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\newtheorem{lemma}{Lemma}[section]
\newtheorem{prop}{Proposition}[section]
\newtheorem{cor}{Corollary}[section]
\newtheorem{thm}{Theorem}[section]
\newtheorem{conj}{Conjecture}[section]
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\newtheorem{project}{Project}[section]
\newtheorem{defn}{Definition}[section]
\newtheorem{example}{Example}[section]


\author{Andrew Timothy Wilson\thanks{\href{mailto:andwils@math.upenn.edu}{andwils@math.upenn.edu}. The author was supported by an NSF Mathematical Sciences Postdoctoral Research Fellowship.}}
\title{Torus link homology and the nabla operator}
\address{Department of Mathematics, University of Pennsylvania}
\keywords{torus links, symmetric functions, Macdonald polynomials, nabla operator}
\received{\today}
\revised{\today}
\accepted{tomorrow}


\abstract{
In recent work, Elias and Hogancamp develop a recurrence for the Poincar\'e series of the triply graded Hochschild homology of certain links, one of which is the $(n,n)$ torus link. In this case, Elias and Hogancamp give a combinatorial formula for this homology that is reminiscent of the combinatorics of the modified Macdonald polynomial eigenoperator $\nabla$. We give a combinatorial formula for the homologies of all links considered by Elias and Hogancamp. Our first formula is not easily computable, so we show how to transform it into a computable version. Finally, we conjecture a direct relationship between the $(n,n)$ torus link case of our formula and the symmetric function $\nabla p_{1^n}$.
}

\begin{document}

\maketitle

\section{Introduction}

We begin by establishing some notation from knot theory, following \cite{elias-hogancamp}. The remaining sections of the paper will take a more combinatorial perspective.

The \emph{braid group on $n$ strands}, denoted $\braidgroup_n$, can be defined by the presentation
\begin{align}
\braidgroup_n &= \left\langle \sg_1, \sg_2, \ldots, \sg_{n-1} \, | \, \sg_i \sg_{i+1} \sg_i = \sg_{i+1} \sg_i \sg_{i+1}, \, \sg_i \sg_j = \sg_j \sg_i \right\rangle
\end{align}
for all $1 \leq i \leq n-2$ and $|i-j| \geq 2$. This group can be pictured as all ways to ``braid'' together $n$ strands, where $\sg_i$ corresponds to crossing string $i+1$ over string $i$ and the group operation is concatenation. One particularly notable braid is the \emph{full twist braid} on $n$ strands, denoted $\fulltwist_n$, which can be written
\begin{align}
\fulltwist_n &=\left( (\sg_1) (\sg_2 \sg_1) \ldots (\sg_{n-1} \sg_{n-2} \ldots \sg_1) \right)^2.
\end{align}
where multiplication is left to right. We will also need an operation $\omega$ on braids which corresponds to rotation around the horizontal axis. We define $\omega$ on $\braidgroup_n$ by $\omega(\sg_i) = \sg_i$ and $\omega(\alpha \beta) = \omega(\beta) \omega(\alpha)$. Then $\omega$ is an anti-involution on $\braidgroup_n$. All of our braids will have the property that the string that begins in column $i$ also ends in column $i$ for all $i$; these are sometimes called \emph{perfect braids}.

Given a braid with $n$ strands, one can form a \emph{link} (i.e.\ nonintersecting collection of knots) by identifying the top of the strand that begins in position $i$ with the bottom of the strand that ends in position $i$ for $1 \leq i \leq n$. The result is called a \emph{closed braid}. Alexander proved that every link can be represented by a closed braid (although this representation is not unique) \cite{alexander}. The closure of a perfect braid is a link that consists of $n$ separate unknots linked together.

