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\title[Combinatorics of Boundary Algebras]{Combinatorics of Boundary Algebras}

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\author{Jonah Berggren\thanks{\href{mailto:jrberggren@uky.edu}{jrberggren@uky.edu}. J. Berggren was partially supported by NSF grant DMS-2054255.}\addressmark{1} \and Jonathan Boretsky\thanks{\href{mailto:jboretsky@math.harvard.edu}{jboretsky@math.harvard.edu}. J. Boretsky was partially supported by Natural Sciences and Engineering Research Council of Canada (NSERC), funding
reference number 557353-2021. J. Boretsky était partiellement soutenu par le Conseil de recherches en sciences naturelles et en génie du
Canada (CRSNG), numéro de référence 557353-2021.
}\addressmark{2} }



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\address{\addressmark{1}Department of Mathematics, University of Kentucky, Lexington, KY \\ 
\addressmark{2}Mathematics Department, Harvard University, Cambridge, MA}

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\abstract{Boundary algebras are an important tool in the categorification, by Jensen--King--Su and by Pressland, of cluster structures on positroid varieties, defined by Scott and by Galashin--Lam. Each connected positroid has a corresponding boundary algebra. We give a combinatorial way to recover a positroid from its boundary algebra. We then describe the set of algebras which arise as the boundary algebra of some positroid. Finally, we give the first complete description of the minimal relations in the boundary algebra. We expect this description to be helpful in extending results known for Grassmannian boundary algebras to more general settings.}

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\keywords{Categorification, Positroids, Cluster Algebras}

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\begin{document}

\maketitle


\section{Introduction}

%\textit{Positroid varieties} were defined by Knutson--Lam--Speyer~\cite{KLS}. %, inspired by work of Postnikov \cite{pos}.
%Their intersections with the totally nonnegative Grassmannian form the cells of a stratification of the totally nonnegative Grassmannian capturing the combinatorics of positroids \cite{pos}. 
An  \textit{open positroid variety}, defined by Knutson--Lam--Speyer~\cite{KLS}, is the variety of points in the Grassmannian realizing a given positroid. They broaden the positroid stratification of the nonnegative Grassmannian \cite{pos} to the full Grassmannian. 
%In this abstract, we will work with \textit{open positroid varieties}, which are subsets of the Grassmannian defined by the vanishing and the non-vanishing of certain \Pl coordinates.  
As conjectured in \cite{MS, Lec} and proven in \cite{Sco, SSBW, GL}, the coordinate ring of an open positroid variety has the structure of a \textit{cluster algebra}. Such a cluster structure is a combinatorially rich algebraic structure that in particular interacts well with nonnegativity \cite{FZ}. For instance, the positivity of the cluster variables in a single cluster is enough to guarantee the positivity of all the other, possibly infinitely many, cluster variables. 

%More stuff about dimer models


\textit{Boundary algebras} appear in the context of \textit{categorification} of the cluster structure on an open positroid variety. Categorification is a process by which structures from other areas of math are realized using category theory, often through module categories. In 2006, Scott~\cite{Sco} showed that the Grassmannian has a cluster structure, which was categorified by Jensen--King--Su~\cite{JKS} as the %\textit{Gorenstein-projective category} of the \textit{circle algebra}. 
category of \textit{Gorenstein-projective modules} over the \textit{circle algebra}. In 2016, Baur--King--Marsh~\cite{BKM} connected this with dimer models by realizing the circle algebra as a completed boundary algebra of a Grassmannian dimer model.  Pressland~\cite{Pre} extended this setup in 2022 by showing that the cluster structure defined by Galashin--Lam on an arbitrary connected positroid variety~\cite{GL} is categorified by the Gorenstein-projective category of the completed boundary algebra of an appropriate dimer model. 
%of a dimer model with the appropriate strand permutation.



This categorification has proven useful in understanding the Galashin--Lam cluster structure on positroid varieties. Pressland~\cite{Pre2} used it to prove a conjecture of Muller--Speyer that two a priori different cluster are closely related \cite{MS}. In particular, he shows that the source-labelled and target-labelled cluster structures on a positroid variety quasi-coincide. Much work is being done on the cluster structures on Grassmannians, including studying the Gorenstein-projective modules over the circle algebra corresponding to rank 2 and 3 cluster variables~\cite{BBE,BBL}. This work is difficult to extend to all positroid varieties because there is no generators-and-relations description in the literature for boundary algebras of general positroid varieties. Forthcoming work of the first author and Khrystyna Serhiyenko contains a combinatorial construction of the boundary algebra, but only describes the relations up to an operation called \textit{cancellative closure}. %In this abstract, we build on this construction and give a combinatorial description of the boundary algebra for any positroid variety, including a complete set of minimal relations. We also characterize which algebras arise as the boundary algebra of some positroid. We further give an inverse map which recovers a positroid from its boundary algebra. Together, these last two results give us a new cryptomorphism of connected positroids.
{We build on this construction and give a combinatorial description of the boundary algebra of a connected positroid variety, including a minimal set of relations. We isolate combinatorial data which determines the boundary algebra, and call it a \textbf{boundary chart}. We characterize boundary charts of connected positroids and provide an explicit bijection between realizable boundary charts and connected positroids. This gives us a new cryptomorphism of connected positroids.}
%We isolate some underlying combinatorial information of the boundary algebra, called its \textbf{boundary chart}. We give a combinatorial characterization for boundary charts of connected positroids and provide an explicit bijection between realizable boundary charts and connected positroids. This gives us a new cryptomorphism of connected positroids.



Our new description of boundary algebras gives additional tools for studying the Gorenstein-projective modules over these algebras. %We plan to use our results to generalize the work mentioned above from the Grassmannian setting to general positroid varieties.
We expect our results to be useful in generalizing the work mentioned above from the Grassmannian setting to general positroid varieties.



% The problem: Positroid varieties have cluster structures. Cluster structures are important because of reasons. One way to understand cluster structures is through categorification. Categorification realizes the cluster structure as Gorenstein projective modules over the boundary algebra. How do you compute this thing? 

\section{Background}
\subsection{Positroids}

A \textit{positroid} is a special type of realizable matroid which reflects the combinatorial structure of the \textit{totally nonnegative Grassmannian}. See \cite{pos} for background on positroids. In this section, we introduce \textit{perfect orientations} and \textit{decorated permutations}, which are two of many equivalent descriptions of positroids.

\begin{defn}
    A \textbf{plabic graph} (\ul{pla}nar \ul{bi-c}olored graph) is an undirected planar graph embedded in a disc with $n$ vertices on the boundary, labelled $b_i$ for $i\in[n]$ in clockwise order. Plabic graphs may have additional vertices in the interior of the disc which are each assigned one of two colors, either white ($\tikz\draw[black,fill=white] (0,0) circle (.5ex);$) or black ($\tikz\draw[black,fill=black] (0,0) circle (.5ex);$). Boundary vertices must be incident to exactly one edge. We consider plabic graphs modulo homotopy.   
\end{defn}

There are moves and reductions that can be applied to plabic graphs which preserve the key combinatorial properties we are interested in. Using these, we may, and will, assume that our plabic graphs are bipartite (for this purpose, we ignore boundary vertices) and \textit{reduced}~\cite[Def. 12.5]{pos}. An example of such a graph is illustrated in \cref{fig:plabic}.

