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\documentclass[finalversion]{FPSAC2024}
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\title{Triangular $(q,t)$-Schr\"oder Polynomials and Khovanov-Rozansky Homology}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
%\author{Longer names me\thanks{\href{mailto:hello@world.c}{hello@world.c}. Longer names me was partially supported by Grant 2017.11.14.$\partial$\;supp.}\addressmark{1}, \and you\addressmark{2}}



\author{Carmen Caprau\addressmark{1}, Nicolle Gonz\'alez\thanks{\href{mailto:nicolles@berkeley.edu}{nicolles@berkeley.edu}. The authors thank AIM for the ``Algebra, geometry, and combinatorics of link homology'' workshop where this collaboration was started.}\addressmark{2}, Matthew Hogancamp\addressmark{3}, \and Mikhail Mazin\addressmark{4}}



\address{
\addressmark{1}Department of Mathematics,  California State University, Fresno, U.S.A.\\
\addressmark{2}Department of Mathematics,  University of California,  Berkeley, U.S.A. \\ 
\addressmark{3}Department of Mathematics, Northeastern University,  U.S.A.\\ 
\addressmark{4}Department of Mathematics,  Kansas State University, U.S.A.
}




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%\address{\addressmark{1}Department of This, University of That, There \\ \addressmark{2}School of Something, University of Something Else, Somewhere}

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\received{\today}

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%% put your English abstract here, or comment this out if you don't have one yet
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\abstract{We define generalized Schr\"oder polynomials $S_\lambda(q,t,a)$ for triangular partitions and prove that these polynomials recover the triangular $(q,t)$-Catalan polynomials of \cite{BHMPS} at $a=0$. Moreover, we show that the Poincar\'e polynomials of the reduced Khovanov-Rozansky homology of Coxeter knots of these partitions are given by $S_\lambda(q,t,a)$. Finally, combined with recent results in \cite{GMO}, we compute the Poincar\'e polynomial of the $(d,dnm+1)$-cable of the $(n,m)$-torus knot, thus proving a special case of the Oblomkov-Rassmusen-Shende conjecture \cite{ORS,OS} for generic unibranched planar curves with two Puiseux pairs. }

%Triangular partitions are those whose diagram consists of all boxes that (weakly) fit under a straight line connecting points $(0,s)$ and $(r,0)$ for some positive real numbers $s$ and $r.$ We prove that the $(q,t)$-Catalan polynomial corresponding to a triangular partition $\tau_{s,r}$ is equal to the $a=0$ part of the Poincar\'e polynomial of the reduced Khovanov-Rozansky homology of the so called Coxeter knot (see \cite{OR}) corresponding to $\tau_{s,r}$. Furthermore, the full Poincar\'e polynomial of the reduced Khovanov-Rozansky homology can be recovered by replacing the $(q,t)$-Catalan polynomial by the corresponding $(q,t,a)$-Schr\"oeder polynomial.
%We further verify that in the case when $s=dn$ and $r=dm,$ where $n$ and $m$ are relatively prime, the corresponding Coxeter knot is the $(d,dnm+1)$-cable of the $(n,m)$-torus knot. 
%Combined with the recent results Gorsky-Mazin-Oblomkov (\cite{GMO}), this proves a special case of the Oblomkov-Rassmunsen-Shende conjecture (see \cite{ORS,OS}) for generic unibranched planar curves with two Puiseux pairs.}

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%\resume{\lipsum[2]}

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%\keywords{Khovanov-Rozansky homology, Catalan polynomials, Recursions}

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\begin{document}

\maketitle
\section{Introduction}

A fundamental pursuit in knot theory for the last century has been the classification of knots and links. One particularly effective method has been the study of certain homology theories that realize the knot as certain chain complexes, whose Poincar\'e polynomials, which compute the graded dimensions of the homology groups, are then used as knot invariants. One such especially celebrated homology theory is (reduced) \emph{Khovanov-Rozansky homology}. This tri-graded theory associates to each link $L$ a polynomial in three variable $P_{L}^{KR}(q,t,a)$. It turns out that computing these polynomials explicitly is very difficult, and the pursuit of a closed form for them has spurned a remarkable volume of deep and surprising results bridging combinatorics with low dimensional topology and algebraic geometry. One of the only cases where these polynomials are explicitly known is the case of \emph{torus knots and links}, where by transforming these knots to certain binary sequences and defining a family of recursions, Hogancamp and Mellit were able to compute explicit solutions \cite{HM, Me}. The connections between these polynomials and $(q,t)$-Catalan combinatorics have been deeply established \cite{GL20, Ho17, GorksyCatalan}, with the $a=0$ specialization of $P_{L}^{KR}(q,t,a)$ for a torus knot recovering the \emph{(q,t)-Catalan polynomial} and higher $a$ powers equaling \emph{(q,t)-Schr\"oder polynomials}. 


One of our first main results is the generalization of {(q,t)-Schr\"oder polynomials} to the context of \emph{triangular partitions}, which are defined as maximal partitions that fit under a line of arbitrary slope (i.e. certain Dyck paths under lines with non-integer intercepts.). These partitions were thoroughly studied by Bergeron and Mazin \cite{BM}. In particular, our triangular Schr\"oder polynomial recovers at $a=0$ the triangular $(q,t)$-Catalan polynomials studied in \cite{BM} that appeared in the generalized shuffle theorem under any line \cite{BHMPS}. Our construction relies on certain recursions introduced by Gorsky-Mazin-Vazirani \cite{GMV} using so called \emph{$(m,n)$-invariant subsets}, which allow us to produce certain binary sequences from the triangular partitions and compute them using the recursions. 

Recently, Oblomkov and Rozansky considered \emph{Coxeter links}, which contain torus knots and links as special cases, and identified their homology with certain sections on the flag Hilbert scheme \cite{OR}. Even more recently, Galashin and Lam introduced a family of knots that arise directly from certain monotone paths on an $m \times n$ grid and proved that all \emph{monotone links} are Coxeter.  Thus, since the Gorsky-Mazin-Vazirani recursions agree with the Hogancamp-Mellit recursions, using the results above we prove that the Poincar\'e polynomial of the reduced Khovanov-Rozansky homology of Coxeter knots arising from triangular partitions is precisely our triangular Schr\"oder polynomial. 


The next natural family of links to try to understand are cabled torus knots. Informally, a \emph{cabled knot} is a knot within a knot, so that the string that makes up the knot locally carries a smaller knot on it. A highly nontrivial and celebrated conjecture due to Oblomkov-Rasmussen-Shende, relates the Khovanov-Rozansky homology of algebraic links to certain plane curve singularities on the Hilbert scheme of points. Combining results in \cite{GMO} with our previous results above, we compute the Poincar\'e polynomial for a certain family of cabled knots, proving a special case of the \emph{Oblomkov-Rasmussen-Shende conjecture} for unibranched planar curves with two Puiseux pairs.  



\section{Background and Definitions}

\subsection{Recursions for the Poincar\'e Series of Link Homology}\label{subsection: recursions for knot hom}

In \cite{HM} the third author and Anton Mellit introduced a recursive method for computing the Poincar\'e series of the reduced Khovanov-Rozansky homology of torus links. Given two finite binary sequences ${\bf u}$ and ${\bf v}$ with the same number of $1$'s, they introduced the power series $R_{{\bf u},{\bf v}}(q,t,a)$ via the following recursive relations:

\begin{align*}
R_{0{\bf u},0{\bf v}} & =t^{-|{\bf u}|}R_{{\bf u}1,{\bf v}1}+qt^{-|{\bf u}|}R_{{\bf u}0,{\bf v}0}, &
R_{1{\bf u},0{\bf v}}&=R_{{\bf u}1,{\bf v}}, & R_{\emptyset,0^n}&=\left(\frac{1+a}{1-q}\right)^n,
\end{align*}
\begin{align*}
R_{1{\bf u},1{\bf v}}&=(t^{|{\bf u}|}+a)R_{{\bf u},{\bf v}},&
R_{0{\bf u},1{\bf v}}&=R_{{\bf u},{\bf v}1},
&
R_{0^m,\emptyset}&=\left(\frac{1+a}{1-q}\right)^m,
\end{align*}
where $|{\bf u}|$ is the number of $1$'s in ${\bf u}$ and $R_{\emptyset, \emptyset}=1$. Let $l({\bf u})$ denote the length of ${\bf u}.$



\begin{theorem}[\cite{HM}]
Let $(n,m)$ be any positive integers. The Poincar\'e series of the reduced Khovanov-Rozansky homology of the $(n,m)$-torus link is given by 
$$
P_{L_{n,m}}^{KR}(q,t,a)=(1-q)R_{0^n,0^m}=R_{0^{n-1}1,0^{m-1}1}.
$$
\end{theorem}

Furthermore, it follows from their construction that for  $|{\bf u}|=|{\bf v}|=1$, the recurrence applied to $R_{{\bf u},{\bf v}}(q,t,a)$ will terminate and %will algorithmically 
compute the Poincar\'e series of the reduced Khovanov-Rozansky homology for the link $L_{{\bf u},{\bf v}}$%. Denote by $\ell$ the length of the sequences. Given sequences $\bf u$ and $\bf v$ the link $L_{{\bf u},{\bf v}}$ is 
constructed as follows:
 
\noindent\textbf{Step 1:} Mark $\ell(\bf{v})$ points on the %left
bottom edge and $\ell(\bf{u})$ points on the %top 
left edge of $[0,1]^2$, labeled with the sequences starting from the %top left 
bottom left corner. Mark also the points on the %right and bottom 
top and right edges directly across from marked points labeled by $0.$  

\noindent\textbf{Step 2:} Starting with the %leftmost
lowest point on the %top
left edge and %topmost
leftmost point on the %left
bottom edge, connect the dots with diagonal non-intersecting lines until all points are matched. 

\noindent\textbf{Step 3:} Erase the tail of the line connected to the point labeled $1$ on the %left 
bottom wall and connect it to %bottom 
right side of $[0,1]^2$, going \emph{above} all other strands in the process. 

\noindent\textbf{Step 4:} Erase the tail of the line connected to the point labeled $1$ on the %top
left wall, pass it \emph{underneath} all other strands %to its left
beneath it, and connect it once again to the %top 
left wall, but now in the first position, directly %above
across the new marked point created in Step 3.  

\noindent\textbf{Step 5:} Close the diagram by identifying the edges in the usual way for a torus.

