:ice_quiver}, we review the notions of quiver mutation and maximal green sequences. We also recall some relevant results in cluster type $\mathbb{A}$. We then state our first main result, which allows one to calculate the length of minimal length maximal green sequences of a quiver with finitely many cluster type $\mathbb{A}$ quivers attached to it (see \cref{Cor:min_length_Qtilde}). 


In \cref{Sec:triangulations}, we recall how triangulations of Riemann surfaces can be used to model maximal green sequences of an important family of quivers. We use this model to present \cref{mullertheorem}, which is crucial to proving \cref{Cor:min_length_Qtilde}. We state \cref{mullertheorem} in restricted generality in order to formulate it in a way that fits with our exposition. 

Finally, in \cref{type_D_and_affineA}, we combine \cref{Cor:min_length_Qtilde} and the combinatorics of surface triangulations to find the length of minimal length maximal green sequences of quivers of {cluster type} $\mathbb{D}_n$ and {cluster type} $\widetilde{\mathbb{A}}_{n-1}$.

\section{Ice quivers and maximal green sequences}\label{Sec:ice_quiver}


A \textbf{quiver} $Q$ is a 4-tuple $(Q_0,Q_1,s,t)$, where $Q_0 = [m] := \{1,2, \ldots, m\}$ is a set of \textbf{vertices}, $Q_1$ is a set of \textbf{arrows}, and two functions $s, t:Q_1 \to Q_0$ defined so that for every $\alpha \in Q_1$, we have $s(\alpha) \xrightarrow{\alpha} t(\alpha)$. An \textbf{ice quiver} is a pair $(Q,F)$ with $Q$ a quiver and $F \subset Q_0$ a set of \textbf{frozen vertices} where any $i,j \in F$ have no arrows of $Q$ connecting them. By convention, we assume $Q_0\backslash F = [n]$ and $F = [n+1,m] := \{n+1, n+2, \ldots, m\}.$ We refer to elements of $Q_0\backslash F$ as \textbf{mutable vertices}. Any quiver $Q$ is regarded as an ice quiver by setting $F = \emptyset$.

If an ice quiver $(Q,F)$ is \textbf{2-acyclic} (i.e., $Q$ has no loops or 2-cycles), we can define a local transformation of it called \textbf{mutation}. The {\bf mutation} of $(Q,F)$ at a mutable vertex $k$, denoted $\mu_k$, produces a new ice quiver $(\mu_kQ,F)$ by the three step process:

(1) For every $2$-path $i \to k \to j$ in $Q$, adjoin a new arrow $i \to j$.

(2) Reverse the direction of all arrows incident to $k$ in $Q$.

(3) Remove any $2$-cycles, and remove any arrows that connect two frozen vertices.

From now on, we will only work with 2-acyclic quivers. We show an example of mutation below with the mutable (respectively, frozen) vertices in black (respectively, blue).

\vspace{-.1in}
\[
\begin{array}{c c c c c c c c c}
\raisebox{-.4in}{$(Q,F)$} & \raisebox{-.4in}{=} & {\begin{xy} 0;<1pt,0pt>:<0pt,-1pt>:: 
(0,30) *+{1} ="0",
(40,0) *+{2} ="1",
(80,30) *+{3} ="2",
(40,60) *+{\textcolor{blue}{4}} ="3",
"0", {\ar@<-.5ex>"1"},
"0", {\ar@<.5ex>"1"},
"1", {\ar"2"},
"3", {\ar"2"},
\end{xy}} & \raisebox{-.4in}{$\stackrel{\mu_2}{\longmapsto}$} & {\begin{xy} 0;<1pt,0pt>:<0pt,-1pt>:: 
(0,30) *+{1} ="0",
(40,0) *+{2} ="1",
(80,30) *+{3} ="2",
(40,60) *+{\textcolor{blue}{4}} ="3",
"2", {\ar"1"},
"0", {\ar@<-.5ex>"2"},
"0", {\ar@<.5ex>"2"},
"1", {\ar@<-.5ex>"0"},
"1", {\ar@<.5ex>"0"},
"3", {\ar"2"},
\end{xy}} & \raisebox{-.4in}{=} & \raisebox{-.4in}{$(\mu_2Q,F)$}
\end{array}
\]


Let Mut($(Q,F)$) denote the collection of ice quivers obtainable from $(Q,F)$ by finitely many mutations where such ice quivers are considered up to an isomorphism of quivers that fixes the frozen vertices. We refer to Mut($(Q,F)$) as the \textbf{mutation class} of $(Q,F)$. The following description of the mutation class of cluster type $\mathbb{A}_n$ quivers will be useful.



\begin{lem}\cite[Prop. 2.4]{bv08}\label{BV}
A connected quiver $Q$ with $n$ vertices is of cluster type $\mathbb{A}_n$ if and only if $Q$ satisfies the following:

$\begin{array}{rl}
      i) & \text{All non-trivial cycles in the underlying graph of $Q$ are oriented and of length 3.}\\
      ii) & \text{Any vertex has degree at most 4.}\\
      iii) & \text{If a vertex has degree 4, then two of its adjacent arrows belong to one 3-cycle, and the}\\ & \text{other two belong to another 3-cycle.}\\
      iv) & \text{If a vertex has degree 3, then two of its adjacent arrows belong to a 3-cycle, and the}\\ & \text{third arrow does not belong to any 3-cycle.}\
     \end{array}$ 
\end{lem}






