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%% define your title in the usual way
\title[Geometry of $C$-Matrices]{Geometry of $C$-Matrices for Mutation-Infinite Quivers}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author[T. J. Ervin, B. Jackson, K. Lee, S. D. Nguyen]{Tucker J. Ervin\addressmark{1}\thanks{\href{mailto:tjervin@crimson.ua.edu}{tjervin@crimson.ua.edu}}, Blake Jackson\addressmark{2}\thanks{\href{mailto:blake.jackson@uconn.edu}{blake.jackson@uconn.edu}}, Kyungyong Lee\addressmark{1}\thanks{\href{mailto:klee94@ua.edu}{klee94@ua.edu}}, \and Son Dang Nguyen\addressmark{1}\thanks{\href{mailto:sdnguyen1@crimson.ua.edu}{sdnguyen1@crimson.ua.edu}}}

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\address{\addressmark{1}Department of Mathematics, University of Alabama, Tuscaloosa, AL \\ \addressmark{2}Department of Mathematics, University of Connecticut, Storrs, CT}

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%% put your English abstract here, or comment this out if you don't have one yet
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\abstract{
 The set of forks is  a class of quivers introduced by M. Warkentin, where every connected mutation-infinite quiver is mutation equivalent to infinitely many forks. Let $Q$ be a fork with $n$ vertices, and $\bw$ be a fork-preserving mutation sequence.  
We show that every $c$-vector of $Q$ obtained from $\bw$ is a solution to a quadratic equation of the form 
$$\sum_{i=1}^n x_i^2 + \sum_{1\leq i<j\leq n} \pm q_{ij} x_i x_j =1,$$
where $q_{ij}$ is the number of arrows between the vertices $i$ and $j$ in $Q$.
From the proof of this result, when $Q$ is a rank 3 mutation-cyclic quiver, every $c$-vector of $Q$ is a solution to a quadratic equation of the same form.
}

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%\resume{\lipsum[2]}

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\keywords{quivers, $c$-vectors, forks, quadratic equations}

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\begin{document}
\maketitle
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\section{Introduction}

The mutation of a quiver $Q$ was discovered by S. Fomin and A. Zelevinsky in their seminal paper \cite{fomin_cluster_2002} where they introduced cluster algebras. It also appeared in the context of Seiberg duality \cite{feng_toric_2002}. 
 The $c$-vectors (and $C$-matrices) of $Q$ were defined through mutations in further developments of the theory of cluster algebras \cite{fomin_cluster_2007}, and together with their companions, $g$-vectors (and $G$-matrices), played  fundamental roles in the study of cluster algebras (for instance, see \cite{derksen_quivers_2010, gross_canonical_2018, lin_two_2023, nakanishi_tropical_2012, plamondon_cluster_2011}). 
 When $Q$ is acyclic, positive $c$-vectors are actually real Schur roots, that is, the dimension vectors of indecomposable rigid modules over $Q$ \cite{chavez_c-vectors_2015, hubery_categorification_2016, speyer_acyclic_2013}. 
 Moreover, they appear as the denominator vectors of non-initial cluster variables of the cluster algebra associated to $Q$ \cite{caldero_triangulated_2006}. 

Due to the multifaceted appearance of $c$-vectors in important constructions, there have been various results related to the description of $c$-vectors (or real Schur roots) of an acyclic quiver \cite{baumeister_note_2014,hubery_categorification_2016,igusa_exceptional_2010,schofield_general_1992,seven_cluster_2014,speyer_acyclic_2013}.
In \cite{lee_geometric_2023}, K.-H. Lee and K. Lee conjectured  a correspondence between real Schur roots of an acyclic quiver  and  non-self-crossing curves on a Riemann surface and proposed a new combinatorial/geometric description.
The conjecture is now proven by A. Felikson and P. Tumarkin \cite{felikson_acyclic_2018} for acyclic quivers with multiple edges between every pair of vertices. Recently, S. D. Nguyen \cite{nguyen_proof_2022} proved the conjecture for an arbitrary acyclic (valued) quiver.

For a given (not necessarily acyclic) quiver $Q$, the set of quivers that are mutation equivalent to $Q$ is called the mutation equivalence class of $Q$ and denoted by $\text{Mut}(Q)$. The quiver $Q$ is said to be \emph{mutation-infinite} if $|\text{Mut}(Q)|$ is not finite, and \emph{mutation-finite} if $|\text{Mut}(Q)|<\infty$. The mutation-finite quivers are completely classified, and relatively well studied. On the other hand,  mutation-infinite quivers still await further investigations.

A reader-friendly version of our main theorem may be stated as follows.
\begin{Thm} \label{main-thm}
Let $n$ be any positive integer. Let $P$ be a mutation-infinite connected quiver with $n$ vertices. Then there exist an infinite number of pairs of a quiver $Q\in \text{Mut}(P)$ and $k\in\{1,...,n\}$ such that every $c$-vector of $Q$ obtained from any mutation sequence not starting with $k$  is a solution to a quadratic equation of the form 
\begin{equation}\label{quad}
\sum_{i=1}^n x_i^2 + \sum_{1\leq i<j\leq n} \pm q_{ij} x_i x_j =1,
\end{equation}
where $q_{ij}$ is the number of arrows between the vertices $i$ and $j$ in $Q$.
There does not seem to be a simple way of determining the exact signs of the $x_i x_j$ terms.
\end{Thm}

To state a more precise theorem, we need to recall the definition of forks.
An \emph{abundant quiver} is a quiver such that there are two or more arrows between every pair of vertices.

\begin{Def}{\cite[Definition 2.1]{warkentin_exchange_2014}} \label{def-forks}
A \textit{fork} is an abundant quiver $F$, where $F$ is not acyclic and where there exists a vertex $r$, called the point of return, such that
\begin{itemize}
    \item For all $i \in F^{-}(r)$ and $j \in F^{+}(r)$ we have $f_{ji} > f_{ir}$ and $f_{ji} > f_{rj}$, where $F^{-}(r)$ is the set of vertices with arrows pointing towards $r$ and $F^{+}(r)$ is the set of vertices with arrows coming from $r$.
    