We may define positroids in terms of \textit{perfect orientations} of plabic graphs. These are orientations $\mathcal{O}$ of the edges of the plabic graph such that each white internal vertex is incident to exactly one incoming edge and each black internal vertex is incident to exactly one outgoing edge. The source set of $\mathcal{O}$ is then $\{i\mid b_i \textnormal{ is a source in } \mathcal{O}\}$. Fix a plabic graph $G$ on $n$ boundary vertices. All perfect orientations have source sets of the same size $k$. We say such a plabic graph is of \textit{type} $(k,n)$. The set consisting of the source sets of all perfect orientations forms a positroid $\mathcal{P}(G)$ of rank $k$ on $[n]$ \cite{pos}.


%definition environment for decorated permutation 

Move equivalent plabic graphs give the same positroid. One way to distinguish move equivalence classes of plabic graphs is by using their \textit{trip permutations}. These are permutations $\pi$ defined as follow: For each $i\in [n]$, construct a trip which starts from $b_i$ and follow the edges of the plabic graph according to the "rules of the road": At each white vertex turn right, and at each black vertex turn left. This trip will end at some boundary vertex $b_j$ and we define $\pi(i)=j$. We obtain a fixed point $\pi(i)=i$ if and only if $b_i$ is connected by a single edge to a leaf. At fixed points, we additionally keep track of the color of this leaf. With this additional data, we refer to $\pi$ as a \textit{decorated permutation}. Decorated permutations of $[n]$ are in bijection with positroids on $[n]$. We denote by $\mathcal{P}_\pi$ the positroid corresponding to a decorated permutation $\pi$. In this abstract, we will be primarily concerned with connected positroids, in which case the decorated permutations are \textit{stabilized-interval-free permutations} \cite{ARW}. These have no fixed points and so, in particular, are undecorated permutations.

Consider $\Gr_{k,n}$, the Grassmannian of $k$-planes in $\mathbb{C}^n$, embedded in $\mathbb{CP}^{{n\choose k}-1}$ by \textit{\Pl coordinates} $\Delta_I$, where $I$ is a $k$-element subset of $[n]$. To define \textit{open positroid varieties}, we will need to label faces $F$ of a plabic graph $G$ of type $(k,n)$ by the set of $i\in [n]$ such that $F$ lies to the left of the trip terminating at $i$. One can show that each such label will have size $k$ and, if $\mathcal{P}(G)=\mathcal{P}(G')$, then the boundary faces of $G$ and $G'$ will have the same labels~\cite{MS}. 
  


\begin{figure}
\centering
\begin{tikzpicture} [scale=0.8]
    
    \draw (0,0) circle (1.5 cm);
    \filldraw[](1.5,0)circle (1pt);
    \node at (1.73,0) {$3$};
    \filldraw[](0.95,1.16)circle (1pt);
    \node at (1.13,1.34) {$2$};
    \filldraw[](-0.95,1.16)circle (1pt);
    \node at (-1.13,1.34) {$1$};
    \filldraw[](-0.95,-1.16)circle (1pt);
    \node at (-1.13,-1.34) {$5$};
    \filldraw[](0.95,-1.16)circle (1pt);
    \node at (1.13,-1.34) {$4$};
    

    \draw (-0.4,0.75)--(0.4,0.75);
    \draw (-0.4,0)--(0.4,0);
    \draw (-0.4,-0.75)--(0.4,-0.75);
    \draw (-0.4,0.75)--(-0.4,-0.75);
    \draw (0.4,0.75)--(0.4,-0.75);
    
    \draw (0.4,0)--(1.5,0);
    \draw (0.4,0.75)--(0.95,1.16);
    \draw (-0.4,0.75)--(-0.95,1.16);
    \draw (-0.4,-0.75)--(-0.95,-1.16);
    \draw (0.4,-0.75)--(0.95,-1.16);
    
    
    \filldraw [draw=black, fill= white] (-0.4,0.75) circle (2pt);
    \filldraw [draw=black, fill= black] (0.4,0.75) circle (2pt);
    \filldraw [draw=black, fill= black] (-0.4,0) circle (2pt);
    \filldraw [draw=black, fill= white] (0.4,0) circle (2pt);
    \filldraw [draw=black, fill= white] (-0.4,-0.75) circle (2pt);
    \filldraw [draw=black, fill= black] (0.4,-0.75) circle (2pt);

    


    
    \filldraw [red] (0,0.37) circle (2pt);
    \filldraw [red] (0,-0.37) circle (2pt);
    \filldraw [red] (-0.07,-1.56) rectangle ++(4pt,4pt);
    \filldraw [red] (-.07,1.44) rectangle ++(4pt,4pt);
    \filldraw [red] (-1.56,0) rectangle ++(4pt,4pt);
    \filldraw [red] (1.26,0.68) rectangle ++(4pt,4pt);
    \filldraw [red] (1.26,-0.74) rectangle ++(4pt,4pt);

    
    \draw [-stealth,red](0,-0.27) -- (0,0.27);
    \draw [-stealth,red](0,-0.47) -- (0,-1.4);
    \draw [-stealth,red](0.105,0.4) -- (1.2,0.7);
    \draw [-stealth,red](-0.105,0.33) -- (-1.36,0.1);
   
    \draw [-stealth,red](0,1.4) -- (0,0.47);
    \draw [-stealth,red](1.22,-0.62) -- (0.1,-0.39);
    \draw [-stealth,red](-1.37,0) -- (-.1,-0.35);

    
    \draw [-stealth,red](0.12,-1.42) -- (1.2,-0.7);
    \draw [-stealth,red](1.31,0.6) -- (1.31,-0.55);
    \draw [-stealth,red](1.19,0.8) -- (0.1,1.43);
    \draw [-stealth,red](-1.38,.15) -- (-0.1,1.43);
    \draw [-stealth,red](-0.1,-1.4) -- (-1.4,-0.05);
    
\end{tikzpicture}
\caption{A plabic graph on 5 boundary vertices, with its quiver indicated in red, where frozen vertices are squares and unfrozen vertices are circles. The decorated permutation, obtained by following the rules of the road, is $\pi=45123$.}\label{fig:plabic}


\end{figure}

%\subsection{Positroid Varieties}


%Independnent of choice of plabic graph. 

\begin{defn}
    Fix a positroid $\mathcal{P}$ of rank $k$ on $[n]$. Let $G$ be a reduced plabic graph such that $\mathcal{P}=\mathcal{P}(G)$. Label the faces of $G$ as above. The \textbf{open positroid variety} $\Pi_\mathcal{P}^{\circ}$ is the subset of $\Gr_{k,n}$ where $\Delta_I=0$ for all $I\notin \mathcal{P}$ and $\Delta_I\neq 0$ for all $I$ which label a boundary face of $G$.
    