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\draw [very thick] (-1,-3) to[in=180,out=-90] (2,-4)--(10,-4) to[in=-90,out=0] (14,0)--(14,10) to[in=0,out=90] (10,14)--(6,14) to[in=90,out=180] (1.5,11);
\draw [green!60!black, very thick] (0,5) to[in=90,out=-125] (-1,3)--(-1,-3);

\draw [very thick] (0,2.5) to[in=180,out=-125] (0,1.5)--(2.6,1.5);
\draw [very thick] (3.2,1.5)--(4.9,1.5);
\draw [very thick] (5.5,1.5)--(8.9,1.5);
\draw [very thick] (9.5,1.5)--(10,1.5) to[in=55,out=0] (10,2.5);

\draw [very thick] (0,7.5) to[in=180,out=-125] (0,6.5)--(1,6.5);
\draw [very thick] (1.5,6.5)--(2.9,6.5);
\draw [very thick] (3.5,6.5)--(5.6,6.5);
\draw [very thick] (6.2,6.5)--(8.9,6.5);
\draw [very thick] (9.5,6.5)--(10,6.5) to[in=55,out=0] (10,7.5);
\end{tikzpicture}\ \ 
    \caption{(Left) Steps 1-4 in constructing the link $L_{\bf{u},\bf{v}}$ for $\bf{u} = 0010$ and $\bf{v}=010$. (Right) Two diagrams for the knot $L_{0010,010}.$ Middle right: the closure of the diagram to its left (step 5). Far right: an equivalent closure considered in \cite{HM}. 
    %On the left one has to glue the square into a torus in the standard way. Equivalently, one can close it into the knot diagram in the middle. Note that in \cite{HM} a slightly different, but equivalent way to close the diagram is used (diagram on the right).
    }\label{figure: L_uv}
\end{figure}


\begin{example}
Consider $\bf{u}=0010$ and $\bf{v}=010$ with lengths $\ell(\bf{u}) = 4$, $\ell(\bf{v}) = 3$, and with $|\bf{u}| = |\bf{v}|=1$. Steps 1 and 2 followed by 3 and 4 will yield the left two diagrams in Figure \ref{figure: L_uv}. It's closure, Step 5, is the third diagram. Iteratively applying the recurrence we see that the Poincare series of the KhR-homology of $L_{\bf u,\bf v}$ is then equal to:
\begin{align*}
    R_{0010,010}&=t^{-1}R_{0101,101}+qt^{-1}R_{0100,100}=\dots\\
   %&= t^{-1}R_{11,11} +qt^{-1}(qt^{-1}R_{010,10}+t^{-1}R_{011,11}) = \dots\\
    &=t^{-1}(t+a)(a)R_{\emptyset, \emptyset} + (qt^{-1})^2(a)R_{\emptyset, \emptyset} + (qt^{-1})(t^{-1})(t+a)(a) R_{0,\emptyset} \\
    &= a (t^{-1}(t+a)+(qt^{-1})^2)+(qt^{-1})(t^{-1})(t+a)(a) \frac{(1+a)}{1-q}.
\end{align*}

    
\end{example}


%On the top edge of a square $T$ mark $l({\bf u})$ points corresponding to the elements of ${\bf u}$ counting from the top-left corner. Similarly, on the left edge of $T$ mark $l({\bf v})$ points corresponding to the elements of ${\bf v},$ also counting from the top-left corner. On the bottom edge of $T$ mark $l({\bf u})-1$ points matching the points on the top edge of $T$ corresponding to $0$'s in ${\bf u}.$ Similarly, on the right edge of $T$ mark $l({\bf v})-1$ points matching the points on the left edge of $T$ corresponding to $0$'s in ${\bf v}.$ Now we connect the $l({\bf u})+l({\bf v})-1$ marked points on the top and right edges of $T$ to the $l({\bf u})+l({\bf v})-1$ marked points on the left and bottom edges of $T$ by non-intersecting straight lines. Now for the marked point on the left edge corresponding to the $1$ in ${\bf u}$ we erase a little bit at the end of the corresponding line and instead of reaching the marked point we turn down crossing all the strands \textbf{above} before reaching the bottom edge at point $A$ next to the bottom-left corner. Similarly, for the marked point on the top edge corresponding to the $1$ in ${\bf v},$ instead of reaching the top edge we turn left, crossing all the strands \textbf{under} before almost reaching the left edge and then turning up to reach the top edge next to the top-left corner exactly on top of the point $A$ on the bottom edge. Finally, on folds the square $T$ into a torus in the standard way (see Figure \ref{figure: L_uv} for an example).

%introduce invariant subsets,catalan objects, and torus knot diagrams. State the main theorems we are generalizing,etc.

\subsection{Invariant Subsets and Dyck Paths}

Given positive integers $m$ and $n$, an \newword{$(n,m)$-Dyck path} is a lattice path from $(m,0)$ to $(0,n)$ consisting exclusively of north and west steps that stays weakly below the diagonal line $y=n-\frac{n}{m}x$. Indexing each cell by its top right lattice point, for any such choice of $(n,m)$ we define an \newword{Anderson filling} on each of the cells of the lattice via the function $\gamma:\Z^2 \to \Z$ by $\gamma(x,y) = mn-nx-my$. 



\begin{definition}
    A subset $\Delta\subset\mathbb{Z}_{\ge 0}$ is called \newword{$(n,m)$-invariant} if $\Delta+n\subset\Delta$ and $\Delta+m\subset\Delta.$ Let $\Inv_{n,m}$ denote the set of all $(n,m)$-invariant subsets. In addition, an $(n,m)$-invariant subset $\Delta$ is called \newword{$0$-normalized} if $0\in\Delta.$ We will use the notation $\Inv_{n,m}^0$ for the set of $0$-normalized $(n,m)$-invariant subsets.
\end{definition}

If $n$ and $m$ are relatively prime then the set of $(n,m)$-Dyck paths is in a natural bijection with the set of $0$-normalized $(n,m)$-invariant subsets. Namely, given an $(n,m)$-Dyck path $D$ let $\Gaps(D)$ be the set of positive Anderson labels corresponding to the cells above $D$ (positivity of a label is equivalent to the cell fitting under the diagonal). The corresponding $0$-normalized $(n,m)$-invariant subset is given by $\Delta(D)=\mathbb{Z}_{\ge 0}\setminus \Gaps(D).$ It is not hard to see that this defines a bijection. 

The rational $(q,t)$-Catalan polynomials and the rational Schr\"oder polynomials are usually defined in terms of the $(n,m)$-Dyck and Schr\"oder paths. However, it is more suitable for us to follow \cite{GM13,GM14} and define these polynomials in terms of the invariant subsets. The two approaches are equivalent due to the bijection described above. To do so, we need the following statistics. Let $\Delta\in\Inv_{n,m}$ and define:  
\begin{itemize}[leftmargin=*] 
\item the \newword{area} to be the number of gaps in $\Delta$,
$ 
\area(\Delta):=\sharp(\mathbb{Z}_{\ge 0}\setminus\Delta).
$
\item the \newword{$n$-generators} of $\Delta$ as the set $\ngen(\Delta):=\Delta\setminus(\Delta+n)=\{a\in\Delta:a-n\notin\Delta\}$.

\item the \newword{codinv} as the number of gaps in length $m$ intervals beginning at $n$-generators:
\begin{align}
\codinv(\Delta)&:=\sum\limits_{a\in\ngen(\Delta)} \sharp\{a\le g< a+m:g\notin\Delta\},\label{def:codinv}
\\
\dinv(\Delta)&:=\delta(n,m) - \codinv(\Delta),\ \ \delta(n,m):= \frac{(n-1)(m-1)}{2}. \nonumber
\end{align}
\end{itemize}

\begin{comment}
\begin{definition}
We set $\area$ to be the number of gaps in $\Delta:$
$ 
\area(\Delta):=\sharp(\mathbb{Z}_{\ge 0}\setminus\Delta).
$
\end{definition}

\begin{definition}
The $n$-generators of $\Delta$ are $\ngen(\Delta):=\Delta\setminus(\Delta+n)=\{a\in\Delta:a-n\notin\Delta\}$.
\end{definition} 

\begin{definition}
\label{def:codinv}
Codinv counts gaps in intervals of length $m$ starting from $n$-generators:
\begin{equation*}
\codinv(\Delta):=\sum\limits_{a\in\ngen(\Delta)} \sharp\{a\le g< a+m:g\notin\Delta\},
\end{equation*}
$$ 
\dinv(\Delta):=\delta(n,m) - \codinv(\Delta),\ \ \delta(n,m):= \frac{(n-1)(m-1)}{2}.
$$
%where $\delta(n,m):=\frac{(n-1)(m-1)}{2}$ is the maximal possible value of $\codinv.$
\end{definition}
\end{comment}

\begin{definition}
   For each coprime pair $(m,n)$, the \newword{rational $(q,t)$-Catalan polynomial}, denote $C_{n,m}(q,t)$, is given by:
    \begin{equation*}
        C_{n,m}(q,t):=\sum_{\Delta\in\Inv_{n,m}^0}q^{\area(\Delta)}t^{\dinv(\Delta)}=(1-q)\sum_{\Delta\in\Inv_{n,m}}q^{\area(\Delta)}t^{\dinv(\Delta)}.
    \end{equation*}
\end{definition}


In order to define Schr\"oder polynomials, we will need a couple more ingredients.
\begin{itemize}[leftmargin=*]
\item Let
$
\Cogen(\Delta):=\{a\in\mathbb{Z}:a\notin\Delta, a+n\in\Delta,a+m\in\Delta\}  
$
be the set of \newword{double co-generators} of $\Delta.$
\item Let $k\in\mathbb{Z}$. Set
$
\lambda_k(\Delta):=\sharp\{a\in\ngen(\Delta): k+n+1\le a\le k+n+m\}.
$
\end{itemize}

\begin{comment}
\begin{definition}
Let
$
\Cogen(\Delta):=\{a\in\mathbb{Z}:a\notin\Delta, a+n\in\Delta,a+m\in\Delta\}  
$
be the set of double co-generators of $\Delta.$
\end{definition}

\begin{definition}
Let $k\in\mathbb{Z}$. Set
$
\lambda_k(\Delta):=\sharp\{a\in\ngen(\Delta): k+n+1\le a\le k+n+m\}.
$
\end{definition}
\end{comment}

\begin{definition}
For each coprime pair $(m,n)$, the \newword{Schr\"oder polynomial} $S_{n,m}(q,t,a)$ is:
\begin{equation*}
S_{n,m}(q,t,a):=\sum\limits_{\Delta\in \Inv_{n,m}^0} q^{\area(\Delta)}t^{\dinv(\Delta)}\prod\limits_{k\in\Cogen(\Delta)}\left(1+at^{-\lambda_k(\Delta)}\right).
\end{equation*}
\end{definition}

\begin{example}
    In Figure \ref{figure: 3,4 Schroder}, the bijection between $(3,4)$-Dyck paths and the $0$-normalized $(3,4)$-invariant subsets is illustrated, complemented with a computation of the $\area$ and $\dinv$ statistics, as well as the factors necessary for the Schr\"oder polynomial, for two out of five invariant subsets in $\Inv_{3,4}^0$. The rest are computed similarly. Summing all together, one obtains the Schr\"oder polynomial:    
    \begin{align*}
    S_{3,4}(q,t,a)=&t^3(1+a)(1+at^{-1})(1+at^{-2})+qt^2(1+a)(1+at^{-1})+qt(1+a)(1+at^{-1})\\
    &\ \ \ \ \ \ \ \ +q^2t(1+a)(1+at^{-1})+q^3(1+a)\\
    =&q^3 + q^2t + qt^2 + t^3 + qt + a(q^3 + q^2t + qt^2 + t^3 + q^2 + 2qt + t^2 + q + t)\\
    &\ \ \ \ \ \ \ \ +a^2(q^2 + qt + t^2 + q + t + 1) + a^3.
    \end{align*}
\end{example}