    \item The full subquivers induced by $F^{-}(r)$ and $F^{+}(r)$ are acyclic.
\end{itemize}
An example of a fork is given by 
\[\begin{tikzcd}
    & r \\
    i & & j
    \arrow[from=2-1, to=1-2, "3"]
    \arrow[from=1-2, to=2-3, "4"]
    \arrow[from=2-3, to=2-1, "5"]
\end{tikzcd},\]
where $r$ is the point of return.
\end{Def}

It is known that "most" quivers in $\text{Mut}(Q)$ of any connected mutation-infinite quiver $Q$ are forks, as Theorem~\ref{thm-connected-fork} and Proposition~\ref{exgraph-connected-fork} imply. 

\begin{Thm}{\textup{\cite[Theorem 3.2]{warkentin_exchange_2014}}} \label{thm-connected-fork}
    A connected quiver is mutation-infinite if and only if it is mutation-equivalent to a fork.
\end{Thm}

\begin{Prop}{\textup{\cite[Proposition 5.2]{warkentin_exchange_2014}}} \label{exgraph-connected-fork}
Let $G$ be the exchange graph of a connected mutation-infinite quiver. A simple random walk on $G$ will almost surely leave the fork-less part and never come back.
\end{Prop}

%The exchange graph contains as its subgraph a tree  consisting of forks.
%\end{Thm}

A \emph{fork-preserving} mutation sequence is a reduced sequence of mutations that starts with a fork and does not mutate at its point of return.
A more precise version of our main theorem is as follows.

\begin{Thm} \label{thm-preserving-fork}
Let $Q$ be a fork, and let $\bw$ be a fork-preserving mutation sequence.  Every $c$-vector of $Q$ obtained from $\bw$  is a solution to a quadratic equation of the form \eqref{quad}.
\end{Thm}

A quiver Q is called \emph{mutation-acyclic} if it is mutation-equivalent to an acyclic quiver, else it is called \emph{mutation-cyclic}.
Notably, we have discovered a counterexample to Theorem \ref{thm-preserving-fork} for truly arbitrary mutation-sequences $\bw$ in the case of quivers on four vertices (to appear in the full version of this abstract \cite{ervin_geometry_nodate}), but the proof of the theorem provides a stronger corollary in the three vertex case. 
Ahmet Seven informed us that he had independently discovered this result.

\begin{Cor} \label{rank 3}
Let $Q$ be a mutation-cyclic quiver with 3 vertices. Then every $c$-vector of $Q$  is a solution to a quadratic equation of the form 
\eqref{quad} with $n=3$.
\end{Cor}

 As a byproduct of our proof, we also obtain the following theorem, which is closely related to a result of Fomin and Neville \cite[Lemma 6.14]{fomin_long_2023}.

\begin{Thm} \label{thm-base-case}
Let $\bw$ be a fork-preserving mutation sequence. 
The sign-vector (see Definition~\ref{C_sign}) of $C^\bw$ depends only on the signs of entries of initial exchange matrix $B$. 
In other words, the sign-vector is independent of the number of arrows between vertices of the initial quiver $Q$. 
\end{Thm}


\begin{Cor} \label{cor-reflection-triple}
 Let $n$ be any positive integer, and let $Q$ be a fork with $n$ vertices. 
For each fork-preserving mutation sequence $\bw$ from $Q$, the corresponding $n$-tuple of reflections $(r_1^\bw, r_2^\bw, \dots,  r_n^\bw)$ (see Definition~\ref{def-r}) depends only on the signs of entries of the initial exchange matrix $B$.
\end{Cor}

From this, we are able to prove that the product of reflections is equal to a Coxeter element. More precisely, we have the following.

\begin{Thm} \label{thm-coxeter-element}
Let $n$ be any positive integer, and let $Q$ be a fork with $n$ vertices. 
For each fork-preserving mutation sequence $\bw$ from $Q$, we have $$r_{\lambda(1)}^\bw ... r_{\lambda(n)}^\bw = r_{\rho(1)}...r_{\rho(n)}$$ for some permutations $\lambda, \rho \in\mathfrak{S}_n$, where $\mathfrak{S}_n$ is the symmetric group on $\{1,...,n\}$ and $r_1,...,r_n$ are the initial reflections, where $\lambda$ is determined by $\bw$ and $\rho$ is fixed by the first mutation of $\bw$.
\end{Thm}


\begin{Cor} \label{cor-non-self-crossing}
 Let $n$ be any positive integer, and let $Q$ be a fork with $n$ vertices. 
For each fork-preserving mutation sequence $\bw$ from $Q$, there exist pairwise non-crossing and non-self-crossing admissible curves $\eta_i^\bw$ (see Definition~\ref{def-adm})  such that $r_i^\bw = \nu(\eta_i^\bw)$ for every $i \in \{1,...,n\}$.
\end{Cor}

The above results are explored more thoroughly in our forthcoming paper \cite{ervin_geometry_nodate}, and they all rely heavily on our use of $l$-vectors and generalized intersection matrices.

\section{Preliminaries} \label{sec-preliminaries}

\subsection{C-matrices} Let $n$ be a positive integer. If $B = [b_{ij}]$ is an $n\times n$ skew-symmetric matrix,
then $B$ is in correspondence with a quiver $Q$ on $n$ vertices: if $b_{ij} > 0$ and $i \neq j$, then $Q$ has $b_{ij}$ arrows from vertex $i$ to vertex $j$.
The statements of some theorems have been formulated in terms of $Q$; however, we prefer to work with $B$ since the description of $c$-vectors is more clear in this setting.
Also, for a nonzero vector $c=( c_1 , \dots , c_n) \in \mathbb Z^n$, we write $c >0$ if all $c_i$ are non-negative, and  $c <0$ if all $c_i$ are non-positive. %Define $|c|=( |c_1|, \dots,  |c_n| )$, and set $\alpha_1=(1,0,\dots , 0), \ \alpha_2=(0,1,0, \dots, 0), \dots, \ \alpha_n=(0, \dots, 0, 1)$.