The \textbf{framed quiver} of $Q$ is the ice quiver $\widehat{Q}$ where $\widehat{Q}_0 := Q_0 \sqcup [n+1, 2n]$, $F = [n+1, 2n]$, and $\widehat{Q}_1 := Q_1 \sqcup \{i \to n+i: i \in [n]\}$. A mutable vertex $i$ of some quiver $\overline{Q} \in \text{Mut}(\widehat{Q})$ is \textbf{green} (respectively, \textbf{red}) if there are no arrows in $\overline{Q}$ of the form $i \leftarrow n+j$ (respectively, $i \rightarrow n+j$) for some $j \in [n]$. Sign-coherence of $\textbf{c}$-vectors  \cite[Theorem 1.7]{dwz10} implies that any mutable vertex of a quiver $\overline{Q} \in \text{Mut}(\widehat{Q})$ is either green or red.


\begin{definition}[\cite{keller2011cluster}]\label{mgsdef}A \textbf{maximal green sequence} of $Q$ is a sequence $\textbf{i} = (i_1, \ldots, i_k)$ of mutable vertices of $\widehat{Q}$ where % \in [n]^{k}

$\begin{array}{rl}
i) & \text{for all $j \in [k]$ vertex $i_j \in [n]$ is green in $\mu_{i_{j-1}}\circ \cdots\circ \mu_{i_1}(\widehat{Q})$, and}\\ 
ii) & \text{each mutable vertex $i \in [n]$ of $\mu_{\textbf{i}}(\widehat{Q})$ is red where $\mu_{\textbf{i}} := \mu_{i_{k}}\circ \cdots\circ \mu_{i_1}$}.\end{array}$ 


\end{definition}

It is an open problem to determine what positive integers can be realized as lengths of maximal green sequences of a quiver $Q$. In \cite[Lemma 2.20]{bdp}, it is shown that if $Q$ is acyclic, then $Q$ has a maximal green sequence of length $\#Q_0$ and there are no shorter maximal green sequences of $Q$. The following theorem is the first to address this open problem for an infinite family of quivers, many of which are not acyclic.



\begin{thm}\label{thmtypeA}\cite{cormier2015minimal}
The length of a minimal length maximal green sequence of a cluster type $\mathbb{A}_n$ quiver $Q$ is $n + \#\{\text{3-cycles of $Q$}\}.$
\end{thm}


Our first main result (\cref{Cor:min_length_Qtilde}) shows that the problem of finding the minimal length of maximal green sequences of a quiver reduces to solving this problem for quivers $Q$ without any cluster type $\mathbb{A}$ quivers attached to $Q$ as in the following definition. The proof of \cref{Cor:min_length_Qtilde} uses properties of the \textbf{consistent scattering diagram} associated with quiver $Q$, which are proven in \cite{gross2014canonical}. We refrain from discussing scattering diagrams here. Instead, we emphasize the applications of \cref{Cor:min_length_Qtilde} in this paper.

\begin{definition}\label{Def:Q_tilde}
Given a quiver $Q$ let $\widetilde{Q}$ be a quiver composed of full connected subquivers $Q,$ $Q^1,$ $Q^2, \dots, Q^k$, such that all of the following conditions hold: 

$\begin{array}{rl}
\bullet & Q_0^i \cap Q_0 = \{x_i\},\\
\bullet & Q_0^i \cap Q_0^j = \Big\{ \begin{array}{ll} \{x_i\} & \text{if } x_i = x_j \\ \varnothing & \text{otherwise} \end{array}, \\
\bullet & \text{for every arrow in $\widetilde{Q}$, whenever one of the endpoints belongs to $Q_0^i \setminus \{x_i\}$ then the other}\\ & \text{endpoint belongs to $ Q_0^i$,}\\
\bullet & \text{for every $i$ the quiver $Q^i$ is of cluster type $\mathbb{A}$.}
\end{array}$

$$
\scalebox{.75}{\begin{tikzpicture}
[scale=.4][
> = stealth, % arrow head style
            shorten > = 4pt, ]
           
\node[draw=none, fill=none] at (-10,2) {$\widetilde{Q}$:};
\node[draw=none, fill=none] at (2,2) {$Q$};
\node[draw=none, fill=none] at (6.5,6) {$Q^1$};
\node[draw=none, fill=none] at (6,-2.5) {$Q^2$};
\node[draw=none, fill=none] at (-4,2) {$Q^k$};


\node[draw, shape=circle, fill=black, scale=0.4] at (-1,2) { };
\node[draw=none, fill=none] at (-1.5,3) {\small ${x_k}$};

\node[draw, shape=circle, fill=black, scale=0.4] at (3.8,4.4) { };
\node[draw=none, fill=none] at (3.3,5.3) {\small ${x_1}$};


\node[draw, shape=circle, fill=black, scale=0.4] at (3.6,-0.7) { };
\node[draw=none, fill=none] at (3,-1.6) {\small ${x_2}$};

\draw (2,2) circle (3cm);

\draw (-4,2) ellipse (3cm and 1cm);
\draw[rotate=30] (8.5,2) ellipse (3cm and 1cm);
\draw[rotate=-35] (6.3,1.5) ellipse (3cm and 1cm);
\draw[dotted][rotate=210]  (2,-1) arc (0:75:5cm);
\end{tikzpicture}}$$

\end{definition}



\begin{thm}\label{Cor:min_length_Qtilde}
Let $\widetilde{Q}$ be a quiver as in \cref{Def:Q_tilde}. Then the minimal length of a maximal green sequence of $\widetilde{Q}$ is 
$l_{\text{min}}+l_{min}^1+l_{min}^2+\dots+ l_{min}^k - k$
where $l_{min}^i$ (respectively, $l_{\text{min}}$) is the minimal length of a maximal green sequence for $Q^i$ (respectively, $Q$).  
\end{thm}




