# Khovanov–Rozansky Homology of Coxeter Knots and Schröder Polynomials for
paths under any line

Carmen Caprau  Department of Mathematics, California State University, Fresno,
CA 93740, USA<EMAIL_ADDRESS>,  Nicolle González  Department of
Mathematics, University of California Berkeley, CA 94720-3840, USA
<EMAIL_ADDRESS>,  Matthew Hogancamp  Department of Mathematics,
Northeastern University, Boston, MA 02115, USA<EMAIL_ADDRESS>and  Mikhail Mazin  Department of Mathematics, Kansas State University,
Manhattan, KS 66506, USA<EMAIL_ADDRESS>
###### Abstract.

We introduce a family of generalized Schröder polynomials $S_{\tau}(q,t,a)$,
indexed by triangular partitions $\tau$ and prove that $S_{\tau}(q,t,a)$
agrees with the Poincaré series of the triply graded Khovanov–Rozansky
homology of the Coxeter knot $K_{\tau}$ associated to $\tau$. For all integers
$m,n,d\geq 1$ with $m,n$ relatively prime, the $(d,mnd+1)$-cable of the torus
knot $T(m,n)$ appears as a special case. It is known that these knots are
algebraic, and as a result we obtain a proof of the $q=1$ specialization of
the Oblomkov–Rasmussen–Shende conjecture for these knots. Finally, we show
that our Schröder polynomial computes the hook components in the Schur
expansion of the symmetric function appearing in the shuffle theorem under any
line, thus proving a triangular version of the $(q,t)$-Schröder theorem.

###### 2020 Mathematics Subject Classification:

Primary 57K18, 05E05; Secondary 05A15, 05A17, 05A19

###### Contents

  1. 1 Introduction
  2. 2 Khovanov–Rozansky Homology and the Hogancamp-Mellit Recursion
  3. 3 Dyck paths, Invariant Subsets, and the Gorsky-Mazin-Vazirani Recursion
  4. 4 Schröder Polynomials for Triangular Partitions
  5. 5 The Knot Associated to a Triangular Partition
  6. 6 A $(q,t)$-Schröder Theorem for Paths Under Any Line
  7. A An Alternative Derivation of the Hook Component
  8. B Sequences for Links

##  1\. Introduction

Triply graded Khovanov–Rozansky homology is a rich invariant of knots and
links, which categorifies the HOMFLY-PT polynomial. During the last fifteen
years a myriad of conjectures have appeared relating the Khovanov–Rozansky
homology of various special families of links to Catalan combinatorics [23,
18, 19], the Hilbert schemes of points on a plane curve singularity [52],
representations of rational DAHA [30, 10, 11, 12], sheaves on the Hilbert
scheme of points in the plane [28, 29], and refined Chern-Simmons theory [1]
(see §1.3 for brief summary of these conjectures). In order to employ a
_compute-both-sides_ approach to such conjectures, one must first overcome the
obvious challenge of computing Khovanov–Rozansky homology of the links or
knots in question.

In [15], the third named author and Ben Elias introduced a technique for
computing Khovanov–Rozansky homology relying on a recursively defined
polynomial $R_{\bf u,v}$ for certain binary sequences $\bf u$ and $\bf v$,
with the torus link $T(n,n)$ being the main example. This technique was then
exploited in subsequent papers by Elias, Hogancamp, and Mellit to compute the
homologies of various families of links: the paper [38] computes the
homologies of $T(m,md)$ and $T(m,md\pm 1)$ for $m,d\geq 1$, followed shortly
thereafter by [49] which computes the homology of $T(m,n)$ for $m,n\geq 1$
coprime, followed by [39], generalizing all the aforementioned, which computes
the homology of $T(m,n)$ for $m,n\geq 1$, not necessarily coprime. Importantly
for us, the aforementioned computations include some non-torus-links as well.

The initial motivation for this paper stemmed from the problem of computing
and giving a combinatorial interpretation of the Khovanov–Rozansky homology of
iterated torus knots of the form $T(m,n)(d,mnd+1)$, where
$m,n,d\in\mathbb{Z}_{\geq 1}$ and $m,n$ are coprime. Along the way, we
realized such knots are special cases of the knots whose homology was computed
in [39], which in turn coincide with a certain subfamily of _Coxeter knots_
$K_{\tau}$, where $\tau$ is a _triangular partition_. Coxeter knots were
previously studied in [53, 19] and triangular partitions in [5].

In this paper we investigate the Khovanov–Rozanksy homology of such knots. We
introduce the new family of _triangular Schröder polynomials_
$S_{\tau}(q,t,a)$ which we prove compute the Khovanov–Rozanksy homology of
Coxeter knots $K_{\tau}$ for triangular partitions. As a consequence of the
knots $T(m,n)(d,mnd+1)$ with $m,n$ coprime being algebraic, we obtain a proof
of a conjecture of Oblomkov–Rasmussen–Shende restricted to these knots in the
special case when $q=1$. Combinatorially, our triangular Schröder polynomials
fit nicely into the existing $(q,t)$-Catalan combinatorics [20, 4, 35, 36, 31,
34, 32, 23, 3, 48, 9, 6]. They generalize both the classical $(q,t)$-Catalan
[20] and Schröder polynomials [13] and correspond to the hook components of
the Schur expansion of the shuffle theorem under any line of Blasiak-Haiman-
Morse-Pun-Seelinger [6], thus providing a triangular generalization of the
classical _$(q,t)$ -Schröder Theorem_ of Haglund [31].

###  1.1. Triangular Schröder Polynomials and Invariant Subsets

Deeply ingrained in the many conjectures surrounding Khovanov–Rozansky
homology is the combinatorics of Dyck paths and related $(q,t)$-Catalan
objects. The $(q,t)$-Catalan polynomials were introduced in [20] and studied
extensively in Haglund’s book [32]. In order to remain consistent with the
notation used throughout this paper, we denote the $(q,t)$-Catalan by
$C_{n,n+1}(q,t)$, where $n\in\mathbb{Z}_{\geq 1}$.

These polynomials appear in a wide variety of contexts. First,
$C_{n,n+1}(q,t)$ admits a combinatorial definition as a generating function
which counts Dyck paths in the $n\times(n+1)$ rectangle (which are certain
lattice paths below the line $(n+1)y+nx=n(n+1)$ consisting of north and west
steps), weighted by some explicit combinatorially defined statistics
$\mathtt{area}$ and $\mathtt{dinv}$. In representation theory,
$C_{n,n+1}(q,t)$ appears as the coefficient of the Schur function
$s_{(1^{n})}$ in the Schur expansion of $\nabla e_{n}$, (which through work of
Haiman [35, 36], coincides with the Frobenius character of the ring of a
diagonal coinvariants). Here, $e_{n}$ is the $n^{\text{th}}$ elementary
symmetric function and $\nabla$ is the Bergeron-Garsia operator on symmetric
functions [4], which acts diagonally on modified Macdonald polynomials. In
topology, $C_{n,n+1}(q,t)$ also appears as the graded dimension of the lowest
$a$-degree component of the Khovanov–Rozansky homology of the $(n,n+1)$-torus
knot (this was conjectured by Gorsky in [23], and proven in [38]).

We now point out two directions in which $(q,t)$-Catalan polynomials have been
generalized. In the first, one replaces the Dyck paths appearing in the
definition of $C_{n,n+1}(q,t)$ with the more general _Schröder paths_ [59].
These are certain lattice paths which are allowed to contain some number of
northwest steps in addition to the usual north or west steps of a Dyck path.
The resulting generating function $S_{n,n+1}(q,t,a)$ is called the _Schröder
polynomial_ [13], defined so that the coefficient of $a^{k}$ in
$S_{n,n+1}(q,t,a)$ is a generating function counting Schröder paths in the
$n\times(n+1)$ rectangle with exactly $k$ diagonal steps. At the level of
symmetric functions, this amounts to extracting the coefficient of hook-
indexed Schur function $s_{(k,1^{n-k})}$ (for a precise statement see §1.4);
at the level of link homology, this amounts to taking the graded dimension of
the higher $a$-degree components of Khovanov–Rozansky homology of $T(n,n+1)$.

The second direction in which we can generalize $C_{n,n+1}(q,t)$ is to allow
more Dyck paths in rectangles of arbitrary sizes. For instance, for a
relatively prime pair of integers $m,n\geq 1$, one has the associated
_rational $(q,t)$-Catalan_ $C_{m,n}(q,t)$ which counts Dyck paths in an
$m\times n$ rectangle, weighted by an analogous pair of statistics
$\mathtt{area}$, $\mathtt{dinv}$. As in the classical case, $C_{m,n}(q,t)$ is
indeed the sign component of an associated symmetric function, in this case
defined by Hikita [37]. At the level of link homology, $C_{m,n}(q,t)$ is the
graded dimension of the lowest $a$-degree component of the Khovanov–Rozansky
homology of $T(m,n)$ (this was conjectured in [10, 28], and proven by Mellit
in [49]). Of course, one can generalize in both directions, obtaining
_rational Schröder polynomial_ $S_{m,n}(q,t,a)$ which calculates the hook
components of Hikita’s symmetric function and also the graded dimension of the
Khovanov–Rozansky homology of $T(m,n)$ in all $a$-degrees.

In [6] Blasiak–Haiman–Morse–Pun–Seelinger provide a further generalization in
the aforementioned second direction by considering set of Dyck paths “under
any line”. More precisely given arbitrary real numbers $r,s>0$, a _Dyck path
under $L_{r,s}$_ is a path in $\mathbb{Z}_{\geq 0}\times\mathbb{Z}_{\geq 0}$
consisting of unit length west and north steps, from $(\lfloor r\rfloor,0)$ to
$(0,\lfloor s\rfloor)$, and staying weakly below $L_{r,s}$. In _loc. cit._ it
is shown how to construct an analogue of the Hikita symmetric function from
the set of Dyck paths under $L_{r,s}$, and a combinatorial interpretation of
the associated $(q,t)$-Catalan is given. From this perspective, the present
paper provides the missing Schröder polynomial and link homology
interpretation for paths “under any line”. In order to delve further into what
this means, we must first introduce the notion of triangular partitions.