\begin{figure}
\center
\begin{tabular}{|c|c|c|c|}
%\hline
%\begin{tikzpicture}[scale=0.5]
%\draw (0,3.2) node {};
%\fill [fill=gray!20!white] (0,0) rectangle (1,2);
%\fill [fill=gray!20!white] (0,0) rectangle (2,1);%
% pick an m and n HERE
% \pgfmathtruncatemacro{\m}{3} 
%  \pgfmathtruncatemacro{\n}{4} 
%%%%%%% DONT TOUCH THIS PART %%%%%%%%%%%%%%%%
% \pgfmathtruncatemacro{\mi}{\m-1} %\mi=m-1 
%    \pgfmathtruncatemacro{\ni}{\n-1} %\ni=n-1
    %
%  \foreach \x in {0,...,\n} % n=5
%    \foreach \y in {1,...,\m}     % m=7  
%       {\pgfmathtruncatemacro{\label}{ - \m * \x - \n *  \y +\m*\n} % - m * \x - n *  \y +mn
      % \node   (\x\y) at (\x,\y){};
%       \node  at (\x-.4,\y-.4) [blue!60!white, scale=.8]{\footnotesize \label};} 
%generates the grid and anderson labels
%\draw[dotted] (0,\m)--(\n,0); %draws the diagonal from (0,m) -- (n,0)
%  \foreach \x in {0,...,\n} % n=5
%    \foreach \y  [count=\yi] in {0,...,\mi}   % m-1=6  
    %\draw (\x\y)--(\x\yi);
%      \draw (\x,\y)--(\x,\yi);
%        \foreach \y in {0,...,\m}     % m=7  
 %      \foreach \x  [count=\xi] in {-1,...,\ni}   % n-1=4
        %\draw (\x\y)--(\xi\y);
%         \draw (\x,\y)--(\xi-1,\y);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %% Draw dyck path by hand below
%\draw [very thick] (4,0)--(0,0)--(0,3); %lower boundary
%\draw[ very thick,red](0,3)--(0,2)--(1,2)--(1,1)--(2,1)--(2,0)--(4,0); %Dyck path

%
%\end{tikzpicture}
%& 
%\begin{tikzpicture}[scale=0.5]
%\draw (0,5) node {$\Gaps=\emptyset$};
%\draw (0,4) node {$\Delta=\mathbb{Z}_{\ge 0}$};
%\draw (0,3) node {$\area(\Delta)=0$}; 
%\draw (0,2) node {};
%\end{tikzpicture}
%& 
%\begin{tikzpicture}[scale=0.5]
%\draw (0,5) node {$3$-$\gen=\{0,1,2\}$};
%\draw (0,4) node {$\codinv(\Delta)=0$};
%\draw (0,3) node {$\dinv(\Delta)=3$};
%\draw (0,2) node {};
%\end{tikzpicture}
%& 
%\begin{tikzpicture}[scale=0.5]
%\draw (0,5) node {$\Cogen=\{-3,-2,-1\}$};     
%\draw (0,4) node {\footnotesize $(1+a)(1+at^{-1})(1+at^{-2})$}; 
%\draw (0,3) node {};
%\end{tikzpicture}
%\\
%\hline
%\begin{tikzpicture}[scale=0.5]
%\draw (0,3.2) node {};
%\fill [fill=yellow!50!white] (0,1) rectangle (1,2);
%\fill [fill=gray!20!white] (0,0) rectangle (2,1);%
% pick an m and n HERE
% \pgfmathtruncatemacro{\m}{3} 
%  \pgfmathtruncatemacro{\n}{4} 
%%%%%%% DONT TOUCH THIS PART %%%%%%%%%%%%%%%%
% \pgfmathtruncatemacro{\mi}{\m-1} %\mi=m-1 
%    \pgfmathtruncatemacro{\ni}{\n-1} %\ni=n-1
    %
%  \foreach \x in {0,...,\n} % n=5
%    \foreach \y in {1,...,\m}     % m=7  
%       {\pgfmathtruncatemacro{\label}{ - \m * \x - \n *  \y +\m*\n} % - m * \x - n *  \y +mn
      % \node   (\x\y) at (\x,\y){};
%       \node  at (\x-.4,\y-.4) [blue!60!white, scale=.8]{\footnotesize \label};} 
%generates the grid and anderson labels
%\draw[dotted] (0,\m)--(\n,0); %draws the diagonal from (0,m) -- (n,0)
%  \foreach \x in {0,...,\n} % n=5
%    \foreach \y  [count=\yi] in {0,...,\mi}   % m-1=6  
    %\draw (\x\y)--(\x\yi);
%      \draw (\x,\y)--(\x,\yi);
%        \foreach \y in {0,...,\m}     % m=7  
%       \foreach \x  [count=\xi] in {-1,...,\ni}   % n-1=4
        %\draw (\x\y)--(\xi\y);
%         \draw (\x,\y)--(\xi-1,\y);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %% Draw dyck path by hand below
%\draw [very thick] (4,0)--(0,0)--(0,3); %lower boundary
%\draw[ very thick,red](0,3)--(0,1)--(2,1)--(2,0)--(4,0); %Dyck path

%
%\end{tikzpicture}
%& 
%\begin{tikzpicture}[scale=0.5]
%\draw (0,5) node {$\Gaps=\{1\}$};
%\draw (0,4) node {$\Delta=\mathbb{Z}_{\ge 0}\setminus\{1\}$};
%\draw (0,3) node {$\area(\Delta)=1$}; 
%\draw (0,2) node {};
%\end{tikzpicture}
%& 
%\begin{tikzpicture}[scale=0.5]
%\draw (0,5) node {$3$-$\gen=\{0,2,4\}$};
%\draw (0,4) node {$\codinv(\Delta)=1$};
%\draw (0,3) node {$\dinv(\Delta)=2$};
%\draw (0,2) node {};
%\end{tikzpicture}
%& 
%\begin{tikzpicture}[scale=0.5]
%\draw (0,5) node {$\Cogen=\{-1,1\}$};     
%\draw (0,4) node {$(1+a)(1+at^{-1})$}; 
%\draw (0,3) node {};
%\end{tikzpicture}
%\\
\hline
\begin{tikzpicture}[scale=0.5]
\draw (0,3.2) node {};
\fill [fill=gray!20!white] (0,0) rectangle (1,2);
\fill [fill=yellow!40!white] (1,0) rectangle (2,1);%
% pick an m and n HERE
 \pgfmathtruncatemacro{\m}{3} 
  \pgfmathtruncatemacro{\n}{4} 
%%%%%%% DONT TOUCH THIS PART %%%%%%%%%%%%%%%%
 \pgfmathtruncatemacro{\mi}{\m-1} %\mi=m-1 
    \pgfmathtruncatemacro{\ni}{\n-1} %\ni=n-1
    %
  \foreach \x in {0,...,\n} % n=5
    \foreach \y in {1,...,\m}     % m=7  
       {\pgfmathtruncatemacro{\label}{ - \m * \x - \n *  \y +\m*\n} % - m * \x - n *  \y +mn
      % \node   (\x\y) at (\x,\y){};
       \node  at (\x-.4,\y-.4) [blue!60!white, scale=.8]{\footnotesize \label};} 
%generates the grid and anderson labels
\draw[dotted] (0,\m)--(\n,0); %draws the diagonal from (0,m) -- (n,0)
  \foreach \x in {0,...,\n} % n=5
    \foreach \y  [count=\yi] in {0,...,\mi}   % m-1=6  
    %\draw (\x\y)--(\x\yi);
      \draw (\x,\y)--(\x,\yi);
        \foreach \y in {0,...,\m}     % m=7  
       \foreach \x  [count=\xi] in {-1,...,\ni}   % n-1=4
        %\draw (\x\y)--(\xi\y);
         \draw (\x,\y)--(\xi-1,\y);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %% Draw dyck path by hand below
\draw [very thick] (4,0)--(0,0)--(0,3); %lower boundary
\draw[ very thick,red](0,3)--(0,2)--(1,2)--(1,0)--(4,0); %Dyck path

%
\end{tikzpicture}
& 
\begin{tikzpicture}[scale=0.5]
\draw (0,5) node {$\Gaps=\{2\}$};
\draw (0,4) node {$\Delta=\mathbb{Z}_{\ge 0}\setminus\{2\}$};
\draw (0,3) node {$\area(\Delta)=1$}; 
%\draw (0,2) node {};
\end{tikzpicture}
& 
\begin{tikzpicture}[scale=0.5]
\draw (0,5) node {$3$-$\gen=\{0,1,5\}$};
\draw (0,4) node {$\codinv(\Delta)=2$};
\draw (0,3) node {$\dinv(\Delta)=1$};
%\draw (0,2) node {};
\end{tikzpicture}
& 
\begin{tikzpicture}[scale=0.5]
\draw (0,5) node {$\Cogen=\{-3,2\}$};     
\draw (0,4) node {$(1+a)(1+at^{-1})$}; 
\draw (0,3) node {};
\end{tikzpicture}
\\
\hline
\begin{tikzpicture}[scale=0.5]
\draw (0,3.2) node {};
\fill [fill=gray!20!white] (0,0) rectangle (1,1);
\fill [fill=yellow!40!white] (0,1) rectangle (1,2);
\fill [fill=yellow!40!white] (1,0) rectangle (2,1);
% pick an m and n HERE
 \pgfmathtruncatemacro{\m}{3} 
  \pgfmathtruncatemacro{\n}{4} 
%%%%%%% DONT TOUCH THIS PART %%%%%%%%%%%%%%%%
 \pgfmathtruncatemacro{\mi}{\m-1} %\mi=m-1 
    \pgfmathtruncatemacro{\ni}{\n-1} %\ni=n-1
    %
  \foreach \x in {0,...,\n} % n=5
    \foreach \y in {1,...,\m}     % m=7  
       {\pgfmathtruncatemacro{\label}{ - \m * \x - \n *  \y +\m*\n} % - m * \x - n *  \y +mn
      % \node   (\x\y) at (\x,\y){};
       \node  at (\x-.4,\y-.4) [blue!60!white, scale=.8]{\footnotesize \label};} 
%generates the grid and anderson labels
\draw[dotted] (0,\m)--(\n,0); %draws the diagonal from (0,m) -- (n,0)
  \foreach \x in {0,...,\n} % n=5
    \foreach \y  [count=\yi] in {0,...,\mi}   % m-1=6  
    %\draw (\x\y)--(\x\yi);
      \draw (\x,\y)--(\x,\yi);
        \foreach \y in {0,...,\m}     % m=7  
       \foreach \x  [count=\xi] in {-1,...,\ni}   % n-1=4
        %\draw (\x\y)--(\xi\y);
         \draw (\x,\y)--(\xi-1,\y);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %% Draw dyck path by hand below
\draw [very thick] (4,0)--(0,0)--(0,3); %lower boundary
\draw[ very thick,red](0,3)--(0,1)--(1,1)--(1,0)--(4,0); %Dyck path

