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\title[Skein relations for punctured surfaces]{Skein relations for punctured surfaces}

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\author{Esther Banaian\thanks{\href{mailto:banaian@math.au.dk}{banaian@math.au.dk}. Supported by Independent Research Fund Denmark (1026-00050B).}\addressmark{1}, Wonwoo Kang\thanks{\href{mailto:wonwook2@illinois.edu}{wonwook2@illinois.edu}}\addressmark{2}, \and Elizabeth Kelley\thanks{\href{mailto:kelleye@illinois.edu}{kelleye@illinois.edu}. Supported by DMS-1937241.}\addressmark{2}}

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\address{\addressmark{1}Institute for Mathematics, Aarhus University \\ \addressmark{2}Department of Mathematics, University of Illinois Urbana Champaign}

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\abstract{We use a combinatorial expansion formula for cluster algebras of surface type via order ideals of posets to give explicit skein relations for elements of a cluster algebra arising from a punctured surfaces. An immediate corollary of this is that the bangles and bracelets of Musiker, Schiffler, and Williams, which are known to provide a basis in the unpunctured case, form a spanning set in the punctured case.}
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\keywords{skein relation, triangulated surfaces, cluster algebras}

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\begin{document}

\maketitle
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\section{Introduction}

Subsequent to the original introduction of cluster algebras by Fomin and Zelevinsky in 2002 \cite{FZ2002}, a significant amount of effort has been devoted to studying cluster algebras of \emph{surface type}, as defined in \cite{FST-I, FT-II}. Such cluster algebras are particularly appealing objects of study because they admit constructions of a variety of combinatorial objects - including snake graphs, $T$-paths, and posets - that can be used to prove important structural results about positivity or the existence of bases. In this extended abstract, we use a cluster expansion formula from \cite{oğuz2024cluster,pilaud2023posets} which expresses elements of a cluster algebra as generating functions of order ideals of certain posets. We use this expansion formula to prove \emph{skein relations}, i.e. relations used to resolve intersections or incompatibilities of arcs. Topologically, a skein relation takes a pair of intersecting arcs or an arc with self-intersection and replaces this configuration with two sets of arcs which avoid the intersection in two different ways. This method gives a generalization and new perspective to \emph{snake graph calculus}, as defined in \cite{ccanakcci2020snake}. Skein relations for unpunctured surfaces were given in \cite{musiker2013, snakegraphcalculus1}. Skein relations on punctured surfaces in the coefficient-free case were discussed in \cite{Mandel2023} and specific forms of skein relations in the principal coefficient case (so called ``tidy exchange relations'') were given in \cite{pilaud2023posets}. Here, we give explicit formulae and show all skein relations on (potentially punctured) surfaces contain a term that is not divisible by any coefficient variable $y_i$.
%Here, we give an explicit formula for skein relation and thereby showing that all skein relations on a (potentially punctured) surfaces contain a term that is not divisible by any coefficient variable $y_i$. 
Consequently, we observe that the \emph{bangles} and \emph{bracelets} defined in \cite{musiker2013bases} form spanning sets and are linearly independent.
% Again, I don't know whether it's okay, please feel free to ask me any question about this modification regarding M-path.
\section{Background}
\label{sec:background}

Cluster algebras are a type of recursively generated commutative ring with distinguished generators, called \emph{cluster variables}, that appear in fixed-size subsets $\mathbf{x} = (x_1, \dots, x_n)$ called \emph{clusters}. Each cluster $\mathbf{x}$ has an associated set of \emph{coefficients} $\mathbf{y} = (y_1, \dots, y_n)$. Clusters can be obtained from each other via an involutive process called \emph{mutation}. A single mutation $\mu_k$ uniquely exchanges a cluster variable $x_k \in \mathbf{x}$ for some $x_k' \not\in \mathbf{x}$. The relation between $x_k$ and $x_k'$ is referred to as an \emph{exchange relation}. Given a cluster $\mathbf{x}$, it is always possible to mutate at every $x_i \in \mathbf{x}$. A single cluster is sufficient to generate the entire cluster algebra. 

Two of the most celebrated properties of cluster algebras are the \emph{Laurent phenomenon} and \emph{positivity}, which together state that every cluster algebra element can be written as a Laurent polynomial with positive integer coefficients in terms of any choice of cluster.

Triangulated surfaces provide a well-known geometric model for \emph{ordinary cluster algebras of surface type} \cite{FST-I, FT-II}. Let $S$ be a surface with (potentially empty) boundary and a non-empty set of \emph{marked points} $M$, where there is at least one marked point on each boundary component. Marked points in the interior of $S$ are referred to as
\emph{punctures}. Every such marked surface $(S,M)$ has an associated cluster algebra $\mathcal{A}_{S}$. Clusters of $\mathcal{A}_S$ correspond to distinct triangulations of $(S,M)$, with individual cluster variables corresponding to individual arcs (i.e., curves with endpoints in $M$ and no self-intersections). Coefficients correspond to \emph{laminations} \cite{FT-II}, i.e. additional collections of curves on $(S,M)$ that meet certain conditions. Following the restrictions in \cite{musiker2011positivity}, we do not allow $(S,M)$  to be a closed surface with exactly two punctures, a monogon with less than two punctures, an unpunctured bigon or triangle, or a sphere with less than four punctures.

In the surface model, mutation at $x_k$ is represented by \emph{flipping} the corresponding arc $\gamma$ in a triangulation $T$ - that is, by replacing $\gamma$ with a different arc $\gamma'$, which corresponds to $x_k'$, such that $T - \{\gamma\} \cup \{\gamma'\}$ is still a valid triangulation. To provide complete geometric models for cluster algebras from punctured surfaces
%in which it is possible to mutate at all cluster variables, 
\cite{FST-I} introduced the more general notion of \emph{tagged arcs}. A \emph{tagged arc} is an arc whose ends have been tagged either \emph{plain} or \emph{notched} such that: the arc does not cut out a once-punctured monogon, any end on $\partial S$ is tagged plain, and both ends of a loop have the same tagging.

If $\eta$ is a tagged arc with endpoints $p$ and $q$, we write $\eta^{0}$ to denote the underlying plain arc. If we wish to emphasize the notching of $\eta$, we will write $\eta^{(p)}$ when $\eta$ has a single notched end at $p$ and $\eta^{(pq)}$ when $\eta$ is notched at both endpoints. Two tagged arcs $\alpha$ and $\beta$ are \emph{compatible} if and only if the following properties hold: the isotopy classes of $\alpha^{0}$ and $\beta^{0}$ contain non-intersecting representatives; if $\alpha^{0} = \beta^{0}$ then at least one end of $\alpha$ has the same tagging as the corresponding end of $\beta$; and if $\alpha^{0} \neq \beta^{0}$ have a shared endpoint, then $\alpha$ and $\beta$ must have the same tagging at that endpoint. A \emph{tagged triangulation} is a maximal collection of pairwise compatible tagged arcs. We will work with clusters associated to triangulations with only plain arcs.

%An ideal triangulation is one where all arcs are plain and arcs cutting out once-punctured monogons are allowed. \esther{do we ever mention ideal triangulations?}



\section{Cluster expansion formula}
\label{sec:expansionFormula}

\subsection{The poset for an arc}
\label{subsec:posetConstruction}

Let $T = \{\tau_1,\ldots, \tau_n\}$ be a triangulation of a surface $(S,M)$. For any arc $\gamma$ on $(S,M)$, we construct a corresponding poset $P_{\gamma}$, following \cite{oğuz2024cluster, pilaud2023posets}.  We note that the posets $P_\gamma$ will be exactly the poset of join-irreducibles in the lattice of perfect matchings of the snake graph $\mathcal{G}_\gamma$, as in \cite{musiker2011positivity,wilson2020surface}.

