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\title{The HOMFLY Polynomial of a Forest Quiver}

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\author{\texorpdfstring{Amanda Schwartz\thanks{\href{mailto:amschw@umich.edu}{amschw@umich.edu}. The author was supported by grants DMS-1840234 and DMS-1953852 from the National Science Foundation.}\addressmark{1}}{Amanda Schwartz}}

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\address{\addressmark{1}Department of Mathematics, University of Michigan, Ann Arbor, U.S.A. }
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\received{November 13th 2024}

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\abstract{
We define the HOMFLY polynomial of a forest quiver $Q$ using a recursive definition on the underlying graph of the quiver. We then show that this polynomial is equal to the HOMFLY polynomial of any plabic link which comes from a connected plabic graph whose quiver is $Q$. We also prove a closed-form expression for the Alexander polynomial of a forest quiver, which is a specialization of the HOMFLY polynomial.
}

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\keywords{plabic graph, plabic link, quiver, HOMFLY polynomial, Alexander polynomial}

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\begin{document}

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\section{Introduction}
There have been many developments in recent years relating cluster algebras and knot theory. In this work, we will define an invariant of forest quivers and relate it to certain link invariants. This gives one way to connect the study of forest quivers considered up to mutation equivalence and the study of associated links considered up to isotopy. Postnikov's plabic graphs, introduced in \cite{postnikov} to study a stratification of the totally nonnegative Grassmannian into positroid cells, will serve as an intermediate object in establishing this connection. 


These plabic graphs and their generalizations have proved useful in establishing several connections between cluster algebras and knot theory. One can associate a plabic link to any plabic graph, as introduced in \cite{Fomin_2022, Shende_2019}. In  \cite{GLplabiclinks} Galashin and Lam studied connections between invariants of these plabic links and invariants of quivers associated to the plabic graphs. In \cite{galashin2024braidvarietyclusterstructures}, the authors defined 3D plabic graphs, generalizations of Postnikov's two-dimensional plabic graphs, and used them to construct a cluster structure on type $A$ braid varieties. The existence of cluster structures on links or related objects such as braid varieties has been the subject of much recent work; see \cite{ casals2024clusterstructuresbraidvarieties, casals2024positroidlinksbraidvarieties, galashin2023braidvarietyclusterstructures, Shende_2019} for several additional examples. 


When a plabic graph is reduced, the associated plabic link is a positroid link and is isotopic to several other links one can obtain from objects in bijection with positroids; see \cite{casals2024positroidlinksbraidvarieties, GLplabiclinks} for more details. In this setting, polynomial invariants associated to these links can provide information about the objects associated to the plabic graph $G$. For example, if $G$ is a reduced plabic graph, then there is a rational function $R(Q;q)$ of the quiver $Q$ of $G$ which, after a normalization, yields the point count of the open positroid variety $\Pi^\circ(G)$ over a finite field $\F_q$; see \cite{GLplabiclinks, lam2021cohomologyclustervarietiesii}. There are also many connections relating positroid varieties and the Khovanov--Rozansky homology of their associated positroid links to Catalan combinatorics \cite{galashin2021positroidcatalannumbers, galashin2023positroidsknotsqtcatalannumbers, Mellit_2022, hogancamp2019toruslinkhomology}. 


 The link invariants which we will study in this abstract are the Alexander polynomial, which is the oldest knot polynomial, and a stronger invariant called the HOMFLY polynomial which specializes to the Alexander polynomial. The Alexander polynomial was discovered in 1928 by J.W. Alexander \cite{Alexander} while the HOMFLY polynomial was introduced in \cite{HOMFLY} and also studied independently in \cite{PT} in the 1980s. Several interesting connections between these link invariants and cluster algebras have been studied. In \cite{baziermatte2024knottheoryclusteralgebras} Bazier--Matte and Schiffler described a way to associate a cluster algebra to any link diagram and related the Alexander polynomial of the link to the $F$-polynomial of modules associated to the cluster algebra. Lee and Schiffler related the Jones polynomial, a different specialization of the HOMFLY polynomial, of a 2-bridge link to a specialization of a cluster variable in \cite{lee2017clusteralgebrasjonespolynomials}. 
 