In \cite{elias-hogancamp}, Elias and Hogancamp assign a complex $C_v$ to every binary word $v$. We describe this assignment here -- see \cref{fig:cv-braid} for an example. Say $v \in \{0,1\}^n$ with $|v| = m$. We begin with two braids, the full twist braid $\fulltwist_{n-m}$ and a certain recursively defined complex $K_m$ \cite{elias-hogancamp}, which sits to the right of $\fulltwist_{n-m}$. For $i = 1$ to $n$, we feed  string $i$ into the leftmost available position in $K_m$ if $v_i = 1$; otherwise, we feed string $i$ into the leftmost available position in $\fulltwist_{n-m}$. All crossings that occur are forced to be ``positive,'' i.e.\ the right strand crosses over the left strand. This induces a braid $\beta_v \in \braidgroup_n$ that occurs before the adjacent $\fulltwist_{n-m}$ and $K_m$. The final complex $C_v$ is obtained by performing $\omega(\beta_v)$, followed by $\beta_v$, followed by the adjacent $\fulltwist_{n-m}$ and $K_m$. We note that $C_{0^n}$ is the full twist braid $\fulltwist_n$ and that the closure of this braid is the $(n,n)$ torus link. The combinatorics of other links, in particular the $(m,n)$ torus link for $m$ and $n$ coprime, has been studied by a variety of authors in recent years \cite{gors-knots, gorsky-negut}. Haglund gives an overview of this work from a combinatorial perspective in \cite{haglund-knots}.

\begin{figure}
\centering
\includegraphics[scale=1]{afigure}
\caption{We have drawn the complex $C_{10101101}$, where $\fulltwist_3$ is the full twist braid and $K_5$ is a certain complex defined recursively in \cite{elias-hogancamp}. This figure is used courtesy of \cite{elias-hogancamp}.}
\label{fig:cv-braid}
\end{figure}


Elias and Hogancamp map each complex $C_v$ to a graded Soergel bimodule and then consider the \emph{Hochschild homology} of this bimodule; this is sometimes called Khovanov-Rozansky homology \cite{khovanov, khovanov-rozansky}.
This homology has three gradings: the bimodule degree (using the variable $Q$), the homological degree ($T$), and the Hochschild degree ($A$). After the grading shifts $q = Q^2$, $t = T^2 Q^{-2}$, and $a = AQ^{-2}$, Elias and Hogancamp give a recurrence for the Poincar\'e series of this triply graded homology, which they denote $f_{v}(q,a,t)$. They also give a combinatorial formula for the special case $f_{0^n}(q,a,t)$. We will give two combinatorial formulas for $f_{v}(q,a,t)$ for every $v \in \{0,1\}^n$. 

In \cref{sec:infinite}, we define a symmetric function $\linkpoly_v(x;q,t)$ which we call the \emph{link symmetric function}. Its definition is reminiscent of the combinatorics of the Macdonald eigenoperator $\nabla$, introduced in \cite{bght-positivity}. We prove that $f_v(q,a,t)$ is equal to a certain inner product with $\linkpoly_v(x;q,t)$. 

The main weakness of our first formula is that it is a sum over infinitely many objects, so it is not clear how to compute using this formula. We address this issue in \cref{sec:finite}, obtaining a finite formula for $L_v(x;q,t)$ using a collection of combinatorial objects we call \emph{barred Fubini words}.

We close by presenting some conjectures in \cref{sec:conjectures}. In particular, we conjecture that
\begin{align}
\linkpoly_{0^n}(x;q,t) &= (1-q)^{-n} \nabla p_{1^n} .
\end{align}
where the terminology is defined in \cref{sec:conjectures}.
A proof of this conjecture would provide the first combinatorial interpretation for $\nabla p_{1^n}$. There has been much recent work establishing combinatorial interpretations for $\nabla e_n$ \cite{carlsson-mellit} and $\nabla p_n$ \cite{leven-square}. We believe that $\linkpoly_v(x;q,t)$ is also related to Macdonald polynomials for general $v$, although we do not have an explicit conjecture in this direction.