Given a partition $\lambda=(\lambda_{1},\dots,\lambda_{k})$ with
$\lambda_{1}\geq\dots\geq\lambda_{k}\geq 0$, we identify $\lambda$ with its
partition diagram in $\mathbb{R}_{\geq 0}\times\mathbb{R}_{\geq 0}$ and index
any cell $\Box\in\lambda$ by its northeast coordinate
$(x,y)\in\mathbb{Z}_{>0}\times\mathbb{Z}_{>0}$. Thus a partition can be viewed
as a subset $\lambda\subset\mathbb{Z}_{>0}\times\mathbb{Z}_{>0}$.

Any pair of positive real numbers $r,s>0$ determines a partition $\tau_{r,s}$
by declaring that the cell $\Box^{x,y}$ with northeast corner at
$(x,y)\in\mathbb{Z}_{>0}\times\mathbb{Z}_{>0}$ is in $\tau_{r,s}$ if and only
if $ry+sx\leq rs$. In other words, $\tau_{r,s}$ consists of all boxes
contained between the line $L_{r,s}:=\\{ry+sx=rs\\}$ and the coordinate axes.
A _triangular partition_ is any partition of the form $\tau_{r,s}$ for some
real numbers $r,s>0$. The combinatorics of triangular partitions were studied
by Bergeron and the fourth named author in [5], and are the underlying
indexing object in the generalized shuffle theorem of
Blasiak–Haiman–Morse–Pun–Seelinger [6].

Observe that if $\pi$ is a Dyck path under the line $L_{r,s}$ then there is an
associated partition $\lambda$ consisting of all cells in the first quadrant
and below $\pi$. By definition, $\lambda$ will be a subpartition of
$\tau_{r,s}$. This correspondence is a canonical bijection between Dyck paths
under $L_{r,s}$ and subpartitions of $\tau_{r,s}$. In light of this
observation, a pair $(\lambda,\tau)$ consisting of a triangular partition
$\tau$ and a subpartition $\lambda\subset\tau$ will be referred to as a
_$\tau$ -Dyck path_. As matter of notation, given real numbers $r,s>0$ we will
write $\mathrm{D}(r,s)$ for the set of all $\tau_{r,s}$-Dyck paths.

Our definition of the Schröder polynomial associated to a triangular partition
will utilize the recursions introduced by Gorsky-Mazin-Vazirani [26, 27] and
thus will be phrased in the language of invariant subsets rather than lattice
paths, which we now briefly recall.

For coprime positive integers $m,n$, a set $\Delta\subset\mathbb{Z}_{\geq 0}$
is _0-normalized and $(m,n)$-invariant_ if it contains $0$ and is closed under
translations by $+m$ and $+n$, i.e. $\Delta+m,\Delta+n\subset\Delta$. Denote
by $\mathrm{Inv}_{m,n}^{0}$ the set of all such subsets. In [26, 27], the
fourth named author alongside Gorsky and Vazirani defined a bijection

$\mathcal{A}:\mathrm{D}(m,n)\to\mathrm{Inv}_{m,n}^{0}\qquad\text{sending}\qquad\pi=(\lambda,\tau_{m,n})\mapsto\mathbb{Z}_{\geq
0}\setminus\mathrm{Gaps}(\pi),$

where $\mathrm{Gaps}(\pi):=\\{mn-my-
nx\;|\;(x,y)\in\tau_{m,n}\setminus\lambda\\}$. More generally,
Gorsky–Mazin–Vazirani also considered invariant subsets indexed by certain
trinary sequences $\bf x,y$ in letters $\\{0,1,\bullet\\}$ and defined the
polynomial

$Q_{{\bf x},{\bf y}}(q,t,a):=\sum_{\Delta\in\mathrm{Inv}_{{\bf x},{\bf
y}}}q^{\mathtt{area}^{\prime}(\Delta)}t^{-\mathtt{codinv}^{\prime}(\Delta)}\prod_{k\in\mathtt{cogen}_{\geq
0}(\Delta)}(1+at^{-\xi_{k}(\Delta)}),$

where $\mathtt{area},\mathtt{codinv}$, and $\mathtt{cogen}$ are certain
combinatorial statistics. In particular, they proved that $Q_{\bf x,y}$
satisfies recursion relations that coincide with those of Hogancamp-Mellit
[39], so that if $\bf u,v$ are the binary sequences obtained from $\bf x,y$ by
omitting the bullets $\bullet$, then $Q_{\bf x,y}=R_{\bf u,v}$ [26]. As a
consequence, they showed that the $(q,t)$-Schröder (and consequently Catalan)
polynomial $S_{m,n}(q,t,a)$ could be obtained directly as a certain
$q,t,a$-graded sum over $\mathrm{Inv}_{m,n}^{0}$.

In order to extend their construction to the setting of $\mathrm{D}(r,s)$ with
$r,s$ positive real numbers, we choose an embedding of $\tau_{r,s}$ into a
larger triangular partition $\tau_{m,n}$, such that $m,n$ are coprime positive
integers with $L_{m,n}$ parallel to $L_{r,s}$. Indeed, such a choice exists
since any given triangular partition $\tau$ may be cut out by lines of various
slopes [5]. In particular, if $\tau_{r,s}\subset\tau_{m,n}$, then there exists
$0<z\leq mn$ such that $(r,s)=(z/n,z/m)$, for which if $\ell=\lfloor z\rfloor$
then $\tau_{r,s}=\tau_{\ell/n,\ell/m}$ and
$\mathrm{D}(r,s)=\mathrm{D}(\ell/n,\ell/m)$. Since the choice of $\tau_{m,n}$
plays a role in the definitions, we modify the notation slightly in order to
make this choice explicit and set,

$\tau_{m,n,\ell}:=\tau_{\ell/n,\ell/m},\qquad\qquad\mathrm{D}(m,n,\ell):=\mathrm{D}(\ell/n,\ell/m),\qquad\text{and}\qquad
L_{m,n,\ell}:=L_{\ell/n,\ell/m}.$

So then, for any such triple $(m,n,\ell)$ we define

$\mathrm{Inv}_{m,n,\ell}^{0}:=\\{\Delta\in\mathrm{Inv}_{m,n}^{0}\;|\;\Delta\cap\mathrm{Gaps}(\tau_{m,n,l},\tau_{m,n})=\emptyset\\},$

and prove that

$\mathrm{Inv}_{{\bf x}(m,n,\ell),{\bf
y}(m,n,\ell)}=(\mathrm{Inv}_{m,n,\ell}^{0}-(mn-\ell)+(n+m))\cap\mathbb{Z}_{\geq
0},$ (1.1)

where ${\bf x}(m,n,\ell),{\bf y}(m,n,\ell)$ are certain trinary sequences
constructed directly from the combinatorial data of $L_{m,n,\ell}$.

The _triangular $(q,t)$-Schröder polynomial_ associated to the triple
$(m,n,\ell)$ is defined by

$S_{m,n,\ell}(q,t,a):=t^{|\tau_{m,n,\ell}|}Q_{{\bf x}(m,n,\ell),{\bf
y}(m,n,\ell)}(q,t,a).$

As before, we have a bijection
$\mathcal{D}:\mathrm{D}(m,n,\ell)\to\mathrm{Inv}_{{\bf w}(m,n,\ell)}$,
allowing us to give a formulation of these polynomials in terms of Schröder
paths (see Theorem 4.26).

######  Theorem A.

We have

$S_{m,n,\ell}(q,t,a)=\sum_{k\geq
0}a^{k}\left(\sum_{\pi\in\mathrm{D}(m,n,\ell)}q^{\mathtt{area}(\pi)}t^{\mathtt{dinv}(\pi)}\sum_{\begin{subarray}{c}L\subset\mathtt{AB}(\pi)\\\
|L|=k\end{subarray}}t^{-\sum_{\Box\in L}\xi(\pi,\Box)}\right),$ (1.2)

where $\mathtt{AB}(\pi)$, with $\pi=(\lambda,\tau_{m,n,\ell})$, is the set of
all addable boxes of $\lambda$ and $\mathtt{area}$, $\mathtt{dinv}$, and $\xi$
are some explicit statistics.

In the language of lattice paths, an addable box of $\lambda$ corresponds to a
west step (W) immediately followed by a north step (N) (reading the path right
to left and bottom to top). Thus, we can form a Schröder path by replacing
each such pair (W,N) in a given subset $L\subset\mathtt{AB}(\pi)$ of addable
boxes with a single northwest pointing diagonal step. In particular, for each
fixed $k$, the sum over pairs $(\pi,L)$ can be interpreted as a sum over
lattice paths $\pi\in\mathrm{D}(m,n,\ell)$ with exactly $k$ diagonal steps.
Consequently, we recover the more familiar rational $(q,t)$-Schröder
polynomial $S_{m,n}(q,t,a)$ in the special case $\ell=mn-1$.

The following result, exhibited in Corollary 4.20, is now immediate (see §4
for details).

######  Theorem B.

The triangular $(q,t)$-Schröder polynomial $S_{m,n,\ell}(q,t,a)$ specializes
to the triangular $(q,t)$-Catalan polynomial of [5, 6] at $a=0$,

$S_{m,n,\ell}(q,t,0)=\sum_{\pi\in\mathrm{D}(m,n,\ell)}q^{\mathtt{area}(\pi)}t^{\mathtt{dinv}(\pi)}=:C_{m,n,\ell}(q,t).$

###  1.2. The Coxeter Knot Associated to a Partition

The family of (positive) Coxeter knots was introduced in the work of Oblomkov
and Rozansky [53] and was recently proved to coincide with the family of
monotone knots independently defined by Galashin–Lam [19]. Coxeter knots
featured prominently in connections between link homology and the geometry of
the Hilbert scheme of points on $\mathbb{C}^{2}$ (see also [29]). We recall
their construction next.