First, suppose that $\gamma$ is an arc with both endpoints tagged plain. Fix an orientation for $\gamma$ and let $\tau_{i_1}, \ldots \tau_{i_d}$ be the list of arcs of $T$ crossed by $\gamma$, in the order determined by our choice of orientation of $\gamma$.
We will place a poset structure on $[d]$ in the following way. Any two consecutive arcs crossed by $\gamma$, $\tau_{i_j}$ and $\tau_{i_{j+1}}$, border a triangle $\Delta_j$ that $\gamma$ passes through between these crossings. Let $s_j$ denote the shared endpoint of $\tau_{i_j}$ and $\tau_{i_{j+1}}$ which is a vertex of $\Delta_j$. If $s_j$ lies to the right of $\gamma$ (with respect to the orientation placed on $\gamma$), then we set $j \gtrdot j+1$. Otherwise, we set $j \lessdot j+1$. The resulting poset is sometimes referred to as a \emph{fence poset} since its Hasse diagram is a path graph. The process is the same if $\gamma$ is a \emph{generalized arc}, so that it has self-intersections.

\begin{figure}[]
    \centering
    \begin{tikzpicture}

    \draw (4.5,-1) to (5,0);
    \draw (4,1.5) to (5,0);
    \draw (6.5,0) to (5,0);
    \draw (3,-1.2) to (4.5,-1);
    \draw (4.5,-1) to (4,1.5);
    \draw (3,-1.2) to (4,1.5);
    \draw (3,-1.2) to (3,1.5);
    \draw (0,1) to (1.5,0);
    \draw (0,1) to (0,-1);
    \draw (0,-1) to (1.5,0);
    \draw (3,1.5) to (1.5,0);
    \draw (3,-1.2) to (1.5,0);
    \draw (3,-1.2) to (0,-1);
    \draw (3,1.5) to (0,1);
    \draw (3,1.5) to (4,1.5);
    \draw[out=0,in=100, looseness=1.5] (4,1.5) to (6.5,0);
    \draw[out=0,in=-100, looseness=1.5] (4.5,-1) to (6.5,0);

    %adding edge labels

    \node[fill=white,scale=0.7] at (0.6,0.5) {$\sigma_2$};
    \node[fill=white,scale=0.7] at (0.8,-0.4) {$\sigma_3$};
    \node[fill=white,scale=0.7] at (2,-0.3) {$\sigma_4$};
    \node[fill=white,scale=0.7] at (2,0.4) {$\sigma_1$};

    \node[fill=white,scale=0.7] at (3,0.5) {$\tau_1$};
    \node[fill=white,scale=0.7] at (3.5,0.2) {$\tau_2$};
    \node[fill=white,scale=0.7] at (4.15,0.3) {$\tau_3$};

    \draw[red,thick] (1.5,0) to (5,0);
    \draw[out=90,in=90, looseness=0.8, blue, thick] (1,0) to (5.5,0);

    \node[red, fill=white, scale=0.8] at (2.8,0.0) {$\gamma_1$};
    \node[blue, fill=white, scale=0.8] at (3.3,1) {$\gamma_2$};

    \node[fill=white,scale=0.7] at (4.7,0.4) {$\eta_3$};
    \node[fill=white,scale=0.7] at (5.8,0) {$\eta_2$};
    \node[fill=white,scale=0.7] at (4.8,-0.3) {$\eta_1$};
    \draw[out=-90,in=-90, looseness=0.5, blue, thick] (1,0) to (5.5,0);

    \node[scale=0.8] at (1.5,0.2) {$p$};
    \node[scale=0.8] at (5.1,0.2) {$q$};

%draw vertices
    \draw[fill=black] (5,0) circle [radius=2pt];
    \draw[fill=black] (6.5,0) circle [radius=2pt];
    \draw[fill=black] (4,1.5) circle [radius=2pt];
    \draw[fill=black] (4.5,-1) circle [radius=2pt];
    \draw[fill=black] (3,-1.2) circle [radius=2pt];
    \draw[fill=black] (3,1.5) circle [radius=2pt];
    \draw[fill=black] (0,1) circle [radius=2pt];
    \draw[fill=black] (0,-1) circle [radius=2pt];
    \draw[fill=black] (1.5,0) circle [radius=2pt];

    \end{tikzpicture}
    \caption{An example of an arc $\gamma_1$ and closed curve $\gamma_2$ on a triangulated surface.}
    \label{fig:arcs}
\end{figure}


Next, suppose that $\gamma^{(p)}$ is notched at its starting point $s(\gamma) = p$. Begin by drawing the fence poset for $\gamma^0$. Suppose the first triangle $\gamma$ passes through  is $\Delta_0$. Necessarily, $\Delta_0$ is bordered by $\tau_{i_1}$ and two spokes at $p$. Label these spokes $\sigma_1,\sigma_{m}$ where $\sigma_1$ is the clockwise neighbor of $\tau_{i_1}$. Label the remaining spokes in clockwise order. Then, we include elements $1^s,\ldots,m^s$ in the poset, and set $m^s \lessdot (m-1)^s \lessdot \cdots \lessdot 1^s$, $1^s \gtrdot 1$ and $m^s \lessdot 1$. If we have an arc which is instead notched at its terminal point, we repeat this process with elements $1^t,\ldots,m^t$, and we combine these processes for an arc tagged at both endpoints.  We call the resulting posets \emph{loop fence posets} as they correspond to the loop graphs given by Wilson in \cite{wilson2020surface}. We say that the elements $1^s,\ldots,m^s$ are in a \emph{loop}. If we wish to refer to a loop fence poset $P$ with the loop portion removed, we will denote this $P^0$, so that $P_{\gamma^0} = P_\gamma^0$.

Finally, suppose that $\gamma$ is a closed curve. Choose an point $a$ of $\gamma$ which is not a point of intersection between $\gamma$ and $T$. Treat $\gamma$ like an arc with $s(\gamma) = t(\gamma) = a$, choose an orientation of $\gamma$, and form the fence poset on $[d]$ associated to this arc. It must be that $\tau_{i_1}$ and $\tau_{i_d}$ share an endpoint which is an endpoint of the triangle containing $a$. If this endpoint is to the right of $\gamma$ with the chosen orientation, we set $d \gtrdot 1$; otherwise we set $d \lessdot 1$. These posets are called \emph{circular fence posets} since the underlying graph of such a Hasse diagram is a cycle. To improve readability, we will often refer to all of these types of posets as fence posets unless the specific type is relevant, in which case we use the specific term. See Table \ref{table:posetExamples} for several examples; note here and for the remainder of the paper, we label the poset elements with the arcs they correspond to and we conflate these two notions when context is clear.

%The construction of these posets underlying theorems are based on \cite{oğuz2024cluster} and \cite{pilaud2023posets} where they constructed the fence poset $P_{\gamma}$ of an arc $\gamma$ and used it to express cluster variable as a sum of order ideals. \cite{oğuz2024cluster} proved this using products of matrices and \cite{pilaud2023posets} shows this result in more general case, arcs that are in $T$, using the exchange relations in $F$-polynomials.