%
\end{tikzpicture}
& 
\begin{tikzpicture}[scale=0.5]
\draw (0,5) node {$\Gaps=\{1,2\}$};
\draw (0,4) node {$\Delta=\mathbb{Z}_{\ge 0}\setminus\{1,2\}$};
\draw (0,3) node {$\area(\Delta)=2$}; 
%\draw (0,2) node {};
\end{tikzpicture}
& 
\begin{tikzpicture}[scale=0.5]
\draw (0,5) node {$3$-$\gen=\{0,4,5\}$};
\draw (0,4) node {$\codinv(\Delta)=2$};
\draw (0,3) node {$\dinv(\Delta)=1$};
%\draw (0,2) node {};
\end{tikzpicture}
& 
\begin{tikzpicture}[scale=0.5]
\draw (0,5) node {$\Cogen=\{1,2\}$};     
\draw (0,4) node {$(1+a)(1+at^{-1})$}; 
\draw (0,3) node {};
\end{tikzpicture}
\\
\hline
%\begin{tikzpicture}[scale=0.5]
%\draw (0,3.2) node {};
%\fill [fill=yellow!40!white] (0,0) rectangle (1,2);
%\fill [fill=yellow!40!white] (0,0) rectangle (2,1);%
% pick an m and n HERE
 %\pgfmathtruncatemacro{\m}{3} 
 % \pgfmathtruncatemacro{\n}{4} 
%%%%%%% DONT TOUCH THIS PART %%%%%%%%%%%%%%%%
% \pgfmathtruncatemacro{\mi}{\m-1} %\mi=m-1 
 %   \pgfmathtruncatemacro{\ni}{\n-1} %\ni=n-1
    %
%  \foreach \x in {0,...,\n} % n=5
    %\foreach \y in {1,...,\m}     % m=7  
       %{\pgfmathtruncatemacro{\label}{ - \m * \x - \n *  \y +\m*\n} % - m * \x - n *  \y +mn
      % \node   (\x\y) at (\x,\y){};
       %\node  at (\x-.4,\y-.4) [blue!60!white, scale=.8]{\footnotesize \label};} 
%generates the grid and anderson labels
%\draw[dotted] (0,\m)--(\n,0); %draws the diagonal from (0,m) -- (n,0)
%  \foreach \x in {0,...,\n} % n=5
%    \foreach \y  [count=\yi] in {0,...,\mi}   % m-1=6  
    %\draw (\x\y)--(\x\yi);
%      \draw (\x,\y)--(\x,\yi);
%        \foreach \y in {0,...,\m}     % m=7  
%       \foreach \x  [count=\xi] in {-1,...,\ni}   % n-1=4
        %\draw (\x\y)--(\xi\y);
%         \draw (\x,\y)--(\xi-1,\y);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %% Draw dyck path by hand below
%\draw [very thick] (4,0)--(0,0)--(0,3); %lower boundary
%\draw [very thick, red] (4,0)--(0,0)--(0,3); %Dyck path

%
%\end{tikzpicture}
%& 
%\begin{tikzpicture}[scale=0.5]
%\draw (0,5) node {$\Gaps=\{1,2,5\}$};
%\draw (0,4) node {$\Delta=\mathbb{Z}_{\ge 0}\setminus\{1,2,5\}$};
%\draw (0,3) node {$\area(\Delta)=3$}; 
%\draw (0,2) node {};
%\end{tikzpicture}
%& 
%\begin{tikzpicture}[scale=0.5]
%\draw (0,5) node {$3$-$\gen=\{0,4,8\}$};
%\draw (0,4) node {$\codinv(\Delta)=3$};
%\draw (0,3) node {$\dinv(\Delta)=0$};
%\draw (0,2) node {};
%\end{tikzpicture}
%& 
%\begin{tikzpicture}[scale=0.5]
%\draw (0,5) node {$\Cogen=\{5\}$};     
%\draw (0,4) node {$(1+a)$}; 
%\draw (0,3) node {};
%\end{tikzpicture}
%\\
\hline
\end{tabular}
\caption{Three out of five $(3,4)$-Dyck paths are on the left, with the cells corresponding to the gaps in yellow. The corresponding $(3,4)$-invariant subsets are in the second column, together with the $\area$, $3$-generators, $\codinv,$ and $\dinv$ in the third column, and the corresponding Schr\"oder factor is in the fourth.}\label{figure: 3,4 Schroder}
\end{figure}


%While it is obvious that to any $(m,n)$-Dyck path we can associate a partition, one can ask given a partition $\lambda$ what the slope of the `closest' line to its associated Dyck path is. This was studied by Bergeron and Mazin \cite{BM} where they defined a \newword{triangular partition} as the unique maximal Dyck path weakly below the line from $(0,r)$ to $(s,0)$, for any $s,r \in \R$. In particular, different values of $s$ and $r$ may yield the same triangular partition. More precisely, 

%\NS{describe the maximal and minimal slopes from Misha's paper here.}

%\MM{I would put the $n,m$-Dyck paths and the bijection to $n,m$-invariant subsets, plus the area and dinv statistics on both sides, so that in the next subsection we can write the recursion for the $n,m$ $q,t$-Catalan polynomial via invariant subsets. And then after that have a subsection for tringular partitions.}

%\NS{Feel free to move, rewrite, or delete anything I write. I'm just trying to put content here and then move it later. }


\subsection{Recursions for Invariant Subsets}\label{subsection: recursions for inv sets}

In \cite{GMV} the fourth author together with Gorsky and Vazirani introduced a recursion computing the rational $(q,t)$-Catalan series and showed that their recursion is equivalent to the Hogancamp-Mellit recursion in the case of the torus link. Hence, in the relatively prime case the Gorsky-Mazin-Vazirani recursion recovers the $(q,t)$-Catalan polynomials.

Let $(m,n)$ be a pair of positive relatively prime integers. In order to define the recursion, one needs to consider subfamilies in the set of invariant subsets $\Inv_{n,m}$ given by fixing the intersection of the subsets with the interval $[0,n+m-1].$ Let ${\bf w}\in \{0,1\}^{n+m}$ be a binary sequence of length $n+m.$

\begin{definition}
    Set
   $
    I_{\bf w}:=\{\Delta\in \Inv_{n,m}:\forall 0\le i<n+m, i\in\Delta \Leftrightarrow w_i=1\}
    $ 
   and define:
    \begin{equation}\label{eq:Pw}
    P_{\bf w}(q,t,a):=\sum_{\Delta\in I_{\bf w}} q^{\area(\Delta)}t^{\codinv(\Delta)}\prod_{k\in\Cogen(\Delta)\cap\mathbb{Z}_{\ge 0}}(1+at^{\lambda_k(\Delta)}).
    \end{equation}
\end{definition}
Then, the Schr\"oder polynomial can be obtained from \eqref{eq:Pw},
\begin{equation*}
    S_{n,m}(q,t,a)=\frac{(1-q)t^{\delta(n,m)}}{q^{n+m}}P_{0^{n+m}}(q,t^{-1},a)=\frac{t^{\delta(n,m)}}{q^{n+m-1}}P_{0^{n+m-1}1}(q,t^{-1},a).
\end{equation*}

The polynomials $P_{\bf w}$ satisfy a recursion, however, in order to match this recursion to the Hogancamp-Mellit recursion, certain adjustments are required.

First, we need to replace the sequence ${\bf w}$ of length $n+m$ by two sequences $({\bf x},{\bf y})$ in the alphabet $\{0,\bullet,1\}$ of lengths $m$ and $n$ respectively. The sequence ${\bf x}$ records gaps (encoded by $0$), $n$-generators (encoded by $1$), and the rest of the elements of $\Delta$ (encoded by $\bullet$) on the interval $[n,n+m-1].$ Similarly, the sequence ${\bf y}$ records gaps, $m$-generators, and the rest of the elements of $\Delta$ on the interval $[m,n+m-1].$ In other words,

\begin{definition}
Let ${\bf x},{\bf y}$ be sequences as above. Let $I_{{\bf x},{\bf y}}$ be the set of $\Delta\in I_{n,m}$ such that:
\[ \forall 0\le k<m
\begin{cases}
x_k=0 \Leftrightarrow k+n\notin\Delta, \\
x_k=1 \Leftrightarrow k+n\in\ngen(\Delta), \\
x_k=\bullet \Leftrightarrow k\in\Delta,
\end{cases}
%\text{and}
\forall 0\le k<n 
\begin{cases}
y_k=0 \Leftrightarrow k+m\notin\Delta, \\
y_k=1 \Leftrightarrow k+m\in\mgen(\Delta), \\
y_k=\bullet \Leftrightarrow k\in\Delta.
\end{cases}
\]
%\begin{tabular}{lll}
%$I_{{\bf x},{\bf y}}=\{\Delta\in I_{n,m}:$& $\forall 0\le k<m$ & $x_k=0 \Leftrightarrow k+n\notin\Delta,$ \\
%&& $x_k=1 \Leftrightarrow k+n\in\ngen(\Delta),$ \\
%&& $x_k=\bullet \Leftrightarrow k\in\Delta,$ \\
%& $\forall 0\le k<n$ & $y_k=0 \Leftrightarrow k+m\notin\Delta,$ \\
%&& $y_k=1 \Leftrightarrow k+m\in\mgen(\Delta),$ \\
%&& $y_k=\bullet \Leftrightarrow k\in\Delta\}.$ 
%\end{tabular}
\end{definition}

\begin{example} \label{ex:4-7sequences}
Let $(n,m)=(4,7)$ and $\Delta=\mathbb{Z}_{\ge 0}\setminus\{0,1,2,3,4,6,7,8,10\}.$
Then, the associated binary sequence $w=00000100010$ yields the ternary sequences %${\bf x}=(\bullet 1 1 \bullet 0 \bullet \bullet)$ and ${\bf y} =(\bullet 1 0 \bullet 1 )$% these are wrong: 0 - gaps, 1 - generators, \bullet - other elements.
${\bf x}=(01000\bullet 0)$ and ${\bf y} =(0010)$,
since the only $4$-generator in $[4,10]$ is $5$ and the only $7$-generator in $[7,10]$ is $9$. In particular, $9$ is not a $4$-generator since $9-4=5 \in \Delta$. Thus, $ \Delta\in I_{00000100010}= I_{01000\bullet 0, 0010}$.