\section{An infinite formula}
\label{sec:infinite}

Let $\mathbb{N} = \{0,1,2,\ldots\}$ and $\mathbb{Z}_{+} = \{1,2,3,\ldots\}$. We begin by defining two statistics. 
 
\begin{defn}
Given words $\gamma \in \mathbb{N}^n$ and $\pi \in \mathbb{Z}_{+}^n$, we define
\begin{align}
\area(\gamma) &= |\gamma| - \# \{1 \leq i \leq n : \gamma_i > 0 \} \\
\dinv(\gamma, \pi) &= \# \{1 \leq i < j \leq n : \gamma_i = \gamma_j, \pi_i > \pi_j \} \\
&+ \#\{1 \leq i < j \leq n : \gamma_i + 1 = \gamma_j, \pi_i < \pi_j \} \nonumber \\
x^{\pi} &= \prod_{i=1}^{n} x_{\pi_i} .
\end{align}
\end{defn}

In \cref{fig:area-dinv-example}, we draw a diagram for $\gamma = 20141022$ and $\pi = 41322231$. Area counts the empty boxes in such a diagram, dinv counts certain pairs of labels, and $x^{\pi}$ records all labels that appear in the diagram. 

\begin{figure}
\begin{center}
\begin{tikzpicture}
\draw [step=0.5cm] (-0.001, 0) grid (0.5, 1);
\draw [step=0.5cm] (0.499, 0) grid (1, 0);
\draw [step=0.5cm] (0.999, 0) grid (1.5, 0.5);
\draw [step=0.5cm] (1.499, 0) grid (2, 2);
\draw [step=0.5cm] (1.999, 0) grid (2.5, 0.5);
\draw [step=0.5cm] (2.499, 0) grid (3, 0);
\draw [step=0.5cm] (2.999, 0) grid (3.5, 1);
\draw [step=0.5cm] (3.499, 0) grid (4, 1);

\node at (0.25, 0.75) {4};
\node at (0.75, -0.25) {1};
\node at (1.25, 0.25) {3};
\node at (1.75, 1.75) {2};
\node at (2.25, 0.25) {2};
\node at (2.75, -0.25) {2};
\node at (3.25, 0.75) {3};
\node at (3.75, 0.75) {1};

\end{tikzpicture}
\end{center}
\caption{We have depicted the example $\gamma = 20141022$ and $\pi = 41322231$ by drawing bottom-justified columns with heights $\gamma_1$, $\gamma_2$, \ldots, $\gamma_8$ and the labels $\pi_i$ are placed as high as possible in each column. In this example, we compute $\area(\gamma) = 6$, $\dinv(\gamma, \pi) = 7$, where the contributing pairs are in columns $(1, 7)$, $(1, 8)$, $(2,3)$, $(2,5)$, $(3, 5)$, $(5, 7)$, $(7, 8)$, and $x^{\pi} = x_1^2 x_2^3 x_3^2 x_4$.}
\label{fig:area-dinv-example}
\end{figure}



\begin{defn}
Given $n \in \mathbb{Z}_{+}$ and $v \in \{0,1\}^n$, define 
\begin{align}
\linkpoly_{v} &= \linkpoly_{v}(x; q, t) = \sum_{\substack{\gamma \in \mathbb{N}^n, \, \pi \in \mathbb{Z}_{+}^n \\ \gamma_i = 0 \Leftrightarrow v_i = 1}}  q^{\area(\gamma)} t^{\dinv(\gamma, \pi)}x^{\pi} .
\end{align}
\end{defn}

Perhaps the first thing to note about $\linkpoly_{v}$ is that it is symmetric in the $x$ variables; one way to see this is to express $\linkpoly_v$ as a sum of LLT polynomials \cite{llt}. More precisely, each $\gamma \in \mathbb{N}^n$ can be associated with an $n$-tuple $\boldsymbol{\lambda}(\gamma)$ of single cell partitions in the plane, where the $i$th cell is placed on diagonal $\gamma_i$ and the order is not changed. Using the notation of \cite{hhl}, the unicellular LLT polynomial $G_{\boldsymbol{\lambda}(\gamma)}(x; t)$ can be used to write
\begin{align}
\linkpoly_{v} &= \sum_{\substack{\gamma \in \mathbb{N}^n \\ \gamma_i = 0 \Leftrightarrow v_i = 1}} q^{\area(\gamma)} G_{\boldsymbol{\lambda}(\gamma)}(x; t) .
\end{align}
Since LLT polynomials are symmetric, every $L_v$ is also symmetric.