\section{Surface triangulations and shear coordinates}\label{Sec:triangulations}


Let $\textbf{S}$ denote an oriented Riemann surface, and let $\textbf{M} \subset \textbf{S}$ be a finite subset of $\textbf{S}$ where we require that for each component $\textbf{B}$ of $\partial \textbf{S}$ we have $\textbf{B}\cap \textbf{M} \neq \emptyset.$ We call the elements of $\textbf{M}$ \textbf{marked points} and the elements of $\textbf{M}\backslash(\textbf{M}\cap \partial \textbf{S})$ \textbf{punctures}. We call the pair $(\textbf{S},\textbf{M})$ a \textbf{marked surface}\footnote{We require that $(\textbf{S},\textbf{M})$ is not a sphere with one, two, or three punctures; a disc with one, two, or three marked points on the boundary; or a punctured disc with one marked point on the boundary.}. We henceforth fix a marked surface $(\textbf{S},\textbf{M}).$


We define an $\textbf{arc}$ on \textbf{S} to be a curve $\gamma$ in \textbf{S} such that 

$\begin{array}{rl} 
\bullet & \text{its endpoints are marked points;}\\
\bullet & \text{$\gamma$ does not intersect itself, except that its endpoints may coincide; }\\
\bullet & \text{except for the endpoints, $\gamma$ is disjoint from $\textbf{M}$ and from the boundary of $\textbf{S}$;}\\
\bullet & \text{$\gamma$ does not cut out an unpunctured monogon or and unpunctured digon. (In}\\ & \text{other words, $\gamma$ is not contractible into $\textbf{M}$ or onto the boundary of $\textbf{S}$.)}
\end{array}$ 

An arc $\gamma$ is considered up to isotopy relative to the endpoints of $\gamma$. 
We say two arcs $\gamma_1$ and $\gamma_2$ on \textbf{S} are \textbf{compatible} if they are isotopic relative to their endpoints to curves that are nonintersecting except possibly at their endpoints. A \textbf{triangulation} of $(\textbf{S},\textbf{M})$, denoted $\textbf{T}$, is defined to be a maximal collection of pairwise compatible arcs. 

One moves between triangulations by local moves called \textbf{flips}. Define the $\textbf{flip}$ of an arc $\gamma \in \textbf{T}$ as the unique arc $\gamma^\prime \neq \gamma$ that produces a triangulation $\textbf{T}^\prime = (\textbf{T}\backslash\{\gamma\}) \sqcup \{\gamma^\prime\}$ (see \cref{msflip}). If $\textbf{M}$ contains punctures, there exist triangulations containing \textbf{self-folded triangles} (e.g., the region of \textbf{S} bounded by $\gamma_3$ and $\gamma_4$ in \cref{taggedarcs} is a self-folded triangle). We refer to the arc $\gamma_3$ (respectively, $\gamma_4$) in \cref{taggedarcs} as a \textbf{loop} (respectively, a \textbf{radius}). 


\begin{figure}
\captionsetup{width=0.45\textwidth}
\centering
\begin{minipage}{.5\textwidth}
  \centering
  \def\svgwidth{120pt}
  $\begin{array}{rcl}\raisebox{-1in}{\includegraphics[scale=1.5]{MS1.pdf}} & \raisebox{-.25in}{$\longleftrightarrow$} & \raisebox{-1in}{\includegraphics[scale=1.5]{MS2.pdf}} \end{array}$
\caption{A flip connecting two triangulations of an annulus.}
  \label{msflip}
\end{minipage}%
\begin{minipage}{.5\textwidth}
\centering
$\begin{array}{rcl}\raisebox{-1in}{\includegraphics[scale=1.5]{MSquiv.pdf}} & \raisebox{-.25in}{$\rightsquigarrow$} & \begin{xy} 0;<1pt,0pt>:<0pt,-1pt>:: 
(0,40) *+{1} ="0",
(60,40) *+{2} ="1",
(30,0) *+{3} ="2",
"0", {\ar"2"},
"2", {\ar"1"},
"1", {\ar@<-.5ex>"0"},
"1", {\ar@<.5ex>"0"},
\end{xy}\end{array}$
\caption{The (cluster type $\widetilde{\mathbb{A}}_2$) quiver $Q_\textbf{T}$ defined by a triangulation \textbf{T}.}
\label{msquiv}
\end{minipage}
\vspace{-2ex}
\end{figure}




As the flip of a radius is not defined, \textbf{tagged arcs} were introduced in \cite{fomin2008cluster} to obtain such a notion. A \textbf{tagged arc} $\iota(\gamma)$ is obtained from an arc $\gamma$ that does not cut out a once-punctured monogon and ``tagging'' its ends either as $\textbf{plain}$ or $\textbf{notched}$ so that: \begin{itemize}\setlength\itemsep{-.25em} \item an end of $\gamma$ lying on the boundary of $\textbf{S}$ is tagged plain; and \item both ends of a loop have the same tagging. \end{itemize} We use the symbol $\bowtie$ to indicate that an end of an arc is notched. We say two tagged arcs $\iota(\gamma_1)$ and $\iota(\gamma_2)$ are \textbf{compatible} if the following hold: \begin{itemize} 
\setlength\itemsep{-.25em}
\item Their underlying arcs $\gamma_1$ and $\gamma_2$ are the same, and the tagged arcs $\iota(\gamma_1)$ and $\iota(\gamma_2)$ have the same tagging at exactly one endpoint. 
\item Their underlying arcs $\gamma_1$ and $\gamma_2$ are distinct and compatible, and any common endpoints of $\iota(\gamma_1)$ and $\iota(\gamma_2)$ have the same tagging. \end{itemize} 
A \textbf{tagged triangulation} of $(\textbf{S},\textbf{M})$ is a maximal collection of pairwise compatible tagged arcs.  It follows from the construction that any arc in a tagged triangulation can be flipped. For example, see \cref{taggedarcs} for an example of how to flip the radius $\gamma_4$.