\end{defn} 

Finally, we construct the \textit{quiver} of a plabic graph.

\begin{defn}
    A \textbf{quiver} $Q=(Q_0,Q_1)$ is a directed graph with vertices $Q_0$ and arrows $Q_1$, with no loops or oriented $2$-cycles. Some vertices $F_0\subset Q_0$ may be marked as frozen.
\end{defn}

For a bipartite plabic graph $G$, define the quiver $Q(G)$ as follows: Place a vertex at each internal face of the plabic graph. Faces incident to the boundary contain frozen vertices. For each edge $e$ of $G$ which is incident to a white vertex, add an arrow $\alpha$ to the quiver between the faces on either side of $e$ such that the white endpoint of $e$ is to the right of $\alpha$. The only other edges of $G$ are those which connect boundary vertices to black vertices. In this case, add an arrow $\alpha$ such that the black vertex is to the left of $\alpha$. The quiver of a plabic graph is illustrated in \cref{fig:plabic}.

%We will label the frozen vertices of $Q(G)$ as $v_i$, so that $v_i$ lies in the boundary face immediately clockwise of $b_i$.

Galashin and Lam have shown that the coordinate ring of an open positroid variety $\Pi_{\mathcal{P}(G)}^{\circ}$ is isomorphic to the cluster algebra with quiver $Q(G)$ \cite{GL}. %\footnote{It is actually an \textit{ice quiver}, since we allow arrows between frozen vertices.} \cite{GL}. 



\subsection{Boundary Algebras}\label{sec:boundaryalgebra}
We now introduce the \textit{boundary algebra}, which is important for the categorification of the cluster structure on an open positroid variety introduced by Pressland \cite{Pre}.

\begin{defn}
    For a quiver $Q$, the \textbf{path algebra} $\mathbb{C}Q$ is spanned by finite paths in $Q$, including empty paths at each vertex. The product in the algebra between paths $p$ and $q$ is the concatenation of $p$ and $q$, if $p$ ends at the start of $q$, and $0$ otherwise. 
\end{defn}

Fix a stabilized-interval-free permutation $\pi$. Let $G$ be any plabic graph with trip permutation $\pi$, and let $Q=Q(G)$. Each internal face of $Q$ is bounded by an oriented cycle. Each edge $d$ not between two frozen boundary vertices is incident to two faces and thus part of oriented cycles $c_d^+$ and $c_d^-$ bounding those faces. Say %$c_d^+=dp_d^+$
$c_d^+$ factors as $dp_d^+$ and %$c_d^-=dp_d^-$
$c_d^-$ factors as $dp_d^-$. Let $e_i$ be the empty path at the boundary vertex $v_i$ of $Q(G)$, and let $e=\sum_{i=1}^n e_i$. 

\begin{defn}
    The \textbf{dimer algebra} $A_Q$ is $\mathbb{C}Q$ modulo the relations $p_d^+=p_d^-$ for all edges $d$ of $Q$ which are not between two frozen vertices. 
\end{defn}


\begin{defn}
    The \textbf{boundary algebra} of the positroid $\mathcal{P}_\pi$ is $B_{\pi}=eA_{Q(G)}e$ for any $G$ such that $\mathcal{P}=\mathcal{P}(G)$.  
    \end{defn}

% This is well The boundary algebra is independent of which plabic graph $G$ satisfying $\mathcal{P}=\mathcal{P}(G)$ we use to define $Q$, up to isomorphism.

It is not obvious, but if $\mathcal{P}(G)=\mathcal{P}(G')$, then $eA_{Q(G)}e=eA_{Q(G')}e$, so this is well defined. Multiplication by $e$ on both sides discards each path which neither originates from nor terminates at a boundary vertex. Thus, $B_{\pi}$ can be thought of as the algebra of paths between boundary vertices, modulo the relations $p_d^+=p_d^-$. %Note that this relation guarantees that the oriented cycles around the faces incident to a given vertex are all the same in $A_Q$. 
%For $v\in Q_0$, denote any such cycle by $c_v\in A_Q$. Let $c=\sum_{v\in Q_0}c_v$. This element is central in $A_Q$, and $t=ece$ is central in $B_{\pi}$.  
%For $v\in Q_0$, denote by $c_v$ any oriented cycle around a face incident to $v$. Let $c=\sum_{v\in Q_0}c_v$, all of which are equivalent by the relations $p_d^+=p_d^-$. This element is central in $A_Q$, and $t=ece$ is central in $B_{\pi}$.

We define the following elements of $B_\pi$: For each $i\in [n]$, let $x_i$ be a minimal path from $v_i$ to $v_{i+1}$ and let $y_{i}$ be a minimal path from $v_{i+1}$ to $v_{i}$, where all indices are taken modulo $n$. Let $x=\sum_{i=1}^nx_i$ and $y=\sum_{i=1}^ny_i$. Let $t=xy$; then $t$ is central in $B_\pi$. Let $p$ be a path in $B_\pi$ between two nonadjacent vertices. Define $\tau(p)$ and $\eta(p)$ by the condition that $p$ is directed from $v_{\tau(p)}$ to $v_{\eta(p)}$. Suppose $p$ does not factor as a product of other paths. We then say $p$ is a \textit{nonadjacent arrow} and define $\reach_p=\eta(p)-\tau(p)$ taken modulo $n$ so that $\reach_p\in[n]$. One can show that $p$ satisfies $pt^{X_p}=x_{\tau(p)}x_{\tau(p)+1}\cdots x_{\eta(p)-1}$ for a suitable positive integer $X_p$. We represent the boundary algebra by putting the vertices $v_i$ in clockwise order around a circle and drawing in the nonadjacent arrows. For example, the second subfigure of \cref{fig:6V} shows the representation of a boundary algebra. with a nonadjacent arrow from $v_4$ to $v_1$ in black. In this figure, the arrows $x_i$ and $y_i$ are shown in gray.