%\elizabeth{Maybe: These posets also appear in \cite{wilson2020surface} and \cite{oğuz2024cluster}, where Oğuz and Yıldırım give a slightly different expansion formula that also requires the definition of a \emph{T-walk}. Similar posets and expansion formulae also appear in \cite{pilaud2023posets}, with modification to allow self-folded triangles.}
%\elizabeth{I think the comment about the proof of some exchange relations in the Pilaud-Schroll-Reading paper actually belongs in a later section where skein relations are being discussed.}

%\elizabeth{There are also some similar posets in \href{https://arxiv.org/pdf/2303.03466.pdf}{this paper} by Weng. I have only skimmed this paper, so I can't say much else about it.}

\begin{table}[]
    \centering
\begin{tabular}{c|c}
$P_{\gamma_1^{(pq)}}$ & $P_{\gamma_2}$ \\
\begin{tikzpicture}
            \node (s4) at (0,0) {$\sigma_4$};
            \node (s3) at (0.5,-1) {$\sigma_3$};
            \node (s2) at (1,-2) {$\sigma_2$};
            \node (s1) at (1.5,-3) {$\sigma_1$};
            \node[blue] (t1) at (2,-2) {$\tau_1$};
            \node[blue] (t2) at (2.5,-3) {$\tau_2$};
            \node[blue] (t3) at (3,-2) {$\tau_3$};
            \node (e1) at (3.5,-3) {$\eta_1$};
            \node (e2) at (4,-2) {$\eta_2$};
            \node (e3) at (4.5,-1) {$\eta_3$};

            \draw (s4) to (t1);
            \draw (s4) to (s3);
            \draw (s3) to (s2);
            \draw (s2) to (s1);
            \draw (s1) to (t1);
            \draw[blue] (t1) to (t2);
            \draw[blue] (t2) to (t3);
            \draw (t3) to (e1);
            \draw (e1) to (e2);
            \draw (e2) to (e3);
            \draw (t3) to (e3);
        \end{tikzpicture}
&
\begin{tikzpicture}
        \node (t2-1) at (0,0) {$\tau_2$};
        \node (t3-1) at (0.5,1) {$\tau_3$};
        \node (s3) at (1,2) {$\eta_3$};
        \node (s2) at (1.5,1) {$\eta_2$};
        \node (s1) at (2,0) {$\eta_1$};
        \node (t3-2) at (2.5,1) {$\tau_3$};
        \node (t2-2) at (3,0) {$\tau_2$};
        \node (t1-2) at (3.5,1) {$\tau_1$};
        \node (e4) at (4,2) {$\sigma_4$};
        \node (e3) at (4.5,1) {$\sigma_3$};
        \node (e2) at (5,0) {$\sigma_2$};
        \node (e1) at (5.5,-1) {$\sigma_1$};
        \node (t1-1) at (6,1) {$\tau_1$};

        \draw (t2-1) to (t3-1);
        \draw (t3-1) to (s3);
        \draw (s3) to (s2);
        \draw (s2) to (s1);
        \draw (s1) to (t3-2);
        \draw (t3-2) to (t2-2);
        \draw (t2-2) to (t1-2);
        \draw (t1-2) to (e4);
        \draw (e4) to (e3);
        \draw (e3) to (e2);
        \draw (e2) to (e1);
        \draw (e1) to (t1-1);
        \draw (t2-1) to (t1-1);
    \end{tikzpicture} \\
$\begin{aligned}
            \mathbf{a}_{\gamma_1^{(pq)}} &= \mathbf{e}_{\sigma_1} + \mathbf{e}_{\tau_2} + \mathbf{e}_{\eta_{1}} \\
            \mathbf{b}_{\gamma_1^{(pq)}} &= \mathbf{e}_{\tau_1} + \mathbf{e}_{\tau_3} \\
            \mathbf{r}_{\gamma_1^{(pq)}} &= 0
\end{aligned}$
&
$\begin{aligned}
        \mathbf{a}_{\gamma_2} &= \mathbf{e}_{\eta_{1}} + \mathbf{e}_{\tau_2} + \mathbf{e}_{\sigma_1} \\
        \mathbf{b}_{\gamma_2} &= \mathbf{e}_{\eta_3} + \mathbf{e}_{\tau_3} + \mathbf{e}_{\sigma_4} \\
        \mathbf{r}_{\gamma_2} &= 0
    \end{aligned}$
\end{tabular}

\caption{The loop fence poset $P_{\gamma_1^{(pq)}}$ and circular fence poset $P_{\gamma_2}$ for the arcs from Figure~\ref{fig:arcs}. Note that the fence poset $P_{\gamma_1} = P_{\gamma_{1}^{(pq)}}^0$ for the plain arc $\gamma_1$ appears as a subposet of $P_{\gamma_1^{(pq)}}$, indicated in blue, and has  $\mathbf{a}_{\gamma_1} = \mathbf{e}_{\tau_2}$, $\mathbf{b}_{\gamma_1} = 0$, and $\mathbf{r}_{\gamma_1} = \mathbf{e}_{\sigma_1} + \mathbf{e}_{\eta_1}$.}

\label{table:posetExamples}
\end{table}

\subsection{Minimal Terms}

Let $\mathbf{a}_\gamma = (a_1,\ldots,a_n)$ where $a_j$ is the number of times there is a minimal element $\tau_{i_k} \in P_\gamma$ such that $\tau_{i_k} = \tau_j$. Let $\mathbf{b}_\gamma = (b_1,\ldots,b_n)$ where $b_j$ is the number of times there is an element $\tau_{i_k} \in P_\gamma$ which covers at least two elements and is not in a loop such that $\tau_{i_k} = \tau_j$. Note that one or both of the elements which $\tau_{i_k}$ covers can be in a loop. %For shorthand, we will refer to minimal elements as ``deeps'' and maximal elements as ``peaks''. Moreover, we will say a deep or peak is ``strict'' if the element covers or is covered by at least two elements.

 Suppose $\gamma$ is an plain arc and there exists $\tau_i, \tau_j \in T$ such that $\tau_i$ follows $\tau_1$ in clockwise order in $\Delta_0$, the first triangle $\gamma$ passes through,  and similarly $\tau_j$ follows $\tau_d$ in clockwise order in $\Delta_d$, the last triangle $\gamma$ passes through. Then we set $\mathbf{r}_\gamma = \mathbf{e}_i + \mathbf{e}_j$ where $\mathbf{e}_i$ is the $i$-th standard basis vector in $\mathbb{R}^n$. If $\gamma$ is instead notched at an endpoint or the clockwise neighbor of $\tau_1$ or $\tau_d$ is on the boundary of $(S,M)$, then we omit its contribution. 


Given any arc or closed curve $\gamma$, we define $\mathbf{g}_\gamma := -\mathbf{a}_\gamma + \mathbf{b}_\gamma + \mathbf{r}_\gamma$. 
We remark that this notation is inspired by the notation for the $g$-vector of a string module, as in \cite{palu2021non}. 

Gei\ss, Labardini-Fragoso, and Schr{\"o}er studied these $\gb$-vectors for plain arcs and closed curves in \cite{geiss2022schemes}. In particular, using Proposition 10.14 and Remark 11.1, they showed that $\mathbf{x}^{\gb_{\gamma}}$ is the unique term in $x_\gamma^T$ which is not divisible by any variable $y_i$. We show the same statement for a notched arc $\gamma$. 

\begin{lem}\label{lem:gvectorminimal}
Let $T$ be a triangulation of a surface without self-folded triangles. The monomial $\mathbf{x}^{\gb_{\gamma}}$ is the unique term in the expansion of $x_\gamma^T$ which is not divisible by any variable $y_i$.
\end{lem}

%\begin{proof}
%When $\gamma$ is a plain arc or closed curve, this comes from Remark 11.1 and Proposition 10.14 in \cite{geiss2022schemes}. When $\gamma$ is a notched arc, this follows from analyzing how maximal and minimal elements in $P_{\gamma}$ correspond to subgraphs in the loop graph $\mathcal{G}_{T,\gamma}$.
%\end{proof}


 Given an arc $\tau_i \in T$, let $x_{CCW}(\tau_i) = x_{\tau_j}x_{\tau_k}$ if there are two arcs $\tau_j,\tau_k \in T$ that are counterclockwise neighbors of $\tau_i$ within the two triangles that it borders. If one or both of those neighbors is a boundary arc, then we ignore its contribution. The monomial $x_{CW}(\tau_i)$ is defined analogously using the clockwise neighbors of $\tau_i$. We set $\widehat{y}_{\tau_i} := \left(x_{CCW}(\tau_i)/x_{CW}(\tau_i) \right) y_{\tau_i}$. Let $J(P)$ denote the poset of lower order ideals of a poset $P_\gamma$. Each $I \in J(P)$ has an associated \emph{weight} $w(I) = \prod_{j \in I} \widehat{y}_{\tau_{i_j}}$.