\begin{figure}[h]
$${\includegraphics[scale=.8]{TaggedTriang_fpsac.pdf}}$$
\caption{The map identifying a triangulation of a punctured disk as a tagged triangulation of a punctured disk and the flip of the tagged arc $\gamma_4$ from that tagged triangulation.}
\label{taggedarcs}
\vspace{-2ex}
\end{figure}



Each triangulation $\textbf{T}$ of \textbf{S} defines a {\bf signed adjacency quiver} $Q_\textbf{T}$ by associating vertices to arcs and arrows based on oriented adjacencies (see \cref{msquiv}). More precisely, given a triangulation $\textbf{T}$ consider a map $\pi : \textbf{T} \to \textbf{T}$ on the set of arcs defined as follows.  If $\gamma$ is a radius of a self-folded triangle, then let $\pi(\gamma)$ be the corresponding loop, otherwise let $\pi(\gamma)=\gamma$.  Then the quiver $Q_\textbf{T}$ consists of vertices $i_{\gamma}$ for every $\gamma\in \textbf{T}$ and arrows $i_{\gamma}\to i_{\gamma'}$ for every non self-folded triangle with sides $\pi(\gamma)$ and $\pi(\gamma')$ such that $\pi(\gamma)$ follows $\pi(\gamma')$ in the clockwise order.  Finally, remove any 2-cycles from $Q_\textbf{T}$ to produce a 2-acyclic quiver\footnote{For the definition of $Q_{\textbf{T}}$ where $\textbf{T}$ is any tagged triangulation, we refer the reader to \cite{fomin2008cluster}.}. The following theorem shows that flips are compatible with mutations.

\begin{thm}\cite{fomin2008cluster}
Given a tagged triangulation ${\bf T}$ and a tagged arc $\gamma \in {\bf T}$, let ${\bf T}'$ be obtained from ${\bf T}$ by flipping $\gamma$. Then 
$\mu_{i_\gamma}(Q_{\bf T}) = Q_{\bf{T}'}.$
\end{thm}



We now show how shear coordinates provide a way to describe maximal green sequences geometrically. Recall that a fixed orientation $\mathcal{O}$ of a surface $\textbf{S}$ induces an orientation $\mathcal{O}$ on each component of $\partial \textbf{S}$ such that the surface $\textbf{S}$ lies to the right of every component. If $\gamma$ is a tagged arc in {\bf S}, the {\bf elementary lamination} $\ell_{\gamma}$ is a curve that runs along $\gamma$ within a small neighborhood of it. If $\gamma$ has an
endpoint $M$ on the boundary $\partial {\bf S}$, then $\ell_{\gamma}$ begins at a
point $M'\not\in {\bf M}$ on $\partial{\bf S}$ located near $M$ in the direction of $\mathcal{O}$, and proceeds along $\gamma$.  If $\gamma$ has an endpoint at a puncture $p$, then $\ell_{\gamma}$ spirals
into $p$: clockwise if $\gamma$ is notched at $p$, and counterclockwise if it is tagged plain (e.g., see \cref{fig:elem_lamination}).  A set of curves $L$ is a {\bf lamination} if it consists of elementary laminations arising from some pairwise compatible tagged arcs. 


\begin{figure}
\captionsetup{width=0.45\textwidth}
\centering
\begin{minipage}{.5\textwidth}
  \centering
  \resizebox{0.5\textwidth}{!}{
  \def\svgwidth{140pt}
  \input{elem_lamination.pdf_tex}
  }
  \caption{Lamination corresponding to the given set of arcs.}
   \label{fig:elem_lamination}
\end{minipage}%
\begin{minipage}{.5\textwidth}
\centering
\includegraphics[scale=1]{lamin.pdf}
\caption{The shear coordinate of $\gamma$ with respect to $\ell_1$ is $+1$, and the shear coordinate of $\delta$ with respect to $\ell_2$ is $-1$.}
\label{lamin_convention}
\end{minipage}
\vspace{-3ex}
\end{figure}



\begin{definition}\label{def:shearcoord}
Let $L$ be a lamination, and let ${\bf T}$ be a
triangulation. For each arc $\gamma \in {\bf T}$, that is not a radius of a self-folded triangle, the {\bf shear coordinate}\footnote{For the definition of shear coordinates where $\textbf{T}$ is any tagged triangulation, we refer the reader to \cite{fomin2012cluster}.} of $L$ with
respect to ${\bf T}$, denoted by $b_{\gamma}({\bf T}, L)$, is defined as a sum of contributions from all
intersections of curves in $L$ with $\gamma$. Specifically, such an intersection contributes +1
(respectively, $-1$) to $b_{\gamma}({\bf T}, L)$ if the corresponding segment of a curve in $L$ cuts through the
quadrilateral surrounding $\gamma$ as shown in \cref{lamin_convention}  on the left (respectively, right).  
\end{definition}


\begin{definition}\label{cvec}
Fix ${\bf T} = \{\gamma_1, \ldots, \gamma_n\}$, ${\bf T}'$ a tagged triangulation obtained from $\textbf{T}$ by a sequence of flips, and $\gamma^\prime \in {\bf T}'$. Define the \textbf{c-vector} of $\gamma^\prime$ to be the integer vector $\textbf{c}(\gamma^\prime, {\bf T}') := (b_{\gamma^\prime}({\bf T}', \ell_{\gamma_j})) \in \mathbb{Z}^{n}$. Define $\gamma^\prime$ to be \textbf{green} (respectively, \textbf{red}) if $b_{\gamma^\prime}({\bf T}', \ell_{\gamma_j}) \geq 0$ (respectively, $b_{\gamma^\prime}({\bf T}', \ell_{\gamma_j}) \leq 0$) for each $\gamma_j \in \textbf{T}$.
\end{definition}