  %  Let $(n,m)=(5,7)$ and $\Delta=\mathbb{Z}_{\ge 0}\setminus\{1,2,4,9\}.$ Then $$\Delta\in I_{100101111011}=I_{\bullet11\bullet0\bullet\bullet,\bullet10\bullet1}.$$
\end{example}

%\begin{definition}\label{Definition: Iv,w Pv,w}
%By slightly abusing notation, we set
%$I_{{\bf u},{\bf v}}:=I_{\bf w}$ \quad and \quad
%$ P_{{\bf u},{\bf v}}(q,t,a):=P_{\bf w}(q,t,a)$.
%\end{definition} 
The statistics on $\Inv_{n,m}$ are modified as follows. Set:
\begin{align*}
\area'(\Delta)&=\sharp\{k\in\mathbb{Z}_{\ge n+m}:k\notin\Delta\}=\sharp\left(\Gaps(\Delta)\cap\mathbb{Z}_{\ge n+m}\right), \\
\codinv'(\Delta)&=\sum\limits_{a\in\ngen(\Delta)}\sharp\{k\in\mathbb{Z}_{\ge n+m}:a\le k<a+m,\ k\notin\Delta\}-\frac{\lambda(\Delta)(\lambda(\Delta)-1)}{2},
\end{align*}
where $\lambda(\Delta):=\lambda_{-1}(\Delta)=\sharp\{a\in\ngen(\Delta): n\le a< n+m\}.$

\begin{definition}
Given sequences $\bf x$ and $\bf y$ as above, let
\begin{equation*}
Q_{{\bf x},{\bf y}}(q,t,a):=\sum_{\Delta\in I_{{\bf x},{\bf y}}} q^{\area'(\Delta)}t^{-\codinv'(\Delta)}\prod_{k\in\Cogen(\Delta)\cap\mathbb{Z}_{\ge 0}}(1+at^{-\lambda_k(\Delta)}).
\end{equation*}
\end{definition}
Note that for any $\Delta\in I_{0^{n+m}}$ one gets $\area'(\Delta)=-n-m + \area(\Delta),$ $\codinv'(\Delta)=\codinv(\Delta),$ and all the double co-generators are non-negative. Therefore,
\begin{equation*}\label{Equation: Q vs P}
Q_{0^m,0^n}(q,t,a)=q^{-n-m}P_{0^{n+m}}(q,t^{-1},a).
\end{equation*}

\begin{theorem}[\cite{GMV}]\label{Theorem: recursion Q_v,w}
The following recursions hold:
\begin{align*}
Q_{0{\bf u},0{\bf v}}&=t^{-|{\bf u}|}Q_{{\bf u}1,{\bf v}1}+qt^{-|{\bf u}|}Q_{{\bf u}0,{\bf v}0}, &
Q_{1{\bf u},0{\bf v}}&=Q_{{\bf u}1,{\bf v}\bullet},&
Q_{\bullet{\bf u},\bullet{\bf v}}&=Q_{{\bf u}\bullet,{\bf v}\bullet},\\
Q_{1{\bf u},1{\bf v}}&=(t^{|{\bf u}|}+a)Q_{{\bf u}\bullet,{\bf v}\bullet},&
Q_{0{\bf u},1{\bf v}}&=Q_{{\bf u}\bullet,{\bf v}1},&
Q_{\emptyset,\emptyset}&=1.
\end{align*}

%where the recursions on the left are equivalent to those in  \eqref{eq-recursion binary N+M}.
\end{theorem}

Finally, notice that in the recursion for $Q$ one can completely ignore all the $\bullet$'s. Also, it follows that the recursion always terminates in $Q_{\emptyset,\emptyset},$ so one doesn't need the normalization conditions for $Q_{\emptyset,0^n}$ and $Q_{0^m,\emptyset}.$

\begin{theorem}[\cite{GMV}]
\label{thm:ignore bullets}
Let $({\bf u},{\bf v})$ be the sequences obtained from the sequences $({\bf x},{\bf y})$ by ignoring all $\bullet$'s. Then
\begin{equation*}
R_{{\bf u},{\bf v}}(q,t,a)=Q_{{\bf x},{\bf y}}(q,t,a).
\end{equation*}
\end{theorem}

\begin{corollary}[\cite{Me,GMV}]
\label{cor:R vs catalan}
The Poincar\'e polynomial of the reduced Khovanov-Rozansky homology of the $(n,m)$-torus knot is given by
\begin{align*}
R_{0^{m-1}1,0^{n-1}1}(q,t,a)=&Q_{0^{m-1}1,0^{n-1}1}(q,t,a)=\frac{P_{0^{n+m-1}1}(q,t^{-1},a)}{q^{n+m-1}}=\frac{S_{n,m}(q,t,a)}{t^{\delta(n,m)}}.
\end{align*}
\end{corollary}

\begin{remark}
    The last formula was first proven by Anton Mellit in \cite{Me}. We follow notations from \cite{HM} and \cite{GMV}, where the result was generalized to torus links. 
\end{remark}

%\begin{theorem}[\cite{GMV}]
%Let ${\bf w}=(w_0,\ldots,w_{n+m-1})$ be a binary sequence. Let also 
%\begin{equation*}
%\begin{aligned}
%&{\bf w'}=(w_1,\ldots,w_{n+m-1},1),\\
%&{\bf w''}=(w_1,\ldots,w_{n+m-1},0).
%\end{aligned}
%\end{equation*}
%The polynomials $P_{\bf w}$ satisfy the following recurrence relation:
%\begin{equation}
%\label{eq-recursion binary N+M}
%P_{\bf w}=
%\begin{cases} 
%q(P_{\bf w'}+P_{\bf w''}),& \text{if} \ \ %w_0=w_n=w_m=0,\\
%qP_{\bf w'},& \text{if} \ \ w_0=0 \ \  \text{and} \ %w_n+w_m>0,\\
%t^{\lambda({\bf w})}P_{\bf w'},& \text{if} \ \ %w_0=w_n=w_m=1,
%\end{cases}
%\end{equation}
%where $\lambda({\bf w}):=\sum\limits_{i=0}^{m-1}(w_{i+n}-w_i).$
%\end{theorem}


\subsection{Monotone and Coxeter Links}\label{subsection: mon and cox}
 
In \cite{GL23} Galashin-Lam study a family of links, called \emph{monotone} that arise from certain curves on the plane. They define a new invariant, the \emph{elliptic Hall algebra superpolynomial}, which they prove recovers the HOMFLY polynomial of $L_C$ and conjecture agrees with the Poincar\'e series of the Khovanov-Rozansky homology of any algebraic link $L_C$. 

\begin{definition}
Let $C$ denote a curve from $(0,n)$ to $(m,0)$. A \newword{monotone link} $L_C$ is a projection onto $\R^2 / \Z^2$  of a curve $C$ such that the $x$- and $y$-coordinates of $C$ are monotone increasing and decreasing, respectively\footnote{This differs slightly from the definition in \cite{GL23} by a flip sending $x \mapsto -x$.}. Trace the projection of $C$ starting from the left top corner, crossing the earlier strand on top.
\end{definition} 

It is well known \cite{HM} that if $C$ is the straight diagonal line, then $L_C$ is the $(m,n)$ torus link, with the special case of $m,n$ relatively prime yielding a knot. 


%\begin{definition}
%A curve $C$ is \newword{$(m,n)$-almost linear} if it passes just above the diagonal; that is all lattice points on the diagonal lie below it and all lattice points just above the diagonal lie above it. 
%\end{definition} 

%It is proven \cite[Lem. 8.1]{GL23} that for $C$ a $(dm,dn)$-almost linear curve with $\gcd(m,n)=1$, the associated link $L_C$ is isotopic in $\R^3$ to the $(d,dmn+1)$-cable of the $(m,n)$ torus knot $T_{m,n}$.

% Misha: I included a reference to this in the very last paragraph of the paper, I feel like we can safe space by skipping this here. 

Let $\mathbb{A}$ denote the annulus. Given a curve $C$, we can construct its annular closure $\beta_C \in \mathbb{A}\times[0,1]$ as follows. Consider the projection of $C$ onto $[0,1]^2$. Now, identify the top and bottom boundaries so that for each point $x \in (0,1)$ for which $(x,0)$ and $(x,1)$ are in $C$, the line connecting them lies \emph{underneath} all other strands. Denote the resulting braid in $ \mathbb{A}\times[0,1]$ by $\beta_C$ (see Figure \ref{fig:annularclosure}).

\begin{figure}
\center
\begin{tikzpicture}[scale=0.5]
% pick an m and n HERE
 \pgfmathtruncatemacro{\m}{3} 
  \pgfmathtruncatemacro{\n}{4} 
%%%%%%% DONT TOUCH THIS PART %%%%%%%%%%%%%%%%
 \pgfmathtruncatemacro{\mi}{\m-1} %\mi=m-1 
    \pgfmathtruncatemacro{\ni}{\n-1} %\ni=n-1
    \pgfmathtruncatemacro{\mj}{\m+1}
    \pgfmathtruncatemacro{\nj}{\n+1}
    %
 % \foreach \x in {0,...,\n} % n=5
  %  \foreach \y in {1,...,\m}     % m=7  
    %   {\pgfmathtruncatemacro{\label}{ - \m * \x - \n *  \y +\m*\n} % - m * \x - n *  \y +mn
      % \node   (\x\y) at (\x,\y){};
      % \node  at (\x-.4,\y-.4) [blue!60!white, scale=.8]{\footnotesize \label};} 
%generates the grid and anderson labels
\draw[dotted] (0,\m)--(\n,0); %draws the diagonal from (0,m) -- (n,0)
  \foreach \x in {0,...,\n} % n=5
    \foreach \y  [count=\yi] in {0,...,\m}   % m-1=6  
    %\draw (\x\y)--(\x\yi);
      \draw (\x,\y)--(\x,\yi);
        \foreach \y in {0,...,\mj}     % m=7  
       \foreach \x  [count=\xi] in {0,...,\ni}   % n-1=4
        %\draw (\x\y)--(\xi\y);
         \draw (\x,\y)--(\xi,\y);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %% Draw dyck path by hand below
\draw[very thick] (0,\n)--(0,0)--(\m,0);
\draw [red, very thick] (0,3)--(0,2)--(1,2)--(1,1)--(2,1)--(2,0)--(4,0);
\begin{scope}[shift={(0,.15)}]
 \draw[thick,blue] (0,\m)--(\n,0);
\node at (0,\m) [blue, scale=.8]{$\bullet$};
\node at (\n,0) [blue,scale=.8]{$\bullet$};
\end{scope}
\node at (0,-.5){};
\end{tikzpicture}
\qquad \qquad
\begin{tikzpicture}[scale=0.2,yscale=1, xscale=1]
%\draw [thin, color=gray!50!white] (0,0) grid +(10,10);
\begin{scope}[xscale=-1]
%annular closure
%\draw[line width = 6pt, white] (2.5,10)..controls ++(1,1.5) and ++(0,0)..(3.5,10)--(3.5,0)..controls ++(0,-1.5) and ++(-.5,-.5)..(2.5,0);
\draw[very thick, green!60!black] (2.5,10)..controls ++(1,1.5) and ++(0,0)..(3.5,10)--(3.5,0)..controls ++(0,-1.5) and ++(-.5,-.5)..(2.5,0);
%
%\draw[line width = 6pt, white] (7.5,10)..controls ++(1,1.5) and ++(0,0)..(8.5,10)--(8.5,0)..controls ++(0,-1.5) and ++(-.5,-.5)..(7.5,0);
\draw[very thick, green!60!black] (7.5,10)..controls ++(1,1.5) and ++(0,0)..(8.5,10)--(8.5,0)..controls ++(0,-1.5) and ++(-.5,-.5)..(7.5,0);
%
%\draw[line width = 6pt, white] (5,10)..controls ++(1,1.5) and ++(0,0)..(6,10)--(6,0)..controls ++(0,-1.5) and ++(-.5,-.5)..(5,0);
\draw[very thick, green!60!black] (5,10)..controls ++(1,1.5) and ++(0,0)..(6,10)--(6,0)..controls ++(0,-1.5) and ++(-.5,-.5)..(5,0);
\end{scope}
\begin{scope}[rotate=90, shift={(0,0)}]
%
\draw (0,0)--(10,0);
\draw (10,10)--(0,10);
\draw [dotted, thick] (10,0)--(10,10);
\draw [dotted,thick] (0,10)--(0,0);
%
\draw[ white,line width = 6pt] (0,7.5)--(2,10);
\draw[ white,line width = 6pt]  (0,5)--(4,10);
\draw[ white,line width = 6pt]  (0,2.5)--(6,10);
\draw[ white,line width = 6pt]  (2,0)--(8,10);
\draw[ white,line width = 6pt]  (4,0)--(10,7.5);
\draw[ white,line width = 6pt]  (6,0)--(10,5);
\draw[ white,line width = 6pt]  (8,0)--(10,2.5);
%
\draw[ blue,very thick] (0,7.5)--(2,10);
\draw[ blue,very thick] (0,5)--(4,10);
\draw[blue, very thick] (0,2.5)--(6,10);
\draw[blue,very thick] (2,0)--(8,10);
\draw[blue,very thick] (4,0)--(10,7.5);
\draw[blue,very thick] (6,0)--(10,5);
\draw[blue,very thick] (8,0)--(10,2.5);
%