 From the cluster algebra perspective, there has also been interest in finding and studying polynomial mutation invariants of quivers. For example, in \cite{fomin2024cyclicallyorderedquivers} Fomin and Neville studied invariants of cyclically ordered quivers, including an Alexander polynomial which they defined as the determinant of a matrix associated to the quiver. In this abstract, we will focus on forest quivers, i.e. quivers whose underlying graphs are forests. This is a subset of the class of acyclic quivers which includes all type $A_n$, $D_n$, $E_6$, $E_7$, and $E_8$ quivers as well as some affine type quivers. We show that given any forest quiver, there is a reduced plabic graph with that forest quiver. Therefore, the set of links which we are studying includes certain positroid links.
 

We will begin by defining the HOMFLY polynomial of a forest quiver using a recursive definition on the underlying graph of the quiver. This can then be specialized to the Alexander polynomial of a forest quiver, just as the Alexander polynomial of a link is a specialization of the link's HOMFLY polynomial. Since the Alexander and HOMFLY polynomials depend only on the underlying graph of a quiver, we may instead refer to the Alexander or HOMFLY polynomial of a (undirected) forest. Some examples of trees and their Alexander and HOMFLY polynomials are shown in \Cref{tab:polynomial examples}. 

\begin{figure}[ht]
    \centering
    \includegraphics[scale=0.7]{tablecorrected.png}
    \caption{Examples of trees with their Alexander and HOMFLY polynomials.}
    \label{tab:polynomial examples}
\end{figure}

In \Cref{thm:welldef}, we show that for any plabic link which arises from a connected plabic graph whose quiver $Q_G$ is a forest, the HOMFLY polynomial of the link is the same as the HOMFLY polynomial of $Q_G$. Therefore, while plabic graphs serve as an intermediate object between quivers and plabic links, this result allows one to go directly from a forest quiver to an associated link invariant. While the HOMFLY polynomial of a forest quiver is defined recursively, we also prove a closed-form formula for the Alexander polynomial of a forest quiver in \Cref{thm:formula}.




\section{Preliminaries}
\subsection{Plabic Graphs and Plabic Links} 


A \emph{plabic graph} $G$ is a planar, bicolored graph which is embedded in the disk and whose vertices are colored black and white. We assume that $G$ has $N$ vertices on the boundary of the disk which are labeled $1,2,\dots, N$ in a clockwise order, are colored black, and each have degree $1$. A face in $G$ is said to be a \emph{boundary} (resp. \emph{interior}) face if it is (resp. is not) adjacent to the boundary of the disk. A \emph{strand} in $G$ is a path which follows the edges in $G$, obeying the \emph{rules of the road}. That is, the path turns maximally right at each black vertex and maximally left at each white vertex. The \emph{strand permutation} $\pi_G$ of $G$ is a permutation on $N$ obtained by setting $\pi_G(i)=j$ if the strand starting at boundary vertex $i$ ends at boundary vertex $j$. A plabic graph without internal leaves is said to be \emph{reduced} if it has the minimal number of faces among all such plabic graphs with the strand permutation $\pi_G$. See \Cref{fig:plabic graph examples} for an example of two plabic graphs, one reduced and one not reduced, which have the same strand permutation. 

\begin{figure}
    \centering
    \includegraphics[scale=0.8]{ReducedA2.pdf}
    \hspace{1in}
    \includegraphics[scale=0.8]{NonreducedSameTrip.pdf}
    \caption{Two plabic graphs, one reduced (left) and one not reduced (right), which both have strand permutation $\pi = (1\; 4\; 2\; 5\; 3)$.}
    \label{fig:plabic graph examples}
\end{figure}