By \cite[Proposition 17.3]{fomin2012cluster}, tagged triangulations and elements of $\text{Mut}(\widehat{Q}_\textbf{T})$ are in bijection. Thus the ice quiver $\overline{Q} \in \text{Mut}(\widehat{Q}_\textbf{T})$ corresponding to ${\bf T}'$ has $Q_{{\bf T}'}$ as its mutable part and its other arrows are given by the equations $b_{\gamma^\prime}({\bf T}', \ell_{\gamma_j}) = \#\{i_{\gamma^\prime} \rightarrow n+j\} - \#\{i_{\gamma^\prime} \leftarrow n+j\}$, one for each $\gamma^\prime \in {\bf T}'$ and $\gamma_j \in \textbf{T}.$ Moreover, tagged arc $\gamma^\prime \in {\bf T}'$ is green (respectively, red) if and only if vertex $i_{\gamma^\prime} \in \overline{Q}_0$ is green (respectively, red). Additionally, \cite{by14} implies that a maximal green sequence $\textbf{i} = (i_1, \ldots, i_k)$ of $Q_\textbf{T}$ is equivalent to the sequence of \textbf{c}-vectors $\textbf{c}(\textbf{i}) := (\textbf{c}(\gamma^{(1)}, \textbf{T}_1), \ldots, \textbf{c}(\gamma^{(k)}, \textbf{T}_k))$ where $\gamma^{(j)}$ is the tagged arc corresponding to vertex $i_j \in (\mu_{i_{j-1}}\cdots \mu_{i_1}(\widehat{Q}_{\textbf{T}}))_0$ and the mutable part of $\mu_{i_{j}}\cdots \mu_{i_1}(\widehat{Q}_{\textbf{T}})$ is $Q_{\textbf{T}_j}$. 




\begin{thm}\label{mullertheorem}
Let $Q^\dagger$ be a full subquiver of $Q_\textbf{T}$ whose vertices correspond to $\gamma_{j_1}, \ldots, \gamma_{j_\ell}$ and let $\textbf{i} = (i_1,\ldots, i_k)$ be a maximal green sequence of $Q_\textbf{T}$. Then the sequence $\textbf{c}^\dagger$ obtained by removing from $\textbf{c}(\textbf{i})$ every $\textbf{c}$-vector with nonzero entries at positions other than $j_1, \ldots, j_\ell$ is the sequence of \textbf{c}-vectors of a maximal green sequence of $Q^\dagger.$
\end{thm}

{The above is essentially proved in \cite{muller2016existence}, using the consistent scattering diagram of $Q_\textbf{T}$}. {Moreover, the same technique proves \cref{mullertheorem} for a general 2-acyclic quiver $Q$ with the $\textbf{c}$-vectors of $Q$ appropriately defined. This more general version of \cref{mullertheorem} is an ingredient in the proof of \cref{Cor:min_length_Qtilde}.}


\section{\text{Quivers of cluster type $\mathbb{D}_n$ and $\widetilde{\mathbb{A}}_{n-1}$}}\label{type_D_and_affineA}


To present our results, we need to recall the classification of cluster type $\mathbb{D}_n$ quivers. By \cite[Theorem 3.1]{vatne2010mutation}, the mutation class of a type $\mathbb{D}_n$ quiver where $n \ge 4$ consists of four families of quivers. Before presenting these, we say a vertex $c$ of $Q$ is a \textbf{connecting} vertex if it has degree at most 2, and if it has degree 2, it belongs to a 3-cycle of the same quiver.

A quiver $Q$ is of \textbf{Type I} (see \cref{type1}) if and only if $Q$ has the following properties: 
\vspace{-.1in}
\begin{itemize}
\setlength\itemsep{-.25em}
\item it has a full subquiver of the form $a \leftrightarrow c \leftrightarrow b$ (the notation $a \leftrightarrow c$ to indicates that there exists a single arrow in $Q$ connecting $a$ and $c$),
\item the full subquiver $Q^{1}$ of $Q$ on the vertices $Q_0\backslash\{a,b\}$ is of cluster type $\mathbb{A}$, and
\item the vertex $c$ is a connecting vertex of $Q^{1}$.
\end{itemize}