We also remark that $\linkpoly_{1^n}$ is equal to the modified Macdonald polynomial $\widetilde{H}_{1^n}(x;q,t)$, which is also equal to the graded Frobenius series of the coinvariants of $\S_n$ with grading in $t$.

Next, we note that the Poincar\'e series $f_v(q,a,t)$ can be recovered as a certain inner product of $\linkpoly_v$. We follow the standard notation for symmetric functions and their usual inner product, as described in Chapter 7 of \cite{ec2}. The following lemma leads to \cref{thm:fv-lv}. We omit the proofs from this extended abstract.

\begin{lemma}
\label{lemma:first1}
\begin{align}
\linkpoly_{0^n} &= \frac{1}{1-q} \linkpoly_{10^{n-1}} .
\end{align} 
\end{lemma}


\begin{thm}
\label{thm:fv-lv}
For any $v \in \{0,1\}^n$,
\begin{align}
f_{v}(q, a, t) &= \sum_{d=0}^{n} \left\langle \linkpoly_{v}, e_{n-d} h_d \right\rangle a^d .
\end{align}
\end{thm}



For the sake of comparison with \cite{elias-hogancamp}, we give a simplified formula that directly computes $f_v(q,a,t)$ from \cref{thm:fv-lv}. Given $\gamma \in \mathbb{N}^n$ and $1 \leq i \leq n$, let 
\begin{align}
\dinv_i(\gamma) &= \# \{j < i : \gamma_j = \gamma_i\} + \# \{j > i : \gamma_j  = \gamma_i + 1 \}.
\end{align}

\begin{cor}
\label{cor:f-infinite}
\begin{align}
f_v(q,a,t) &= \sum_{\substack{\gamma \in \mathbb{N}^n \\ \gamma_i = 0 \Leftrightarrow v_i = 1}} q^{\area(\gamma)} \prod_{i=1}^{n} \left(a + t^{\dinv_i(\gamma)} \right) 
\end{align}
where, as before, $\area(\gamma) = |\gamma| - \# \{1 \leq i \leq n : \gamma_i > 0 \}$. 
\end{cor}

If $v=0^n$ and $a=0$, this is exactly Theorem 1.9 in \cite{elias-hogancamp}. 

Let us use \cref{cor:f-infinite} to compute $f_{00}(q,a,t)$. We consider three cases: either $\gamma_2 \in \{\gamma_1, \gamma_1 +1\}$, $\gamma_2 > \gamma_1 + 1$, or $\gamma_2 < \gamma_1$. In the first case, we get a contribution of $(1+a)(a+t)$ from the dinv product in \cref{cor:f-finite}; in the other cases, we get $(1+a)^2.$ 

In the first case, our minimum area is either 0 or 1, depending on if $\gamma_2 = \gamma_1$ or $\gamma_1 + 1$. Either way, we can choose any nonnegative integer $k$ and add $k$ to both $\gamma_1$ and $\gamma_2$. This yields an area contribution of  $(1+q)(1-q^2)$. In the second case, the minimum area is 2 and not only can we add area to both $\gamma_1$ and $\gamma_2$ equally but we can add area to $\gamma_2$ by itself. The resulting contribution is $q^2/((1-q^2)(1-q))$. Similarly, in the third case we get a contribution of $q/((1-q^2)(1-q))$. 

Summing up all these weights, we get
\begin{align}
f_{00}(q,a,t) &= \frac{(1+q)(1+a)(a+t)}{1-q^2} + \frac{q^2(1+a)^2}{(1-q^2)(1-q)} + \frac{q(1+a)^2}{(1-q^2)(1-q)} \\
&= \frac{q+t-qt + a(1+q+t-qt) +a^2}{(1-q)^2} .
\end{align}
This process can become difficult for $v$ with length as short as 3. The difficulty comes mostly from considering how to add (possibly infinite amounts of) area without affecting the dinv vectors. We address this issue in the next section by deriving a formula that allows us to restrict to a finite set of $\gamma$ vectors.