%top dots
\filldraw (2,10) [blue] circle (2mm);
\filldraw (4,10) circle (1mm);
\filldraw (6,10) circle (1mm);
\filldraw (8,10) circle (1mm);
%left side dots
\filldraw (0,7.5) circle (1mm);
\filldraw (0,5) circle (1mm);
\filldraw (0,2.5) circle (1mm);
%bottom dots
\filldraw (2,0) [blue] circle (2mm);
\filldraw (4,0) circle (1mm);
\filldraw (6,0) circle (1mm);
\filldraw (8,0) circle (1mm);
%right side dots
\filldraw (10,7.5) circle (1mm);
\filldraw (10,5) circle (1mm);
\filldraw (10,2.5) circle (1mm);
\end{scope}
%\draw[dashed] (current bounding box.north west) rectangle (current bounding box.south east);
\end{tikzpicture}
\caption{The curve $C$ on the left and its projection onto $[0,1]^2$ and annular closure $\beta_C$, on the right. The associated triangular partition is displayed in red, so that $\mu = (3,2,1,0,0)$ with $\mathbf{b}= (0,1,1,1)$ and $\mathbf{e}=(1,1,1)$. It is straightforward to verify that the braid $\beta_{\bf b, \bf e}^{cox} = JM_2^1 JM_3^1 JM_4^1 \sigma_1\sigma_2\sigma_3$ is isotopic to $\beta_C$. }\label{fig:annularclosure}
\end{figure}

Denote by $\sigma_i \in B_n$, with $B_n$ the braid group, the positive crossing of the $i^{th}$ and $i+1^{st}$ strands, i.e. the $i^{th}$ strand is above the $i+1^{st}$ strand. 

%Given sequences $\mathbf{b}=(b_1,\dots, b_m) \in \Z^m_{\geq 0}$ and $\mathbf{e}=(\epsilon_1,\dots, \epsilon_{m-1}) \in \{ 0,1\}^{m-1}$, let:
%\begin{align}
%cox(\mathbf{e})&:= \sigma_1^{\epsilon_1}\dots \sigma_{m-1}^{\epsilon_{m-1}}\\
%JM_i &:= \sigma_{i-1}\dots \sigma_1\sigma_1 \dots \sigma_{i-1}.
%\end{align}

\begin{definition}
Given sequences $\mathbf{b}=(b_1,\dots, b_m) \in \Z^m_{\geq 0}$ and $\mathbf{e}=(\epsilon_1,\dots, \epsilon_{m-1}) \in \{ 0,1\}^{m-1}$, the \newword{Coxeter braid} $\beta_{\mathbf{b},\mathbf{e}}^{cox}$ is given by:
\begin{equation}
\beta_{\mathbf{b},\mathbf{e}}^{cox}:= JM_1^{b_1}\dots JM_{m}^{b_m} \sigma_1^{\epsilon_1}\dots \sigma_{m-1}^{\epsilon_{m-1}}, 
\end{equation}
where $JM_i:= \sigma_{i}\dots \sigma_{m-1}\sigma_{m-1} \dots \sigma_{i}$
%where $JM_i:= \sigma_{i-1}\dots \sigma_1\sigma_1 \dots \sigma_{i-1}$ 
for each $1\leq i\leq m$. 
\end{definition}

To any curve $C$, we can assign a Coxeter braid $\beta_C^{cox}$ in the following way. Let $\mu = (n, \mu_1,\dots, \mu_m)$ be such that $(\mu_1,\dots,\mu_m)$ is the transpose of the triangular partition corresponding to the curve $C$. Set $b_{m-i+1} = \mu_{i-1}-\mu_i$ (with $\mu_0=n)$ and for each $1\leq i \leq m-1$, set $\epsilon_i =0$ if $C$ passes through the lattice point $(i,j)$ for some $j \in \Z$, with $\epsilon_i =1$ otherwise.  Then for $\mathbf{e}_C=(\epsilon_1,\dots,\epsilon_{m-1})$ and $\mathbf{b_C}=(b_1.\dots,b_m)$, let $\beta_C^{cox} := \beta_{\mathbf{b}_C,\mathbf{e}_C}^{cox}$.

\begin{theorem}[\cite{GL23}]
The braid $\beta_{C}^{cox}$ is conjugate to the annular closure $\beta_C$ of $C$. In particular, all monotone links in $\mathbb{A} \times [0,1]$ are Coxeter links, and all Coxeter links arise in this way.     
\end{theorem}

\section{Main Results}

\subsection{Schr\"oder Polynomials for Triangular Partitions}

\begin{definition} A partition $\lambda$ is called \newword{triangular} if there exist two not necessarily integral points $(0,s)$ and $(r,0)$ such that $\lambda$ consists of all the boxes below the line connecting these points, in which case we denote $\lambda$ by $\tau_{r,s}$.
%Given two not necessarily integral points $(0,s)$ and $(r,0)$, the \newword{Triangular partition} $\tau_{r,s}$ is the partition consisting of all the boxes that fit under the line connecting these points. 
\end{definition}

Evidently, for any given triangular partition $\lambda$, there exist many choices for positive real numbers $r$ and $s$ such that $\lambda=\tau_{r,s}$ (see \cite{BM} for details). In particular, one can always choose $r$ and $s$ in such a way that $\lambda=\tau_{r,s},$ and $r/s=n/m,$ where $(n,m)$ are positive relatively prime integers with $r\le n$ (equivalently, the line connecting $(r,0)$ to $(0,s)$ is below the line connecting $(n,0)$ to $(0,m)$). Generalized $(q,t)$-Catalan polynomials corresponding to triangular partitions appeared in the generalized shuffle theorem \cite{BHMPS}. 

We claim that to any triangular partition $\tau_{r,s}$ one can associate a pair of binary sequences ${\bf u}(s,t)$ and ${\bf v}(s,t)$ \cite{CGHM24}, which we explain how to construct in Example \ref{example: tringular Schroder} below. With this in hand, we extend the Schr\"oder polynomial to the triangular setting. 

\begin{definition}
Let $\lambda=\tau_{r,s}$ and $(n,m)$ be as above with associated sequences ${\bf u}(r,s)$ and ${\bf v}(r,s)$ as in Example \ref{example: tringular Schroder}. The \newword{(q,t)-Schr\"oder polynomial} for triangular partitions is defined as $S_{\lambda}(q,t,a):=t^{|\lambda|}R_{{\bf u}(r,s),{\bf v}(r,s)}(q,t,a).$
\end{definition}
%In particular, at $a=0$, the Schr\"oder polynomial $S_{\lambda}(q,t,a)$ specializes to the corresponding $(q,t)$-Catalan polynomial. 

%Below, we will define the Schr\"oder polynomial $S_{\tau_{r,s}}(q,t,a)$ using invariant subsets, which at $a=0$, specialize to the corresponding $(q,t)$-Catalan polynomials. To do so we must construct two sequences ${\bf u}(s,t)$ and ${\bf v}(s,t)$ which we explain via an example. 


\begin{figure}
\center
\begin{tikzpicture}[scale=0.6]
\fill [fill=gray!20!white] (-2,0) rectangle (0,4);
%\fill [fill=gray!20!white] (-1,-1) rectangle (7,0);
\draw[line width = 27pt, yellow](-2,3.5)--(-1,3.5)--(.5,3.5)--(.5,2.5)--(2.5,2.5)--(2.5,1.5)--(3.5,1.5)--(3.5,.5)--(6,.5);\
%
% pick an m and n HERE
 \pgfmathtruncatemacro{\m}{4} 
  \pgfmathtruncatemacro{\n}{7} 
%%%%%%% DONT TOUCH THIS PART %%%%%%%%%%%%%%%%
 \pgfmathtruncatemacro{\mi}{\m-1} %\mi=m-1 
    \pgfmathtruncatemacro{\ni}{\n-1} %\ni=n-1
    %
  \foreach \x in {-1,...,\n} % n=5
    \foreach \y in {1,...,\m}     % m=7  
       {\pgfmathtruncatemacro{\label}{ - \m * \x - \n *  \y +\m*\n} % - m * \x - n *  \y +mn
      % \node   (\x\y) at (\x,\y){};
       \node  at (\x-.3,\y-.3) [blue!60!white, scale=.8]{\label};} 
%generates the grid and anderson labels
\draw[dotted] (0,\m)--(\n,0); %draws the diagonal from (0,m) -- (n,0)
  \foreach \x in {-1,...,\n} % n=5
    \foreach \y  [count=\yi] in {0,...,\mi}   % m-1=6  
    %\draw (\x\y)--(\x\yi);
      \draw (\x,\y)--(\x,\yi);
        \foreach \y in {0,...,\m}     % m=7  
       \foreach \x  [count=\xi] in {-2,...,\ni}   % n-1=4
        %\draw (\x\y)--(\xi\y);
         \draw (\x,\y)--(\xi-1,\y);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %% Draw dyck path by hand below
\draw[ very thick,red](0,2)--(2,2)--(2,1)--(3,1)--(3,0); %Dyck path
\draw[ thick, blue] (-1.4,4)--(-1,3.77)--(0,3.2)--(5.6,0); %shifted diagonal
\draw [very thick] (7,0)--(0,0)--(0,4); %lower boundary
%
\end{tikzpicture}
\caption{The triangular partition $\lambda=(2,3)=\tau_{3.2,5.6}$ with $(m,n)=(4,7)$ and the Anderson labels denoted in blue.}\label{fig:4-7}
\end{figure}


\begin{example}\label{example: tringular Schroder}
Let $\lambda = \tau_{3.2,5.6} = (2,3)$ as in Figure \ref{fig:4-7} and observe that the line connecting $(0,3.2)$ and $(5.6,0)$ has the same slope as the diagonal line connecting $(0,4)$ and $(7,0)$. We call the line connecting $(0,3.2)$ and $(5.6,0)$ the \newword{shifted diagonal}, with $(n,m)=(4,7)$ denoting the closest line above it with equal slope and integer $x$ and $y$ -intercepts. 