\begin{figure}
\captionsetup{width=0.45\textwidth}
\centering
\begin{minipage}{.5\textwidth}
  \centering
  \def\svgwidth{120pt}
  \scalebox{0.9}{
  \begin{tikzpicture}[scale=.35][
> = stealth, % arrow head style
            shorten > = 4pt, ]
\node[draw=none, fill=none] at (-5,0) {$Q$:};
\node[draw=none, fill=none] at (5,0) {$Q^{1}$};
\node[draw, shape=circle, fill=black, scale=0.4] at (2,0) { };
\node[draw=none, fill=none] at (2, .6) {$c$};
\node[draw, shape=circle, fill=black, scale=0.4] at (0,1.5) { };
\node[draw=none, fill=none] at (0,2) {$a$};
\node[draw, shape=circle, fill=black, scale=0.4] at (0,-1.5) { };
\node[draw=none, fill=none] at (0,-2.1) {$b$};
\draw [<->] (0.2,1.3) -- node { } (1.8,0.2);
\draw [<->] (0.2,-1.3) -- node { } (1.8,-0.2);
\draw (5,0) ellipse (3cm and 1cm);
\end{tikzpicture}
}
  \caption{Type I quivers}
   \label{type1}
\end{minipage}%
\begin{minipage}{.5\textwidth}
\centering
\scalebox{0.9}{
\begin{tikzpicture}[scale=.35][
> = stealth, % arrow head style
            shorten > = 4pt, ]         
\node[draw=none, fill=none] at (-11,0) {$Q$:};
\node[draw=none, fill=none] at (-5,0) {$Q^{1}$};
\node[draw=none, fill=none] at (5,0) {$Q^{2}$};
\node[draw, shape=circle, fill=black, scale=0.4] at (2,0) { };
\node[draw=none, fill=none] at (1.9, .7) {$d$};
\node[draw, shape=circle, fill=black, scale=0.4] at (-2,0) { };
\node[draw=none, fill=none] at (-2, .6) {$c$};
\node[draw, shape=circle, fill=black, scale=0.4] at (0,1.5) { };
\node[draw=none, fill=none] at (0,2) {$a$};
\node[draw, shape=circle, fill=black, scale=0.4] at (0,-1.5) { };
\node[draw=none, fill=none] at (0,-2.1) {$b$};
\draw [<-] (0.2,1.3) -- node { } (1.8,0.2);
\draw [<-] (0.2,-1.3) -- node { } (1.8,-0.2);
\draw [<-] (-1.8,0.2) -- node { } (-0.2,1.3);
\draw [<-] (-1.8,-0.2) -- node { } (-0.2,-1.3);
\draw [->] (-1.7,0) -- node { } (1.7,0);
\draw (5,0) ellipse (3cm and 1cm);
\draw (-5,0) ellipse (3cm and 1cm);
\end{tikzpicture}
}
\caption{Type II quivers}
\label{type2}
\end{minipage}
\end{figure}

A quiver $Q$ is of \textbf{Type II} (see \cref{type2}) if and only if $Q$ has the following properties:
\vspace{-.1in}
\begin{itemize}
\setlength\itemsep{-.25em}
\item it has a full subquiver of the form shown in \cref{type2} whose vertices are $a, b, c, d$,
\item the subquiver $Q^\prime$ obtained by removing vertices $a$ and $b$ and the arrow $c \to d$ consists of two cluster type $\mathbb{A}$ quivers $Q^{1}$ and $Q^{2}$, and
\item the vertex $c$ (respectively, $d$) is a connecting vertex of $Q^{1}$ (respectively, $Q^{2}$).
\end{itemize}



A quiver $Q$ is of \textbf{Type III} (see \cref{type3}) if and only if $Q$ has the following properties:
\vspace{-.1in}
\begin{itemize}
\setlength\itemsep{-.25em}
\item it has a full subquiver as shown in \cref{type3} whose vertices are $a,b,c,d$,
\item the full subquiver $Q^\prime$ of $Q$ on the vertices $Q_0\backslash\{a,b\}$ consists of two connected quivers $Q^{1}$ and $Q^{2}$, each of which is of cluster type $\mathbb{A}$, and
\item the vertex $c$ (respectively, $d$) is a connecting vertex of $Q^{1}$ (respectively, $Q^{2}$).
\end{itemize}


\begin{figure}
\captionsetup{width=0.45\textwidth}
\centering
\begin{minipage}{.5\textwidth}
  \centering
  \def\svgwidth{120pt}
  \scalebox{0.9}{
  \begin{tikzpicture}[scale=.35][
> = stealth, % arrow head style
            shorten > = 4pt, ]
\node[draw=none, fill=none] at (-11,0) {$Q$:};
\node[draw=none, fill=none] at (-5,0) {$Q^{1}$};
\node[draw=none, fill=none] at (5,0) {$Q^{2}$};
\node[draw, shape=circle, fill=black, scale=0.4] at (2,0) { };
\node[draw=none, fill=none] at (1.9, .7) {$d$};
\node[draw, shape=circle, fill=black, scale=0.4] at (-2,0) { };
\node[draw=none, fill=none] at (-2, .6) {$c$};
\node[draw, shape=circle, fill=black, scale=0.4] at (0,1.5) { };
\node[draw=none, fill=none] at (0,2) {$a$};
\node[draw, shape=circle, fill=black, scale=0.4] at (0,-1.5) { };
\node[draw=none, fill=none] at (0,-2.1) {$b$};
\draw [<-] (0.2,1.3) -- node { } (1.8,0.2);
\draw [->] (0.2,-1.3) -- node { } (1.8,-0.2);
\draw [<-] (-1.8,0.2) -- node { } (-0.2,1.3);
\draw [->] (-1.8,-0.2) -- node { } (-0.2,-1.3);
\draw (5,0) ellipse (3cm and 1cm);
\draw (-5,0) ellipse (3cm and 1cm);
\end{tikzpicture}
}
  \caption{Type III quivers}
   \label{type3}
\end{minipage}%
\begin{minipage}{.5\textwidth}
\centering
\scalebox{0.9}{
\begin{tikzpicture}[scale=.35][
> = stealth, % arrow head style
            shorten > = 4pt, ]
\node[draw=none, fill=none] at (-11,0) {$Q$:};
\node[draw=none, fill=none] at (0,4.5) {{\small $Q^{1}$}};
\node[draw=none, fill=none] at (7,2) {{\small $Q^{2}$}};
\node[draw=none, fill=none] at (-6.5,2) {{\small $Q^{k}$}};
\node[draw, shape=circle, fill=black, scale=0.4] at (2,0) { };
\node[draw=none, fill=none] at (2.3, .7) {$a_2$};
\node[draw, shape=circle, fill=black, scale=0.4] at (-2,0) { };
\node[draw=none, fill=none] at (-2.2, .6) {$a_1$};
\node[draw, shape=circle, fill=black, scale=0.4] at (0,1.5) { };
\node[draw=none, fill=none] at (1,1.5) {$b_1$};
\node[draw, shape=circle, fill=black, scale=0.4] at (4.5,-2.5) { };
\node[draw=none, fill=none] at (5,-3) {$a_3$};
\node[draw, shape=circle, fill=black, scale=0.4] at (4.5,0) { };
\node[draw=none, fill=none] at (5.2,-0.6) {$b_2$};
\node[draw, shape=circle, fill=black, scale=0.4] at (-4.5,-2.5) { };
\node[draw=none, fill=none] at (-5,-3) {$a_k$};
\node[draw, shape=circle, fill=black, scale=0.4] at (-4.5,0) { };
\node[draw=none, fill=none] at (-5,-0.6) {$b_k$};
\draw [<-] (0.2,1.3) -- node { } (1.8,0.2);
\draw [<-] (-1.8,0.2) -- node { } (-0.2,1.3);
\draw [->] (-1.7,0) -- node { } (1.7,0);
\draw [->] (4.2,0) -- node { } (2.2,0);
\draw [<-] (4.3,-2.3) -- node { } (2.2,-0.2);
\draw [<-] (4.5,-0.3) -- node { } (4.5,-2.2);
\draw [<-] (-4.3,0) -- node { } (-2.2,0);
\draw [->] (-4.3,-2.3) -- node { } (-2.2,-0.2);
\draw [->] (-4.5,-0.3) -- node { } (-4.5,-2.2);
\draw[rotate=90] (4, 0) ellipse (2.5cm and 1cm);
\draw[rotate=40] (6, -2.8) ellipse (2.5cm and 1cm);
\draw[rotate=-40] (-6, -2.8) ellipse (2.5cm and 1cm);
\draw[dotted][rotate=240]  (4.5,-2.5) arc (0:65:8.5cm);
\end{tikzpicture}
}
\caption{Type IV quivers}
\label{type4}
\end{minipage}
\end{figure}