Assume that $M=[m_{ij}]$ is an $n \times 2n$ matrix with integer entries. Let $\mathcal I:= \{ 1, 2, \dots, n \}$ be the set of indices. For $\bw=[i_1, i_2, \dots , i_\ell]$, $i_j \in \mathcal I$, we define the matrix $M^\bw=[m_{ij}^\bw]$ inductively: the initial matrix is $M$ for $\bw=[\,]$, and assuming we have $M^\bw$, define the matrix $M^{\bw[k]}=[m_{ij}^{\bw[k]}]$ for $k \in \mathcal I$ with $\bw[k]:=[i_1, i_2, \dots , i_\ell, k]$
by  
\begin{equation} \label{eqn-mmuu} m_{ij}^{\bw[k]} = \begin{cases} -m_{ij}^\bw & \text{if  $i=k$ or $j=k$}, \\ m_{ij}^\bw + \mathrm{sgn}(m_{ik}^\bw) \, \max(m_{ik}^\bw m_{kj}^\bw,0) & \text{otherwise}, \end{cases}
\end{equation}
where  $\mathrm{sgn}(a) \in \{1, 0, -1\}$ is the signature of $a$. The matrix $M^{\bw[k]}$ is called the {\em mutation of $M^\bw$} at index (or label) $k$, $\bw$ and $\bw[k]$ are called {\em mutation sequences}, and $n$ is the {\em rank}.
%If the mutation sequence $[i_i, i_2, \dots , i_\ell]$ satisfies $i_j\neq i_{j+1}$ for all $j\in\{1,...,\ell-1\}$, then it is said to be \emph{reduced}.


Let $B$ be a $n\times n$ skew-symmetric matrix. 
Consider the $n\times 2n$ matrix $\begin{bmatrix}B&I\end{bmatrix}$ and
 a mutation sequence $\bw=[i_1, \dots ,i_\ell]$. 
 After the mutations at the indices $i_1, \dots , i_\ell$ consecutively, we obtain $\begin{bmatrix}B^\bw&C^\bw\end{bmatrix}$. 
 Write their entries as 
\begin{equation} \label{eqn-bcbc} B^\bw= \begin{bmatrix} b_{ij}^\bw \end{bmatrix}, \qquad
C^\bw= \begin{bmatrix} c_{ij}^\bw \end{bmatrix} = \begin{bmatrix} c_1^\bw \\ \vdots \\ c_n^\bw \end{bmatrix},\end{equation}
where $c_i^\bw$ are the row vectors.
\begin{Def}\label{C_sign}
The matrix $C^\bw$ is called a $C$-matrix of $B$ for any $w$
\footnote{This is slightly different from the original definition by Fomin and Zelevinsky}.
The row vectors $c_i^\bw$ are called {\em $c$-vectors} of $B$ for any $i$ and $\bw$. 
Each non-zero entry of $c_i^\bw$ will share the same sign \cite{derksen_quivers_2008}, allowing us to define the \emph{sign-vector} of $C^\bw$, where the $i$-th entry is 1 if $c_i^\bw > 0$ and $-1$ if $c_i^\bw < 0$.
\end{Def}


\subsection{Reflections and L-matrices}

In order to prove Theorem \ref{thm-preserving-fork}, we needed to study the $L$-matrices arising from reflections and a particular generalized intersection matrix associated to our exchange matrix.

\begin{Def} \label{def-gim-and-coxeter}
    A {\em generalized intersection matrix} (GIM) is a square matrix ${A}=[a_{ij}]$ with integral entries such that
    (1) for diagonal entries, $a_{ii}=2$;
    (2) $a_{ij}>0$ if and only if $a_{ji}>0$;
    (3) $a_{ij}<0$ if and only if $a_{ji}<0$.
    
    Let $\mathcal A$ be the (unital) $\mathbb Z$-algebra generated by $s_i, e_i$, $i=1,2, \dots, n$, subject to the following relations:
    $$ s_i^2=1, \quad \sum_{i=1}^n e_i =1, \quad s_ie_i = -e_i, \quad e_is_j=  \begin{cases} s_i+e_i-1 &\text{if } i =j, \\ e_i &\text{if } i \neq j, \end{cases}  \quad e_ie_j=  \begin{cases} e_i &\text{if } i =j, \\ 0 & \text{if } i \neq j. \end{cases}$$
    Let $\mathcal W$ be the subgroup of the units of $\mathcal{A}$ generated by $s_i$, $i=1, \dots, n$. Note that $\mathcal W$ is (isomorphic to) the universal Coxeter group. %Thus the algebra $\mathcal A$ can be considered as the algebra generated by the reflections and projections (?) of the universal Coxeter group.  
    An element $r\in \mathcal W$ is called a reflection if $r^2=1$. Let $\mathfrak{R}\subset \mathcal W$ be the set of reflections.
\end{Def}
From now on, let ${A}=[a_{ij}]$ be an $n\times n$ symmetric GIM.
Let $\Gamma = \sum_{i=1}^n \mathbb{Z}\alpha_i$ be the lattice generated by the formal symbols $\alpha_1,...,\alpha_n$. Define a representation $\pi:\mathcal A \rightarrow \mathrm{End}(\Gamma)$ by
\[ \pi(s_i)(\alpha_j) = \alpha_j - a_{ji} \alpha_i \quad \text{ and } \quad \pi(e_i)(\alpha_j) =\delta_{ij} \alpha_i , \quad\text{ for }i,j\in\{1,...,n\}. \] We suppress $\pi$ when we write the action of an element of $\mathcal A$ on $\Gamma$.


Given a skew-symmetric matrix $B$, for  each  linear ordering $\prec$ on $\{1,...,n\}$, we  define the associated GIM ${A}=[a_{ij}]$ by
\begin{equation} 
a_{ij}= \begin{cases}  b_{ij}  & \text{ if } i \prec j , \\
2 & \text{ if } i =j, \\ -b_{ij} & \text{ if } i \succ j . \end{cases}\ 
\label{eqn-gim}
\end{equation}
An ordering $\prec$ provides a certain way for us to regard the skew-symmetric matrix $B$ as acyclic even when it is not. 