In forthcoming work, the first author and Khrystyna Serhiyenko prove the following two results showing how to calculate the boundary algebra $B_\pi$ directly from a stabilized-interval-free permutation $\pi$.
In order to state these results, we must represent the permutation $\pi$ as a directed graph on vertices $w_i$ with directed edges from $w_i$ to $w_{\pi(i)}$ for $i\in [n]$. We will draw the permutation graph such that $w_i$ lies between vertices $v_i$ and $v_{i+1}$. We refer to edges of the permutation graph as \textit{strands}. \cref{fig:6V} shows three examples of permutation graphs, in red. 

\begin{defn}
    For $i\in[n]$, we define the \textbf{$i$-shifted linear order} $<_i$ on $[n]$ by $i<_ii+1<_i\cdots<_in<_i1<_i\cdots <_i i-1$. 
%
    We say $(a_1\leq \cdots\leq a_m)\in[n]^m$ is a \textbf{cyclic ordering} if there exists some $i\in [n]$ such that $a_1\leq_ia_2\leq_i\cdots \leq_i a_m$. We will allow ourselves to replace some or all of the inequalities with strict inequalities if consecutive terms are not allowed to be equal.
\end{defn}

\begin{defn}
    Define $[i,j]=\{l\mid (i\leq l\leq j) \textnormal{ is a clockwise ordering}\}$ to be the (closed) clockwise interval between $i$ and $j$. We similarly define the clockwise intervals $(v_i,v_j]$, $[v_i,v_j)$, and $(v_i,v_j)$ by excluding one or both of the endpoints. 
\end{defn}

\begin{defn}
    Let $v_j$ be a boundary vertex of $Q$. Consider a strand $\alpha$ from $r$ to $\pi(r)$ in the permutation graph of $\pi$. We say that $v_j$ is \textbf{to the right of $\alpha$} if $j\in[\pi(r),r)$ and otherwise $v_j$ is \textbf{to the left of $\alpha$.}
    Let $p$ be an arrow between nonadjacent vertices $v_i$ and $v_j$. %(for our purposes, $p$ may or may not be a nonadjacent arrow of the boundary algebra). 
    %We say that the strand $\alpha$ \textbf{right-supports} $p$ if both $\eta(p)$ and $\tau(p)$ are to the right of $\alpha$. We say that $\alpha$ \textbf{left-supports} $p$ if both $\eta(p)$ and $\tau(p)$ are to the left of $\alpha$. In either case, we say that this strand \textbf{supports} $p$. 
    We say that $\alpha$ is \textbf{left-supporting} to $p$ if $(\pi(r)\leq r\leq \eta(p) \leq \tau(p))$ is a cyclic ordering. We say that the strand $\alpha$ is \textbf{right-supporting} to $p$ if $(r\leq \pi(r)\leq \tau(p)\leq \eta(p))$ is a cyclic ordering. In either case, we say that this strand is \textbf{supporting} to $p$.
\end{defn}
%\begin{rem}
    Informally, a strand left (resp. right) supports an arrow $p$ if it points in the opposite direction of $p$ and lies to its left (resp. right). 
%\end{rem}
%\begin{rem}
%    Observe that $X_p$ (resp. $Y_p$), as defined in \cref{permtochart}, enumerates the left (resp. right) supporting strands of $p$. 
%\end{rem}

\begin{thm}\label{permtochart}
	Fix a connected positroid $\mathcal{P}$ with permutation $\pi$. Fix distinct nonadjacent boundary vertices $v_i$ and $v_j$. The arrow $p$ from $v_i$ to $v_j$ is a nonadjacet arrow of $B_{\pi}$ if and only if 
    \begin{enumerate}
        \item for all $l\in(j,i)$, there is a right-supporting strand $\alpha$ of $p$ with $v_l$ to its left, and
        \item for all $l\in(i,j)$, there is a left-supporting strand $\alpha$ of $p$ with $v_l$ to its right.
    \end{enumerate}
% \begin{enumerate}
%		\item
%  there is a sequence   
%    $(j=z_1,z_2,\dots,z_m)$ such that $\pi(z_m)=i-1$ and $(j\leq z_r<z_{r+1}\leq \pi(z_r)<\pi(z_{r+1})\leq i-1)$ is a cyclic ordering for each $r\in[m-1]$, and
%		\item there is a sequence $(j-1=z_1,z_2,\dots,z_m)$ such that $\pi(z_m)=i$ and $(i\leq\pi(z_{r+1})<\pi(z_r)\leq z_{r+1}<z_r\leq j-1)$ is a cyclic ordering for each $r\in[m-1]$.
%	\end{enumerate}
	In this case, the relations
%\begin{equation*}
     $pt^{X_p}=x_{i}x_{i+1}\cdots x_{j-1}$ and $pt^{Y_p}=y_{i-1}y_{i-2}\cdots y_{j}$
% \end{equation*}
  hold in $B_\pi$, where $X_p$ is the number of left-supporting strands and $Y_p$ is the number of right-supporting strands of $p$. We call $X_p$ the \textbf{left relation number} of $p$, and we call $Y_p$ the \textbf{right relation number} of $p$.
\end{thm}

\begin{defn}
    An ideal $I$ of a path algebra $\mathbb C Q$ (generated by commutation relations) is \textbf{cancellative} if, for any $a,p,q,b\in I$ with $\eta(a)=\tau(p)=\tau(q)$ and $\eta(p)=\eta(q)=\tau(b)$, we have $apb-aqb\in I\iff p-q\in I$. The \textbf{cancellative closure} of an ideal $I$, denoted $\textup{CancClos}(I)$, is the smallest cancellative ideal containing $I$.
\end{defn}

%\begin{defn}
%    The \textbf{cancellative closure} of an ideal $I$, denoted $\textup{CancClos}(I)$, is the smallest ideal containing $I$ and all relations obtained from relations in $I$ by cancelling common leading or trailing terms.  
    %polynomial? Expression
%\end{defn}

\begin{thm} \label{generatorrelation}
    Let $Q^\circ_\pi$ be the quiver on $v_i$, for $i\in [n]$, whose arrows are $x_i$, $y_i$ and the nonadjacent arrows of $B_\pi$.
    Let $I_{\pi}^\circ$ be the cancellative closure of the ideal $I$ generated by the relations given in \cref{permtochart}, the relations $xy-yx$ and $x^{k}-y^{n-k}$:
    \begin{equation*}
        I_{\pi}^\circ=\textup{CancClos}\left(\left<xy-yx,x^{k}-y^{n-k}, \sum_{p \textnormal{ nonadjacent arrow}}pt^{X_p}-x_{\tau(p)}x_{\tau(p)+1}\cdots x_{\eta(p)-1}\right>\right).
    \end{equation*}
    Then $B_{\pi}\equiv\mathbb{C}Q_\pi^\circ/I_{\pi}^\circ$.
    %Point is: We can get the boundary algebra from the boundary chart
\end{thm}