One can associate a plabic link $L_G^\plab$ to a plabic graph $G$ as follows. For more details, including an alternative description in terms of divides, see \cite{Fomin_2022, GLplabiclinks}. Draw all the strands of $G$. When two strands $S_1$ and $S_2$ cross at a point $p$, consider the arguments $\theta_1$ and $\theta_2$ of their tangent vectors at $p$, respectively, when considered in the complex plane. We assume that $0< \theta_1\neq \theta_2 <2\pi$. If $\theta_1$ is greater than (resp. less than) $\theta_2$ then $S_1$ goes under (resp. over) $S_2$. 
\begin{figure}
    \centering
    \includegraphics[]{adjust concave up.pdf}
    \hspace{0.5in}
    \includegraphics[]{adjust concave down.pdf}
    \caption{Modifications at a point $p$ on a strand where the tangent vector has argument 0.}
    \label{fig:arg0}
\end{figure}Note that we sometimes mark such points with rectangles. Taking the union of all these strands after these adjustments and connecting strands which start and end at the same boundary vertex gives a link diagram for $L_G^\plab$. For an example, see \Cref{fig:link example} for the plabic link of the leftmost graph in \Cref{fig:plabic graph examples}.
\begin{figure}
    \centering
    \includegraphics[scale=0.9]{plabiclinkex.pdf}
    \caption{A plabic graph $G$ and its plabic link $L_G^\plab$.}
    \label{fig:link example}
\end{figure}
At any point where the tangent vector to a strand has argument 0, the strand must be adjusted as follows. If there is a point $p$ on a strand $S$ where the argument changes from being just below $2\pi$ to being just above $0$ as one travels through $p$ along $S$, then we break $S$ at $p$, sending it to the boundary just before reaching $p$ and then back to continue along its original path after $p$. Along the way to the boundary, it passes under all other strands it crosses, and on the way back to the strand $S$, it passes over all other strands. If the argument instead changes at $p$ from being just above 0 to being just below $2\pi$, the procedure is the same except the strand crosses above all other strands when heading to the boundary and under all others when returning from the boundary. This is demonstrated in \Cref{fig:arg0}.
\begin{figure}
    \centering
    \includegraphics[scale=0.7]{square move.pdf}\\
    \vspace{0.25in}
    \includegraphics[scale=0.7]{contractmove.pdf}
   \hspace{0.5in}
    \includegraphics[scale=0.7]{contractmoveblack.pdf}\\
    \vspace{0.25in}
    \includegraphics[scale=0.7]{degree2delete.pdf}\\
    \vspace{0.25in}
    \includegraphics[scale=0.7]{tailremoval.pdf}
    \caption{Local moves on plabic graphs: (a) square move, (b) contraction/un\-contraction, (c) middle vertex insertion/removal, and (d) tail addition/removal.}
    \label{fig:local moves}
\end{figure}
  %Applying local moves (a) - (c) does not affect the property of being reduced since they do not change the strand permutation nor the number of faces in the graph. %We will refer to the set of all plabic graphs obtained from a plabic graph $G$ using moves (a) - (c) as the \emph{move-equivalence class} of $G$. 
  %A graph is reduced if and only if no graph in its move-equivalence class contains either of the following, pictured in \Cref{fig:nonreduced}:
  %\begin{enumerate}[label=(\roman*)]
    %  \item two trivalent vertices of opposite color which are connected by two edges, or
     % \item an interior leaf which is connected to another interior vertex of the opposite color and degree at least 3.
  %\end{enumerate}
 \begin{comment}
    
  \begin{figure}
      \centering
      \begin{tikzpicture}
         \begin{scope}
          \node at (-1.5, 0) {(i)};
             \node[circle, scale =0.5, fill=white, draw=black] (w) at (-0.5,0) {};
             \node[circle, scale =0.5, fill=black, draw=black] (b) at (0.5,0) {};

             \draw[thick] (-1,0) -- (w) to [out=70, in = 110] (b) -- (1,0);
             \draw[thick] (b) to [out=250, in=290] (w);
         \end{scope} 
         
         \begin{scope}[xshift = 4cm]
          \node at (-1.5, 0) {(ii)};
             \node[circle, scale =0.5, fill=white, draw=black] (w) at (-0.5,0) {};
             \node[circle, scale =0.5, fill=black, draw=black] (b) at (0.5,0) {};