\begin{prop}\label{prop:PosetExpansion}
 Let $\gamma$ be an arc or closed curve on a marked surface $(S,M)$ with triangulation $T$ such that $\gamma \notin T$. Then, the associated element $x_\gamma$ of the cluster algebra $\mathcal{A}(S,M)$ written with respect to the cluster corresponding to $T$ can be expressed by \[
x_\gamma^T = \mathbf{x}^{\mathbf{g}_\gamma} \sum_{I \in J(P_\gamma)} w(I).
 \]
\end{prop}

\begin{proof}
If $\gamma$ is not an arc such that $\gamma^0 \in T$, then this follows from combining Proposition 3.2 in \cite{oğuz2024cluster} with Lemma \ref{lem:gvectorminimal}. If $\gamma \neq \gamma^0$ and $\gamma^0 \in T$, we prove this expansion formula by using the algebraic identities that relate a singly-notched arc to plain arc and Theorem 12.9 in \cite{musiker2011positivity}, which relates a doubly-notched arc to plain and singly-notched arcs. 
\end{proof}

\begin{example}
Applying Proposition~\ref{prop:PosetExpansion} to the arc $\gamma_1^{(pq)}$ from Table~\ref{table:posetExamples} produces
\[ x_{\gamma_1^{(pq)}} = \frac{x_{\tau_1}x_{\tau_3}}{x_{\sigma_4}x_{\tau_2}x_{\eta_3}} \left[ \frac{x_{\sigma_3}y_{\sigma_4}y_{\tau_2}}{x_{\sigma_1}x_{\tau_1}^2x_{\tau_3}} + \frac{x_{\eta_1}y_{\eta_3}}{x_{\tau_1}x_{\tau_3}^2x_{\eta_2}} + \frac{x_{\sigma_3}x_{\eta_1}y_{\sigma_4}y_{\eta_3}}{x_{\tau_1}x_{\sigma_1}x_{\tau_3}x_{\eta_2}} + \frac{y_{\eta_2}y_{\eta_3}}{x_{\tau_3}} + \frac{x_{\sigma_2}x_{\sigma_3}y_{\sigma_3}}{x_{\sigma_1}x_{\sigma_4}} + \cdots \right] \]
where we have explicitly shown only the terms arising from order ideals of size two.
\end{example}

\section{Skein Relations}

Let $\gamma_1$ and $\gamma_2$ be two curves with a point of incompatibility $s$; by this, we mean that either $\gamma_1$ and $\gamma_2$ intersect, or $\gamma_1^0 \neq \gamma_2^0$ share an endpoint and have opposite taggings at the endpoint. In some cases, $\gamma_1$ and $\gamma_2$ cross the same set of arcs before or after passing through $s$; if $s$ is an intersection point, as we vary the representatives of $\gamma_1$ and $\gamma_2$ in their isotopy classes, the point $s$ can lie on any of these arcs. We call such a configuration a \emph{crossing overlap}. When the set-up is understood, we refer to this set of commonly crossed arcs as $R$.  

%Then, natural question is, what is $s$ if it's the case of incompatible at a puncture? We understand tagged arc $\gamma$ as an \emph{associated hooked arc} $\widetilde{\gamma}$. Check the example at Table~\ref{tab:r_and_s}.

If two arcs cross and this point of intersection is near the endpoint of one arc, then when we form some of the arcs in the resolution, these will \emph{pivot} at this endpoint. For example, in the left diagram in Table \ref{tab:r_and_s}, the left arc $C^-$ pivots across $\sigma_2$ in counterclockwise direction and the right arc $C^-$ pivots across $\sigma_6$ in clockwise direction. Some of these pivots will also affect the $y$-monomial in the resolution. For a pair of crossing curves, we define the \emph{sweep set}, denoted $Sw$, to be the set of arcs that an arc in the resolution pivots past in in clockwise (resp counterclockwise) direction at a plain (resp notched) endpoint.  Now suppose we instead have two arcs with incompatible taggings at a puncture $p$. Suppose $\gamma_1^{(p)}$ is tagged at $p$ and $\gamma_2$ is not. Then, we define the sweep set to be the set of arcs from $T$ which lie counterclockwise of $\gamma_1$ and clockwise of $\gamma_2$. See Table \ref{tab:r_and_s} for examples. 

Given two arcs with an incompatibility and associated sets $R \cup Sw$, one can show that one of the sets of arcs in the resolution at the incompatibility will not cross any of the arcs in $R \cup Sw$. We will label the sets of arcs (called \emph{multicurves}) in the resolution as  $C^{+}$ and $C^{-}$ where $C^-$ is the set which does not cross any arcs in $R$ and $Sw$.


\begin{table}[h]
    \centering
    \begin{tabular}{c|c}
        \includegraphics[scale=0.9]{Diagrams/ex2.pdf}
         &  
        \includegraphics[scale=0.4]{Diagrams/ex1.pdf}
         \\
        \hline
        $R=\{\sigma_3,\sigma_4,\sigma_5\}$ & $R=\emptyset$\\
        $Sw=\{\sigma_6\}$ & $Sw=\{\sigma_1,\sigma_2,\sigma_3,\sigma_4\}$
    \end{tabular}
    \caption{Examples of $R$ and $Sw$ for a transverse crossing (left) and an incompatibility at a puncture (right).}
    \label{tab:r_and_s}
\end{table}



\begin{thm}\label{thm:FPSACMain}
\begin{enumerate}
    \item Let $\{\gamma_1, \gamma_2\}$ be a multicurve of arcs or closed curves which are incompatible.  Choose one point of incompatibility and let $C^+$ and $C^-$ be the resolution at this intersection. Then, $x_{\gamma_1}x_{\gamma_2} = x_{C^+} + Y_R Y_{Sw} x_{C^-}$.
    \item Let $\gamma_1$ be an arc or closed curve which is incompatible with itself. Choose one point of incompatibility and let $C^+$ and $C^-$ be the resolution at this intersection.\\
    Then, ${x_{\gamma_1}= x_{C^+} + Y_R Y_{Sw} x_{C^-}}$.
\end{enumerate}
\end{thm}

We prove Theorem \ref{thm:FPSACMain} in cases. In section \ref{sec:GeneralProofideaFPSAC}, we will explain our proof method which can be used for all cases. In Sections \ref{sec:IncompatibleFPSAC} and \ref{sec:TransverseFPSAC}, respectively, we will explicitly prove this Theorem for a pair of arcs with incompatible taggings at a puncture and a pair of arcs with a transverse crossing. The relations that we explicitly discuss in these sections are helpful in unifying some of the cases outlined in \cite{musiker2013bases}. For example, the relation discussed in \ref{sec:IncompatibleFPSAC} can be used to handle cases 6-11 from \cite{musiker2013bases}. For the sake of brevity, we only include explicit proofs for these two examples.