Given an integer $M\geq 1$, let $\mathrm{Br}_{M}$ denote the braid group on
$M$ strands with $\sigma_{1},\ldots,\sigma_{M-1}$ the usual Artin generators.
Let $\lambda$ be a partition with $\lambda_{1}<M$, and let
$\mu_{1}\geq\mu_{2}\geq\cdots\geq\mu_{M}=0$ be the lengths of the columns of
$\lambda$, i.e. $\mu$ is the transpose partition of $\lambda$, extending by
zeros to obtain a sequence of length $M$. We define the _Coxeter knot
associated to $\lambda$_, denoted $K_{\lambda}$, to be the closure of the
$M$-strand braid

$\beta^{cox}_{M,\lambda}:=\mathrm{JM}_{M,2}^{\mu_{1}-\mu_{2}}\mathrm{JM}_{M,3}^{\mu_{2}-\mu_{3}}\dots\mathrm{JM}_{M,M}^{\mu_{M-1}-\mu_{M}}\sigma_{1}\sigma_{2}\cdots\sigma_{M-1},$

where $\mathrm{JM}_{M,i}\in\mathrm{Br}_{M}$ is the $M$-strand _Jucys-Murphy_
braid

$\mathrm{JM}_{M,i}=(\sigma_{M-i+1}\sigma_{M-i+2}\cdots\sigma_{M-1})(\sigma_{M-1}\cdots\sigma_{M-i+2}\sigma_{M-i+1})$

(these Jucys-Murphy braids are obtained from the more standard ones
$(\sigma_{i-1}\cdots\sigma_{2}\sigma_{1})(\sigma_{1}\sigma_{2}\cdots\sigma_{i-1})$
by conjugating with the half-twist). It is straightforward to prove (see
Proposition 5.2) that the closure of $\beta^{cox}_{M,\lambda}$ depends only on
$\lambda$ (and not on $M$) up to isotopy.

We are particularly interested in knots of the form $K_{\tau}$ where $\tau$ is
a triangular partition. A few important examples include the following:

  * (1)

If $\tau=\tau_{m,n}$ with $m,n\geq 1$ coprime integers, then
$K_{\tau}=T(n,m)$. This is most easily seen by translating into the language
of monotone knots using a result of Galashin–Lam [19].

  * (2)

If $\tau=\tau_{M,N}$ with $M,N\geq 1$ arbitrary positive integers (not
necessarily coprime), then $K_{\tau}$ is a certain cabled torus knot.
Specifically let $d=\gcd(M,N)$ and write $(M,N)=(md,nd)$, so that $m,n\geq 1$
are coprime. Then $K_{\tau}=T(m,n)(d,mnd+1)$, where $T(m,n)(d,mnd+1)$ is the
$(d,mnd+1)$-cable of the $(m,n)$-torus knot, $T(m,n)$. This fact was observed
in Galashin–Lam [18], and we give an alternate proof in Appendix B.

To compute the Khovanov–Rozansky homology of $K_{\tau}$, we will recognize
that the family of knots $K_{\tau}$ (in which $\tau$ is a triangular
partition) coincides with the family of knots whose homology is already
computed by Hogancamp and Mellit in [39]. Let us recall this latter family
now.

Given $\mathbf{u}\in\\{0,1\\}^{m}$ and $\mathbf{v}\in\\{0,1\\}^{n}$ binary
sequences with exactly one ‘1’, we write $\mathbf{u}=(0^{i}10^{j})$ and
$\mathbf{v}=(0^{k}10^{l})$. Denote by $K_{\mathbf{u},\mathbf{v}}$ the closure
of the braid

$(\sigma_{1}\cdots\sigma_{i})(\sigma_{1}\sigma_{2}\cdots\sigma_{m-1})^{k}(\sigma_{2}\cdots\sigma_{m-1})^{l}.$

For an example diagram of such a link, see Figure 5. The main result of [39]
is a computation of the Khovanov–Rozansky homology of
$K_{\mathbf{u},\mathbf{v}}$, via a recursion involving general pairs of binary
sequences with the same number of occurences of ‘1’. Specifically, [39]
defines series $R_{\mathbf{u},\mathbf{v}}(q,t,a)$ indexed by pairs of binary
sequences $\mathbf{u}\in\\{0,1\\}^{m}$, $\mathbf{v}\in\\{0,1\\}^{n}$ with
$\sum_{i}u_{i}=\sum_{j}v_{j}$, and proves that

$(1-q)(at^{-1/2}q^{-1/2})^{-\delta(\mathbf{u},\mathbf{v})}P_{K_{\mathbf{u},\mathbf{v}}}^{\text{KR}}(q,t,a)=t^{\delta(\mathbf{u},\mathbf{v})}R_{\mathbf{u},\mathbf{v}}(q,t,a),$
(1.3)

where $\delta(0^{i}10^{j},0^{k}10^{l})=\frac{1}{2}((i+j)(k+l)-j-l+c-1)$, where
$c$ is the number of components of $K_{0^{i}10^{j},0^{k}10^{l}}$ (see §2.6).
Note that $K_{\mathbf{u},\mathbf{v}}$ may be a link of more than one
component. For the case of $K_{\mathbf{u},\mathbf{v}}$ a knot, we prove in §5
the following (see Lemma 5.18 and Theorem 5.19):

######  Theorem C.

For any triangular partition $\tau$, the knot $K_{\tau}$ is isotopic to a knot
of the form $K_{\mathbf{u},\mathbf{v}}$. Conversely, if
$K_{\mathbf{u},\mathbf{v}}$ is a knot (as opposed to a link with more than one
component), then $K_{\mathbf{u},\mathbf{v}}=K_{\tau}$ for some triangular
partition, and moreover $\delta(\mathbf{u},\mathbf{v})=|\tau|$.

Furthermore, if $(m,n,\ell)$ is a triple of integers such that
$\tau=\tau_{m,n,\ell}$, then the Khovanov–Rozansky homology of $K_{\tau}$ is
free over $\mathbb{Z}$ with Poincaré series given by the triangular
$(q,t)$-Schröder polynomial up to normalization. Specifically:

$S_{m,n,\ell}(q,t,a)=(1-q)(at^{-1/2}q^{-1/2})^{-|\tau|}P_{K_{\tau}}^{\text{KR}}(q,t,a).$

In the above theorem (and throughout the rest of the paper), we write
$P_{K}^{\text{KR}}(q,t,a)$ for the Poincaré series of the triply graded
Khovanov–Rozansky homology of $K$, and for brevity refer to this as the _KR
series_ of $K$.

The mapping from triangular partitions to pairs of binary sequences is many-
valued, and requires us to choose an expression of $\tau=\tau_{m,n,\ell}$,
similar to the Schröder polynomials themselves (see §5.4). Nonetheless, the
knot $K_{\tau}$ really only depends on $\tau$ up to isotopy, and well-
definedness of Khovanov–Rozansky homology establishes the following.

######  Corollary 1.1.

The triangular Schröder polynomial $S_{m,n,\ell}(q,t,a)$ depends only on the
triangular partition $\tau=\tau_{m,n,\ell}$.

Below we will state some of the most important special cases of Theorem C. For
combinatorial purposes, it is best to state the result directly in terms of
the series $R_{\mathbf{u},\mathbf{v}}$, using (1.3).

  * (1)

The torus knot $T(m,n)$ is $K_{\tau}$ where $\tau=\tau_{m,n}$ (with $\ell$
taken to be the maximum value available, $\ell=mn-1$). The associated pair of
binary sequences is $(\mathbf{u},\mathbf{v})=(0^{m-1}1,0^{n-1}1)$, and we
recover the known identity

$S_{m,n}(q,t,a)=t^{\delta(m,n)}R_{0^{m-1}1,0^{n-1}1}(q,t,a),$

where $S_{m,n}(q,t,a)=S_{m,n,mn-1}(q,t,a)$ is the usual rational Schröder
polynomial and $\delta(m,n)$ is as in Definition 2.14.

  * (2)

The cabled torus knot $T(m,n)(d,mnd+1)$ is $K_{\tau}$, where
$\tau=\tau_{md,nd}$ (again, $\ell$ is taken to be its maximum value
$\ell=(md)(nd)-1$). The associated pair of binary sequences is
$(\mathbf{u},\mathbf{v})=(0^{d-1}10^{(m-1)d},0^{nd}1)$ Given integers
$m,n,d\geq 1$ with $m,n$ coprime, the $(d,mnd+1)$-cable of $T(m,n)$ is the
closure of the braid $\beta(,0^{nd}1)\in\mathrm{Br}_{md}$, thus

$S_{md,nd}(q,t,a)=t^{\delta(md,nd)}R_{0^{d-1}10^{(m-1)d},0^{nd}1}(q,t,a).$

###  1.3. The Oblomkov-Rasmussen-Shende Conjecture

Let $Z\subset\mathbb{C}^{2}$ be a complex plane curve with a singularity at
the origin $0\in\mathbb{C}^{2}$, and let $K$ denote the _link of the
singularity_ $0\in Z$ (i.e. $K$ is the intersection of $Z$ with the 3-sphere
of radius $\epsilon$ around 0, for sufficiently small $\epsilon>0$. A link is
_algebraic_ if it arises in this way. Algebraic links have been widely studied
for past century [7, 40, 14], with their associated plane curve singularities
particularly receiving a lot of attention in recent years [54, 50, 46, 52, 29,
25, 53].

The _Oblomkov-Rasmussen-Shende conjecture_ asserts that the lowest $a$-degree
part of the Khovanov–Rozansky homology of an algebraic link $K$ is isomorphic
to the cohomology of the Hilbert scheme of points on the plane curve
singularity whose link is $K$ (with the higher $a$-degree components obtained
by considering nested Hilbert schemes). We may refer to this conjecture as the
ORS1 conjecture, referring to [52] for the precise statement.