             \draw[thick] (-1,0.5) -- (w) -- (-1, -0.5);
             \draw[thick] (w)--(b);
         \end{scope} 
         \begin{scope}[xshift = 6.5cm]
          \node at (-1.5, 0) {,};
             \node[circle, scale =0.5, fill=black, draw=black] (b) at (-0.5,0) {};
             \node[circle, scale =0.5, fill=white, draw=black] (w) at (0.5,0) {};

             \draw[thick] (-1,0.5) -- (b) -- (-1, -0.5);
             \draw[thick] (w)--(b);
         \end{scope} 
      \end{tikzpicture}
      \caption{A plabic graph is reduced if and only if no graph in its move-equivalence class contains either configurations of type (i) or (ii).}
      \label{fig:nonreduced}
  \end{figure} \end{comment}
  We will allow for certain local moves on plabic graphs, pictured in \Cref{fig:local moves}. All four local moves (a) - (d) result in isotopic plabic links and therefore will not affect any of the polynomial invariants we study in this abstract. %The \emph{bipartite reduction} of a plabic graph $G$ refers to the result of using local move (b) to contract all edges whose endpoints are interior vertices of the same color. 
  The \emph{tail reduction} of a plabic graph $G$ refers to the graph which results from applying tail removal to $G$ until it is no longer possible to do so. An interior face $F$ of $G$ is said to be a \emph{boundary leaf face} if in the tail reduction of $G$ its boundary consists of two vertices of different colors which are connected by two edges, one of which separates $F$ from a boundary face and one of which separates $F$ from an interior face. For an example, see \Cref{ex: reduced and blf}.

  

One can associate a directed, planar graph $Q_G$ to each plabic graph $G$ as follows. There will be one vertex $v_F$ placed inside each interior face $F$ of $G$. An edge is placed between vertices $v_F$ and $v_{F'}$ for each edge $e$ in $G$ with opposite colored endpoints such that $F$ and $F'$ are adjacent to $e$. The edge in $Q_G$ is oriented so that as one travels along it, the white vertex in $e$ is to the left. If this graph $Q_G$ contains no loops or directed 2-cycles, then $G$ is said to be \emph{simple}. We will refer to $Q_G$ as the quiver associated to $G$. Throughout the rest of this abstract, we will work only with simple plabic graphs $G$ since we will assume that $Q_G$ is a forest quiver. Note that local moves (b) - (d) will not change the quiver of a plabic graph. If $G$ is a connected, simple plabic graph, then $G$ can be assumed to be trivalent, i.e. we can assume that all internal vertices have degree 3 after applying these local moves. 



Our focus in this abstract will be on plabic links arising from simple plabic graphs $G$ whose quivers $Q_G$ are orientations of forests. Given any forest quiver $Q$, it is possible to find a plabic graph whose quiver is $Q$. In fact, one can choose such a graph to be reduced so that the resulting link is a positroid link. It was known to Lam and Speyer that one can find a reduced plabic graph whose quiver is $Q$ for any tree quiver $Q$, as mentioned in \cite{GLplabiclinks}, but their proof has not been published. We have an algorithm for constructing a reduced plabic graph $G$ with $Q_G=Q$ for any forest quiver $Q$. The algorithm proceeds by first constructing reduced plabic graphs $G_i$ for each connected component $Q_i$ of $Q_G= Q_1 \sqcup Q_2 \sqcup \dots \sqcup Q_k$, then placing each of these graphs $G_i$ inside one larger disk, and finally adding an edge to connect the graph $G_{i}$ to the graph $G_{i+1}$ for each $i=1,2,\dots, k-1$.
\begin{prop}
  Let $Q$ be a forest quiver. Then there exists a connected, reduced plabic graph $G$ with $Q_G=Q$.  
\end{prop}


\begin{exmp}\label{ex: reduced and blf}
    See \Cref{fig:forest reduced} for an example of a reduced plabic graph $G$ whose quiver is the pictured orientation of $E_6 \sqcup A_2$. We also make the following observation. Consider, for example, the face corresponding to the topmost leaf in the quiver as pictured in \Cref{fig:forest reduced}. We note that tail removal can be applied four times at this face, resulting in a face whose boundary consists of two vertices joined by two edges. In particular, the face corresponding to this leaf is a boundary leaf face.
\end{exmp}

\begin{figure}
    \centering
    \includegraphics[scale=0.7]{reducedforestex.pdf}
    \caption{A reduced plabic graph with the pictured forest quiver.}
    \label{fig:forest reduced}
\end{figure}