\subsection{General approach}\label{sec:GeneralProofideaFPSAC}

 Let $\gamma_1$ and $\gamma_2$ be two curves with a point of incompatibility and resolutions $C^+$ and $C^-$. Set $P_i:= P_{\gamma_i}$ and $\gb_i:= \gb_{\gamma_i}$. In light of Proposition \ref{prop:PosetExpansion}, we can write $x_{\gamma_1}x_{\gamma_2}$ as \[
\mathbf{x}^{\gb_{1} + \gb_{2}} \sum_{(I_1,I_2) \in J(P_{1}) \times J(P_{2})} w(I_1)w(I_2).
\]
  We set $w(I_1,I_2)$ to be the product of the weights of the components $w(I_1)w(I_2)$. If $C^+ = \{\gamma_3,\gamma_4\}$, then we set $J(C^+) = J(P_{3}) \times J(P_{4})$; otherwise, $C^+$ is a singleton $\{\gamma_3\}$ and we set $J(C^+) = J(P_{3})$. We define $J(C^-)$ similarly. Our method of proof centers on finding a partition of $J(\calP_1) \times J(\calP_2) = A \sqcup B$ such that $(\emptyset,\emptyset) \in A$, and bijections $\Phi_A$ between $A$ and $J(C^+)$ and $\Phi_B$ between $B$ and $J(C^-)$. Moreover, we require that the bijection between $A$ and $J(C^+)$ is weight-preserving, so that $w(I_1)w(I_2) = w(\Phi_A(I_1,I_2))$ and that the bijection between $B$ and $J(C^-)$ is weight preserving up to a unique monomial, so that for some monomial $Z$ in $x$ and $y$ variables, $w(I_1)w(I_2) = Zw(\Phi_B(I_1,I_2))$. Let $Z = XY$ be the decomposition of $Z$ into $x$ and $y$ variables. The final step of each proof is to show that $\gb_{1}+\gb_{2}$ is equal to the sum of the $\gb$-vectors for the posets in $C^+$ (denoted $\gb_{C^+}$) and $\gb_1 + \gb_2 + \deg(X)$ is equal to the the sum of the $\gb$-vectors for the posets in $C^-$ (denoted $\gb_{C^-}$) where $\deg(X) = (\deg_{x_{\tau_1}}(X),\ldots,\deg_{x_{\tau_n}}(X))$. Then, we can rewrite $x_{\gamma_1}x_{\gamma_2}$ as 
\begin{align*}
&=\mathbf{x}^{\gb_1 + \gb_2} \sum_{(I_1,I_2) \in A} w(\Phi_A(I_1,I_2)) + \mathbf{x}^{\gb_1 + \gb_2} \sum_{(I_1,I_2) \in B} Zw(\Phi_B(I_1,I_2)) \\
&=\mathbf{x}^{\gb_{C^+}} \sum_{\boldsymbol{I} \in J(C^+)} w(\boldsymbol{I}) + \mathbf{x}^{\gb_{C^-}}Y \sum_{\boldsymbol{I} \in J(C^-)} w(\boldsymbol{I}) = x_{C^+} + Y x_{C^-},
\end{align*}
where $x_{C^+}$ is the product of $x$ variables associated to the arcs in $C^+$
In each example, $Z$ will be a product of $\hat{y}$-variables that corresponds to the preimage of a tuple of emptysets in $J(C^-)$.
For part (2) of Theorem \ref{thm:FPSACMain}, we have similar statements with just one poset $I_1$. When resolving a self-intersection, it is possible for one arc to have a contractible kink, in which case we remove the kink and multiply the associated expression by $-1$; in this case, the bijections are adjusted to account for the difference in sign.

\subsection{Incompatibility at punctures}\label{sec:IncompatibleFPSAC}


Consider two arcs, $\gamma_1^{(p)}$ and $\gamma_2$ which are incompatible at a puncture $p$ as on the right hand side of Table \ref{tab:r_and_s}. Recall from Section~\ref{sec:background} that this means  $\gamma_1^{(p)}$ and $\gamma_2$ have opposite taggings at $p$ and $\gamma_1^0 \neq \gamma_2^0$. Orient $\gamma_1^{(p)}$ and $\gamma_2$ to both begin at $p$. Let the spokes at $p$ from $T$ be $\sigma_1,\ldots,\sigma_m$, labeled in counterclockwise order such that the first triangle that $\gamma_1^{(p)}$ passes through is bounded by $\sigma_1$ and $\sigma_m$. If $\gamma_2 \notin T$, let $1 \leq k \leq m$ be such that the first triangle $\gamma_2$ passes through is bounded by $\sigma_k$ and $\sigma_{k+1}$, where we interpret $\sigma_{m+1}$ as $\sigma_1$. If $\gamma_2 \in T$, then we let $k$ be such that $\gamma_2 = \sigma_k$.

Draw a small circle $h$ that encompasses $p$ and does not cross any arcs of $T$ except the spokes at $p$. We define $\gamma_1^{-1} \circ_{CCW} \gamma_2$ as the arc which results from following $\gamma_1$ from $t(\gamma_1)$ with reverse orientation until its intersection with $h$, following $h$ counterclockwise until its intersection with $\gamma_2$, and then following $\gamma_2$ until $t(\gamma_2)$. We define $\gamma_1^{-1} \circ_{CW} \gamma_2$ similarly.  Set $\gamma_3:= \gamma_1^{-1} \circ_{CCW} \gamma_2$ and $\gamma_4:= \gamma_1^{-1} \circ_{CW} \gamma_2$ and note that $\gamma_3$ crosses $\sigma_1,\ldots,\sigma_k$ and $\gamma_4$ crosses $\sigma_{k+1},\ldots,\sigma_m$. On the right hand side of Table \ref{tab:r_and_s}, $k = 4$, $\gamma_3$ is the arc denoted $C^+$ and $\gamma_4$ is the arc denoted $C^-$.


When $k = m$ and $\gamma_2 \notin T$, so that the first triangles $\gamma_1^{(p)}$ and $\gamma_2$ pass through are the same, then we have two additional cases based on whether $\gamma_1^{(p)}$ is clockwise or counterclockwise of $\gamma_2$ at $p$. Since these cases produce different sets $Sw$, we differentiate them. We refer to the case where $\gamma_1^{(p)}$ lies clockwise from $\gamma_2$ as the $k=0$ case. 

It is only in the $k=0$ and $k=m$ cases when $\gamma_2 \notin T$ that we will have a crossing overlap.  If $\tau_{i_1},\ldots,\tau_{i_{d_1}}$ and $\tau_{j_1},\ldots,\tau_{j_{d_2}}$ are the ordered sequences of arcs from $T$ crossed by $\gamma_1$ and $\gamma_2$ respectively, and $w \geq 1$ is the largest number such that $\tau_{i_r} = \tau_{j_r}$ for all $1 \leq r \leq w$, then $R = \{\tau_{i_1},\ldots,\tau_{i_w}\}$, regarded as a multiset. When $\gamma_2 \in T$, then there is no possible case for $k =0$, and $R = \emptyset$ in the $k = m$ case.

\begin{prop}
\label{prop:incompatible_puncture_singly_tagged}
Let $\gamma_1^{(p)}$ and $\gamma_2$ be arcs which are incompatible at a puncture $p$. For $k$ and $R$ as defined above, set \[
Y_R = \prod_{\tau \in R} y_{\tau} \quad \text{and} \quad Y_{Sw} = \prod_{\sigma_i \in Sw} y_{\sigma_i} = \prod_{i=1}^{k} y_{\sigma_i}.
\]

Then, we have $x_{\gamma_1^{(p)}}x_{\gamma_2}=C^{+}+C^{-}$ where $C^{+}$ and $C^{-}$ are defined as follows:
\begin{center}
\begin{tabular}{ |c | c | c| } 
 \hline
  & $C^{+}$ & $C^{-}$ \\
 \hline
 $k\neq0, m$ & $x_{\gamma_3}$ & $Y_{Sw} x_{\gamma_4}$ \\ 
 \hline
 $k=0$ & $x_{\gamma_4}$ &  $Y_Rx_{\gamma_3}$\\ 
 \hline
 $k=m$ & $x_{\gamma_3}$ & $Y_{Sw}Y_Rx_{\gamma_4}$ \\ 
 \hline
\end{tabular} 
\end{center}

\end{prop}
\begin{proof}
We detail the $k \neq 0,m$ and $\gamma_2 \notin T$ case; the special cases follow from various modifications to these overarching ideas. The posets $P_{\gamma_1^{(p)}}, P_{\gamma_3}$ and $P_{\gamma_4}$ are provided in Table \ref{tab:PosetsKIsNot0orM}; we suppress the poset $P_{\gamma_2}$ as its structure is not important for the proof.