\begin{Def} \label{def-r}
When $\bw=[\,]$, we let $r_i = s_i\in \mathfrak{R}$ for each  $i\in\{1,...,n\}$. For each  mutation sequence $\bw$ and each $i\in\{1,...,n\}$, define $r_i^\bw \in \mathfrak{R}$ inductively as follows: 
\begin{equation} \label{def-sx_i-1} r_i^{\bw [k]}= \begin{cases} r_k^\bw r_i^\bw  r_k^\bw & \text{ if } \ b_{ik}^\bw c_{k}^\bw>0, \\ r_i^\bw & \text{ otherwise.} \end{cases} \end{equation}
Clearly, each $r_i^\bw$ is written in the form
\[ r_i^\bw = g_i^\bw s_i {(g_i^\bw)}^{-1}, \quad g_i^\bw \in \mathcal W, \quad i\in\{1,...,n\}.\]
\end{Def}

\begin{Def}\label{def-ell}
Let $\sgn=\{1,-1\}$ be the group of order 2, and
consider the natural group action $\sgn\times \mathbb{Z}^n\longrightarrow \mathbb{Z}^n$, where we identify $\Gamma$ with $\mathbb Z^n$. 
Choose an ordering $\prec$ on $\{1,...,n\}$ to fix a GIM ${A}$, and define
\[   l_i^\bw = g_i^\bw (\alpha_i) \in \mathbb Z^n/\sgn, \qquad i\in\{1,...,n\}, \]
where we set $\alpha_1=(1,0,...,0), ..., \ \alpha_n=(0,..., 0, 1)$. Then the {\em $L$-matrix} $L^{\bw}$ associated to ${A}$ is defined to be the $n \times n$ matrix whose $i^\text{th}$ row is $l_i^\bw$ for $i\in\{1,...,n\}$, i.e., $L^{\bw} =  \begin{bmatrix} l_1^{\bw}  \\  \vdots \\ l_n^{\bw} \end{bmatrix}$, and the vectors $l_i^{\bw}$ are called the {\em $l$-vectors of ${A}$}. Note that the $L$-matrix and $l$-vectors associated to a GIM ${A}$ implicitly depend on the representation $\pi$ which is suppressed from the notation. 
\end{Def}

With the above machinery, we show the following, which further implies Theorem \ref{thm-preserving-fork}.

\begin{Thm} \label{thm-l-equals-c}
Let $Q$ be a fork with $n$ vertices, and let $\bw$ be a fork-preserving mutation sequence. For each $i\in\{1,...,n\}$,
 there exists a diagonal matrix $D_i^\bw$ such that $(D_i^\bw)^2=1$ and $l_i^{\bw}= c_i^{\bw} D_i^\bw$. In other words, the entries of $l$-vectors are equal to the entries of $c$-vectors up to sign.
\end{Thm}

\subsection{Geometry of reflections}
Here we review the definition of admissible curves \cite{lee_geometric_2023, lee_correspondence_2021}. 

Let $\mathcal{Q}$ be a fork with $n$ vertices labeled by $I:=\{1,...,n\}$ and point of return $r$. 
Let $\sigma$ be the linear ordering given by $r \prec a_{n-1} \prec a_{n-2} \prec \dots \prec a_1$, where $a_1, a_2, \dots, a_{n-1}$ are the vertices of $Q \setminus \{r\}$ and $a_i \prec a_j$ if and only if there is an arrow from $j$ to $i$. 


We define a labeled Riemann surface $\Sigma_{\sigma}$\footnote{The punctured discs appeared in Bessis' work \cite{bessis_dual_2006}.  For better visualization, here we prefer to use an alternative description using compact Riemann surfaces with one or two marked points.} as follows.
Let $G_1$ and $G_2$ be two identical copies of a regular $n$-gon. Label the edges of each of the two  $n$-gons
 by $T_{\sigma(1)}, \dots ,  T_{\sigma(n)}$ counter-clockwise.    Fix the orientation of every edge of $G_1$ (resp.  $G_2$) to be 
 counter-clockwise (resp. clockwise) as in the following picture. 
 \begin{center}
 \begin{tikzpicture}[scale=0.45]
\node at (2.4,-2.2){\tiny{$\sigma(n)$}};
\node at (1.5,2.6){\tiny{$\sigma(2)$}};
\node at (3.8,0){\tiny{$\sigma(1)$}};
\node at (-2.0,-2.7){\tiny{$\sigma(n-1)$}};
\node at (-2.0,2.5){\tiny{$\sigma(3)$}};
\node at (-3.2,0){\vdots};
\draw (0,0) +(30:3cm) -- +(90:3cm) -- +(150:3cm) -- +(210:3cm) --
+(270:3cm) -- +(330:3cm) -- cycle;
\draw [thick] (2.4,-0.2) -- (2.6,0)--(2.8,-0.2);
\draw [thick] (1.4,1.95) -- (1.3,2.25)--(1.6,2.25);  %arrow for sigma(2)
\draw [thick] (-1.0,2.2) -- (-1.3,2.2)--(-1.2,2.5);  %arrow for sigma(3)
\draw [thick] (-2.4,0.2) -- (-2.6,0)--(-2.8,0.2);   %arrow for ...
\draw [thick] (1.0,-2.2) -- (1.3,-2.2)--(1.2,-2.5);  %arrow for sigma(N)
\draw [thick] (-1.4,-1.95) -- (-1.3,-2.25)--(-1.6,-2.25);  %arrow for sigma(N-1)
%\draw [thick] (0,0)--(2.6,-1.5);  %L_1
%\node at (1.3,-1.1){\tiny{$L_1$}};
\end{tikzpicture}
\begin{tikzpicture}[scale=0.45]
%\node at (-1.3,1.2){\tiny{$L_2$}};
%\draw [thick] (0,0)--(-2.6,1.5);  %L_2
\node at (2.4,-2.2){\tiny{$\sigma(3)$}};
\node at (1.5,2.9){\tiny{$\sigma(n-1)$}};
\node at (3.3,0){\vdots};
\node at (-2.0,-2.7){\tiny{$\sigma(2)$}};
\node at (-2.0,2.5){\tiny{$\sigma(n)$}};
\draw (0,0) +(30:3cm) -- +(90:3cm) -- +(150:3cm) -- +(210:3cm) --
+(270:3cm) -- +(330:3cm) -- cycle;
\draw [thick] (-2.4,-0.2) -- (-2.6,0)--(-2.8,-0.2);
\draw [thick] (-1.4,1.95) -- (-1.3,2.25)--(-1.6,2.25);  %arrow for sigma(2)
\draw [thick] (1.0,2.2) -- (1.3,2.2)--(1.2,2.5);  %arrow for sigma(3)
\draw [thick] (2.4,0.2) -- (2.6,0)--(2.8,0.2);   %arrow for ...
\draw [thick] (-1.0,-2.2) -- (-1.3,-2.2)--(-1.2,-2.5);  %arrow for sigma(N)
\draw [thick] (1.4,-1.95) -- (1.3,-2.25)--(1.6,-2.25);  %arrow for sigma(N-1)
\end{tikzpicture}
 \end{center}
 