\section{A finite formula}
\label{sec:finite}

Although the combinatorial definition of $\linkpoly_{v}$ is straightforward, it is not computationally effective\footnote{There are also infinitely many $\pi \in \mathbb{Z}_{+}^n$, but this problem can be addressed with standardization \cite{haglund-book}.} since it is a sum over infinitely many words $\gamma \in \mathbb{N}^n$. We rectify this issue in \cref{thm:link-finite} below. The idea is to compress the vectors $\gamma$ while altering the statistics so that the link polynomial $\linkpoly_v$ is not changed.

\begin{defn}
A word $\gamma \in \mathbb{N}^n$ is a \emph{Fubini word} if every integer $0 \leq k \leq \max(\gamma)$ appears in $\gamma$. \end{defn}

For example, $41255103$ is a Fubini word but $20141022$ is not a Fubini word, since it contains a 4 but not a 3. We call these Fubini words because they are counted by the Fubini numbers (\cite{oeis}, A000670), which also count ordered partitions of the set $\{1,2,\ldots,n\}$. We will actually be interested in certain decorated Fubini words.

\begin{defn}
Given $v \in \{0,1\}^n$, we say that a Fubini word $\gamma$ is \emph{associated} with $v$ if either
\begin{itemize}
\item $v = 0^n$ and the only zero in $\gamma$ occurs at $\gamma_1$, or
\item $v \neq 0^n$ and $\gamma_i = 0$ if and only if $v_i = 1$. 
\end{itemize}
\end{defn}

\begin{defn}
A \emph{barred Fubini word} associated with $v$ is a Fubini word $\gamma$ associated with $v$ where we may place bars over certain entries. Specifically, the entry $\gamma_j$ may be barred if 
\begin{enumerate}
\item $\gamma_j > 0$, 
\item $\gamma_j$ is unique in $\gamma$, and
\item for each $i<j$ we have $\gamma_i < \gamma_j$, i.e.\ $\gamma_j$ is a left-to-right maximum in $\gamma$. 
\end{enumerate}
We denote the collection of barred Fubini words associated with $v$ by $\fubini_v$. 
\end{defn}

For example, 
\begin{align}
\fubini_0 &= \{0\} \\
\fubini_{00} &= \{01, 0\overline{1}\} \\
\fubini_{000} &= \{011, 012, 0\overline{1}2, 01\overline{2}, 0\overline{1}\overline{2}, 021, 0\overline{2}1 \} .
\end{align}
The sequence $|\fubini_{0^n}|$ for $n \in \mathbb{N}$ begins $1, 1, 2, 7, 35, 226,\ldots$ and seems to appear in the OEIS as A014307 \cite{oeis}. One way to define sequence A014307 is that it has exponential generating function
\begin{align}
\sqrt{\frac{e^z}{2 - e^z}} .
\end{align}
This sequence is given several combinatorial interpretations in \cite{ren}. It would be interesting to obtain a bijection between $\fubini_{0^n}$ and one of the collections of objects in \cite{ren}. See \cref{fig:fubini-words} for more examples of barred Fubini words.

\begin{figure}
\begin{center}
\begin{tabular}{c|c}
$v$	& $\fubini_v$ \\\hline
111	& $000$ \\
011	& $100, \overline{1}00$\\
101	& $010, 0\overline{1}0$\\
110	& $001, 00\overline{1}$\\
001 & $110, 120, 1\overline{2}0, \overline{1}20, \overline{1}\overline{2}0, 210, \overline{2}10 $\\
010	& $101, 102, 10\overline{2}, \overline{1}02, \overline{1}0\overline{2}, 201, \overline{2}01$\\
100	& $011, 012, 0\overline{1}2, 01\overline{2}, 0\overline{1}\overline{2}, 021, 0\overline{2}1  $ \\
000	& $011, 012, 0\overline{1}2, 01\overline{2}, 0\overline{1}\overline{2}, 021, 0\overline{2}1  $
\end{tabular}
\caption{We have listed the barred Fubini words $\fubini_v$ for each $v \in \{0,1\}^3$.}
\label{fig:fubini-words}
\end{center}
\end{figure}