When $K$ is a knot, the associated plane curve singularity is called a
_unibranch singularity_ ; in this case ORS propose a second conjecture, which
we may refer to as ORS2, which asserts that the lowest $a$-degree part of
Khovanov–Rozansky homology of $K$ is isomorphic to the cohomology of the
_compactified Jacobian_ of the singularity $0\in Z$, with its perverse
filtration (the higher $a$-degree components are excluded from this
conjecture; they were incorporated in a recent preprint by Oscar Kivinen and
Minh-Tâm Quang Trinh [51]). Work of Migliorini and Shende [50], and
independently Maulik and Yun [46], establishes the equivalence of ORS1 and
ORS2 (for knots, and only the lowest $a$-degree part).

Ignoring the perverse filtration corresponds to setting $q=1$ on the level of
Poincaré series. Specialized at $q=1$, the ORS2 conjecture can be stated at
the level of Poincaré series as

$\Big{(}(1-q)\mathcal{P}^{\text{KR}}_{K}(q,t,a)\Big{)}|_{\begin{subarray}{c}q=1\\\
t=u^{-2}\end{subarray}}=a^{\delta}u^{-\delta}\mathcal{P}_{\overline{JC}}(u)+(\text{terms
with $a$-degree $>\delta$}),$

where $\delta$ is the Milnor number of the plane curve singularity whose link
is $K$, and $\overline{JC}$ is its compactified Jacobian.

In a recent article [25], the fourth named author jointly with Gorsky and
Oblomkov proved that the compactified Jacobian of (the singularity at the
origin of) the plane curve $(x(t),y(t))=(t^{nd},t^{md}+\lambda
t^{md+1}+\dots)$ has an affine paving with cells indexed by
$\tau_{md,nd}$-Dyck paths, with Poincaré series computed by a specialization
of the triangular $(q,t)$-Catalan polynomial $C_{\tau_{md,nd}}(q,t)$. Combined
with the realization that the link of the singularity in this case is
$T(m,n)(d,mnd+1)$, we obtain a proof of the ORS2 conjecture at $q=1$ for these
knots.

######  Theorem D.

The second Oblomkov-Rasmussen-Shende conjecture at $q=1$ holds for the
$(d,mnd+1)$-cable of the $(m,n)$-torus knot; that is, for $\tau=\tau_{md,nd}$
we have:

$\displaystyle\Big{(}(1-q)\mathcal{P}^{\text{KR}}_{K_{\tau}}(q,t,a)\Big{)}|_{\begin{subarray}{c}q=1\\\
t=u^{-2}\end{subarray}}$
$\displaystyle=a^{\delta}u^{-\delta}u^{2\delta}S_{\tau}(q,t,0)+(\text{terms
with $a$-degree $>\delta$})$
$\displaystyle=a^{\delta}u^{-\delta}\mathcal{P}_{\overline{JC}}(u)+(\text{terms
with $a$-degree $>\delta$}).$

This theorem is restated and proven as Theorem 5.23 in §5.5.

###  1.4. The $(q,t)$-Schröder Theorem

The classical _shuffle theorem_ , conjectured by Haglund-Haiman-Loehr-Remmel-
Ulyanov [34] and proven by Carlsson-Mellit [9], gives a combinatorial formula
in terms of $\tau_{n,n+1}$-Dyck paths for $\nabla e_{n}$. The _$(q,t)$
-Schröder Theorem_ was conjectured by Egge, Haglund, Killpatrick and Kremer
[13] and proven by Haglund [31] over ten years before Carlsson and Mellit’s
proof of the shuffle theorem. It states that the hook components of the Schur
expansion of $\nabla e_{n}$ coincide with the classical Schröder polynomials
$S_{n,n+1}(q,t,a)$, i.e. the following equality holds:

$S_{n,n+1}(q,t,a)=\sum_{k\geq 0}\langle\nabla
e_{n},h_{k}e_{n-k}\rangle\;a^{k}.$

Motivated by their study of certain superpolynomials in the double affine
Hecke algebra and their connection to Khovanov–Rozansky homology of torus
knots, Gorsky and Negut [28] conjectured a rational generalization of the
shuffle theorem (which Mellit [48] proved), wherein $\nabla e_{n}$ is replaced
by the action of certain operator $P_{m,n}$ in the _elliptic Hall algebra_ on
the element $1$ in the ring of symmetric functions and the combinatorial side
now ranges over $\tau_{m,n}$-Dyck paths with $m,n$ relatively prime integers.
The action of the elliptic Hall algebra $\mathcal{E}$ on symmetric functions
and its connection to Macdonald and $(q,t)$-Catalan combinatorics is well
documented with both algebraic and geometric formulations studied by Feigin-
Tsymbaliuk [17] and Burban, Schiffman, and Vasserot [8, 55, 56, 57, 58].
Inspired by this construction, Blasiak-Haiman-Morse-Pun-Seelinger [6] proved
that a similar equality could be given by considering certain operators
$\mathsf{D}_{\tau_{r,s}}\in\mathcal{E}$ indexed by triangular partitions
$\tau_{r,s}$, with $r,s$ arbitrary positive real numbers:

$\mathsf{D}_{\tau_{r,s}}(1)=\sum_{\pi\in\mathrm{D}(r,s)}q^{\mathtt{area}(\pi)}t^{\mathtt{dinv}(\pi)}\omega(\mathcal{G}_{\nu(\pi)}(X;t^{-1})),$

where $\omega$ is the standard Weyl involution on symmetric functions and
$\mathcal{G}_{\nu(\pi)}(X;t^{-1})$ is the LLT-polynomial for some particular
tuple of skew shapes $\nu(\pi)$. Although this so-called _shuffle theorem
under any line_ fully generalized the combinatorial side of the previous
shuffle theorems, its connections to Khovanov–Rozansky homology and torus
links remained completely open. In this article, we make considerable progress
towards bridging this gap by showing that the hook components of the Schur
expansion of $\mathsf{D}_{\tau}(1)$ are exactly given by our triangular
$(q,t)$-Schröder polynomial $S_{\tau}(q,t,a)$ and thus the KR series of the
Coxeter knot $K_{\tau}$. Using a superization procedure for LLT-polynomials,
we prove:

######  Theorem E.

The triangular $(q,t)$-Schröder theorem holds:

$S_{\tau_{r,s}}(q,t,a)=\sum_{k\geq
0}\langle\mathcal{H}_{\tau_{r,s}}(X;q,t),h_{k}e_{\lfloor
s\rfloor-k}\rangle\;a^{k}.$

This result is stated and proven as Corollary 6.12. Note that
$\mathcal{H}_{\tau}(X;q,t)$ is the symmetric function obtained by the action
of particular elliptic Hall algebra elements indexed by a triangular partition
$\tau$ on $1$.

###  1.5. Further Directions

There are many open questions that remain to be studied. Although we were able
to compute the KR series for all Coxeter _knots_ , ideally one would like such
a result for all Coxeter _links_ and general monotone curves. It is our hope
that an extension of our work might yield light in this direction.

Combinatorially there are also several directions worth pursuing. In a recent
article, the second named author alongside Simental and Vazirani [22] gave a
‘finite’ analog of the rational shuffle theorem in which the elliptic Hall
algebra is replaced the double affine Hecke algebra (DAHA). Conjecturally,
their construction coincides with the bigraded $GL_{r}$-character of certain
quantized Gieseker variety representations. On the other hand, Galashin and
Lam’s superpolynomial for monotone links is built from the DAHA. It it unknown
for exactly which monotone links the superpolynomial agrees with the KR
series. Thus, one can ask whether a generalization to the triangular setting
of the finite shuffle theorem of González-Simental-Vazirani could yield light
on a geometric interpretation of our triangular Schröder polynomials, which
combined with the results in this paper might provide a partial answer to when
the KR series and the superpolynomial coincide.

###  1.6. Structure of the paper:

In §2, we recall the necessary topological notions of knots and cables, the
construction of Khovanov–Rozansky homology and binary recursion of Hogancamp
and Mellit. §3 is similarly laying the combinatorial background in the form of
invariant subsets, the bijection onto Dyck paths, and the Gorsky-Mazin-
Vazirani recursions that form the building blocks of the upcoming definitions.
§4 is the combinatorial heart of the article and where Theorems A and B are
proved. Here, triangular partitions are further explored and the
$(q,t)$-Schröder polynomials are finally defined. It is in §5 that most of the
main results come together. We recall the work of Galashin-Lam relating
monotone and Coxeter knots, and follow up by connecting them further to the
binary knots of Hogancamp-Mellit, ending with the proofs of Theorems C and D.
Finally, in §6 we relate the triangular $(q,t)$-Schröder polynomial with the
existing Catalan combinatorics. We review the shuffle theorems, LLT
polynomials, and superization, before proving Theorem E. Lastly, in Appendix A
we provide an alternative self contained derivation of the standardization
procedure of §6 and in Appendix B we give a way of extending the binary
sequences in §5 to certain links.

## Acknowledgments

We are grateful to the American Institute of Mathematics for hosting the
workshop “Algebra, Geometry, and Combinatorics of Link Homology” where this
work was initiated. We also thank Eugene Gorsky for fruitful conversations. C.
Caprau was partially supported by NSF RUI grant, DMS-2204386.

##  2\. Khovanov–Rozansky Homology and the Hogancamp-Mellit Recursion

We begin by recalling some basic topological notions.

###  2.1. Framed Knots and Cables

Let $D^{2}=\\{z\in\mathbb{C}\>|\>|z|\leq 1\\}$ denote the unit disk in the
plane $\mathbb{C}=\mathbb{R}^{2}$, and let $S^{1}=\partial D^{2}$ be the unit
circle. Let $\mathbb{T}=S^{1}\times S^{1}$ denote the 2-dimensional torus. We
identify $S^{1}=\mathbb{R}/\mathbb{Z}$ via the mapping
$\mathbb{R}\twoheadrightarrow S^{1}$ sending $t\mapsto e^{2\pi it}$, which
induces an identification of $\mathbb{T}$ with $\mathbb{R}^{2}/\mathbb{Z}^{2}$
(restricting to the unit square $[0,1]^{2}$ gives the familiar realization of
$\mathbb{T}$ as the square with opposite edges identified). The _longitude_ of
$\mathbb{T}$ is the copy of $S^{1}$ identified as the image of the $x$-axis
$\mathbb{R}\times\\{0\\}$ with _leftward orientation_. The _meridian_ of
$\mathbb{T}$ is the copy of $S^{1}$ identified as the image of the $y$-axis
$\\{0\\}\times\mathbb{R}$ with its _upward_ orientation.