\subsection{The HOMFLY and Alexander Polynomials}
The Alexander polynomial, named after its discoverer J.W. Alexander \cite{Alexander}, was the first knot polynomial invariant to be discovered. John Conway showed that the Alexander polynomial satisfies a skein relation
\begin{equation}\label{AlexanderSkein}
  \Delta(L_+)-\Delta(L_-)=\left(t^{1/2}-t^{-1/2}\right)\Delta(L_0)  
\end{equation}
where $L_+$, $L_-$, and $L_0$ are links whose diagrams are the same except locally at one location where they are related as follows:
\begin{center}
    \includegraphics[]{skeinpositive.pdf}\hspace{0.5in}
\includegraphics[]{skeinnegative.pdf}\hspace{0.5in}
\includegraphics[]{skeinresolved.pdf}
    \label{skeincrossings}
\end{center}
 Setting $\Delta(\textrm{unknot})=1$ fixes a specific choice of the Alexander polynomial for each oriented link, although typically the polynomial is defined up to multiplication by $\pm t^k$ for some $k$.

The Alexander polynomial is also a specialization of a stronger invariant called the HOMFLY polynomial, introduced in \cite{HOMFLY} and also studied independently in \cite{PT}. The HOMFLY polynomial is a Laurent polynomial in $a$ and $z$ defined by the skein relation
\begin{equation}\label{Homflyskein}
aP(L_+)-a^{-1}P(L_-)=zP(L_0)
\end{equation}
and setting $P(\textrm{unknot})=1$. Setting $a=1$ and $z=t^{1/2}-t^{-1/2}$ in the HOMFLY polynomial recovers the Alexander polynomial. 
 The following proposition is a well-known fact about the HOMFLY polynomial of a connected sum of two links which we will use to prove our first main result. See \cite{lickorishintroknottheory} for a proof of this fact.
\begin{prop} \label{prop:connected-sum}
 Let $L$ and $L'$ be two oriented links, and let $L\#L'$ be a connected sum of these two links. Then $P(L\# L')=P(L)\cdot P(L')$.   
\end{prop}

\section{The HOMFLY Polynomial of a Forest Quiver}

\subsection{Defining the HOMFLY polynomial of a forest quiver}

\begin{defn}\label{def: HOMFLY of forest}
    Let $Q$ be a quiver whose underlying graph is a forest. The \emph{HOMFLY polynomial of $Q$}, denoted $f(Q)$, is defined recursively by setting
    \begin{itemize}
        \item $f(Q)=1$ if $Q$ is empty,
        \item $f(Q) = \frac{z+z^{-1}}{a}-\frac{z^{-1}}{a^3}$ if $Q$ is a single vertex,
        \item $f(Q) = \frac{z}{a}f(Q-\{v\})+\frac{1}{a^2}f(Q-\{v,\tilde{v}\})$ if $v$ is a leaf which is adjacent to $\tilde{v}$, and
        \item $f(Q) = f(Q_1)\cdot f(Q_2)$ if $Q=Q_1 \sqcup Q_2$.
    \end{itemize}
      