Together, \cref{permtochart} and \cref{generatorrelation} give a way to calculate the boundary algebra of a positroid from its decorated permutation. However, this is obfuscated by the necessity of taking a cancellative closure. We address this in~\cref{sec:mr}.
%XXX
%As an example: The boundary algebra of the Grassmannian case can be explicitly described as follows: Circle algebra \cite{}

\begin{example}
%    Shown in Figure~\ref{fig:6V} are three permutations with $(n,k)=(6,3)$ along with the nonadjacent arrows and relation numbers of the resulting boundary algebra.
%    \textcolor{violet}{Well, we can't really call them boundary charts yet. Maybe this figure should be after Definition 3.1, but we forward-reference it from this example here.}
    Let $\pi$ be the permutation depicted in the middle of \cref{fig:6V}. We see that there is an arrow $p$ from $v_1$ to $v_4$. Since $p$ has one left-supporting strand (from 6 to 4) and one right-supporting strand (from 1 to 3), it is labelled with $X_p:Y_p=1:1$. By \cref{generatorrelation}, the boundary algebra is $B_\pi\cong\mathbb{C}Q_\pi^\circ/I_\pi^\circ$, where the arrows of $Q_\pi^\circ$ are $p$ along with the greyed-out arrows (representing $x_i$ and $y_i$) and the ideal $I_\pi^\circ$ is generated up to cancellative closure by
    \[\{xy-yx,\ x^k-y^{n-k},\ pt-x_4x_5x_6\}.\]
%    \textbf{\textcolor{violet}{I'm having trouble writing this example up succinctly without being able to use ``strand'' or ``relation number'' or ``supporting.'' Maybe we should say that elements of the permutation graph can be called strands, and we have a notion of supporting strands to an arrow, earlier?}}

\end{example}


\section{Combinatorial construction of the boundary algebra}

\subsection{From Boundary Algebras to Permutations}

Our work involves a new combinatorial object, which we call a \textit{boundary chart}. 

\begin{defn}
    A \textbf{boundary chart} consists of data $C=(k,n,S,X)$ as follows, where $k$ and $n$ are integers satisfying $1\leq k\leq n$ and $S$ is a set of arrows on vertices $v_i$, for $i\in[n]$, such that arrows are not between $v_i$ and $v_{i\pm 1}$, with indices taken cyclically, and there is at most one arrow from $v_i$ to $v_j$ for any $i,j\in [n]$. Finally,  $X\in \mathbb{Z}_{> 0}^{|S|}$ gives a positive integer for each arrow.
%
   
\end{defn}

Given a positroid, the boundary chart is precisely the data determined in \cref{permtochart}. Here, we start with a connected positroid of rank $k$ on $[n]$ with permutation $\pi$, and the set $S$ consists of the nonadjacent arrows of $B_\pi$. The numbers $X_p$ are as in the theorem. Observe that it is implicit in \cref{permtochart} that there is at  most one arrow between any two vertices. By \cref{generatorrelation}, this information fully determines $B_\pi$. 
    \begin{figure}
            \centering
    	\def\svgscale{0.21}\input{7V1.pdf_tex}
    	\hspace{1.5cm}
    	\def\svgscale{0.21}\input{7V2.pdf_tex}
    	\hspace{1.5cm}
    	\def\svgscale{0.21}\input{7V3.pdf_tex}
    	\caption{Three boundary charts on six vertices with $k=3$ and their corresponding decorated permutations depicted in red. The arrows $x_i$ and $y_i$ are greyed out. For clarity, we use a red $i$ in place of $w_i$ and a black $i$ in place of $v_i$.}%; they are part of the information of the boundary algebra, but not of the boun}
    	\label{fig:6V}
    \end{figure}

\begin{defn}
    A boundary chart is \textbf{realizable} if it is obtained from a positroid via this process.
\end{defn}

We introduce the following auxiliary piece of data for a boundary chart: Let $Y\in \mathbb{Z}_{>0}^{|S|}$ be a vector of positive integers, indexed by the arrows in $S$, such that $Y_p=X_p+k-\reach_p$
for each $p\in S$. Using $xy=yx$ and $x^k=y^{n-k}$ from \cref{generatorrelation}, one can show that the $Y_p$ here coincide with the $Y_p$ in \cref{permtochart}. 

We represent the data in a boundary chart by placing the vertices $v_i$ around a circle, in clockwise order. We draw in the arrows and mark each arrow $p\in S$ with the pair of integers $(X_p:Y_p)$. We refer to these as the left and right relation numbers of $p$, respectively. Note that knowing $X_p$ and $Y_p$ suffices to recover $k$ when the set $S$ of nonadjacent arrows is nonempty. %For realizable boundary charts, these representations are similar to the representations of boundary algebras discussed earlier. The only difference is that we add the labels $(X_p:Y_p)$ to the arrows.
Three examples are illustrated in black in \cref{fig:6V} (the red and grey parts are not in the boundary chart). For the realizable boundary chart obtained from $\mathcal{P}=\mathcal{P}_\pi$, this coincides with the representation of $B_\pi$ described in section \cref{sec:boundaryalgebra}. We will need the following terminology: 

\begin{defn}
    Let $p$ and $q$ be two arrows in $S$. 
    
    \begin{itemize}
        \item If $(\tau(p)< \tau(q)< \eta(p)< \eta(q))$ is a cyclic ordering, we say $p$ and $q$ \textbf{cross}.
        \item If $(\tau(p)\leq\tau(q)<\eta(q)\leq\eta(p))$ is a cyclic ordering, then $p$ lies to the right of $q$. We say that $p$ and $q$ are \textbf{parallel}, with $p$ \textbf{right-parallel} of $q$. Define \textbf{left-parallel} similarly.  
        \item If %$(\tau(p)<\eta(p)\leq\tau(q)<\eta(q))$
        $(\eta(p)\leq\tau(q)<\eta(q)\leq\tau(p))$ is a cyclic ordering, then $p$ lies to the right of $q$. We say $p$ and $q$ are \textbf{antiparallel}, with $p$ \textbf{right-antiparallel} of $q$. Define \textbf{left-antiparallel} similarly.
    \end{itemize} 

\end{defn}

    Visually, crossing arrows are arrows that intersect on their interiors, like the arrow from $9$ to $4$ and the arrow from $7$ to $2$ in \cref{fig:CROSS}. Parallel arrows are, roughly, arrows that go in the "same direction", like the arrow from $9$ to $4$ and the arrow from $10$ to $1$ in \cref{fig:CROSS}, while antiparallel arrows are arrows which neither cross nor are parallel. 