We may embed $\mathbb{T}$ in $\mathbb{R}^{3}$ in a standard fashion with image
the set of points of distance $1/2$ from $\\{x^{2}+y^{2}=1\\}$, so that the
meridian of $\mathbb{T}$ maps to the circle with parametrization
$\theta\mapsto(0,1,0)+\frac{1}{2}(0,-\cos(\theta),\sin(\theta))$, and the
longitude of $\mathbb{T}$ maps to the circle parametrized by
$\theta\mapsto\frac{1}{2}(\cos(\theta),\sin(\theta),0)$.

Given relatively prime integers $m,n$, we consider the line segment in
$\mathbb{R}^{2}$ from $(m,0)$ to $(0,n)$. The image of this segment in
$\mathbb{T}$ is a closed curve, denoted $T(m,n)^{\mathbb{T}}$. The image of
$T(m,n)^{\mathbb{T}}$ under the standard embedding
$\mathbb{T}\hookrightarrow\mathbb{R}^{3}$ is the _$(m,n)$ -torus knot_,
denoted $T(m,n)$. Given $d\geq 1$, let $T(md,nd)^{\mathbb{T}}$ denote $d$
parallel copies of $T(m,n)^{\mathbb{T}}$ in $\mathbb{T}$, whose image in
$\mathbb{R}^{3}$ is the torus link $T(md,nd)$.

We will be interested in knots which are obtained as cables of torus knots. In
order to discuss these, we must introduce the notion of framed knots. A
_framed knot_ $\mathcal{K}$ is an embedding of the solid torus $S^{1}\times
D^{2}\hookrightarrow\mathbb{R}^{3}$, regarded up to isotopy. Given any point
$p\in D^{2}$ we obtain an ordinary knot $K$ by restricting $\mathcal{K}$ to an
embedding $S^{1}\times\\{p\\}\hookrightarrow\mathbb{R}^{3}$. The knot $K$ does
not depend on $p$, up to isotopy, and we say that $\mathcal{K}$ is a framing
of $K$.

If $\mathcal{K}$ is a framed knot and $p_{1}\neq p_{2}\in D^{2}$ are distinct
points, then we may consider knots $K_{i}$ obtained as the images of
$S^{1}\times\\{p_{i}\\}$ under $\mathcal{K}$. Let $r$ denote the _linking
number_ of the 2-component link $K_{1}\cup K_{2}$ (upon choosing a diagram for
$K_{1}\cup K_{2}$, the linking number is one half the number of crossings
between $K_{1}$ and $K_{2}$, counted with signs). This linking number does not
depend on the choice of $p_{1}$ and $p_{2}$, and is called the _framing
coefficient of $\mathcal{K}$_. Equivalently (but perhaps less precise), the
framing coefficient of $\mathcal{K}$ is the number of twists in the embedded
annulus obtained by taking the image of $S^{1}\times a$, where $a\subset
D^{2}$ is an embedded arc connecting the points $p_{1}$ and $p_{2}$.

Up to isotopy, a framed knot is completely determined by its framing
coefficient $r$, and we will abuse notation and write $\mathcal{K}=(K,r)$. The
_Seifert framing_ of $K$ is the unique framing with coefficient $r=0$.

Given a framed knot $\mathcal{K}=(K,r)$, restricting from the solid torus to
its boundary yields embedding
$\phi_{K,r}\colon\mathbb{T}\hookrightarrow\mathbb{R}^{3}$. When $K=U$ is the
unknot with Seifert framing, we obtain the standard embedding
$\phi_{U,0}\colon\mathbb{T}\hookrightarrow\mathbb{R}^{3}$. For more general
$r\in\mathbb{Z}$, the embedding $\phi_{U,r}$ sends the closed curve
$T(p,q)^{\mathbb{T}}$ to $T(p,q+rp)$.

######  Definition 2.1.

Let $K$ be a knot. The _$(p,q)$ -cable of $K$_, denoted $K(p,q)$, is defined
to be the image of $T(p,q)^{\mathbb{T}}$ under the embedding
$\phi_{K,0}\colon\mathbb{T}\hookrightarrow\mathbb{R}^{3}$ introduced above.

######  Remark 2.2.

We can also realize $K(p,q)$ as the image of $T(p,q-rp)$ under $\phi_{K,r}$
for any $r\in\mathbb{Z}$, since the effect of changing the framing from 0 to
$r$ is cancelled by the insertion of $r$ negative full twists (in the meridian
direction) into $T(p,q)^{\mathbb{T}}$, obtaining $T(p,q-rp)^{\mathbb{T}}$.

###  2.2. Braid Presentations of Some Special Knots

In this paper we are particularly interested in the $(d,mnd+1)$-cable of the
torus knot $T(m,n)$, denoted $T(m,n)(d,mnd+1)$, where $m,n,d$ are positive
integers with $m,n$ relatively prime. These knots are the simplest instances
of algebraic knots (see §5.5). Note that this family includes positive torus
knots as the special case $d=1$. In this section we will focus on finding a
braid presentation of these knots.

Let $\mathrm{Br}_{n}$ denote the _braid group_ on $n$ strands. Let
$\mathbbm{1}_{n}\in\mathrm{Br}_{n}$ denote the identity braid. For $1\leq
i\leq n-1$ let $\sigma_{i}$ denote the standard Artin generator, and let
$\mathrm{cox}_{n}:=\sigma_{1}\cdots\sigma_{n-1}\in\mathrm{Br}_{n}$ denote the
_standard Coxeter braid_ on $n$ strands. These are pictured in Figure 1.

$\sigma_{i}\ \ =\ \ \leavevmode\hbox to53.43pt{\vbox
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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Figure 1. The standard Coxeter generators of the braid group (left) and the
standard Coxeter braid (right).

Given braids $\beta_{1}\in\mathrm{Br}_{n_{1}}$ and
$\beta_{2}\in\mathrm{Br}_{n_{2}}$, let
$\beta_{1}\sqcup\beta_{2}\in\mathrm{Br}_{n_{1}+n_{2}}$ denote the union of the
braids $\beta_{1}$ and $\beta_{2}$. In terms of the diagrams above,
$\beta_{1}\sqcup\beta_{2}$ consists of $\beta_{2}$ above $\beta_{1}$. Given a
braid $\beta$, let $\hat{\beta}$ denote the oriented link obtained by closing
$\beta$.

######  Example 2.3.

For integers $M,N$ with $M\geq 1$, the torus link $T(M,N)$ is the closure of
the braid $\mathrm{cox}_{M}^{N}$.

######  Lemma 2.4.

For $m,n$ coprime and $d\in\mathbb{Z}_{\geq 1}$, we have
$T(md,nd)=T(m,n)(d,mnd)$.

###### Proof.

By definition, $T(md,nd)$ consists of $d$ parallel copies of $T(m,n)$, where
the notion of “parallel” comes from the embedding in the standard torus
$\mathbb{T}$. On the other hand, $T(m,n)(d,mnd)$ is the image of
$T(d,0)^{\mathbb{T}}$ under the embedding $\phi_{T(m,n),mn}$; this consists of
$d$ parallel copies of $T(m,n)$, where the notion of “parallel” comes from the
framing coefficient $mn$, so that two distinct copies of $T(m,n)$ in
$\phi_{T(m,n),mn}(T(d,0)^{\mathbb{T}})$ have linking number $mn$. To prove the
lemma, we just need to see that any two distinct copies of $T(m,n)$ in
$T(md,nd)$ have linking number $mn$. To see this, assume without loss of
generality that $d=2$. Then we have a presentation of $T(2m,2n)$ as the
closure of the braid $(\mathrm{cox}_{2m})^{2n}$. Of the $2n(2m-1)$ crossings,
$2mn$ of these involve the two distinct components, so the linking number
between these components is $mn$, which proves the lemma. ∎

Figure 2. Realizing $T(6,8)$ as the $(2,24)$ cable of $T(3,4)$.

From Lemma 2.4 we deduce the following.

######  Lemma 2.5.

$T(m,n)(d,mnd+1)$ can be represented as the closure of
$(\mathrm{cox}_{d}\sqcup\mathbbm{1}_{(m-1)d})(\mathrm{cox}_{md})^{nd}$.

###### Proof.

Lemma 2.4 tells us that $T(m,n)(d,mnd)=T(md,nd)$. It is known that $T(md,nd)$
can be represented as the closure of the braid $(\mathrm{cox}_{md})^{nd}$. To
obtain $T(m,n)(d,mnd+1)$ from $T(m,n)(d,mnd)$, insert one more copy of the
braid $\mathrm{cox}_{d}$, and the statement follows. ∎

###  2.3. Soergel Bimodules and Rouquier Complexes

We briefly recall the main ideas needed for the construction of
Khovanov–Rozansky homology and refer the reader to [42, 41, 43, 15, 39, 16,
49] for detailed exposure on this topic.

Fix an integer $n\geq 0$ and let $\mathbb{K}$ be a commutative ring. Let
$R=\mathbb{K}[x_{1},\ldots,x_{n}]$ regarded as a graded ring with
$\deg(x_{i})=2$. If $B$ is a graded $(R,R)$-bimodule and
$k\in\frac{1}{2}\mathbb{Z}$, then we denote by $q^{k}B$ the bimodule $B$ with
grading “shifted up by $2k$ units”; that is, $(q^{k}B)_{i}=B_{i-2k}$. Let
$\star=\otimes_{R}$ denote the monoidal operation on bimodules, and let
$\boxtimes=\otimes_{\mathbb{K}}$ denote the external tensor product.

If $X$ is a complex of graded $(R,R)$-bimodules, then we let $\Omega X$ denote
the cohomological shift; that is, the $i$-th chain group of $\Omega X$ is
$X^{i-1}$, with negated differential $d_{\Omega X}=-d_{X}$.