Given a barred Fubini word $\gamma$ and a word $\pi \in \mathbb{Z}_{+}^n$, we use the same area statistic and modify the dinv statistic slightly:
\begin{align}
\area(\gamma) &= |\gamma| - \# \{1 \leq i \leq n : \gamma_i > 0\} \\
\dinv(\gamma, \pi) &= \# \{1 \leq i < j \leq n : \gamma_i = \gamma_j, \pi_i > \pi_j \} \\
&+ \#\{1 \leq i < j \leq n : \gamma_i + 1 = \gamma_j, \pi_i < \pi_j, \gamma_j \text{ is not barred} \} \nonumber.
\end{align}
We also let $\barstat(\gamma)$ be the number of barred entries in $\gamma$. We have the following result.

\begin{thm}
\label{thm:link-finite}
For $v \in \{0,1\}^n$, 
\begin{align}
\linkpoly_{v} &= \sum_{\substack{\gamma \in \fubini_v \\ \pi \in \mathbb{Z}_{+}^n}} q^{\area(\gamma) + \barstat(\gamma)} t^{\dinv(\gamma, \pi)} (1-q)^{-\barstat(\gamma)-\chi(v=0^n)} x^{\pi}
\end{align}
where $\chi$ of a statement is 1 if the statement is true and 0 if it is false.
\end{thm}

As in \cref{sec:infinite}, we give a formula for computing $f_v(q,a,t)$ directly. Given a barred Fubini word $\gamma$, we define
\begin{align}
\dinv_i(\gamma) &= \# \{j < i : \gamma_j = \gamma_i\} + \# \{j > i : \gamma_j  = \gamma_i + 1, \text{$\gamma_j$ is not barred} \}.
\end{align}

\begin{cor}
\label{cor:f-finite}
\begin{align}
f_v(q,a,t) &= \sum_{\gamma \in \fubini_v} q^{\area(\gamma) + \barstat(\gamma)} (1-q)^{-\barstat(\gamma) - \chi(v=0^n)} \prod_{i=1}^{n} \left(a + t^{\dinv_i(\gamma)} \right) 
\end{align}
\end{cor}

Let us use \cref{cor:f-finite} to compute $f_{00}(q,a,t)$ again. We note that $\fubini_{00} = \{01, 0\overline{1}\}$. Computing the weights for these vectors and then adding, we get
\begin{align}
f_{00}(q,a,t) &= \frac{(1+a)(a+t)}{1-q} + \frac{q(1+a)^2}{(1-q)^2} \\
&= \frac{q+t-qt + a(1+q+t-qt) + a^2}{(1-q)^2}
\end{align}
which is the same answer we computed at the end of \cref{sec:infinite}.

\section{Conjectures}
\label{sec:conjectures}

So far, we have used the inner product $\langle \linkpoly_v, e_{n-d} h_d\rangle$ to compute $f_v(q,a,t)$; one might wonder if there is any value in studying the full symmetric function $L_v$. In this section, we conjecture that the link symmetric function $L_v$ is closely related to the combinatorics of Macdonald polynomials, hinting at a stronger connection between Macdonald polynomials and link homology. Following \cite{elias-hogancamp}, we must first define a ``normalized'' version of the link symmetric function $\linkpoly_{v}$.

\begin{defn}
\begin{align}
\linknorm_{v} = \linknorm_v(x;q,t) = (1-q)^{n-|v|} \linkpoly_{v}(x; q, t). 
\end{align}
\end{defn}

We could also define $\linknorm_{v}$ in terms of diagrams; each box that contains a label from $\pi$ contributes an additional factor of $1-q$. \cref{thm:link-finite} implies that $\linknorm_{v}$ has coefficients in $\mathbb{Z}[q,t]$, whereas the coefficients of $\linkpoly_{v}$ are elements of $\mathbb{Z}[[q,t]]$. We conjecture that the normalized link symmetric function $\linknorm_{v}$ is closely connected to the Macdonald eigenoperators $\nabla$ and $\Delta$. 