Given a realizable boundary chart $C=(k,n,S,X)$, we describe how to recover the permutation of its associated boundary algebra. We will construct the permutation as a permutation graph on vertices $w_i$ such that $w_i$ lies between vertices $v_i$ and $v_{i+1}$. We refer to the vertices $w_i$ as strand vertices.
First, we must discuss the ideas of visibility and of influence in a boundary chart. We do this inductively. We will give an example of these definitions in \cref{ex:on}.
We start with a reformulation of the right and left relation numbers which will make the following construction easier to state. 

\begin{defn}
   Let $C$ be a boundary chart with left and right relation numbers $X$ and $Y$, respectively. For $p\in S$, let $L(p)$ (resp. $R(p)$) be the set of arrows left-parallel (resp. right-parallel) to $p$. Then the \textbf{adjusted left relation numbers} are defined inductively by $X'_p=X_p-\sum_{q\in L(p)}X'_q$ and the \textbf{adjusted right relation numbers} are defined inductively by $Y'_p=Y_p-\sum_{q\in R(p)}Y'_q$.  
\end{defn}

\begin{defn}\label{defn:head-influence}
    Let $\alpha$ be an arrow of a realizable boundary chart $C=(k,n,S,X)$. Inductively define the following:
    % \begin{enumerate}
    %     \item If no arrow is to the right of $\alpha$, then every strand vertex to the right of $\alpha$ is \textbf{right-head-visible} to $\alpha$. Then $Y(\alpha)=Y'(\alpha)$. We define the $Y'(\alpha)$ strand vertices most immediately clockwise of $\h(\alpha)$ to be \textbf{right-head-influenced} by $\alpha$. %\BAD{Since there are $\reach_y(\alpha)$ strand vertices right-visible to $\alpha$ and $Y'(\alpha)<\reach_y(\alpha)$ by Condition~\eqref{cond1}, all of these strand vertices are to the right of $\alpha$.}
    %     %\JRB{ARGUE THAT ALL OF THESE ARE WITHIN $Y(\alpha)$ TO THE RIGHT OF $h(\alpha)$!}
    %     \item If no arrow is to the left of $\alpha$, then every strand vertex to the left of $\alpha$ is \textbf{left-head-visible} to $\alpha$. Then $X(\alpha)=X'(\alpha)$. We define the $X'(\alpha)$ strand vertices most immediately counterclockwise of $\h(\alpha)$ to be \textbf{left-head-influenced} by $\alpha$.
    % \end{enumerate}
    % We now suppose $\alpha$ is a general arrow of $C$.
    \begin{enumerate}
        \item The \textbf{right-head-visible} strand vertices to $\alpha$ are those strand vertices to the right of $\alpha$ which are not right-head-influenced by a right-parallel arrow of $\alpha$ or left-head-influenced by a right-antiparallel arrow of $\alpha$ (this condition is vacuous if there are no arrows to the right of $\alpha$). The \textbf{right-head-influenced} strand vertices of $\alpha$ are the $Y'_\alpha$ right-head-visible strand vertices most immediately clockwise of $\eta(\alpha)$.
        \item The \textbf{left-head-visible} strand vertices of $\alpha$ are those strand vertices to the left of $\alpha$ which are not left-head-influenced by a left-parallel arrow of $\alpha$ or right-head-influenced by a left-antiparallel arrow of $\alpha$. The \textbf{left-head-influenced} strand vertices of $\alpha$ are the $X'_\alpha$ left-head-visible strand vertices most immediately counterclockwise of $\eta(\alpha)$.
    \end{enumerate}
    We will use the phrase \textbf{head-influenced} to mean either left- or right-head-influenced. We define \textbf{left-tail visible} and \textbf{right-tail-visible} strand vertices as above, swapping ``head'' with ``tail,'' ``$\eta(\alpha)$'' with ``$\tau(\alpha)$,'' and ``clockwise'' with ``counter-clockwise.''
\end{defn}

\begin{con}\label{reversemapN}
Let $C=(k,n,S,X)$ be a realizable boundary chart.
Let $\alpha$ be an arrow of $C$. Let $w_{\sigma_1},\dots,w_{\sigma_{X'(\alpha)}}$ be the right-head-influenced strand vertices of $\alpha$, ordered clockwise so that $w_{\sigma_1}$ is most immediately clockwise of $\eta(\alpha)$. Let $w_{\sigma'_1},\dots,w_{\sigma'_{X'(\alpha)}}$ be the right-tail-influenced strand vertices of $\alpha$, ordered clockwise so that $w_{\sigma_{X'(\alpha)}}$ is most immediately counter-clockwise of $\tau(\alpha)$. Then define $\phi_\alpha(\sigma_j)=\sigma'_j$ for $j\in X'(\alpha)$. Symmetrically define $\phi_\alpha$ on the left-head-influenced vertices of $\alpha$.
We define a function (indeed, we will see, a permutation) $\pi$ on $[n]$ by \[\pi(j)=\begin{cases}
    \phi_\alpha(j)&w_j\textup{ is head-influenced by an arrow }\alpha\in C\\
    j-k&\textup{ otherwise.}
\end{cases}\]
\end{con}

It is not immediately obvious that $\pi$ is well-defined; for example, some strand vertex $w_j$ may be head-influenced by two arrows $\alpha$ and $\beta$. In fact, whenever this happens, $\phi_\alpha(j)=\phi_\beta(j)$. 
\begin{thm}\label{reversemapthmN}
    The map $\pi$ is a well-defined stable-interval-free permutation, and $C$ is the boundary chart of $B_\pi$.
\end{thm}

This process is most easily understood visually, in an example. We denote nonadjacent arrows in the boundary algebra from $v_i$ to $v_j$ by $p_{i\to j}$.

\begin{figure}
        \centering
	\def\svgscale{0.21}\input{BCHART.pdf_tex} \hspace{1cm}
%	\def\svgscale{0.21}\input{TWV2.pdf_tex}
%	\def\svgscale{0.21}\input{7VS.pdf_tex}
	\def\svgscale{0.21}\input{TWV3.pdf_tex}\hspace{1cm}
	\def\svgscale{0.21}\input{TWV4.pdf_tex}
	\caption{Application of \cref{reversemapN} to a boundary chart with $12$ vertices and $k=5$. For clarity, we use a red $i$ in place of $w_i$ and a black $i$ in place of $v_i$.}
	\label{fig:CROSS}
\end{figure}
\begin{example}\label{ex:on}
 We look at \cref{fig:CROSS}. The first subfigure shows a boundary chart on $12$ vertices. We have $Y'_{p_{7\to2}}=1$, hence the only right-head-influenced strand vertex of $p_{7\to2}$ is $w_2$ and its only right-tail-influenced strand vertex is $w_6$ and we see that $\pi(2)=6$. Similarly, the right-influence of $p_{9\to4}$ induces $\pi(4)=8$. We have $Y'_{p_{10\to1}}=Y_{p_{10\to1}}-Y_{p_{7\to2}}-Y_{p_{9\to4}}=2$, so the right-head-influence of $p_{10\to1}$ is $\{w_1,w_3\}$ (skipping over $w_2$, which is right-head-influenced by the right-parallel arrow $p_{7\to2}$) and the right-tail-influence of $p_{10\to1}$ is $\{w_7,w_9\}$ (skipping over $w_8$, which is right-tail-influenced by $p_{9\to4}$). Then we see $\pi(1)=7$ and $\pi(3)=9$. The middle of \cref{fig:CROSS} shows in red all strands induced by the influence of an arrow; the right completes the permutation graph by adding in blue the remaining strands from $w_j$ to $w_{j-k}$. %VVV maybe say something about the left?
 \end{example}