A quiver $Q$ is of \textbf{Type IV} (see \cref{type3}) if and only if $Q$ has the following properties:
\vspace{-.1in}
\begin{itemize}
\setlength\itemsep{-.25em}
\item it has a full subquiver $R$ that is an oriented $k$-cycle where $k \ge 3$, $R_0 = \{a_1,a_2, \ldots, a_k\}$, and $R_1 = \{a_i \to a_{i+1}: i \in [k-1]\}\sqcup \{a_k\to a_1\}$,
\item for each arrow $\alpha \in R_1$, there may be a vertex $b_i \in Q_0\backslash R_0$ that is in a 3-cycle $b_i \to a_i \stackrel{\alpha}{\to} a_{i+1} \to b_i$, which is a full subquiver of $Q$, but there are no other vertices in $Q_0\backslash R_0$ that are connected to vertices of $R$,
\item the full subquiver $Q^\prime$ obtained from $Q$ by removing the vertices and arrows of the subquiver $R$ consists of the quivers $\{Q^{i}\}_{i \in [k]}$ some of which may be empty quivers, and where each quiver $Q^{i}$ is of cluster type $\mathbb{A}$ and has $b_i$ as a connecting vertex.
\end{itemize}


The cluster type $\widetilde{\mathbb{A}}_{n-1}$ quivers can be described in a very similar way to the Type IV quivers (see \cite{bastian2012mutation}). However, the only oriented cycles in cluster type $\widetilde{\mathbb{A}}_{n-1}$ quivers are 3-cycles. Another important distinction between $\mathbb{D}_n$ and $\widetilde{\mathbb{A}}_{n-1}$ quivers is that $\text{Mut}(\widehat{Q})$ is finite (respectively, infinite) if $Q$ is of cluster type $\mathbb{D}_n$ (respectively, $\widetilde{\mathbb{A}}_{n-1}$).

\begin{thm}\label{Thm:main}
Let $l$ denote the length of a minimal length maximal green sequence of a quiver $\widetilde{Q}$.

$\begin{array}{rl}
$i)$ & \text{If $\widetilde{Q}$ is of Type I or is of cluster type $\widetilde{\mathbb{A}}_{n-1}$, then $l = n + \#\{\text{3-cycles in } \widetilde{Q}\}$.}\\
$ii)$ & \text{If $\widetilde{Q}$ is of Type II, then $l = n + 1 + \#\{\text{3-cycles in } Q^{1}\} + \#\{\text{3-cycles in } Q^{2}\}.$}\\
$iii)$ & \text{If $\widetilde{Q}$ is of Type III, then $l = n + 2 + \#\{\text{3-cycles in } \widetilde{Q}\}.$}\\
$iv)$ & \text{If $\widetilde{Q}$ is of Type IV, then $l = n + k - 2 + \#\{a_i: \text{deg}(a_i) = 4\} + \sum_{i = 1}^{k}\#\{\text{3-cycles in } Q^{i}\}$.}
\end{array}$
\end{thm}

To prove \cref{Thm:main}, first, observe that $\widetilde{Q}$ satisfies \cref{Def:Q_tilde}. One thus applies \cref{Cor:min_length_Qtilde} to reduce the problem of calculating $l$ to calculating the minimal length of a maximal green sequence of $Q$, which is obtained by removing all vertices of $\widetilde{Q}$ belonging to $Q^i_0\backslash\widetilde{Q}_0$ for some $i$. If $\widetilde{Q}$ is of Type I, II, or III, the resulting family of quivers $Q$ consists of exactly six quivers. Thus, it is a finite calculation to verify the theorem. 

On the other hand, if $\widetilde{Q}$ is of Type IV or of cluster type $\widetilde{\mathbb{A}}_{n-1}$, the same family of quivers $Q$ is infinite. However, one can parameterize this family of quivers by triangulations of the once-punctured disk, in the former case, and triangulations of the unpunctured annulus, in the latter case, with certain conditions on the triangles. We then use the corresponding triangulation to construct a minimal length maximal green sequence of each quiver $Q$. We will sketch this approach when $Q$ is a Type IV quiver.