    
\end{defn}

\begin{prop}\label{prop: HOMFLY well-defined}
    The function $f$ is well-defined.
\end{prop}

\begin{rem}
    Since the definition does not depend on the orientation of the edges in $Q$, we may occasionally write $f(Q)$ where $Q$ is an undirected forest.
\end{rem}





\begin{exmp}\label{ex: star HOMFLY}
  Fix $n \geq 3$. Let $S_n$ be the star graph on $n$ vertices which has one vertex of degree $n-1$ connected to $n-1$ leaves.
   One can show using induction that the HOMFLY polynomial of $S_n$ for $n \geq 3$ is
    \begin{align*}
        P(S_n)
        &=\frac{z^{n-2}}{a^n} + \frac{z^{n-1}+z^{n-3}}{a^{n-1}}\left( \frac{z+z^{-1}}{a}-\frac{z^{-1}}{a^3}\right) + \sum_{k=2}^{n-2}\left( \frac{z^{k-2}}{a^{k}}\left( \frac{z+z^{-1}}{a}-\frac{z^{-1}}{a^3}\right)^{n-k}\right).
    \end{align*}  
\end{exmp}

\subsection{Connections to the HOMFLY polynomial of a plabic link}

In \cite{GLplabiclinks}, Galashin and Lam made the following conjecture.
\begin{conj}[{\cite{GLplabiclinks}}]
    Let $G, G'$
be two connected simple plabic graphs. Assume that the
quivers $Q_G$ and $Q_{G'}$ are mutation equivalent. Then $P(L_G^\plab)=P(L_{G'}^\plab).$
\end{conj}

Our first main result represents partial progress towards proving this conjecture. We show that given a forest quiver $Q$ and any connected plabic graph $G$ with $Q_G=Q$, $P(L_G^\plab)=f(Q)$. This implies that given any two connected plabic graphs $G, G'$ whose quivers are orientations of the same forest $Q$, $P(L_G^\plab)=f(Q)=P(L_{G'}^\plab)$. Additionally, this result gives a way to go directly from a forest quiver to a corresponding link invariant. We will sketch the proof of this result. For more details, see \cite{schwartz2024homflypolynomialforestquiver}. The proof relies on the following results. 

\begin{lem}[{\cite{GLplabiclinks}}]\label{skeinrelation}
    Let $G$ be a simple plabic graph with a boundary leaf face $F$. Let $x$ and $y$ be the vertices on the boundary of $F$ and $e$ be the edge separating $F$ from a boundary face. Let $G'=G-e$ and $G''=G-\{x,y\}$. Then the HOMFLY polynomials of their plabic links satisfy
    \begin{equation}\label{blfskein}
        aP(L_G^\plab)-a^{-1}P(L_{G''}^\plab)=zP(L_{G'}^\plab).
    \end{equation}
\end{lem}

\begin{prop}\label{prop: disconnected-quiver-connected-sum}
    Let $G$ be a connected plabic graph whose quiver $Q_G$ is a disjoint union of non-empty tree quivers $Q_1,Q_2,\dots, Q_k$ for some $k \geq 2$. Then $L_G^\plab$ is isotopic to a connected sum of links $L_{G_1}^\plab, \dots L_{G_k}^\plab$ for some choice of connected plabic graphs $G_i$ with $Q_{G_i}=Q_i$ for $i=1,\dots, k$.
\end{prop}
\begin{proof}[Proof Sketch]
    The main idea of the proof is as follows. If an edge in $G$ separates two (not necessarily distinct) boundary faces, we say that it is a \emph{dividing edge}. Deleting such an edge divides the plabic graph $G$ into two smaller plabic graphs $G_1$ and $G_2$.
    Since $Q_G$ is disconnected, then after potentially after applying local moves one can always find a dividing edge $e$ in $G$ such that $G_1$ and $G_2$ have non-empty quivers whose disjoint union is $Q_G$. Further, one can verify that $L_G^\plab$ is isotopic to a connected sum of $L_{G_1}^\plab$ and $L_{G_2}^\plab$; see \Cref{fig:connected-sum}. 