Given $k\in\frac{1}{2}\mathbb{Z}$, we let $t^{k}X:=\Omega^{2k}q^{-k}X$.
Observe that $q,t,\Omega$ are not independent, but satisfy the relation
$q^{\frac{1}{2}}t^{\frac{1}{2}}=\Omega.$

Figure 3. A braid whose closure is $T(3,4)(3,37)$, shown here as embedded in a
pair-of-pants surface.

We let $\mathbb{S}\mathrm{Bim}_{n}$ denote the category of _Soergel bimodules_
for the symmetric group $S_{n}$ [16]. This is a monoidal subcategory of the
category of graded $(R,R)$-bimodules. We denote by
$\operatorname{Ch}^{b}(\mathbb{S}\mathrm{Bim}_{n})$ the category of bounded
chain complexes in $\mathbb{S}\mathrm{Bim}_{n}$.

The grading shifts interact with $\operatorname{Hom}$ spaces as follows:

$\operatorname{Hom}_{\mathbb{S}\mathrm{Bim}_{n}}(B,q^{k}B^{\prime})=q^{k}\operatorname{Hom}_{\mathbb{S}\mathrm{Bim}_{n}}(B,B^{\prime})\
,\qquad\operatorname{Hom}_{\mathbb{S}\mathrm{Bim}_{n}}(q^{l}B,B^{\prime})=q^{-l}\operatorname{Hom}_{\mathbb{S}\mathrm{Bim}_{n}}(B,B^{\prime}).$

Now, for $1\leq i\leq n-1$, we let

$B_{i}:=q^{-\frac{1}{2}}R\otimes_{R^{s_{i}}}R,$

where $s_{i}$ is the simple transposition $(i,i+1)$ in the symmetric group
$S_{n}$, and $R^{s_{i}}\subset R$ is the graded subalgebra of
$s_{i}$-invariant polynomials. We have canonical degree zero maps

$B_{i}\rightarrow q^{-\frac{1}{2}}R\ \text{given by}\ f\otimes g\mapsto fg$

and

$q^{\frac{1}{2}}R\rightarrow B_{i}\ \text{given by}\ 1\mapsto x_{i}\otimes
1-1\otimes x_{i+1}.$

######  Definition 2.6.

Let $T_{i}^{\pm 1}=T(\sigma_{i}^{\pm 1})$ be following two-term complexes:

$T_{i}:=\left(t^{-1/2}B_{i}\rightarrow R\right)\quad\text{and}\qquad
T_{i}^{-1}:=\left(R\rightarrow t^{1/2}B_{i}\right).$

The _Rouquier complex_ associated to a braid
$\beta=\sigma^{\pm}_{i_{1}}\dots\sigma^{\pm}_{i_{k}}\in\mathrm{Br}_{n}$ is the
chain complex $T(\beta)\in\operatorname{Ch}^{b}(\mathbb{S}\mathrm{Bim}_{n})$
obtained by tensoring

$T(\sigma^{\pm}_{i_{1}}),\dots,T(\sigma^{\pm}_{i_{k}}).$

######  Remark 2.7.

The above conventions for $q$ and $t$ will enable a direct comparison between
our computations of $\operatorname{HHH}$ and the $(q,t)$-Catalan numbers. It
is more traditional in Soergel bimodule literature to denote the grading and
homological shifts of $X$ by $X(1)$ and $X[1]$, respectively; these are
related to our $q,t,\Omega$ by

$X(1)=q^{-\frac{1}{2}}X\ ,\qquad X[1]=\Omega^{-1}X.$

###  2.4. Khovanov–Rozansky Homology

Let $\mathbbm{k}\text{-Mod}^{\mathbb{Z}\times\mathbb{Z}}$ denote the category
of bigraded $\mathbb{K}$-modules. Let
$\operatorname{HH}^{\cdot}\colon\mathbb{S}\mathrm{Bim}_{n}\to\mathbbm{k}\text{-Mod}^{\mathbb{Z}\times\mathbb{Z}}$
be the $\mathbb{K}$-linear functor sending a graded $(R,R)$-bimodule to its
Hochschild cohomology. Extending to complexes gives us a functor

$\operatorname{HH}^{\cdot}\colon\operatorname{Ch}^{b}(\mathbb{S}\mathrm{Bim}_{n})\to\operatorname{Ch}^{b}(\mathbbm{k}\text{-Mod}^{\mathbb{Z}\times\mathbb{Z}}).$

The complex $\operatorname{HH}^{\cdot}(X)$ is triply graded via

$\operatorname{HH}^{i,j,k}(X):=\text{Soergel degree-$i$ component of
}\operatorname{HH}^{j}(X^{k}).$

The homology of $\operatorname{HH}^{\cdot}(X)$ is denoted by

$\operatorname{HHH}^{\cdot}(X):=H^{\cdot}(\operatorname{HH}^{\cdot}(X)).$

On triply graded $\mathbb{K}$-modules, we will use the grading shifts $q,a,t$
defined by

$(qM)^{i,j,k}=M^{i-2,j,k}\ ,\qquad(aM)^{i,j,k}=M^{i+2,j-1,k}\
,\qquad(tM)^{i,j,k}=M^{i+2,j,k-2},$

and the cohomological shift

$(\Omega M)^{i,j,k}=M^{i,j,k-1}.$

Note that $qt=\Omega^{2}$. Note also that $a$ acts by shifing up by 1 in the
Hochschild degree, together with a shift down by 2 in the Soergel bimodule
degree.

######  Definition 2.8.

Let $\beta\in\mathrm{Br}_{n}$ and $e=e(\beta)$ be the number of positive
crossings minus the number of negative crossings, and let $c=c(\beta)$ denote
the number of components of the link represented by $\beta$. Define the
_normalizing exponent_ of the braid $\beta$ by the formula

$\delta(\beta):=\frac{1}{2}(e(\beta)+c(\beta)-n(\beta)).$ (2.1)

The _Khovanov–Rozansky complex_ of $\beta$ is

$\displaystyle C_{\mathrm{KR}}(\beta)$
$\displaystyle:=(at^{1/2}q^{-1/2})^{\delta(\beta)}\operatorname{HH}^{\cdot}(T(\beta)).$

Its homology is the _Khovanov–Rozansky_ homology of the link $\widehat{\beta}$
represented by $\beta$, and is denoted by
$H_{\mathrm{KR}}(\beta):=(at^{1/2}q^{-1/2})^{\delta(\beta)}\operatorname{HHH}(T(\beta))$.

######  Remark 2.9.

If $K$ is a positive knot and $\beta$ is a positive braid representing $K$,
then the normalizing exponent $\delta(\beta)$ agrees with the _genus_ of $K$
(in general, $\delta(\beta)$ is the genus of the Seifert surface produced by
Seifert’s algorithm). In particular, for positive braids $\delta(\beta)$
depends only on the braid closure, hence is actually a well-defined knot
invariant. Note that all the braids of interest in this paper are positive, so
this remark applies.

Recall that for any tri-graded $\mathbb{K}$-module
$M=\bigoplus_{i,j,k}M^{i,j,k}$, the _Poincaré series_ of $M$ computes the sum
of the graded ranks over $\mathbb{K}$; namely,

$\mathcal{P}(M)=\sum_{i,j,k}q^{i}t^{j}a^{k}\mathrm{rank}_{\mathbb{K}}(M^{i,j,k}).$

######  Definition 2.10.

The _Khovanov–Rozansky (KR) series_ of a link $L=\widehat{\beta}$ is defined
to be
$\mathcal{P}_{L}^{\text{KR}}(q,t,a):=\mathcal{P}(H_{\mathrm{KR}}(\beta))$.

######  Theorem 2.11 ([42, 41]).

If $\beta$ and $\beta^{\prime}$ represent isotopic links, then there is a
homotopy equivalence $C_{\mathrm{KR}}(\beta)\simeq
C_{\mathrm{KR}}(\beta^{\prime})$, and consequently
$\mathcal{P}_{\widehat{\beta}}^{\text{KR}}(q,t,a)=\mathcal{P}_{\widehat{\beta^{\prime}}}^{\text{KR}}(q,t,a)$.

######  Example 2.12.

For non-negative integers $M,N\geq 0$, the torus link $T(M,N)$ is the closure
of $\mathrm{cox}_{M}^{N}$. For this braid we have $e=(M-1)N$ and
$c=\gcd(M,N)$, hence

$\delta=\frac{MN-M-N+\gcd(M,N)}{2}.$

######  Example 2.13.

For integers $m,n,d\geq 0$ with $m,n$ coprime, we consider the
$(d,mnd+1)$-cable of $T(m,n)$ as the closure of
$(\mathrm{cox}_{d}\sqcup\mathbbm{1}_{(m-1)d})\mathrm{cox}_{md}^{nd}$, as in
Lemma 2.5. For this braid we have $e=(d-1)+(md-1)(nd)$, with $c=1$ and number
of strands $md$, hence

$\delta=\frac{(md)(nd)-md-nd+d}{2}.$

The integers $\delta$ appearing in the above examples will occur frequently in
this paper, and we will abbreviate them.

######  Definition 2.14.

For integers $M,N\geq 0$, we let $\delta(M,N):=\frac{MN-M-N+\gcd(M,N)}{2}$.