% We look at \cref{fig:CROSS}. The first subfigure shows a boundary chart on $12$ vertices. The conditions on $\prec$ are that $p_{7\to2}\prec p_{10\to 1}$ and $p_{9\to4}\prec p_{10\to 1}$. We choose $p_{7\to 2}\prec p_{10\to1}\prec p_{9\to 4}$. Each arrow is labelled with numbers $X_p:Y_p$, giving the number of left-supporting and right-supporting strands to $p$, respectively. The strands in the second subfigure are those added in step \ref{step2} of \cref{permtochart}: We draw one strand that right-supports $p_{7\to 2}$, from $w_2$ to $w_6$, and one strand that right-supports $p_{9\to 4}$, from $w_2$ to $w_6$. We then note that there are already $2$ strands right-supporting $p_{10\to 1}$, so we need only $2$ more, which are from $w_1$ to $w_7$ and from $w_3$ to $w_9$. In the third subfigure, we add left-supporting strands, as in step \ref{step3} of \cref{permtochart}. Finally, in the last subfigure, we add, in blue, the strands specified in step \ref{step4}. In this case, $k=5$. Thus, we add, for example, a strand from $w_8$ to $w_{8-5}=w_3$. 

 %Observe that strands added in step \ref{step2} can additionally left-support other arrows. For example, the strand from $w_1$ to $w_7$ left-supports $p_{7\to 2}$.


\subsection{Realizable Boundary Charts}

We next classify realizable boundary charts. 

\begin{thm}\label{thm:chartcharac}
Let $C=(k,n,S,X)$ be a boundary chart with left and right relation numbers $X$ and $Y$, respectively, and with adjusted left and right relation numbers $X'$ and $Y'$, respectively. For $p\in S$, let $R_{\nparallel}(p)$ and $L_{\nparallel}(p) $ denote the sets of arrows right and left-antiparallel to $p$, respectively. Then $C$ is realizable if and only if the following hold. 
\begin{enumerate}
 		\item \label{cond1}For all $p\in S$, $X_p+\sum_{q\in L_\nparallel(p)}Y'_q<\reach_p$ and $Y_p+\sum_{q\in R_\nparallel(p)}X'_q<n-\reach_p$.
       \item \label{cond2} For all $p\in S$, $X'_p\geq 0$ and $Y'_p\geq 0$.
       \begin{enumerate} 
           \item \label{relequal0}If $X'_p=0$ (resp. $Y'_p=0$), there must be crossing arrows $q$ and $r$, both left (rep. right) parallel to $p$, such that $\tau(p)=\tau(q)$ and $\eta(p)=\eta(r)$.
       \end{enumerate}
	%\item \label{cond3} Let $p,q\in S$ be crossing such that $(v_i=\eta(p)< v_j=\eta(q)< \tau(p)< \tau(q))$ is a cyclic ordering. Let $A_p$, respectively, $A_q$, denote the set of arrows $r$ (necessarily right-)parallel to $p$, respectively, (necessarily left-)parallel to $q$, such that $\eta(r)\in[\eta(p),\eta(q))$, respectively, $\eta(r)\in(\eta(p),\eta(q)]$. Then, $Y'_p+X'_q+\sum_{r\in A_p}Y'_r+\sum_{r\in A_q}X'_r\leq j-i$, where the right side is taken modulo $n$ so that it lies in $[n]$.	
 	\item \label{cond3} Let $p,q\in S$ be crossing such that $(v_i=\eta(p)< v_j=\eta(q)< \tau(p)< \tau(q))$ is a cyclic ordering. Let $A_p$ denote the set of arrows $r$ right-parallel to $p$ such that $\eta(r)\in[\eta(p),\eta(q))$. Let $A_q$ denote the set of arrows $r$ left-parallel to $q$ such that $\eta(r)\in(\eta(p),\eta(q)]$. Then $Y'_p+X'_q+\sum_{r\in A_p}Y'_r+\sum_{r\in A_q}X'_r\leq j-i$, where the right side is taken modulo $n$ so that it lies in $[n]$.
  %\item \label{cond3} Let $p,q\in S$ be crossing such that $(\eta(p)<\eta(q)<\tau(p)<\tau(q))$ is a cyclic ordering. Let $A_p$ (resp. $A_q$) denote the set of arrows lying entirely between $\eta(p)$ and $\eta(q)$ or between $\tau(q)$ and $\tau(p)$ which are parallel to $p$ (resp. $q$). Then, $Y'_p+X'_q+\sum_{r\in A_p}Y'_r+\sum_{s\in A_q}X'_s\leq\eta(q)-\eta(p)$, where the right side is taken modulo $n$ so that it lies in $[n]$.
  		\item \label{cond4} If $p,q \in S$ form an oriented digon, then $X_p+X_q+Y_p+Y_q<n$.
	\end{enumerate}
\end{thm}
One may attempt to apply \cref{reversemapN} to general boundary algebras. However, the conditions of \cref{thm:chartcharac} are necessary in order for the result to be a well-defined stable-interval-free permutation. For example, condition \ref{cond1} ensures that there are enough strand vertices to the right of any $p\in S$ to count out $Y'_p$ right-influenced strand vertices. In the first subfigure of \cref{fig:nonexample}, there are not enough red vertices for $p_{4\to1}$ to have three left-supporting strands.
The second subfigure in \cref{fig:nonexample} violates condition \ref{cond2}, as $X'(p_{6\to2})=-1$. The left-supporting strands of $p_{8\to12}$ also left-support $p_{6\to2}$, so $p_{6\to2}$ has too many left-supporting strands.  
%\textcolor{red}{we are sort of using here that the numbers are counting left-supporting and right-supporting strands, but this isn't really stressed anymore???}
Condition \ref{cond3} ensures that, if $p$ and $q$ are arrows of $C$ which both influence $w_j$, then $\phi_p(j)=\phi_q(j)$. See the third subfigure of \cref{fig:nonexample}, where the two crossing arrows are pulling the strand starting at $w_1$ in different directions. Condition \ref{cond4} ensures the permutation constructed in \cref{reversemapN} is stable-interval-free; see the fourth subfigure of \cref{fig:nonexample}, where the permutation fixes $[1,3]$.
    \begin{figure}
        \centering
        \def\svgscale{0.21}\input{BAD1.pdf_tex}
    	\def\svgscale{0.21}\input{BAD11.pdf_tex}
        \def\svgscale{0.21}\input{BAD3.pdf_tex}
        \def\svgscale{0.21}\input{BAD44.pdf_tex}
        \caption{Four \textit{nonrealizable} boundary charts violating the conditions of \cref{thm:chartcharac}. For clarity, we use a red $i$ in place of $w_i$ and a black $i$ in place of $v_i$.}
        \label{fig:nonexample}
    \end{figure}

The sufficiency of these conditions is more surprising. We prove sufficiency by showing that \cref{reversemapN} and the map of \cref{permtochart} compose to the identity on the boundary charts satisfying the conditions of \cref{thm:chartcharac}. Hence, we may view the combinatorial conditions of \cref{thm:chartcharac} as a new cryptomorphism for connected positroids.