The modified Macdonald polynomials $\widetilde{H}_{\mu}$ form a basis for the ring of symmetric functions with coefficients in $\mathbb{Q}(q,t)$. They can be defined via triangularity relations or combinatorially \cite{hhl, haglund-book}. Given a partition $\mu$, let $B_{\mu}$ be the alphabet of monomials $q^{i} t^j$ where $(i,j)$ ranges over the coordinates of the cells in the Ferrers diagram of $\mu$. We compute an example in \cref{fig:bmu}.

\begin{figure}
\begin{center}
\begin{displaymath}
\begin{ytableau}
t^2 \\
t	& qt	& q^2t  \\
1	&  q 	&  q^2	&  q^3
\end{ytableau}
\end{displaymath}
\caption{This is the Ferrers diagram of the partition $\mu = (4,3,1)$. In each cell we have written the monomial $q^i t^j$ that corresponds to the cell, yielding $B_{\mu} = \{1, q, q^2, q^3, t, qt, q^2 t, t^2 \}$.}
\label{fig:bmu}
\end{center}
\end{figure}

Given a symmetric function $F$ and a set of monomials $A = \{a_1, a_2, \ldots, a_n\}$, we let $F[A]$ be the result of setting $x_i = a_i$ for $1 \leq i \leq n$ and $x_i = 0$ for $i > n$. Then we define two operators on symmetric functions by setting, for $\mu \vdash n$,
\begin{align}
\Delta_F \widetilde{H}_{\mu} &= F \left[ B_{\mu} \right] \widetilde{H}_{\mu} \\
\nabla \widetilde{H}_{\mu} &= \Delta_{e_n} \widetilde{H}_{\mu}
\end{align}
and expanding linearly. Note that, for $\mu \vdash n$, $e_n[B_{\mu}]$ is simply the product of the $n$ monomials in $B_{\mu}$, which is sometime written $T_{\mu}$ in the literature.

\begin{conj}
\label{conj:p1^n}
\begin{align}
\label{p1^n1}
\nabla p_{1^n} &= \linknorm_{0^n} \\
\label{p1^n2}
\Delta_{e_{n-1}} p_{1^n} &= \sum_{\substack{ v \in \{0,1\}^n \\  |v| = 1}} \linknorm_{v}
\end{align}
In fact, both conjectures follow from the conjecture that
\begin{align}
\label{p1^n3}
\linknorm_{v0} &= \nabla p_1 \nabla^{-1} \linknorm_v.
\end{align}
\end{conj}

We should mention that Eugene Gorsky first noticed that the identity
\begin{align}
\sum_{a=0}^{d} \left\langle \nabla p_{1^n}, e_{n-d} h_d \right\rangle a^d &= (1-q)^n f_{0^n}(q,a,t)
\end{align}
seemed to hold and communicated this observation to the author via Jim Haglund. Gorsky's conjectured identity is a special case of \cref{conj:p1^n}. It is also interesting to note that the operator in \eqref{p1^n3} appears in the setting of the Rational Shuffle Conjecture as $-\mathbf{Q}_{1,1}$ \cite{rational-shuffle}.

As an example of our conjecture, we can use Sage to compute
\begin{align}
\left\langle \nabla p_{1,1}, p_{1,1} \right\rangle &= 1 + q + t - qt.
\end{align}
This expression should equal $\left\langle \linknorm_{00}, p_{1,1}\right\rangle$ by \cref{conj:p1^n}.  To compute this inner product using \cref{thm:link-finite}, we consider the barred Fubini words $01$ and $0\overline{1}$, each of which can receive labels $\pi = 12$ or $21$. The corresponding diagrams are

\vspace{10pt}

\begin{center}
\begin{tikzpicture}
\draw [step=0.5cm] (-0.001, 0) grid (0.5, 0);
\draw [step=0.5cm] (0.499, 0) grid (1, 0.5); 

\node at (0.25, -0.25) {1};
\node at (0.75, 0.25) {2};

\draw [step=0.5cm] (1.999, 0) grid (2.5, 0);
\draw [step=0.5cm] (2.499, 0) grid (3, 0.5); 