\begin{table}[]
    \centering
\begin{tabular}{c|c|c}
    $P_{\gamma_1^{(p)}}$ & {$P_{\gamma_3}$} & {$P_{\gamma_4}$} \\
    \begin{tikzpicture}[scale = 0.6]
    \node[](dots) at (8,0){$\cdots$};
    \node[](alpha) at (6.5,0){$\tau_{i_1}$};
    \node[](sigma1) at (5,-1){$\sigma_1$};
    \node[](sigma2) at (5,0) {$\sigma_2$};
    \node[scale=0.75,yshift=9](dots1) at (5,0.5){$\vdots$};
    \node[](sigmak) at (5,1.2){$\sigma_k$};
    \node[scale=0.75,yshift=7](dots2) at (5,1.7){$\vdots$};
    \node[](sigmas) at (5,2.6){$\sigma_m$};
    %\draw(alpha)--(dots);
    \draw(alpha)--(sigma1);
    \draw(sigma1) -- (sigma2);
    %\draw (sigma2) -- (dots1);
    %\draw (dots1) -- (sigmak);
    %\draw(sigmak) -- (dots2);
    %\draw(dots2) -- (sigmas);
    \draw(sigmas) -- (alpha);
    \end{tikzpicture} &  
    \begin{tikzpicture}[scale = 0.6]
    \node[](dots) at (-1.24,0){$\cdots$};
    \node[](tau) at (0,0){$\tau_{i_1}$};
    \node[](sigma1) at (1.24,-1){$\sigma_1$};
    \node[](sigma2) at (2.5,0){$\sigma_2$};
    \node[rotate=10,yshift=6] (dots3) at (3.3,0.7){$\iddots$};
    \node[](sigmak) at (4,1.5){$\sigma_k$};
    \node[](etam) at (5,0.5){$\tau_{j_1}$};
    \node[](dots2) at (6,0.5){$\cdots$};
    %\draw(tau) -- (dots);
    \draw(tau) -- (sigma1);
    \draw (sigma1) -- (sigma2);
    %\draw(sigma2) -- (dots3);
    %\draw(dots3) --(sigmak);
    \draw(sigmak) -- (etam);
    %\draw(dots2) -- (etam);
    \end{tikzpicture}
    &  
    \begin{tikzpicture}[scale = 0.6]
    \node[](dots) at (-1,0){$\cdots$};
    \node[](tau) at (0,0){$\tau_{i_1}$};
    \node[](sigma1) at (1,1){$\sigma_m$};
    \node[](sigma2) at (2,0){$\sigma_{m-1}$};
    \node[rotate=-15,yshift=5] (dots3) at (2.6,-0.75){$\ddots$};
    \node[](sigmak) at (3.3,-1.5){$\sigma_{k+1}$};
    \node[](etam) at (4.3,-0.5){$\tau_{j_1}$};
    \node[](dots2) at (5.3,-0.5){$\cdots$};
   % \draw(tau) -- (dots);
    \draw(tau) -- (sigma1);
    \draw (sigma1) -- (sigma2);
    %\draw(sigma2) -- (dots3);
    %\draw(dots3) --(sigmak);
    \draw(sigmak) -- (etam);
    %\draw(dots2) -- (etam);
    \end{tikzpicture}\\
\end{tabular}
\caption{Posets for a resolution of an incompatibility for the $k \neq 0,m$ cases. Recall that $\tau_{i_1}$ is the first arc crossed by $\gamma_1$ and $\gamma_2$ is the first arc crossed by $\gamma_2$.}\label{tab:PosetsKIsNot0orM}
\end{table}

Let $A_1 \subseteq J(P_{\gamma_1^{(p)}}) \times J(P_{\gamma_2})$ consist of all pairs $(I_1,I_2)$ such that $\sigma_k \notin I_1$ and let $A_2$ consist of all pairs such that $\sigma_k \in I_1, \sigma_{k+1} \notin I_1$, and $\tau_{j_1} \in I_2$. Let $B$ be the complement of $A_1 \sqcup A_2$; in other words, $B$ consists of pairs $(I_1,I_2)$ such that $\tau_{j_1} \in I_2$ only if $\sigma_{k+1} \in I_1$. 

 It is clear that $A_1$ is in bijection with $\{I_3 \in J(P_3): \sigma_k \notin I_3\}$ and $A_2$ is in bijection with $\{I_3 \in J(P_3): \sigma_k \in I_3\}$, where this bijection sends each element to its image in $P_{\gamma_3}$. Similarly, we have a bijection $B \cong P_{\gamma_4}$ which sends $(I_1,I_2) \in B$ to $(I_1 \backslash \langle \sigma_k \rangle ) \cup I_2$. The description of $B$ ensures that this set is an order ideal so that this map is well-defined. 

We now compare the $g$-vectors. Let $\delta_{\tau_{i_1} > \tau_{i_2}} = 1$ if $\tau_{i_2}$ exists and $\tau_{i_1} > \tau_{i_2}$. We have that 
$\gb_{\gamma_1^{(p)}} = -\eb_{\sigma_1} + \delta_{\tau_{i_1} > \tau_{i_2}} \eb_{\tau_{i_1}} + \gb_1'$ where $\gb_1'$ involves contributions from $\tau_{i_\ell}$ for $ \ell > 1$. For simplicity, suppose $\tau_{j_1} < \tau_{j_2}$. Then, $\gb_{\gamma_2} = \eb_{\sigma_k} - \eb_{\tau_{j_1}} + \gb_2'$ for similarly defined $\gb_2'$.
We see immediately that 
$\gb_{\gamma_3} = -\eb_{\sigma_1} + \eb_{\sigma_k} + \delta_{\tau_{i_1} > \tau_{i_2}} \eb_{\tau_{i_1}} + \gb_1' - \eb_{\tau_{j_1}} + \gb_2' = \gb_1 + \gb_2$. Now, we compute  $\gb_{\gamma_4} = -\eb_{\sigma_{k+1}} + \eb_{\sigma_m} - (1 - \delta_{\tau_{i_1} > \tau_{i_2}} ) \eb_{\tau_{i_1}} + \gb_1' + \gb_2'$, so that $\gb_{\gamma_4} - (\gb_{\gamma_1^{(p)}} + \gb_{\gamma_2}) =  \eb_{\sigma_m} + \eb_{\sigma_1} + \eb_{\tau_{j_1}} - \eb_{\sigma_k} - \eb_{\sigma_{k+1}} - \eb_{\tau_{i_1}}$. Let $\sigma_{[i]}$ denote the third arc in the triangle formed by $\sigma_i$ and $\sigma_{i+1}$. Then, from the definition, we have $\hat{y}_{\sigma_i} = y_{\sigma_i} \frac{x_{\sigma_{i-1}} x_{\sigma_{[i]}}}{x_{\sigma_{i+1}}x_{\sigma_{[i-1]}}}$. One can see that $\hat{y}_{\sigma_1} \cdots \hat{y}_{\sigma_k} = (y_{\sigma_1} \cdots y_{\sigma_k}) \frac{x_{\sigma_m}x_{\sigma_{1}}x_{\sigma_{[k]}}}{x_{\sigma_{k}}x_{\sigma_{k+1}} x_{\sigma_{[0]}}}$, and the claim follows after noting that $\sigma_{[k]} = \tau_{j_1}$ and $\sigma_{[0]} = \sigma_{[m]} = \tau_{i_1}$. One can repeat similar calculations if $\tau_{j_1} > \tau_{j_2}$.

\end{proof}

\subsection{Transverse Crossings}\label{sec:TransverseFPSAC}

Here, we consider two arcs, $\gamma_1$ and $\gamma_2$ that have a point of intersection. For brevity, here we will assume these arcs have a crossing overlap, so that $R \neq \emptyset$.  If not, we have two more cases based on the fact that the point of intersection must occur in the first or last triangle of one or both of the arcs.