    \begin{figure}
        \centering
        \includegraphics[width=0.9\linewidth]{connectedsumb-bedge.pdf}
        \caption{A dividing edge in $G$ means $L_G^\plab$ is a connected sum of two other plabic links $L_{G_1}^\plab$ and $L_{G_2}^\plab$. The isotopy between $L_{G_1}^\plab\#L_{G_2}^\plab$ and $L_G^\plab$ corresponds to rotating the portion of the connected sum coming from $L_{G_1}^\plab$ by 360$^\circ$ about the horizontal axis in the paper.}
        \label{fig:connected-sum}
    \end{figure}
\end{proof}

\begin{thm}\label{thm:welldef}
Let $G$ be a connected plabic graph, and suppose the quiver $Q_G$ of $G$ is a forest quiver. Then $P(L_G^\plab) = f(Q_G)$.
\end{thm}
\begin{proof}[Proof Sketch]

The proof proceeds by induction on the number of vertices in $Q_G$. Recall that $f(Q_G)$ can be computed recursively by removing leaves using the relation
$$f(Q) = \frac{z}{a}f(Q-\{v\})+\frac{1}{a^2}f(Q-\{v,\tilde{v}\})$$ if $v$ is a leaf which is adjacent to $\tilde{v}$
or by taking the product over connected components using the relation
$$f(Q) = f(Q_1)\cdot f(Q_2)$$ if $Q=Q_1 \sqcup Q_2$. The inductive step relies on finding analogues of these relations between the HOMFLY polynomials of plabic links. 

In particular, if $Q_G$ is disconnected, then by \Cref{prop: disconnected-quiver-connected-sum} there are connected plabic graphs $G_1$ and $G_2$ with non-empty quivers such that $Q_G=Q_{G_1}\sqcup Q_{G_2}$ and $P(L_G^\plab)$ is isotopic to a connected sum of $L_{G_1}^\plab$ and $L_{G_2}^\plab$. Using \Cref{prop:connected-sum}, it follows that
\begin{align*}
        P(L_G^\plab) &= P(L_{G_1}^\plab)\cdot P(L_{G_2}^\plab) \\
        &= f(Q_{G_1})\cdot f(Q_{G_2})\\
        &= f(Q_G).
    \end{align*}

If $Q_G$ is connected, a leaf $v$ in $Q_G$ can be assumed to correspond to a boundary leaf face in $G$, possibly after applying local moves.
Further, $Q_{G'}=Q_G-\{v\}$ and $Q_{G''}=Q_G-\{v,\tilde{v}\}$. From the inductive hypothesis and \Cref{skeinrelation}, it follows that 
\begin{align*}
        P(L_G^\plab) &= \frac{z}{a}P(L_{G'}^\plab)+\frac{1}{a^2}P(L_{G''}^\plab)\\
        &= \frac{z}{a}f(Q_G-\{v\}) +\frac{1}{a^2}f(Q_G-\{v,\tilde{v}\})\\
        &= f(Q_G).
    \end{align*}
\qedhere
\end{proof}




\section{The Alexander Polynomial of a Forest Quiver}

Just as one can specialize the HOMFLY polynomial of a link to obtain the link's Alexander polynomial using the specialization $a=1$ and $z=t^{1/2}-t^{-1/2}$, we can use the same specialization to obtain the Alexander polynomial $\Delta(Q)$ of a forest quiver $Q$. 
\begin{exmp}
    One can prove by induction that for the type $A_n$ quiver $$\Delta(A_n) = t^{-n/2}\cdot \sum\limits_{k=0}^n (-1)^{n-k} t^k.$$
\end{exmp}

\begin{exmp}
    Similarly, one can prove via induction that $$\Delta(D_n) = t^{-n/2}\left(t^n-t^{n-1} + (-1)^{n-1}t+(-1)^n \right) = t^{-n/2}(t-1)(t^{n-1}+(-1)^{n-1}).$$
\end{exmp}

\noindent Our second main result is a proof of a closed formula for the Alexander polynomial for any forest quiver $Q$. 