%Consider $\lambda=(2,3)=\tau_{3.2,5.6}.$ Let $(n,m)=(4,7)$ so that the line connecting points $(0,3.2)$ and $(5.6,0)$ is parallel to the line connecting $(0,n)$ and $(m,0),$ and is located below it 
%(see Figure \ref{fig:4-7}). We call the line connecting $(0,3.2)$ and $(5.6,0)$ the \newword{shifted diagonal}.  %This condition can be reformulated in the to align with constructions of Section \ref{subsection: recursions for inv sets}.
%The associated set of $(4,7)$-invariant subsets $S$ can be constructed as follows:


Let $W=\{-5, -4 \dots, 5\}=[-5,5]$ be the window of labels of all cells intersected by the shifted diagonal (shaded yellow in Figure \ref{fig:4-7}). The 
subdiagrams of $\lambda$ are in bijection with the subfamily $I_{3.2,5.6}^0 \subset I_{4,7}^0$ consisting of subsets $\Delta$, such that $\{1,2,3,5\}\cap\Delta=\emptyset$, where $\{1,2,3,5\} = \Gaps(\lambda)$. 
This is equivalent to saying that $\Delta\cap W=\{0,4\}$. Hence,
%Consider the window of all the values in the cells intersected by the shifted diagonal $W=\{-5, -4 \dots, 5\}=[-5,5]$ (highlighted in yellow in Figure \ref{fig:4-7}). The subdiagrams of $\lambda$ are then in bijection with the subfamily $I_{3.2,5.6}^0 \subset I_{4,7}^0$ consisting of subsets $\Delta$, such that $\{1,2,3,5\}\cap\Delta=\emptyset.$ (Here $\{1,2,3,5\}$ are exactly the positive labels that are above the boundary of $\lambda.$) Equivalently, we get $\Delta\cap W=\{0,4\}.$ Therefore,
\begin{align*}
I_{3.2,5.6}^0=&\{\Delta\in I_{4,7}^0: \{1,2,3,5\}\cap\Delta=\emptyset\}\\
=&\{\Delta\in I_{4,7}^0: \Delta\cap W=\{0,4\}\}=\{\Delta \in I_{4,7}^0: \Delta+5\in I_{00000100010}\}.
\end{align*}
That is, the family $I_{3.2,5.6}^0$ is simply $I_{00000100010}=I_{01000\bullet0,0010}$ from Example \ref{ex:4-7sequences} shifted down by $5$. So setting ${\bf u}(3.2,5.6)=010000$ and ${\bf v}(3.2,5.6)=0010$, we obtain
$S_{\lambda}(q,t,a)=t^5R_{010000,0010}(q,t,a)$. 
 %   S_{\lambda}(q,t,a)=t^{|\lambda|}Q_{01000\bullet0,0010}(q,t,a)=t^{|\lambda|}R_{010000,0010}(q,t,a)=t^5R_{010000,0010}(q,t,a).
%We will use the notation ${\bf u}(3.2,5.6)=010000$ and ${\bf v}(3.2,5.6)=0010$ for the binary sequences obtained through the construction. 
In Figure \ref{figure: triangular Schroder} we illustrate the computation of the contributions towards $S_{\lambda}(q,t,a)$ of three of the nine invariant subsets in $I_{3.2,5.6}^0.$ All together:
\begin{align*}
S_{\lambda}(q,t,a)=&q^5(1+a)+q^4t(1+a)(1+at^{-1})+q^3t^2(1+a)(1+at^{-1})\\
+&q^3t(1+a)(1+at^{-1})+q^2t^3(1+a)(1+at^{-1})(1+at^{-2})\\
+&q^2t^2(1+a)(1+at^{-1})+qt^3(1+a)(1+at^{-1})\\
+&qt^4(1+a)(1+at^{-1})(1+at^{-2})+t^5(1+a)(1+at^{-1})(1+at^{-2}).    
\end{align*}
Note that plugging in $a=0$ we recover the corresponding $(q,t)$-Catalan polynomial:
$$
C_{\lambda}(q,t)=S_\lambda(q,t,0)=q^5+q^4t+q^3t^2+q^2t^3+qt^4+t^5+q^3t+q^2t^2+qt^3.
$$
\end{example}

\begin{figure}
    \centering
\begin{tabular}{|l|l|l|}
%\hline
%$\Gaps\cap\mathbb{Z}_{\ge 6}=\{6,9,10,13,17\}$&$4$-$\gen=\{0,7,14,21\}$& $\Cogen=\{17\}$  \\
%$\area=5$&$\codinv=5,$ $\dinv=0$& $1+a$  \\
%\hline
%$\Gaps\cap\mathbb{Z}_{\ge 6}=\{6,9,10,13\}$&$4$-$\gen=\{0,7,14,17\}$& $\Cogen=\{10,13\}$  \\
%$\area=4$&$\codinv=4,$ $\dinv=1$& $(1+a)(1+at^{-1})$  \\
%\hline
%$\Gaps\cap\mathbb{Z}_{\ge 6}=\{6,9,10\}$&$4$-$\gen=\{0,7,13,14\}$& $\Cogen=\{9,10\}$  \\
%$\area=3$&$\codinv=3,$ $\dinv=2$& $(1+a)(1+at^{-1})$  \\
\hline
$\Gaps\cap\mathbb{Z}_{\ge 6}=\{6,9,13\}$&$4$-$\gen=\{0,7,10,17\}$& $\Cogen=\{3,13\}$  \\
$\area=3$&$\codinv=4,$ $\dinv=1$& $(1+a)(1+at^{-1})$  \\
\hline
$\Gaps\cap\mathbb{Z}_{\ge 6}=\{6,9\}$&$4$-$\gen=\{0,7,10,13\}$& $\Cogen=\{3,6,9\}$  \\
$\area=2$&$\codinv=2,$ $\dinv=3$& $(1+a)(1+at^{-1})(1+at^{-2})$  \\
\hline
$\Gaps\cap\mathbb{Z}_{\ge 6}=\{6,10\}$&$4$-$\gen=\{0,7,9,14\}$& $\Cogen=\{5,10\}$  \\
$\area=2$&$\codinv=3,$ $\dinv=2$& $(1+a)(1+at^{-1})$  \\
%\hline
%$\Gaps\cap\mathbb{Z}_{\ge 6}=\{9\}$&$4$-$\gen=\{0,6,7,9\}$& $\Cogen=\{3,9\}$  \\
%$\area=1$&$\codinv=2,$ $\dinv=3$& $(1+a)(1+at^{-1})$  \\
%\hline
%$\Gaps\cap\mathbb{Z}_{\ge 6}=\{6\}$&$4$-$\gen=\{0,7,9,10\}$& $\Cogen=\{3,5,6\}$  \\
%$\area=1$&$\codinv=1,$ $\dinv=4$& $(1+a)(1+at^{-1})(1+at^{-2})$  \\
%\hline
%$\Gaps\cap\mathbb{Z}_{\ge 6}=\emptyset$&$4$-$\gen=\{0,6,7,9\}$& $\Cogen=\{2,3,5\}$  \\
%$\area=0$&$\codinv=0,$ $\dinv=5$& $(1+a)(1+at^{-1})(1+at^{-2})$  \\
\hline
\end{tabular}
    \caption{For three of the nine $\Delta\in I_{3.2,5.6}^0$ we record the gaps that are greater than $5,$ since only those contribute to $\area$ and $\codinv$ (respectively, $\area'$ and $\codinv'$ on $I_{01000\bullet0,0010}$). There cannot be any double co-generators below the interval $W=[0,n+m-1]-5,$ therefore all co-generators are used for the Schr\"oder factors.}
    \label{figure: triangular Schroder}
\end{figure}



%The same triangular partition $\lambda$ can be cut by different lines. In other words, there are many choices for positive real numbers $r$ and $s$ such that $\lambda=\tau_{r,s}$ (see \cite{BM} for many details). In particular, one can always choose numbers $r$ and $s$ in such a way that $\lambda=\tau_{r,s},$ $\frac{r}{s}=\frac{n}{m},$ where $(n,m)$ are positive relatively prime integers with $r\le n,$ or, equivalently, the line connecting $(r,0)$ to $(0,s)$ is below the line connecting $(n,0)$ to $(0,m).$ In such a case one can repeat the construction from Example \ref{example: tringular Schroder}.


With the definition of the Schr\"oder polynomial established for any triangular partition, we can now state our first main theorem. 

\begin{theorem} \cite{CGHM24}
The triangular Schr\"oder polynomial $S_{\lambda}(q,t,a)$ at $a=0$ specializes to the triangular $(q,t)$-Catalan polynomial of \cite{BM} and \cite{BHMPS}. Hence, for $\lambda=\tau_{r,s}$ as before, we obtain that
\begin{equation*}
    C_{\lambda}(q,t)=S_{\lambda}(q,t,0)=t^{|\lambda|}R_{{\bf u}(r,s),{\bf v}(r,s)}(q,t,0).
\end{equation*}
\end{theorem}
By construction, the polynomial $S_\lambda(q,t,a)$ depends on a choice of a shifted diagonal. At $a=0$ this corresponds to choosing an appropriate slope in the definition of the $\dinv$ statistic \cite{BM, BHMPS}. The Catalan polynomial doesn't depend on that choice (this follows from the shuffle theorem of \cite{BHMPS}, see also \cite{LW, Ma}). The shuffle theorem argument can be generalized to show that the full Schr\"oder polynomial also depends only on the partition. Nonetheless, this will also follow from our results below.