We orient $\gamma_1$ and $\gamma_2$ so that they pass through the arcs in $R$ in the same direction. With our fixed point of intersection $s$, let $\gamma_1 \circ \gamma_2$ denote the arc given by following $\gamma_1$ along its orientation until $s$ and then following $\gamma_2$. Let $\gamma_3 = \gamma_1\circ \gamma_2, \gamma_4 = \gamma_2\circ \gamma_1, \gamma_5 = \gamma_1 \circ \gamma_2^{-1},$ and $\gamma_6 = \gamma_2^{-1} \circ \gamma_1$, where $-1$ denotes using the reverse orientation. Note that $\gamma_3$ and $\gamma_4$ both pass through $R$, though they do not have a crossing overlap here, while $\gamma_5$ and $\gamma_6$ avoid the intersections with the arcs in $R$. Therefore, $C^+ = \{\gamma_3,\gamma_4\}$ and $C^- = \{\gamma_5,\gamma_6\}$.


%The general case is where there is a set of arcs which  $\gamma_1$ and $\gamma_2$ both cross near their intersection point. If this is not the case, then this intersection point will be in the first or last triangle which at least one of the arcs passes through. In their treatment of intersections of plain arcs, \canakci and Schiffler also differentiate between these crossings. We borrow their language in calling the former case a "crossing overlap" and the latter case "grafting." \esther{i think this will change based on how we set up the general theorem, which will already mention R and sweep} 




\begin{prop}\label{prop:GraftingType0}
Let $\gamma_1$ and $\gamma_2$ be two arcs which intersect in a crossing overlap $R$.  Let the resolution be $\{\gamma_3, \gamma_4\} \cup \{\gamma_5,\gamma_6\}$. Then,\[
x_{\gamma_1} x_{\gamma_2} = x_{\gamma_3}x_{\gamma_4} + Y_R Y_{Sw} x_{\gamma_5} x_{\gamma_6}
\]
where $Y_R = \prod_{\tau \in R} y_{\tau}$ and $Y_{Sw} = \prod_{\tau \in Sw} y_{\tau}$.

%\[
%Y_{S_s} = \begin{cases} 
%Y_{\tau_{i_1}-1}^{CCW} & \tau_{j_1} \in R \text{ and } s(\gamma_2) \text{ is notched} \\ 
%Y_{\tau_{j_1}-1}^{CW} & \tau_{i_1} \in R \text{ and } s(\gamma_1) \text{ is plain} \\ 
%1 & \text{ otherwise,}\\ 
%\end{cases}
%\quad 
%Y_{S_s} = \begin{cases} 
%Y_{\tau_{i_{d_1}}+1}^{CCW} & \tau_{j_{d_2}} \in R \text{ and t(\gamma_2) \text{ is notched} \\ 
%Y_{\tau_{j_{d_2}}+1}^{CW} & \tau_{i_{d_1}} \in R \text{ and } t(\gamma_1) \text{ is plain} \\ 
%1 & \text{ otherwise.}\\ 
%\end{cases}
%\]
\end{prop}

In the proof, we will use a poset-theoretic version of a tool from \cite{snakegraphcalculus1}. Let posets $P_1$ and $P_2$ have a crossing overlap in a region $R$. Index the elements in $P_1 \cap R$ as $P_1(1),\ldots,P_1(m)$ such that $P_1(i)$ only has cover relations with $P_1(i-1)$ and $P_1(i+1)$, when these exist, and index the elements in $P_2 \cap R$ analogously such that $P_1(i)$ and $P_2(i)$ are equivalent for each $i$. Given $I_1 \in J(P_1)$ and $I_2 \in J(P_2)$, let the \emph{switching position} be the smallest value $j$ such that $P_1(j) \in I_1$ if and only if $P_2(j) \in I_2$. One can show that a switching position exists unless $R \subseteq I_1$ and $R \cap I_2 = \emptyset$ or vice versa. 
%Since our construction for a fence poset $P_\gamma$ began with a chosen orientation of $\gamma$, there is a natural "chronological" ordering on the elements of $P_{\gamma}^0$. 
%We will occasionally use this idea in the following proof.

\begin{proof}

We say that a subset $R$ of a poset $P$ is on \emph{top} if there is no $j \in P \backslash R$ such that $j$ is larger than an element in $R$ and define a subset being on \emph{bottom} similarly. One can show that,  when $\gamma_1$ and $\gamma_2$ have a crossing overlap, up to relabeling, $R_1$ is on top and $R_2$ is on bottom. 
In the following, suppose that $\gamma_1$ crosses arcs $\alpha_{1},\ldots,\alpha_{d_1}$ in $T$ and $\gamma_2$ crosses $\eta_{1},\ldots,\eta_{d_2}$. We assume that these arcs have a crossing overlap in regions $R_1 \subseteq P_1$ and $R_2 \subseteq P_2$. Let $1 \leq s \leq t \leq d_1$ and $1 \leq s' \leq t' \leq d_2$ be such that $R_1 = \{\alpha_s,\ldots,\alpha_{t}\}$ and $R_2 = \{\eta_{s'},\ldots,\eta_{t'}\}$. 

We focus on one case which includes a nonempty set $Sw$ as an illustrative proof. We will omit discussion of $\gb$-vectors as the previous proof already illustrated all relevant ideas. Suppose $s' = 1$ and $s(\gamma_2)$ is notched. It must be that $s > 1$ in order for $\gamma_1$ and $\gamma_2$ to have an intersection. Necessarily, the arc $\alpha_{s-1}$ is a spoke incident to the puncture $s(\gamma_2)$. Index this set of spokes as $\sigma_1,\ldots,\sigma_m$ in counterclockwise order such that $\alpha_{s-1} = \sigma_1$. Suppose that $\gamma_1$ crosses $\sigma_1,\ldots,\sigma_k$ and let $\beta$ be the arc which $\gamma_1$ crosses right before crossing $\sigma_k$, if it exists.  We will assume $t < d_1$ and $t' < d_2$; we can repeat these arguments two times if we also have one of these cases. Table \ref{tab:PosetsForTransverse} provides the posets $P_1,P_2,P_5$, and $P_6$. If $\beta$ does not exist, then $P_5$ is the chain between $\sigma_{k+1}$ and $\sigma_{k-1}$, with order as in the Table. The poset $P_3$ is obtained by taking $P_1$ and replacing $R_1 > \alpha_{t+1}$ with $R_3 < \eta_{t'+1}$ and $P_4$ is obtained dually from $P_2$.