###  2.5. Links from Binary Sequences

######  Definition 2.15.

If $\mathbf{u}=(u_{1},\ldots,u_{m})\in\\{0,1\\}^{m}$, we let
$|\mathbf{u}|=\sum_{i}u_{i}$. Given sequences $\mathbf{u}\in\\{0,1\\}^{m}$,
$\mathbf{v}\in\\{0,1\\}^{n}$ with $|\mathbf{u}|=|\mathbf{v}|$, we define a
complex
$C(\mathbf{u},\mathbf{v})\in\operatorname{Ch}^{b}(\mathbb{S}\mathrm{Bim}_{n})$
as follows. Abbreviate by writing $l=|\mathbf{u}|=|\mathbf{v}|$. Let
$\pi_{\mathbf{u}}\in S_{m}$ denote the shuffle permutation corresponding to
$\mathbf{u}$, i.e. the permutation given in one-line notation by
$(i_{1},\ldots,i_{k},j_{1},\ldots,j_{l})$ with $k+l=m$, where
$i_{1}<\cdots<i_{k}$ are the indices for which $u_{i}=0$, and
$j_{1}<\cdots<j_{l}$ are the indices for which $u_{j}=1$. Let
$\beta_{\mathbf{u}}$ denote the positive braid lift of
$\pi_{\mathbf{u}}^{-1}$. Next, write $\mathbf{v}$ as
$\mathbf{v}=(0^{r_{0}}10^{r_{1}}1\cdots 10^{r_{l}})$, and set

$\gamma_{m,\mathbf{v}}:=(\mathrm{cox}_{m})^{r_{0}}(\mathbbm{1}_{1}\boxtimes\mathrm{cox}_{m-1})^{r_{1}}\cdots(\mathbbm{1}_{l}\boxtimes\mathrm{cox}_{m-l})^{r_{l}}.$

Finally, let
$\beta(\mathbf{u},\mathbf{v})=\beta_{\mathbf{u}}\gamma_{m,\mathbf{v}}$, and

$C(\mathbf{u},\mathbf{v}):=T(\beta(\mathbf{u},\mathbf{v}))\star(K_{l}\boxtimes\mathbbm{1}_{n-l}),$

where $K_{l}$ is the complex constructed in [15, 39]. When $l=1$ we let
$K_{\mathbf{u},\mathbf{v}}$ denote the closure of the braid
$\beta(\mathbf{u},\mathbf{v})$.

######  Example 2.16.

If $\mathbf{u}=(10001)$ and $\mathbf{v}=(011000)$, then
$C(\mathbf{u},\mathbf{v})$ may be pictured as in Figure 4.

$C(\mathbf{u},\mathbf{v})\ \ \ =\ \ \ \leavevmode\hbox to208.31pt{\vbox
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{90.92462pt}{1.15863pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$ Figure 4. The
complex $C({\bf u,v})$ for $\mathbf{u}=(10001)$ and $\mathbf{v}=(011000)$.

The following is proven in [39].

######  Lemma 2.17.

If $|\mathbf{u}|=|\mathbf{v}|=1$, then
$\operatorname{HHH}(\beta(\mathbf{u},\mathbf{v}))\cong\operatorname{HHH}(C(\mathbf{u},\mathbf{v}))\otimes\mathbb{K}[x_{1}]$.

######  Lemma 2.18.

For $\mathbf{u}=0^{i}10^{j}$ and $\mathbf{v}=0^{k}10^{l}$ we have that
$K_{\mathbf{u},\mathbf{v}}$ is the closure of the braid

$\beta(0^{i}10^{j},0^{k}10^{l})=(\sigma_{1}\cdots\sigma_{i})(\sigma_{1}\sigma_{2}\cdots\sigma_{m-1})^{k}(\sigma_{2}\cdots\sigma_{m-1})^{l}$

with normalizing exponent (2.1) given by

$\delta(\beta(0^{i}10^{j},0^{k}10^{l}))=\frac{(i+j)(k+l)-j-l+c-1}{2}.$ (2.2)

###### Proof.

Immediate from the definitions. ∎

######  Lemma 2.19.

Given integers $m,n,d\geq 0$ with $m,n$ coprime, the $(d,mnd+1)$-cable of
$T(m,n)$ coincides with $K_{0^{d-1}10^{(m-1)d},0^{nd}1}$. Moreover, the
normalizing exponent (2.2) agrees with $\delta(md,nd)$ from Definition 2.14.

###### Proof.

From the definitions we have
$\beta(0^{d-1}10^{(m-1)d},0^{nd}1)=(\mathrm{cox}_{d}\sqcup\mathbbm{1}_{d(m-1)})\mathrm{cox}_{md}^{nd}$,
whose closure is the $(d,mnd+1)$-cable of the torus knot $T(m,n)$, by Lemma
2.5. Comparison of the normalizing exponent with $\delta(md,nd)$ is
elementary. ∎

######  Construction 2.20.

Let $\mathbf{u}\in\\{0,1\\}^{M}$, $\mathbf{v}\in\\{0,1\\}^{N}$ be binary
sequences with $|\mathbf{u}|=|\mathbf{v}|=1$. We provide a description of the
link $K_{\mathbf{u},\mathbf{v}}$ from Definition 2.15. First, choose real
numbers $0<y_{1}<\cdots<y_{M}<1$ and $0<x_{1}<\cdots<x_{N}<1$. Consider the
unit square $[0,1]^{2}$ with the following set of points on its boundary:

$S=\bigcup_{1\leq i\leq M}(0,y_{i})\cup\bigcup_{1\leq j\leq
N}(x_{j},0)\cup\bigcup_{\begin{subarray}{c}1\leq i\leq M\\\ i\neq
a\end{subarray}}(1,y_{i})\cup\bigcup_{\begin{subarray}{c}1\leq j\leq N\\\
j\neq b\end{subarray}}(x_{j},1),$

where $a$ is the index with $u_{a}=1$ and $b$ is the index with $v_{b}=1$. Let
$D$ be union of arcs of the following types:

  1. (1)

Consider the ordered tuples of boundary points

$\displaystyle(p_{1},\ldots,p_{M+N-1})$
$\displaystyle=\Big{(}(0,y_{1}),\ldots,(0,y_{M}),(x_{1},1),\ldots,\widehat{(x_{b},1)},\ldots,(x_{N},1)\Big{)},$
$\displaystyle(q_{1},\ldots,p_{M+N-1})$
$\displaystyle=\Big{(}(x_{1},0),\ldots,(x_{N},0),(1,y_{1}),\ldots\widehat{(1,y_{a})},\ldots,(1,y_{M})\Big{)}.$

For $1\leq k\leq M+N-1$, let $A_{k}$ denote the straight line segment from
$p_{k}$ to $q_{k}$.

  2. (2)

For each $1\leq i\leq M$ with $i\neq a$, let $H_{i}$ denote an arc connecting
$(0,y_{i})$ to $(1,y_{i})$, disjoint from the interior of $[0,1]^{2}$. Assume
that the $H_{i}$, for $1\leq i\leq M$, are pairwise disjoint.

  3. (3)

For each $1\leq j\leq N$ with $j\neq b$, let $V_{j}$ denote an arc connecting
$(x_{j},0)$ to $(x_{j},1)$, beneath the arcs $A_{k}$, and disjoint from each
$H_{i}$. Assume that the $V_{j}$, for $1\leq j\leq N$, are pairwise disjoint.

  4. (4)

The only boundary points that remain unpaired are $(0,y_{a})$ and $(x_{b},0)$.
Choose $\epsilon>0$ and let $B_{-}$ be an arc connecting $(0,y_{a})$ to
$(-\epsilon,-\epsilon)$ below the all arcs $H_{i}$ (and disjoint from all
other arcs); let $B_{+}$ be an arc connecting $(-\epsilon,-\epsilon)$ to
$(x_{b},0)$ above all arcs $V_{j}$ (and disjoint from all other arcs). Let
$B=B_{-}\cup B_{+}$.

The union of all arcs of the form $A_{k},H_{i},V_{j}$ and $B$ is a closed,
piecewise linear diagram which represents $K_{\mathbf{u},\mathbf{v}}$.

######  Example 2.21.

Construction 2.20 is illustrated for $\mathbf{u}=0010$ and $\mathbf{v}=01000$
in Figure 5. The arc $B$ has been drawn in orange in order to facilitate the
comparison with the monotone link.

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Construction 2.20 illustrated for $\mathbf{u}=0010$ and $\mathbf{v}=01000$.

###  2.6. The Hogancamp-Mellit Recursion

In [39], the third named author and Mellit presented a recursive computation
of $\operatorname{HHH}(C(\mathbf{u},\mathbf{v}))$ (for
$(C(\mathbf{u},\mathbf{v})$ as defined in Definition 2.15), which we now
recall.

######  Definition 2.22.

Let ${\bf u}\in\\{0,1\\}^{m+\ell}$ and ${\bf v}\in\\{0,1\\}^{n+\ell}$ be two
binary sequences with $|{\bf u}|=|{\bf v}|=\ell$. Define $R_{{\bf u},{\bf
v}}\in\mathbb{N}[q,t^{\pm},a,(1-q)^{-1}]$ as the power series uniquely
determined by the following recursions:

$\displaystyle R_{0{\bf u},0{\bf v}}$ $\displaystyle=t^{-|{\bf u}|}R_{{\bf
u}1,{\bf v}1}+qt^{-|{\bf u}|}R_{{\bf u}0,{\bf v}0},$ $\displaystyle R_{1{\bf
u},0{\bf v}}$ $\displaystyle=R_{{\bf u}1,{\bf v}},$ $\displaystyle
R_{\emptyset,0^{n}}$ $\displaystyle=\left(\frac{1+a}{1-q}\right)^{n},$
$\displaystyle R_{1{\bf u},1{\bf v}}$ $\displaystyle=(t^{|{\bf u}|}+a)R_{{\bf
u},{\bf v}},$ $\displaystyle R_{0{\bf u},1{\bf v}}$ $\displaystyle=R_{{\bf
u},{\bf v}1},$ $\displaystyle R_{0^{m},\emptyset}$
$\displaystyle=\left(\frac{1+a}{1-q}\right)^{m},$

where $R_{\emptyset,\emptyset}:=1$.

######  Theorem 2.23 ([39]).

For any two binary sequences ${\bf u}$ and ${\bf v}$ with $|{\bf u}|=|{\bf
v}|$, we have

$\mathcal{P}(\operatorname{HHH}^{\cdot}(C(\mathbf{u},\mathbf{v})))=R_{\mathbf{u},\mathbf{v}}.$

In particular, if $|\mathbf{u}|=|\mathbf{v}|=1$, then the KR series of the
link $K_{\mathbf{u},\mathbf{v}}:=\hat{\beta}(\mathbf{u},\mathbf{v})$ is given
by

$\mathcal{P}_{K_{\mathbf{u},\mathbf{v}}}^{\text{KR}}(q,t,a)=\frac{(at^{1/2}q^{-1/2})^{\delta(\beta(\mathbf{u},\mathbf{v}))}}{1-q}R_{\bf
u,v}(q,t,a),$ (2.3)

where $\delta(\beta)$ is the normalizing exponent (2.1).