 
 Let $\Sigma_{\sigma}$ be the (compact) Riemann surface of genus $\lfloor \frac{n-1}{2}\rfloor$
obtained by gluing together the two $n$-gons with all the edges of the same label identified according 
to their orientations.  The edges of the $n$-gons become $N$ different curves in $\Sigma_\sigma$. If $n$ is odd, all the vertices of the two $n$-gons 
are identified to become one point in $\Sigma_\sigma$ and the curves obtained from the edges are loops. If $n$ is even, two distinct
 vertices are shared by all curves. Let $\mathcal{T}$ be the set of all curves, i.e., $\mathcal{T}={T}_1\cup\cdots\cup{T}_n\subset \Sigma_\sigma$, and $V$ be the set of the vertex (or vertices) on $\mathcal{T}$.  
\iffalse

  Let $\mathfrak W$ be the set of words $\mathfrak w=i_1i_2 \cdots i_k$ from the alphabet $I$ such that no two consecutive letters $i_p$ and $i_{p+1}$ are the same.   For each element $w\in W$, let $R_w\subset \mathfrak W$ be the set of words $i_1i_2 \cdots i_k$ such that $w=s_{i_1}s_{i_2} \cdots s_{i_k}$. Recall that the set of positive real roots and the set of reflections in $W$ are in one-to-one correspondence. Also note that if $\mathcal{Q}$ is 2-complete, i.e., $W=\langle s_1,...,s_N\ : \ s_1^2=\cdots=s_N^2=e \rangle$, then there is a unique expression for every element $w\in W$ 
as a product of simple reflections (with no two consecutive simple reflections being the same), hence $R_w$ contains a unique element.  Define $$
R:=\bigcup_{\tiny{\begin{array}{c}w\in W\\w:\text{reflection}\end{array}}}R_w\subset \mathfrak W.
$$  

\begin{Def} \label{def-adm} Let $\sigma\in P_{\mathcal{Q}}$.
A $\sigma$-\emph{admissible} curve is a continuous function $\eta:[0,1]\longrightarrow \Sigma_{\sigma}$ such that

1) $\eta(x)\in V$ if and only if  $x\in\{0,1\}$;

2) there exists $\epsilon>0$ such that $\eta([0,\epsilon])\subset L_1$ and $\eta([1-\epsilon,1])\subset L_2$;

3) if $\eta(x)\in \mathcal{T}\setminus V$ then $\eta([x-\epsilon,x+\epsilon])$ meets $\mathcal{T}$ transversally for sufficiently small $\epsilon>0$;

4) and   $\upsilon(\eta)\in R$, where $\upsilon(\eta):={i_1}\cdots {i_k}$ is given by 
$$\{x\in(0,1) \ : \ \eta(x)\in \mathcal{T}\}=\{x_1<\cdots<x_k\}\quad \text{ and }\quad \eta(x_\ell)\in T_{i_\ell}\text{ for }\ell\in\{1,...,k\}.$$ 

If $\sigma$ is clear from the context, a $\sigma$-admissible curve will be just called an admissible curve.
\end{Def}
\fi

For simplicity, here we give a precise definition of an admissible curve for rank 3 quivers only, but it is straightforward to generalize to quivers of higher rank.
For our geometric model on rank 3 quivers, we consider the (triangulated) torus with one marked point along with admissible curves (see Definition \ref{def-adm}). The key point here is that there is  a map from the set of admissible curves to $\mathfrak R$.

For each $\sigma\in \mathfrak{S}_3$, let $\Sigma_\sigma$ be the closed Riemann surface of genus $1$ with a single marked point $V$, and let $\widetilde{\Sigma_{\sigma}}$ be the universal cover of $\Sigma_\sigma$, which can be regarded as $\mathbf{R}^2$. 
Let $\alpha=\sigma(1)$, $\beta=\sigma(2)$, and $\gamma=\sigma(3)$. Fix three arcs $T_\alpha, T_\beta,$ and $T_\gamma$ on $\Sigma_\sigma$ and the projection $p:\widetilde{\Sigma_{\sigma}}\longrightarrow \Sigma_{\sigma}$ such that  $p^{-1}(T_\alpha)=\mathbf{Z}\times \mathbf{R}\subset \mathbf{R}^2$, $p^{-1}(T_\beta)=\{(x,y) \, : \, x+y\in \mathbf{Z}\}\subset \mathbf{R}^2$, $p^{-1}(T_\gamma)=\mathbf{R}\times \mathbf{Z}\subset \mathbf{R}^2$, and $p^{-1}(V)=\mathbf{Z}^2\subset \mathbf{R}^2$.
Hence $T_\alpha$ is the vertical line segment, $T_\beta$ is the diagonal, and $T_\gamma$ is the horizontal line segment. 
Let ${T}={T}_1\cup T_2 \cup T_3$. See Figure 1.

\begin{figure}\label{Figure-1}
\centering
\begin{tikzpicture}[scale=0.7mm]
\draw [help lines] (0,0) grid (3,2);
\draw [help lines] (0,1)--(1,0);
\draw [help lines] (0,2)--(2,0);
\draw [help lines] (3,0)--(1,2);
\draw [help lines] (3,1)--(2,2);
\foreach \i in {0, ..., 3}
    \foreach \j in {0, ..., 2}{
        \draw (\i,\j)node{$\bullet$};}
\draw[thick, red] (1,1)node[below left]{$V$};
%\draw [thick, green] (3.5-1, 2.5-1)node{$\bullet$}; %R1
%\draw [thick, blue] (2.1,0.95) -- (5.9,4.05); %R3
%\draw [thin, blue] (1,0) -- (7,5); %R3
%\draw [thick, red] (2.1+1+3,1.85+2) -- (4.9+1+3,4.15+2); %R2
%\draw [thick,red] (1,1)--(1.1,1.1);  %endpoint
%\draw [thick,red] (6,4)--(5.9,3.9);  %endpoint
%\draw [thick,red] (1.36,1.22)--(6-0.36,4-0.22);  %mid-line
%\draw[thick,red,scale=0.5,domain=0.75:13.12,smooth,variable=\t]
%plot ({(0.2+0.05*\t)*cos(\t r)+2},{(0.2+0.05*\t)*sin(\t r)+2});
%\draw[thick,red,scale=0.5,domain=3.89:16.26,smooth,variable=\t]
%plot ({(0.03+0.05*\t)*cos(\t r)+12},{(0.03+0.05*\t)*sin(\t r)+8});
\draw [thick, red] (1.5,1.1)node{$T_\gamma$};
\draw [thick, red] (1.6,1.6)node{$T_\beta$};
\draw [thick, red] (1.1,1.5)node{$T_\alpha$};
\end{tikzpicture}
\caption{This picture illustrates a portion of the universal cover $\Sigma_\sigma$, and the three arcs $T_\alpha, T_\beta,$ and $T_\gamma$.}
\end{figure}