\begin{table}[]
    \centering
\begin{tabular}{c|c|c|c}
$P_1$& $P_2$ & $P_5$ & $P_6$\\
   \begin{tikzpicture}[scale=0.75]
   \node[] at (-3.3,1){$\cdots$};
    \node(sk) at (-1.8,2) {$\sigma_k$};
    \node(ddots) at (-1.3,1.4) {$\ddots$};
    \node(s2) at (-0.7,0.7) {$\sigma_2$};
    \node(s1) at (0,0) {$\sigma_{1}$};
    \node(R1) at (1,1) {$R_1$};
    \node(at1) at (2,0){$\alpha_{t+1}$};
    \node(beta) at (-2.8,1){$\beta$};
    \node[] at (2.85,0){$\cdots$};
    \draw(beta) -- (sk);
    \draw (s2) -- (s1);
    \draw (s1) -- (R1);
    \draw (R1) -- (at1);
   \end{tikzpicture}& 
    \begin{tikzpicture}[scale = 0.9]
    \node(s1) at (0,0){$\sigma_1$};
    \node(s2) at (0,1){$\sigma_2$};
    \node(vdots) at (0,1.6){$\vdots$};
    \node(sm) at (0,2){$\sigma_m$};
    \node(R2) at (1,1) {$R_2$};
    \node(taut1) at (2,2){$\eta_{t'+1}$};
    \node[] at (3,2){$\cdots$};
    \draw (R2) -- (taut1);
    \draw(R2) -- (s1);
    \draw (s1) -- (s2);
    \draw(R2) -- (sm);
   \end{tikzpicture}& 
   \begin{tikzpicture}
    \node(sk1) at (0,0){$\sigma_{k+1}$};
    \node[] at (0,0.55){$\vdots$};
    \node(sm) at (0,0.9){$\sigma_m$};
    \node(s1) at (0,1.6){$\sigma_1$};
    \node[] at (0,2.25){$\vdots$};
    \node(sk) at (0,2.6){$\sigma_k$};
    \node(beta) at (1,1) {$\beta$}; 
    \node[] at (1.6,1){$\cdots$};
    \draw(sm) -- (s1);
    \draw(beta) -- (sk);
    \draw(beta) -- (sk1);
   \end{tikzpicture}&
   \begin{tikzpicture}[scale = 0.8]
   \node(at) at (0,0){$\alpha_{t+1}$};
   \node(eta) at (1,-1){$\eta_{t'+1}$};
   \draw (at) -- (eta);
   \node[] at (-1,0){$\cdots$};
   \node[] at (2,-1){$\cdots$};
   \end{tikzpicture}\\
    %\begin{tikzpicture}[scale=0.8]
    %\node(sk) at (-1.8,2) {$\sigma_k$};
    %\node(ddots) at (-1.3,1.4) {$\ddots$};
    %\node(s2) at (-0.6,0.6) {$\sigma_2$};
    %\node(s1) at (0.1,0.1) {$\sigma_{1}$};
    %\node(R3) at (1,1) {$R_3$};
    %\node(et1) at (2,2){$\eta_{t'+1}$};
   % \node(beta) at (-2.8,1){$\beta$};
    %\draw(beta) -- (sk);
   % \draw (s2) -- (s1);
   % \draw (s1) -- (R3);
   % \draw (R3) -- (et1);
  % \end{tikzpicture}& 
    %\begin{tikzpicture}[scale = 0.9]
    %\node(s1) at (0,0){$\sigma_1$};
   % \node(s2) at (0,1){$\sigma_2$};
   % \node(vdots) at (0,1.6){$\vdots$};
  %  \node(sm) at (0,2){$\sigma_m$};
  %  \node(R4) at (1,1) {$R_4$};
  %  \node(at1) at (2,0){$\alpha_{t+1}$};
  %  \draw (R4) -- (at1);
  %  \draw(R4) -- (s1);
  %  \draw (s1) -- (s2);
  %  \draw(R4) -- (sm);
\end{tabular}
\caption{Some of the posets for a resolution of  a transverse crossing between $\gamma_1$ and $\gamma_2$}\label{tab:PosetsForTransverse}
\end{table}

%For $(I_1,I_2) \in J(P_1) \times J(P_2)$, if we do not have $I_1 \cap R_1 = \emptyset$ and $R_2 \subseteq I_2$ or vice versa, then we have a switching position between $I_1$ and $I_2$. Let this position be $j$ in $R_1$ and $j'$ in $R_2$. Then, we let $I_3$ be the result of $I_1 \backslash R \cup  (\{\alpha_s,\ldots,\alpha_j\} \cap I_1) \cup (\{\eta_{j'+1},\ldots, \eta_{t'}\} \cap I_2)$ and define $I_4$ dually. Since $\alpha_j \in I_1$ if and only if $\eta_j \in I_1$, we have that $(I_3,I_4) \in J(P_3) \times J(P_4)$. The image of the map on this subset of $J(P_1) \times J(P_2)$ is the set of $(I_3,I_4) \in J(P_3) \times J(P_4)$ which have a switching position, and the switching operation gives an inverse map for this portion of the map. 

We set $A$ to be the union of pairs $(I_1,I_2)$ such that one of the following holds: (1) there is a switching position between $R_1$ and $R_2$, (2) $R_1 \subseteq I_1$ and $R_2 \cap I_2 = \emptyset$, (3) $R_2 \subseteq I_2, R_1 \cap I_1 = \emptyset, \alpha_{t+1} \in I_1$ and $\eta_{t'+1} \notin I_2$, or (4) $R_2 \subseteq I_2, R_1 \cap I_1 = \emptyset,\alpha_{t+1} \notin I_1$, $\sigma_k \in I_2$ only if $\beta \in I_1$ and $\sigma_{k+1} \notin I_2$ if the highest element $\sigma_m$ is in $I_1$. If $\beta$ does not exist, the condition involving $\beta$ is removed. We define $\Phi_A$ as follows. If $(I_1,I_2)$ has a switching position, which is $j$ in $R_1$ and $j'$ in $R_2$, then we set $\Phi_A(I_1,I_2) = (I_3,I_4)$ where $I_3$ is the result of taking all elements of $I_1$ up to $\alpha_j$ and all elements of $I_2$ after $\eta_{j'}$ and $I_4$ is the result of taking all elements of $I_2$ up to $\eta_{j'}$ and all elements of $I_1$ after $\alpha_j$. Since $\alpha_j \in I_1$ if and only if $\eta_{j'} \in I_2$, these form order ideals. If a pair $(I_1,I_2)$ is from item (2) we send $R_1$ to $R_3$, if from item (3) we send $R_2$ to $R_4$, and if from item (4) we send $R_2$ to $R_3$. Some of the elements $\sigma_i$ do not have one clear image in $P_3 \times P_4$, so care is taken in these latter items to send them to appropriate places so that the resulting sets are still order ideals.

We let $B$ be the complement of $A$ in $J(P_1) \times J(P_2)$; explicitly, $B$ is the set of tuples such that $R_1 \cap I_1 = \emptyset$, $R_2 \cup \langle \sigma_k \rangle \subseteq I_2$, $\alpha_{t+1} \in I_1$ only if $\eta_{t'+1} \in I_2$, and $\beta \in I_1$ only if $\sigma_{k+1} \in I_2$. Our definition of $B$ implies that the restrictions of $I_1 \sqcup (I_2 \backslash (R_2 \cup \langle \sigma_k \rangle))$ to $P_5$ and $P_6$ are order ideals. This defines our bijection $\Phi_B$.
\end{proof}

%\subsection{Self-Crossings}

%\subsection{Closed Curves}

\section{Implications}

In \cite{musiker2013bases}, given a surface $(S,M)$, Musiker, Schiffler, and Williams define two sets of arcs, bangles $\mathcal{C}^\circ$ and bracelets $\mathcal{C}$, and show that the set of elements of $\mathcal{A}_S$  arising from each ($\mathcal{B}^\circ$ and $\mathcal{B}$ respectively) forms a basis of $\mathcal{A}_S$. They leave as a question whether these sets could also give the basis of $\mathcal{A}_S$ when $(S,M)$ has punctures; the lack of skein relations in the punctured setting is a large reason why they did not extend their basis to this case. 

Our skein relations show that a product $x_{\gamma_1}x_{\gamma_2}$ of incompatible arcs can be written in terms of $\mathcal{B}^\circ$ and of $\mathcal{B}$, which shows that these sets are still spanning in the punctured case. Moreover, because our relations are always of the form $x_{\gamma_1}x_{\gamma_2} = x_{C^+} + Y x_{C^-}$, we know that Lemma 6.3 from \cite{musiker2013bases} remains true. As explained in Section 8.5 of the same article, this will show that these sets are also linearly-independent.


\begin{lem}
    Let $\gamma_1$ and $\gamma_2$ be multicurves with at least one point of incompatibility on $(S,M)$. Then the expansion
    \[ x_{\gamma_1}x_{\gamma_2} = \sum_{i} Y_iM_i, \]
    where $M_i \in \mathcal{B}^{\circ}$ and the $Y_i$ represent monomials in the coefficient variables, has a unique index $j$ such that $Y_j = 1$. 
\end{lem}

As future work, it remains for us to verify that $\mathcal{B}$ and $\mathcal{B}^{\circ}$ are still subsets of $\mathcal{A}_{S}$. Although we expect this to be true, it is non-trivial to prove and will, as a consequence, complete the proof that $\mathcal{B}$ and $\mathcal{B}^{0}$ remain bases in the punctured setting. 

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