\begin{thm}\label{thm:formula}
    Let $Q$ be a be a forest quiver with $n$ vertices. Then the Alexander polynomial of $Q$ is given by
    \begin{equation}\label{eqn:formula}
        \Delta(Q) =\frac{(-1)^{n}}{t^{n/2}}\sum\limits_{k=0}^{\lfloor{n/2}\rfloor} c_k(Q) t^k(1-t)^{n-2k}
    \end{equation}
    where $c_k(Q)$ is the number of independent sets of size $k$ in the line graph of $Q$, i.e. the number of subsets of $k$ distinct edges in $Q$ which do not share any endpoints.
\end{thm}

\begin{rem}
    In \cite{lam2021cohomologyclustervarietiesii}, Lam and Speyer studied the point count of acyclic cluster varieties over finite fields. They proved that for an acyclic quiver $Q$ with $n$ vertices the point count over $\F_q$ of the associated cluster variety $\mathcal{A}$ is given by
    \begin{equation}\label{eqn:point count}
        \#\mathcal{A}(\F_q) = \sum\limits_{k \geq 0} a_k(q-1)^{n-2k}q^k
    \end{equation}
where $a_k$ is the number of independent sets in the underlying graph of the quiver. It is interesting to note the similarity between this formula and the formula in (\ref{eqn:formula}). In an upcoming updated version of \cite{schwartz2024homflypolynomialforestquiver}, we will prove a result about the HOMFLY polynomial which recovers both equations (\ref{eqn:formula}) and (\ref{eqn:point count}) in the case where $Q$ is a forest quiver.
\end{rem}

\begin{exmp}
    Let $Q$ be the Dynkin diagram $E_6$. Then there are 5 ways to pick a single edge in $Q$, $5$ ways to pick two distinct edges which do not share a vertex, and 1 way to pick three distinct edges, none of which share any vertices; see \Cref{fig:E6 formula example}. Therefore,
    \begin{align*}
        \Delta(E_6) &= t^{-3} \left(1\cdot t^0(1-t)^6 +5 \cdot t^1(1-t)^4+5\cdot t^2(1-t)^2 + 1 \cdot t^3(1-t)^0 \right)\\
        &= t^{-3}\left(t^6 - t^5 + t^3 - t + 1 \right).
    \end{align*}
\end{exmp}
    \begin{figure}
        \centering
        \includegraphics[scale=0.8]{E6 Example Updated.pdf}
        \caption{The sets counted by the coefficients in (\ref{eqn:formula}) when $Q$ is $E_6$. The edges which are used are bolded and drawn in red.}
        \label{fig:E6 formula example}
    \end{figure}
\begin{exmp}
    Fix $n \geq 3$. Recall from \Cref{ex: star HOMFLY} that $S_n$ is the star graph on $n$ vertices which has one vertex of degree $n-1$ connected to $n-1$ leaves. Any edge in $S_n$ must have the degree $n-1$ vertex as one of its endpoints, so it is impossible to pick multiple edges in $S_n$ which do not share an endpoint. It follows that 
    \begin{align*}
        \Delta(S_n) &= \frac{(-1)^n}{t^{n/2}}\left((1-t)^n + (n-1)\cdot t(1-t)^{n-2} \right)\\
        &=  \frac{(-1)^n}{t^{n/2}}\left(\left(1-(n-3)t+t^2\right)(1-t)^{n-2} \right).
    \end{align*}
\end{exmp}

%\section{Future Work}
%With regard to future work, there are several ways in which our results could possibly be extended. For example, one could search for formulas which are similar to that given in \Cref{thm:formula} for other link invariants. In particular, we have shown how one can compute the HOMFLY polynomial $P(Q)$ for a forest quiver $Q$ recursively. A natural question is whether one can find a closed formula, expressed in terms of the forest quiver $Q$, for this invariant. We hope to explore this more in future work. Alternatively, one could attempt to generalize the formula in \Cref{thm:formula} to a wider class of quivers. It would also be interesting to study link homologies like Khovanov--Rozansky homology and knot Floer homology - which categorify the HOMFLY and Alexander polynomials, respectively - of links arising from plabic graphs whose quivers are forests.



\acknowledgements{I thank my advisor, Thomas Lam, for his guidance and comments. I also thank Eugene Gorsky for helpful discussions and his suggestion of the formula which is proved in \Cref{thm:formula}.}

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