%\begin{Def} 
%Define the function $\mathfrak{z}: T\setminus\{V\} \longrightarrow \{s_i : i \in [1,3]\}$ as follows: if $x\in T_i$ then $\mathfrak{z}(x)=s_i$.
%\end{Def}


\begin{Def} \label{def-adm} 
An \emph{admissible curve} is a pair consisting of a continuous function $\eta:[0,1]\longrightarrow \Sigma_\sigma$ and a sequence $\{i_\ell\}_{\ell = 1}^k$ of entries with in $i_\ell\in\{1,2,3\}$ such that

1) $\eta(x)= V$ if and only if  $x\in\{0,1\}$;

\iffalse
2) there exists $\epsilon>0$ such that $\eta([0,\epsilon])\subset L_1$ and $\eta([1-\epsilon,1])\subset L_2$;
\fi

2) if $\eta(x)\in {T}\setminus \{V\}$ then $\eta([x-\epsilon,x+\epsilon])$ meets ${T}$ transversally for sufficiently small $\epsilon>0$;

3) $\eta(x_\ell)\in T_{i_\ell}$ and $\ell\in\{1,...,k\},$ where
$$\{x_1<\cdots<x_k\} = \{x\in(0,1) \ : \ \eta(x)\in {T}\}$$

4) $\upsilon(\eta)\in \mathfrak R$, where $\upsilon(\eta):=r_{i_1}\cdots r_{i_k}\in \mathcal W$.
\end{Def}
\begin{Exa}In Example~\ref{examp_curve}, when $\bw=[1,2,3]$, the admissible curve
     $\eta_2^w$  has  $$\upsilon(\eta_2^w)=r_2r_1r_3r_1r_2r_1r_3r_1r_2.$$
Note that $\eta_2^w$ crosses $T_2$, $T_1$, $T_3$, $T_1$, $T_2$, $T_1$, $T_3$, $T_1$, $T_2$ in this order.
\end{Exa}

\section{Examples}\label{ex_sec_first}
In this section, we will consider the following two quivers to demonstrate our theorems: 
    \[P = \begin{tikzcd}
    & 2\\
    1 & & 3
    \arrow[from=2-1, to=1-2, "3"]
    \arrow[from=1-2, to=2-3, "3"]
    \arrow[from=2-3, to=2-1, "6"]
    \end{tikzcd}
    \text{ and  }\quad
    Q = \begin{tikzcd}
    & 2\\
    1 & & 3
    \arrow[from=2-1, to=1-2, "3"]
    \arrow[from=1-2, to=2-3, "4"]
    \arrow[from=2-3, to=2-1, "5"]
    \end{tikzcd}\]
Both quivers are mutation-cyclic \cite{beineke_cluster-cyclic_2011}.
Also, $P$ and $Q$ are forks and are mutation-equivalent to only forks. 
In this section, we will consider the c-vectors of both $P$ and $Q$ under three mutation sequences, namely, $\bw=[1]$,  $\bw=[1,2]$, and $\bw=[1,2,3]$.




\begin{Exa} \label{exa-first-theorem}
    An example of Theorem~\ref{thm-base-case} is given in the table below:   
    \begin{center}
    \resizebox{\columnwidth}{!}{
    \begin{tabular}{|c | c | c|}
    \hline
         Mutation Sequence & $[B^\bw| C^\bw]$-matrix for $P$ & $[B^\bw| C^\bw]$-matrix for $Q$  \\
         \hline
         $\bw = [1]$     & $\begin{bmatrix}[ccc|ccc] 
    0 & -3 & 6  & -1 & 0 & 0 \\
    3 & 0 & -15   & 0 & 1 & 0\\
    -6 & 15 & 0   & 6 & 0 & 1
    \end{bmatrix}$&  $\begin{bmatrix}[ccc|ccc] 
    0 & -3 & 5  & -1 & 0 & 0 \\
    3 & 0 & -11   & 0 & 1 & 0\\
    -5 & 11 & 0   & 5 & 0 & 1
    \end{bmatrix}$\\
    \hline
         $\bw=[1,2]$   & $\begin{bmatrix}[ccc|ccc] 
    0 &  3 & -39  & -1 & 0 & 0 \\
    -3 & 0 & 15   & 0 & -1 & 0\\
    39 & -15 & 0   & 6 & 15 & 1
    \end{bmatrix}$&  $\begin{bmatrix}[ccc|ccc]  
    0 &  3 & -28  & -1 & 0 & 0 \\
    -3 & 0 & 11   & 0 & -1 & 0\\
    28 & -11 & 0   & 5 & 11 & 1
    \end{bmatrix}$\\
    \hline
         $\bw=[1,2,3]$& $\begin{bmatrix}[ccc|ccc] 
    0 &  -582 & 39  & -1 & 0 & 0 \\
    582 & 0 & -15   & 90 & 224 & 15\\
    -39 & 15 & 0   & -6 & -15 & -1
    \end{bmatrix}$ &  $\begin{bmatrix}[ccc|ccc] 
    0 & - 305 & 28  & -1 & 0 & 0 \\
    305 & 0 & -11   & 55 & 120& 11\\
    -28 & 11 & 0   & -5 & -11 & -1
    \end{bmatrix}$\\
    \hline
    \end{tabular}}
    \end{center}

    For each quiver, the sign vector of the $C$-matrix for $\bw=[1]$, $\bw=[1,2]$, and $\bw=[1,2,3]$ is $(-1,1,1)$, $(-1,-1,1)$, and $(-1,1,-1)$).
    