\node at (2.25, -0.25) {2};
\node at (2.75, 0.25) {1};

\draw [step=0.5cm] (3.999, 0) grid (4.5, 0);
\draw [step=0.5cm] (4.499, 0) grid (5, 0.5); 

\node at (4.25, -0.25) {1};
\node at (4.75, 0.25) {$\overline{2}$};

\draw [step=0.5cm] (5.999, 0) grid (6.5, 0);
\draw [step=0.5cm] (6.499, 0) grid (7, 0.5); 

\node at (6.25, -0.25) {2};
\node at (6.75, 0.25) {$\overline{1}$};
\end{tikzpicture}
\end{center}
where we have moved the bars from $\gamma_i$ to the corresponding $\pi_i$. The weights of these diagrams coming from \cref{thm:link-finite} are 
\begin{align}
\frac{t}{1-q} \qquad \quad \frac{1}{1-q} \qquad \quad  \frac{q}{(1-q)^2} \qquad \quad \frac{q}{(1-q)^2} 
\end{align}
respectively. After multiplying by the normalizing factor $(1-q)^2$ to go from $\linkpoly_{00}$ to $\linknorm_{00}$, we sum the resulting weights to get 
\begin{align}
(1-q)t + 1-q + q + q = 1 + q + t - qt
\end{align}
as desired.

After reading an earlier version of this paper, Fran\c{c}ois Bergeron contacted the author with the following additional conjectures.


\begin{conj}[Bergeron, 2016]
\label{conj:bergeron}
\begin{align}
\label{bergeron-append-0}
\linkpoly_{v0} &= \linkpoly_{1v} + q \linkpoly_{0v} \\
\label{bergeron2}
\linkpoly_{0^n} &= \sum_{v \in \{0,1\}^k} q^{n-|v|} \linkpoly_{v0^{n-k}} \\
t \left(\linkpoly_{u011v} - \linkpoly_{u101v} \right) &= \linkpoly_{u101v} - \linkpoly_{u110v} \\
\linknorm_{0^a 1^b 0^c} &= \nabla p_{1^c} \nabla^{-1} \widetilde{H}_{1^b} \nabla p_{1^a}  \\
\linkpoly_{1^a 0 1^b} &= \frac{t^a-1}{t^{a+b}-1} \left[ \nabla p_1 \nabla^{-1}, \widetilde{H}_{1^{a+b}} \right] + \widetilde{H}_{1^{a+b}} p_1 
\end{align}
where the  bracket represents the Lie bracket and operators are applied to 1 if nothing is explicitly specified. Bergeron also observed that $\linkpoly_v(x;q,1+t)$ is $e$-positive. (For more context on this last statement, see Section 4 of \cite{bergeron-open}.) 
\end{conj}

It is clear that \eqref{bergeron-append-0} implies \eqref{bergeron2}. We do not know of any other relations between these conjectures. We close with more open questions.
\begin{enumerate}
\item Is there a Macdonald eigenoperator expression for $\linknorm_{v}$ for other $v$?  Perhaps we can use ideas from the Rational Shuffle Conjecture \cite{rational-shuffle}, recently proved by Mellit \cite{mellit-rational}. In particular, Mellit's proof relies heavily on facts about toric braids, which seems relevant to the appearance of torus links in our setting.

\item Can we generalize our conjecture for $\nabla p_{1^n}$ to ``interpolate'' between our conjecture and the Shuffle Theorem \cite{carlsson-mellit}, or maybe the Square Paths Theorem \cite{leven-square}?
\end{enumerate}

\section{Acknowledgements}

The author would like to Ben Elias and Matt Hogancamp for their exciting paper and for use of \cref{fig:cv-braid}; Lyla Fadali for reading an earlier draft; Jim Haglund for editing and feedback; Eugene Gorsky for his comments and for the idea that Elias and Hogancamp's work could be related to Macdonald polynomials; and Fran\c{c}ois Bergeron for \cref{conj:bergeron} along with other helpful suggestions.

\printbibliography

\end{document}