Let $(\textbf{S}, \textbf{M})$ be the once-punctured disk with $n$ marked points on the boundary and unique puncture $p$. A triangulation $\textbf{T}$ defining $Q$ is determined by the following properties (see the top left triangulation in \cref{fig:plot}): 

$\begin{array}{rl} \bullet & \text{if $\gamma$ is not connected to $p$, there is a triangle whose only internal arc is $\gamma$, and}\\ \bullet & \text{if $\gamma$ is connected to $p$, it is tagged plain at $p$.} \end{array}$ 

\noindent The triangulation \textbf{T} is the unique triangulation satisfying $Q_{\textbf{T}} = Q$, up to the action of the mapping class group of the surface and up to simultaneously changing the tagging of all ends of arcs connected to $p$.

The maximal green sequence we construct has five components, which we denote by $\textbf{i}_1$, $\textbf{i}_2$, $\textbf{i}_3$, $\textbf{i}_4$, and $\textbf{i}_5$. We construct the sequence in the context of \cref{fig:plot} wherein $\gamma_{a_s}, \gamma_{b_t} \in \textbf{T}$ are the arcs corresponding to vertices $a_s, b_t \in Q_\textbf{T}$, respectively. We do not justify it here, but the length of $\textbf{i}$ is $n + k - 2 + \#\{a_i: \text{deg}(a_i) = 4\}$\footnote{To show that this number is the minimal length of a maximal green sequence requires its own intricate argument. Our technique is to use the combinatorics of tagged triangulations to show that at least $n + k - 2 + \#\{a_i: \text{deg}(a_i) = 4\}$ tagged arcs must be flipped in any maximal green sequence.}. 

The first components, $\textbf{i}_1 = (a_7, b_6, a_4, b_3, b_2, a_2, b_1)$ and $\textbf{i}_2 = (a_3, a_5, a_6, a_1)$, perform flips at each of the arcs in the initial triangulation $\textbf{T}$ exactly once so that the total length of these two sequences is $n$. The vertices in $\textbf{i}_j$ with $j = 1, 2$ are ordered so that if two vertices in $\textbf{i}_j$ are connected by an arrow $\alpha \in (Q_\textbf{T})_1$, then $s(\alpha)$ is mutated before $t(\alpha)$ in $\textbf{i}_j$.

The sequence $\textbf{i}_3$ consists of all green vertices of $\mu_{\textbf{i}_2}\mu_{\textbf{i}_1}(\widehat{Q}_\textbf{T})$ where flipping the corresponding arcs produces ones that can be obtained from arcs of $\textbf{T}$ not connected to $p$ by moving their endpoints clockwise along the boundary to the next two marked points. As shown in \cref{fig:plot}, we have $\textbf{i}_3 = (i_\alpha)$. Moreover, the new arc $\alpha^\prime$ is obtained by moving the endpoints of $\gamma_{b_2}$ clockwise along the boundary to the next two marked points. 

The sequence $\textbf{i}_4$ consists of all vertices of $\mu_{\textbf{i}_3}\mu_{\textbf{i}_2}\mu_{\textbf{i}_1}(\widehat{Q}_\textbf{T})$ whose corresponding tagged arcs are connected to $p$ and is tagged plain at $p$. Similar to $\textbf{i}_2$, we also require that if two vertices in $\textbf{i}_4$ are connected by an arrow $\alpha \in \mu_{\textbf{i}_3}\mu_{\textbf{i}_2}\mu_{\textbf{i}_1}(\widehat{Q}_\textbf{T})$, then $s(\alpha)$ is mutated before $t(\alpha)$ in $\textbf{i}_4$. As shown in \cref{fig:plot}, we have $\textbf{i}_4 = (i_\delta)$. It turns out that the length of $\textbf{i}_3$ plus the length of $\textbf{i}_4$ is $\#\{a_i: \ \text{deg}(a_i) = 4\}$.

Lastly, the sequence $\textbf{i}_5$ is defined inductively. First, mutate at all green vertices of $\mu_{\textbf{i}_4}\mu_{\textbf{i}_3}\mu_{\textbf{i}_2}\mu_{\textbf{i}_1}(\widehat{Q}_\textbf{T})$ whose corresponding tagged arcs appear in a triangle whose other two sides are tagged arcs notched at $p$. In the context of \cref{fig:plot}, one mutates at the vertices corresponding to ${\sigma^{(1)}_1}$ and ${\sigma^{(1)}_2}$. Next, repeat this process, each time with the new ice quiver, until there are no remaining green vertices. In the example in \cref{fig:plot}, this process must be repeated twice: in the second (respectively, third) iteration one mutates at the vertices corresponding to $\sigma^{(2)}_1$ and $\sigma^{(2)}_2$ (respectively, $\sigma^{(3)}_1$). The length of $\textbf{i}_5$ is $k - 2$.


\begin{figure}
\centering
\includegraphics[scale=.9]{Punct_disk_1_fpsac.pdf}
\caption{The maximal green sequence $\textbf{i} = \textbf{i}_1\circ \textbf{i}_2\circ \textbf{i}_3\circ \textbf{i}_4\circ\textbf{i}_5$ for a Type IV $\mathbb{D}_{11}$ quiver.}
\label{fig:plot}
\end{figure}


\vspace{-.1in}
\acknowledgements{A. Garver received support from an RTG grant DMS-1148634, NSERC, and the Canada Research Chairs program. K. Serhiyenko was supported by the NSF Postdoctoral Fellowship MSPRF-1502881. {The authors are grateful to the referees for their careful comments.}}

\begingroup\emergencystretch=1em
\printbibliography
\endgroup



\end{document}