\subsection{The Monotone Knot of a Triangular Partition}

Let $\lambda=\tau_{r,s}$ be a triangular partition, and $(r,s)$ be as in the previous section: $r/s=n/m,$ where $n,m\in\mathbf{Z}_{>0}$ are relatively prime and $r\le n.$ The monotone curve $C$ from $(0,\lceil r\rceil)$ to $(\lceil s \rceil,0)$ is constructed by augmenting the shifted diagonal connecting $(0,r)$ to $(s,0)$ by adding an almost vertical segment at the top and an almost horizontal segment at the bottom (see Figure \ref{figure: monotone to hogancamp-mellit}). Let $K_{r,s}$ be the closure of the corresponding monotone braid $\beta_C$ (see Section \ref{subsection: mon and cox}). It follows from \cite[Prop. 7.5]{GL23} that $K_{r,s}$ is isotopic to the closure of the Coxeter braid $\beta_C^{cox}$, which only depends on the partition $\lambda$ and not on the choice of the shifted diagonal. We will call it the \newword{Coxeter knot} of the partition $\lambda$ and denote it $K_\lambda$. 

\begin{theorem}[\cite{CGHM24}]\label{theorem: schroder=poincare}
    The monotone knot $K_{r,s}$ is isotopic to the knot $L_{{\bf u}(r,s),{\bf v}(r,s)}$ (see Section \ref{subsection: recursions for knot hom} for a definition). In particular, the Poincar\'e polynomial of the reduced Khovanov-Rozansky homology of the Coxeter knot $K_\lambda$ is given by $P_{K_\lambda}^{KR}(q,t,a)=R_{{\bf u}(r,s),{\bf v}(r,s)}(q,t,a)=t^{-|\lambda|}S_\lambda(q,t,a).$
\end{theorem}

\begin{remark}
    Theorem \ref{theorem: schroder=poincare} implies that the Schr\"oder polynomial $S_{\lambda}(q,t,a)$ does not depend on the choice of the shifted diagonal $(0,r)-(s,0),$ but only on the triangular partition $\lambda=\tau_{r,s}.$
\end{remark}

\begin{example}
    In Figure \ref{figure: monotone to hogancamp-mellit} we illustrate the construction of the monotone knot $K_{r,s}$ and its isotopy to $L_{{\bf u}(r,s),{\bf v}(r,s)}$ for $(r,s)=(3.2,5.6),$ continuing Example \ref{example: tringular Schroder}.
\begin{figure}
    \centering
    \begin{tikzpicture}[scale=0.6]
        \draw [gray] (0,0) grid +(6,4);
        \draw [thick, blue!50!black] (0.14,3.12)--(5.39,0.12);
        \draw [thick, green!50!black] (0.14,3.12)--(0,4);
        \draw [thick, green!50!black] (6,0)--(5.39,0.12);
        \draw[ very thick,red](0,2)--(2,2)--(2,1)--(3,1)--(3,0);
        \draw[white] (0,-0.8)--(0,-0.8);
    \end{tikzpicture}\ 
    \begin{tikzpicture}[scale=2.5]
        %\draw[step=0.1, gray!20!white] (0,0) grid (1,1);
        \draw [thick, green!50!black] (0.14,0.12)--(0,1);
        \filldraw[white] (0.02,0.91) circle (1pt);
        \filldraw[white] (0.07,0.59) circle (1pt);
        \filldraw[white] (0.08,0.43) circle (1pt);
        \filldraw[white] (0.12,0.28) circle (1pt);
        \draw[thick, blue!50!black] (0.349,0)--(0.14,0.12);
        \draw[thick, blue!50!black] (1,0.628)--(0.349,1);
        \draw[thick, blue!50!black] (1,0.057)--(0,0.628);
        \draw[thick, blue!50!black] (0.1,0)--(0,0.057);
        \draw[thick, blue!50!black] (1,0.4857)--(0.1,1);
        \draw[thick, blue!50!black] (0.85,0)--(0,0.4857);
        \filldraw[white] (0.76,0.04) circle (1pt);
        \draw [thick, blue!50!black] (1,0.914)--(0.85,1);
        \draw [thick, blue!50!black] (1,0.343)--(0,0.914);
        \draw [thick, green!50!black] (1,0)--(0.39,0.12);
        \draw [thick, blue!50!black] (0.39,0.12)--(0,0.343);
        \draw[thick] (0,0) grid (1,1);
        \draw[red] (0.95,-0.05)--(0.95,1.05);
        \draw[white] (0,-0.1)--(0,-0.1);
    \end{tikzpicture}\ 
    \begin{tikzpicture}[scale=2.5]
        %\draw[step=0.1, gray!20!white] (-0.1,0) grid (1,1);
        \draw[green!50!black] (0,1) to[in=0,out=90] (-0.01,1.02) to[in=90,out=180] (-0.02,1)--(-0.02,0) to[in=180,out=-90] (-0.01,-0.02) to[in=-90,out=0] (0,0);
        \draw [thick, green!50!black] (0.14,0.12)--(0,1);
        \filldraw[white] (0.02,0.91) circle (1pt);
        \filldraw[white] (-0.02,0.92) circle (1pt);
        \filldraw[white] (0.07,0.59) circle (1pt);
        \filldraw[white] (-0.02,0.64) circle (1pt);
        \filldraw[white] (0.08,0.43) circle (1pt);
        \filldraw[white] (-0.02,0.5) circle (1pt);
        \filldraw[white] (0.12,0.28) circle (1pt);
        \filldraw[white] (-0.02,0.35) circle (1pt);
        \filldraw[white] (-0.02,0.07) circle (1pt);
        \draw[thick, blue!50!black] (0.349,0)--(0.14,0.12);
        \draw[thick, blue!50!black] (0.95,0.657)--(0.349,1);
        \draw[thick, blue!50!black] (0.95,0.057+0.0286)--(-0.05,0.657);
        \draw[thick, blue!50!black] (0.1,0)--(-0.05,0.057+0.0286);
        \draw[thick, blue!50!black] (0.95,0.4857+0.0286)--(0.1,1);
        \draw[thick, blue!50!black] (0.85,0)--(-0.05,0.4857+0.0286);
        \filldraw[white] (0.76,0.04) circle (1pt);
        \draw [thick, blue!50!black] (0.95,0.914+0.0286)--(0.85,1);
        \draw [thick, blue!50!black] (0.95,0.343+0.0286)--(-0.05,0.914+0.0286);
        \draw [thick, green!50!black] (0.95,0.01)--(0.39,0.12);
        \draw [thick, green!50!black] (-0.05,0.01)--(0,0);
        \draw [thick, blue!50!black] (0.39,0.12)--(-0.05,0.343+0.0286);
        \draw[thick] (-0.05,0)--(0.95,0)--(0.95,1)--(-0.05,1)--(-0.05,0);
        \draw[white] (0,-0.1)--(0,-0.1);
    \end{tikzpicture}
    \begin{tikzpicture}[scale=2.5]
        %\draw[step=0.1, gray!20!white] (-0.1,0) grid (1,1);
        %\draw[green!50!black] (0,1) to[in=0,out=90] (-0.01,1.02) to[in=90,out=180] (-0.02,1)--(-0.02,0) to[in=180,out=-90] (-0.01,-0.02) to[in=-90,out=0] (0,0);
        \draw [thick, green!50!black] (0.14,0.12) to[in=90,out=145] (-0.02,0.1)--(-0.02,0.03) to[in=0,out=-90] (-0.05,0.01);
        \filldraw[white] (-0.02,0.07) circle (1pt);
        \draw[thick, blue!50!black] (0.349,0)--(0.14,0.12);
        \draw[thin, dashed, blue!50!black] (0.14,0.12)--(-0.05,0.2286);
        \draw[thick, blue!50!black] (0.95,0.657)--(0.349,1);
        \draw[thick, blue!50!black] (0.95,0.0856)--(-0.05,0.657);
        \draw[thick, blue!50!black] (0.1,0)--(-0.05,0.0856);
        \draw[thick, blue!50!black] (0.95,0.5143)--(0.1,1);
        \draw[thick, blue!50!black] (0.85,0)--(-0.05,0.5143);
        \filldraw[white] (0.76,0.04) circle (1pt);
        \draw [thick, blue!50!black] (0.95,0.9426)--(0.85,1);
        \draw [thick, blue!50!black] (0.95,0.3716)--(-0.05,0.9426);
        \draw [thick, green!50!black] (0.95,0.01)--(0.39,0.12);
        \draw [thick, blue!50!black] (0.39,0.12)--(-0.05,0.3716);
        \draw [thin, dashed, blue!50!black] (0.5995,0)--(0.39,0.12);
        \draw[thick] (-0.05,0)--(0.95,0)--(0.95,1)--(-0.05,1)--(-0.05,0);
        \filldraw[blue!50!black] (-0.05,0.3716) circle (0.01);
        \filldraw[blue!50!black] (-0.05,0.9426) circle (0.01);
        \filldraw[blue!50!black] (-0.05,0.5143) circle (0.01);
        \filldraw[blue!50!black] (-0.05,0.0856) circle (0.01);
        \filldraw[blue!50!black] (-0.05,0.657) circle (0.01);
        \filldraw[orange] (-0.05,0.2286) circle (0.01);
        \filldraw[blue!50!black] (0.349,0) circle (0.01);
        \filldraw[blue!50!black] (0.1,0) circle (0.01);
        \filldraw[blue!50!black] (0.85,0) circle (0.01);
        \filldraw[orange] (0.5995,0) circle (0.01);
        \draw[blue!50!black] (-0.1,0.3716) node {\scriptsize $0$};
        \draw[blue!50!black] (-0.1,0.9426) node {\scriptsize $0$};
        \draw[blue!50!black] (-0.1,0.5143) node {\scriptsize $0$};
        \draw[blue!50!black] (-0.1,0.0856) node {\scriptsize $0$};
        \draw[blue!50!black] (-0.1,0.657) node {\scriptsize $0$};
        \draw[orange] (-0.1,0.2286)  node {\scriptsize $1$};
        \draw[blue!50!black] (0.349,-0.05) node {\scriptsize $0$};
        \draw[blue!50!black] (0.1,-0.05) node {\scriptsize $0$};
        \draw[blue!50!black] (0.85,-0.05) node {\scriptsize $0$};
        \draw[orange] (0.5995,-0.05) node {\scriptsize $1$};
    \end{tikzpicture}
    \caption{Left: $\lambda=(3,2)$ together with the monotone curve $C$ obtained by augmenting the shifted diagonal. Second: the monotone knot $K_{r,s}$ drawn on a torus. We cut a vertical strip on the right (the red line) and reattach it on the left for the third picture. We also close the vertical green strand. Finally, we pull the green strand from under the blue ones to obtain the picture on the right, which is the knot $L_{010000,0010}.$}
    \label{figure: monotone to hogancamp-mellit}
\end{figure}
\end{example}

In the case when $(r,s)=(dn,dm),$ where $d,n,m$ are integers and $n$ and $m$ are relatively prime, Galashin and Lam in \cite[Lem. 8.1]{GL23} proved that the monotone knot $K_{\tau_{dn,dm}}$ is the $(d,dnm+1)$-cable of the $(n,m)$-torus knot, which is an algebraic knot: it can be obtained as the intersection of the planar curve $(x=t^{dn},y=t^{dm}+t^{dm+1})$ with a small $3$D sphere around the origin in $\mathbb{C}^2.$ Such curves were studied by the fourth author, Gorsky, and Oblomkov in \cite{GMO}, were they showed that the Poincar\'e polynomial $P_{\overline{JC}}(t)$ of the Compactified Jacobian of such a curve is a specialization of the $(dn,dm)$ $(q,t)$-Catalan polynomial. Combining this with our Theorem \ref{theorem: schroder=poincare}, we obtain
\begin{equation*}
    P_{\overline{JC}}(t)=t^{2\delta}C_{nd,md}(1,t^{-2})=t^{2\delta}S_{\tau_{dn,dm}}(1,t^{-2},0)=P_{K_{\tau_{dn,dm}}}^{KR}(1,t^{-2},0),
\end{equation*}
which is a special case of the Oblomkov-Rasmussen-Shende conjecture for such curves.


%\acknowledgements{\lipsum[20]}

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\end{document}