\end{Exa}

\begin{Exa}
    The quadratic equation for the quiver $P$ is given by
    $$x^2 + y^2 + z^2 - 3xy - 6xz + 3yz = 1.$$

    and the quadratic equation for $Q$ is given by
    $$x^2 + y^2 + z^2 - 3xy - 5xz + 4yz = 1.$$

    It is easy to verify that the c-vectors 
    $$(x,y,z) = (90,224,15) \text{ and } (x,y,z) = (-6,-15,-1)$$
    both satisfy the quadratic equation for $P$ and that the c-vectors 
    $$(x,y,z) = (55,120,11) \text{ and } (x,y,z) = (-5,-11,-1)$$
    both satisfy the quadratic equation for $Q$.
\end{Exa}


\begin{Exa} \label{exa-second-corollary}
    In this example, we demonstrate Corollary~\ref{cor-reflection-triple}.
    If we mutate the reflections for both of $P$ and $Q$ with $\bw = [1]$, then we arrive at 
    $$\aligned 
    r^\bw_1&=r_1, \quad
    r^\bw_2=r_2, \quad
    r^\bw_3=r_1r_3r_1.
\endaligned$$
        If we mutate both of them with $\bw=[1,2]$, then we arrive at 
      $$\aligned 
    r^\bw_1&=r_1, \quad
    r^\bw_2=r_2, \quad
    r^\bw_3=r_2r_1r_3r_1r_2.
\endaligned$$   
    If we mutate both of them with $\bw=[1,2,3]$, then we arrive at
    $$\aligned 
    r^\bw_1&=r_1, \quad
    r^\bw_2=r_2r_1r_3r_1r_2r_1r_3r_1r_2, \quad
    r^\bw_3=r_2r_1r_3r_1r_2.
\endaligned$$

We can see that both of these are fork-preserving mutation sequences with the same initial orientation for the $B$ matrix. 
%As our theorem predicts \beb cite which theorem \eb, both of the mutations of reflections are equal for $P$ and $Q$.
\end{Exa}



\begin{Exa} \label{exa-fourth-theorem}
    In this example, we demonstrate Theorem~\ref{thm-coxeter-element}.
    If we take the three mutated reflections from Example~\ref{exa-second-corollary} for $\bw=[1]$, then 
    $$r_1^\bw r_3^\bw r_2^\bw = r_3 r_1 r_2.$$
    For $\bw=[1,2]$, we have 
    $$r_1^\bw r_2^\bw r_3^\bw = r_3 r_1 r_2.$$
    Finally, for $\bw=[1,2,3]$, we have
    $$r_1^\bw r_3^\bw r_2^\bw = r_3 r_1 r_2.$$
\end{Exa}


\begin{Exa}\label{examp_curve}
In this example, we demonstrate Corollary~\ref{cor-non-self-crossing}.
If we take the three mutated reflections from Example~\ref{exa-second-corollary} for $\bw=[1]$, then we get the following admissible curves: 
%%%% PICTURE
\begin{center}
\begin{tikzpicture}[scale=0.5mm]
\draw [help lines] (-1,-1) grid (3,2);
\draw [help lines] (0,-1)--(-1,0);
\draw [help lines] (1,-1)--(-1,1);
\draw [help lines] (2,-1)--(-1,2);
\draw [help lines] (3,-1)--(0,2);
\draw [help lines] (3,0)--(1,2);
\draw [help lines] (3,1)--(2,2);
%\draw [thick,red] (1,1)--(1.1,1.1);  %endpoint
%\draw [thick,red] (6,4)--(5.9,3.9);  %endpoint
\draw [thick,red] (0,0)--(1,1);  % r_1
\draw [thick, red] (1,0).. controls (1.5,0.7) and (0.5,0.3) ..(1,1); % r_2
\draw [thick,red] (0.2,-0.1)--(0.8,0.1);  % r_3
\draw [thick, red] (0.8,0.1).. controls (0.95,0.2) .. (1,0); % r_3 spiral top
\draw [thick, red] (0.2,-0.1).. controls (0.05,-0.2) .. (0,0); % r_3 spiral bottom
%labels
\draw [thick, red] (0.5,0.75)node{$\eta_1^w$};
\draw [thick, red] (1.3,0.3)node{$\eta_2^w$};
\draw [thick, red] (0.5,-0.2)node{$\eta_3^w$};
%\triangulation labels
\draw [thick, red] (2.5,0.1)node{$T_3$};
\draw [thick, red] (2.6,0.6)node{$T_1$};
\draw [thick, red] (2.1,0.5)node{$T_2$};
\end{tikzpicture}
\end{center}
For $\bw=[1,2]$ and $\bw=[1,2,3]$, we get the following admissible curves respectively:
%%%% PICTURE
\begin{center}
\begin{tikzpicture}[scale=0.5mm]
\draw [help lines] (-1,-1) grid (2,2);
\draw [help lines] (0,-1)--(-1,0);
\draw [help lines] (1,-1)--(-1,1);
\draw [help lines] (2,-1)--(-1,2);
\draw [help lines] (0,2)--(2,0);
\draw [help lines] (1,2)--(2,1);
%\draw [thick,red] (1,1)--(1.1,1.1);  %endpoint
%\draw [thick,red] (6,4)--(5.9,3.9);  %endpoint
\draw [thick,red] (0,0)--(1,1);  % r_1
\draw [thick, red] (1,0).. controls (1.5,0.7) and (0.5,0.3) ..(1,1); % r_2
\draw [thick,red] (0.2,-0.1)--(0.8,0.1);  % r_3
\draw [thick, red] (0.8,0.1).. controls (0.8,0.2) and (1.2,0.2) .. (1,0); % r_3 spiral top
\draw [thick, red] (0.2,-0.1).. controls (0.2,-0.2) and (-0.3,-0.3) .. (0,0); % r_3 spiral bottom
%labels
\draw [thick, red] (0.5,0.75)node{$\eta_1^w$};
\draw [thick, red] (1.3,0.3)node{$\eta_2^w$};
\draw [thick, red] (0.5,-0.2)node{$\eta_3^w$};
\end{tikzpicture}
\begin{tikzpicture}[scale=0.5mm]
\draw [help lines] (-1,-1) grid (3,2);
\draw [help lines] (0,-1)--(-1,0);
\draw [help lines] (1,-1)--(-1,1);
\draw [help lines] (2,-1)--(-1,2);
\draw [help lines] (0,2)--(3,-1);
\draw [help lines] (1,2)--(3,0);
\draw [help lines] (2,2)--(3,1);
%\draw [thick,red] (1,1)--(1.1,1.1);  %endpoint
%\draw [thick,red] (6,4)--(5.9,3.9);  %endpoint
\draw [thick,red] (0,0)--(1,1);  % r_1
\draw [thick,red] (0.2,0.1)--(1.8,0.9);  % r_2
\draw [thick, red] (1.8,0.9).. controls (1.8,1.15) and (2.25,1.25) .. (2,1); % r_2 spiral top
\draw [thick, red] (0.2,0.1).. controls (0.2,-0.15) and (-0.25,-0.25) .. (0,0); % r_2 spiral bottom
\draw [thick,red] (1.2,0.9)--(1.8,1.1);  % r_3
\draw [thick, red] (1.8,1.1).. controls (1.8,1.2) and (2.3,1.3) .. (2,1); % r_3 spiral top
\draw [thick, red] (1.2,0.9).. controls (1.2,0.8) and (0.7,0.7) .. (1,1); % r_3 spiral bottom
%labels
\draw [thick, red] (0.5,0.75)node{$\eta_1^w$};
\draw [thick, red] (1.2,0.3)node{$\eta_2^w$};
\draw [thick, red] (1.5,1.2)node{$\eta_3^w$};
%\triangulation labels
\end{tikzpicture}
\end{center}