% By definition, realizable boundary charts are in bijection with connected positroids. With \cref{thm:chartcharac}, we have a combinatorial description of realizable boundary charts. We can view this as a new cryptomorphism for connected positroids.

\subsection{Minimal relations}\label{sec:mr}
Recall the presentation of the boundary algebra $B_\pi\cong \mathbb C Q_\pi^\circ/I_\pi^\circ$ given in~\cref{generatorrelation}, which has the drawback of the ideal $I_\pi^\circ$ being defined using a cancellative closure.
In this section, we give a description of the minimal relations of the ideal $I_\pi^\circ$ using the information of the permutation $\pi$ and the boundary chart $C=(k,n,S,X)$.

%\begin{defn}
%A strand is \textbf{visible} to $p$ if it neither right-supports an arrow right-parallel to $p$, nor left-supports an arrow right-antiparallel to $p$, nor left-supports an arrow left-parallel to $p$, nor right-supports an arrow left-antiparallel to $p$.
%\end{defn}

\begin{defn}
Given an arrow $p\in S$ of $C$, let $R_p$ (respectively $T_p$) be the vertex $w_i$ of the permutation graph most immediately clockwise of $\eta(p)$ (resp. $\tau(p)$) which is the start (resp. end) of a strand which crosses $p$ (i.e., which starts to the right of $p$ and ends to the left, or vice versa).
\begin{itemize}
    \item Two (necessarily parallel) arrows $p$ and $q$ are \textbf{stitch-equivalent} if $R_p=R_q$ and $T_p=T_q$.
    \item A strand is \textbf{relation-defining} if it does not travel from $R_p$ to $T_p$ for any arrow $p$.
\end{itemize} %VVV Un-itemize this for a space boost

\end{defn}
\begin{example}
    In \cref{fig:CROSS}, the arrows $p_{10\to1}$ and $p_{9\to4}$ are stitch-equivalent to each other with $T_{10\to1}=T_{9\to4}=12$ and $R_{10\to1}=R_{9\to4}=5$, but not to the arrow from 7 to 2, with $T_{7\to2}=8$ and $R_{7\to2}=3$. Every strand is relation-defining with the exception of the strand from 5 to 12. 
\end{example}

\begin{defn}
    For $v_a$ and $v_b$ boundary vertices of $B_\pi$, define the \textbf{aggressive clockwise path $\textup{ACL}(v_a,v_b)$ from $v_a$ to $v_b$} by starting at $v_a$ and repeatedly taking the non-$y_j$ arrow which ends most immediately counter-clockwise of $v_b$. Similarly define the \textbf{aggressive counter-clockwise path $\textup{ACC}(v_a,v_b)$}. When these paths are equivalent, we say that the \textbf{aggressive relation from $v_a$ to $v_b$} is $[\textup{ACL}(v_a,v_b)]-[\textup{ACC}(v_a,v_b)]$.
\end{defn}

%Intuitively, $\textup{AGCL}(v_a,v_b)$ and $\textup{AGCC}(v_a,v_b)$ represent the most greedy ways of getting from $v_a$ to $v_b$ by travelling clockwise or counter-clockwise, respectively, and an aggressive relation equates these two paths.

\begin{thm}\label{thm:relations}
	The following relations of $\mathbb{C}Q_\pi^\circ$, along with the relation $x_iy_i=y_{i-1}x_{i-1}$ for each $i\in[n]$, form a minimal generating set for $I_\pi^\circ$:
	\begin{enumerate}
% \item The relation xy=yx.
		\item For every relation-defining strand from $w_a$ to $w_b$, take the aggressive relation from $v_b$ to $v_a$. % which equates the most direct path composed of left-supported arrows to the most direct path composed of right-supported arrows.
		\item Let $\{p_1,\ldots,p_m\}$ be a stitch-equivalence class, ordered left to right, with $T_{p_i}=T$ and $R_{p_i}=R$ for all $i\in[m]$. Define $v_{a_{m+1}}:=T$, $v_{b_0}:=R$, and $v_{a_i}:=\tau(p_i)$, $v_{b_i}:=\eta(p_i)$ for $i\in [m]$. Then, take the aggressive relation from $v_{a_{i}}$ to $v_{b_{i-1}}$ for each $i\in[m+1]$.
	\end{enumerate}
%	These relations form a minimal generating set for $I_\pi^\circ$.
\end{thm}

\begin{example}
	Consider the boundary chart of Figure~\ref{fig:CROSS}. The strand from $w_1$ to $w_7$ is relation-defining and yields the relation $p_{7\to2}y_1=x_7x_8x_9p_{10\to1}$. All of the strands $w_j\mapsto w_{j-k}$, drawn in blue, give relations composed only of $x$'s and $y$'s. For example, the strand from $w_5$ to $w_{12}$ gives $x_{12}x_1\cdots x_4=y_{11}y_{10}\cdots y_5$. There are two stitch-equivalence classes $\{p_{10\to1},p_{9\to4}\}$ and $\{p_{7\to2}\}$. The former gives $\{y_{11}y_{10}p_{10\to1}=x_{12},\ p_{10\to1}x_1x_2x_3=y_9p_{9\to4},\ p_{9\to4}x_4=y_8y_7y_6\}$ and the latter gives $\{p_{7\to2}x_2=y_6y_5y_4y_3,\ x_8x_9p_{10\to1}x_1=y_7p_{7\to2}\}$. These five relations and those given by relation-defining strands make up all minimal relations of $I_\pi^\circ$.
\end{example}
Note that \cref{thm:relations} uses both the boundary chart and the permutation obtained from it by \cref{reversemapN}. It would be hard to rephrase the theorem in terms of one or the other. This indicates that boundary charts and stabilized-interval-free permutations, while both cryptomorphisms of connected positroids, highlight different information.

\acknowledgements{J. Boretsky thanks Bernhard Keller for discussions that led to the foundations of this work.}

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\end{document}