Note that these curves are pairwise non-crossing as well as non-self-crossing.
Also, using the labeling of the 3 arcs from the picture for the first set of non-crossing curves, we can recover the sequence of reflections from the curves in each picture and confirm the correspondence.

\end{Exa}


\begin{Exa} \label{exa-third-theorem}
    To demonstrate how to calculate $l$-vectors, we consider $l_2^\bw$ for the quiver $Q$ with $\bw = [1,2,3]$ and linear ordering $2 \prec 1 \prec 3$.
    First, construct the GIM
    $$A = \begin{bmatrix}
        2 & -3 & -5 \\
        -3 & 2 & 4 \\
        -5 & 4 & 2
    \end{bmatrix}.$$
    Then consider the following matrices in $M_{3\times 3}(\mathbb{Z})$.
    $$S_1 = \begin{bmatrix}
    -1 & 0 &  0 \\
    3  & 1 &  0 \\
    5  & 0 &  1
    \end{bmatrix},\quad \quad 
    S_2 = \begin{bmatrix}
    1 & 3 &  0 \\
    0  & -1 &  0 \\
    0  & -4 &  1
    \end{bmatrix},\quad \quad  
    S_3 = \begin{bmatrix}
    1 & 0 &  5 \\
    0  & 1 &  -4 \\
    0  & 0 &  -1
    \end{bmatrix}.$$
    Using the sequence of reflections from Example~\ref{exa-second-corollary} and the definition of $l$-vectors, we know that
    $$\aligned 
    l_2^\bw &= s_2s_1s_3s_1(\alpha_2)\\
    &= (S_2^TS_1^TS_3^TS_1^T (\alpha_2^T) )^T\\
    &= \alpha_2 S_1 S_3 S_1 S_2 \\
    &= (\alpha_2 ) S_1 S_3 S_1 S_2\\
    &= (3\alpha_1 + \alpha_2 ) S_3 S_1 S_2 \\
    &= (3\alpha_1 + \alpha_2 + 11 \alpha_3) S_1 S_2\\
    &= (55\alpha_1 + \alpha_2 + 11 \alpha_3) S_2\\
    &= 55\alpha_1 +120\alpha_2  + 11 \alpha_3.
    \endaligned$$


  \iffalse    
    If we look at $l_2^\bw$ for the quiver $Q$ with $\bw = [1,2,3]$ and linear ordering $2 \prec 1 \prec 3$.
    First, construct the GIM:
    $$A = \begin{bmatrix}
        2 & 2 & -3 \\
        2 & 2 & -3 \\
        -3& -3 & 2
    \end{bmatrix}.$$

  
    From Example \ref{exa-second-corollary}, we know that
    $$l_2^\bw = s_2s_1s_3s_1(\alpha_2)$$
    $$= s_2s_1s_3(- 2\alpha_1 +\alpha_2 )$$
    $$= s_2s_1(- 2\alpha_1 +\alpha_2  - 3 \alpha_3)$$
    $$= s_2(- 9\alpha_1 +\alpha_2  - 3 \alpha_3)$$
    $$= - 9\alpha_1 + 8\alpha_2  - 3 \alpha_3$$
\fi
    
    These calculations can then be used to demonstrate Theorem~\ref{thm-l-equals-c}. Compare the table below with the one given in Example~\ref{exa-first-theorem}.
    
    \begin{center}  
    \begin{tabular}{|c | c | c|}
    \hline
         Mutation Sequence & $L$-matrix for $P$ & $L$-matrix for $Q$  \\
         \hline
         $\bw = [1]$     & $\begin{bmatrix}
    1 & 0 &  0 \\
     0 &   1 &  0 \\
    6 & 0 &  1
    \end{bmatrix}$ & $\begin{bmatrix}
    1 & 0 &  0 \\
     0 &   1 &  0 \\
    5 & 0 &  1
    \end{bmatrix}$\\
    \hline
         $\bw = [1,2]$   & $\begin{bmatrix}
    1 & 0 &  0 \\
     0 &   1 &  0 \\
    6 & 15 &  1
    \end{bmatrix}$ & $\begin{bmatrix}
    1 & 0 &  0 \\
     0 &   1 &  0 \\
    5 & 11 &  1
    \end{bmatrix}$\\
    \hline
         $\bw = [1,2,3]$ & $\begin{bmatrix}
    1 & 0 &  0 \\
     90 &   224 &  15 \\
    6 & 15 &  1
    \end{bmatrix}$ & $\begin{bmatrix}
    1 & 0 &  0 \\
     55 & 120 &  11 \\
    5 & 11 &  1
    \end{bmatrix}$\\
    \hline
    \end{tabular}
    \end{center}
\end{Exa}

\acknowledgements{We thank Ahmet Seven and Sergey Fomin for their correspondences. The authors were supported by the NSF grants DMS 2042786 and DMS 2302620.}

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\end{document}