######  Remark 2.24.

The denominator $\frac{1}{1-q}$ in (2.3) comes from the tensor factor
$\mathbb{K}[x_{1}]$ in Lemma 2.17.

######  Remark 2.25.

The special case ${\bf u}=0^{M-1}1$, ${\bf v}=0^{N-1}1$ yields the KR series
of $T(M,N)$. One of the central aims of this paper is to give a combinatorial
interpretation of $K_{\bf u,v}$ and its KR series, when $K_{\bf u,v}$ is a
knot.

This together with Lemma 2.19 gives the following.

######  Corollary 2.26.

For integers $m,n,d\geq 0$ with $m,n$ coprime, the KR series of the cable knot
$T(m,n)(d,mnd+1)$ is given by

$\mathcal{P}_{T(m,n)(d,mnd+1)}=\frac{(at^{1/2}q^{-1/2})^{\delta(md,nd)}}{1-q}R_{(0^{d-1}10^{(m-1)d},0^{nd}1)}.$

∎

Here we are using $\delta(md,nd)$ from Definition 2.14, and the computation
from Example 2.13.

##  3\. Dyck paths, Invariant Subsets, and the Gorsky-Mazin-Vazirani Recursion

In this article a _box_ or _cell_ will be a unit square of the form
$[x-1,x]\times[y-1,y]$ in $\mathbb{R}^{2}$, with $x,y\in\mathbb{Z}_{\geq 1}$.
The point $(x,y)$ is the _northeast (NE) corner_ of this box, while
$(x-1,y-1)$ is the _southwest (SW) corner_. We will generally identify a box
$\Box$ with its NE corner, thereby labeling boxes by elements of
$\mathbb{Z}_{\geq 1}\times\mathbb{Z}_{\geq 1}$.

Given two real numbers $r,s\in\mathbb{R}_{>0},$ consider the line
$L_{r,s}:=\\{x/r+y/s=1\\}$ connecting points $(r,0)$ and $(0,s)$ on the
coordinate axes.

######  Definition 3.1.

Given a line $L_{r,s}$, we say that a box $\Box=(x,y)$ _lies on $L$_, if we
have inequalities $(x-1)/r+(y-1)/s\leq 1$ (which expresses that the SW corner
of $\Box$ lies weakly below $L_{r,s}$) and $x/r+y/s>1$ (which expresses that
the NE corner of $\Box$ lies strictly above $L_{r,s}$). We say that $\Box$ is
below $L$ if $x/r+y/s\leq 1$, and above $L_{r,s}$ if $(x-1)/r+(y-1)/s>1$.

Given an integer $m$, let $[m]:=\\{1,\ldots,m\\}$. We will identify a
partition $\lambda=(\lambda_{1},\ldots,\lambda_{\ell})$ with its associated
Young diagram (written in the French style) so that
$\lambda_{1}\geq\lambda_{2}\geq\dots\geq\lambda_{\ell}$.

######  Definition 3.2 ([5]).

A partition $\tau$ is _triangular_ if there exists a line
$L_{r,s}=\\{x/r+y/s=1\\}$ such that $\tau$ consists of all the boxes contained
between $L_{r,s}$ and the coordinate axes. If so, we write $\tau=\tau_{r,s}$
and say that $L_{r,s}$ _cuts out_ $\tau_{r,s}.$ Note that the same triangular
partition can be cut out by different lines (see Figure 6).

Figure 6. The partition $(4,2,1)$ is triangular and is cut out by different
lines; for instance $(4,2,1)=\tau_{3.9,5.9}=\tau_{4.6,5.2}=\tau_{3.6,6.6}.$

######  Definition 3.3.

A _Dyck path_ is a pair $\pi=(\lambda,\tau)$, where $\tau$ is a triangular
partition and $\lambda\subset\tau$ is a subpartition. We refer to
$(\lambda,\tau)$ as a _$\tau$ -Dyck path_, and write $\mathrm{D}(\tau)$ for
the set of all such pairs. When $\tau=\tau_{r,s}$ we write
$\mathrm{D}(r,s):=\mathrm{D}(\tau_{r,s})$ and refer to elements of this set as
_$r,s$ -Dyck paths_.

The _area_ of a Dyck path $\pi=(\lambda,\tau)$ is defined as

$\mathtt{area}(\pi):=|\tau|-|\lambda|.$

###  3.1. Rational Dyck Paths and Anderson Fillings

For the remainder of this section we mostly specialize to the case when $r,s$
are relatively prime integers and will resume the general triangular setting
in §4. For the most part, the results in this subsection are well known and
straightforward but we include the proofs for completeness.

So then, let $m,n\in\mathbb{Z}_{>0}$ be relatively prime integers and consider
the rectangular diagram $[m]\times[n]$. We may decompose $[m]\times[n]$ as a
disjoint union of three subsets:

$[m]\times[n]=\tau_{m,n}\sqcup\mathrm{diag}_{m,n}\sqcup\tau_{m,n}^{\vee},$

where $\mathrm{diag}_{m,n}$ is the set of boxes that lie on the diagonal line
$L_{m,n},$ $\tau_{m,n}$ is the corresponding triangular partition, and
$\tau_{m,n}^{\vee}$ consists of those boxes in $[m]\times[n]$ which lie above
the diagonal.

######  Remark 3.4.

Since $m,n$ are relatively prime, there are no integer solutions to $nx+my=nm$
with $(x,y)\in[m]\times[n]$. This implies that the strict inequalities
$nx+my<nm$ and $nx^{\prime}+my^{\prime}>nm$ are equivalent to the
corresponding weak inequalities for boxes in $[m]\times[n]$.

Note that the cardinality of $[m]\times[n]$ is $mn$, the cardinality of
$\mathrm{diag}_{m,n}$ is $n+m-1$, and $|\tau_{m,n}|=|\tau_{m,n}^{\vee}|$ by
symmetry, which forces

$|\tau_{m,n}|=\frac{(m-1)(n-1)}{2}.$

The following lemma is elementary but quite useful.

######  Lemma 3.5.

Let $m,n\geq 1$ be coprime integers. Suppose we are given integers
$x,x^{\prime},y,y^{\prime}\in\mathbb{Z}$ such that
$nx+my=nx^{\prime}+my^{\prime}$. Then $y,y^{\prime}\in[n]$ implies
$(x,y)=(x^{\prime},y^{\prime})$ (by symmetry $x,x^{\prime}\in[m]$ implies
$(x,y)=(x^{\prime},y^{\prime})$ as well).

###### Proof.

Suppose $nx+my=nx^{\prime}+my^{\prime}$ for some boxes $(x,y)$ and
$(x^{\prime},y^{\prime})$ in $Z\times[n]$. This implies that $|y-y^{\prime}|$
is divisible by $n$, and $1\leq y,y^{\prime}$ forces $|y-y^{\prime}|<n$, which
then forces $y-y^{\prime}=0$. This then implies $nx=nx^{\prime}$, hence
$x=x^{\prime}$. ∎

######  Definition 3.6 ([2]).

For coprime integers $m,n\geq 1$, the _$(m,n)$ -Anderson filling_ is the
function $\gamma_{m,n}:\mathbb{Z}^{2}\to\mathbb{Z}$ defined by

$\gamma_{m,n}(x,y)=mn-nx-my.$

Given a box $\Box$ with NE corner $(x,y)$, we refer to
$\gamma_{m,n}(\Box):=\gamma_{m,n}(x,y)$ as the _$(m,n)$ -Anderson label of
$\Box$_. When $m,n$ are understood, we will write $\gamma=\gamma_{m,n}$ and
use the terms Anderson filling and label unambigiously.

######  Definition 3.7.

Let $\Gamma_{m,n}$ denote the additive submonoid of $\mathbb{Z}$ generated by
$m$ and $n$. That is, $\Gamma_{m,n}=\\{k\in\mathbb{Z}\>|\>k=nx+my\text{ for
some $x,y\in\mathbb{Z}_{\geq 0}$}\\}$. $\Gamma_{m,n}$ is also called the
_semigroup generated by $m$ and $n.$_

The main relevant properties of the Anderson filling are stated in the lemma
below. Before we state it, we make some very simple observations. First, the
fact that $\gamma(x-m,y+n)=\gamma(x,y)$ means that $\gamma$ descends to a
well-defined function on the quotient
$\gamma^{\prime}\colon\mathbb{Z}\times\mathbb{Z}/\langle(-m,n)\rangle\to\mathbb{Z}$.
Secondly, modulo the equivalence relation $(x,y)\sim(x-m,y+n)$, every cell in
$\mathbb{Z}\times\mathbb{Z}$ is equivalent to a unique cell in
$\mathbb{Z}\times[n]$. That is to say, the obvious composition

$\mathbb{Z}\times[n]\hookrightarrow\mathbb{Z}\times\mathbb{Z}\twoheadrightarrow(\mathbb{Z}\times\mathbb{Z})/\langle(-m,n)\rangle$
(3.1)

is a bijection.

######  Lemma 3.8.

For coprime integers $m,n\geq 1$, the Anderson filling $\gamma=\gamma_{m,n}$
satisfies:

  1. (1)

$\gamma(\Box)>0$ if and only if the box $\Box$ is strictly below the line
$nx+my=nm$.

  2. (2)

$\gamma$ induces a bijection
$\gamma^{\prime}\colon\mathbb{Z}\times\mathbb{Z}/\langle(-m,n)\rangle\to\mathbb{Z}$.

  3. (3)

$\gamma$ restricts to a bijection $\mathbb{Z}\times[n]\to\mathbb{Z}$.

  4. (4)

we have $(x,y)\in\mathbb{Z}_{\leq 0}\times[n]$ if and only if
$\gamma(x,y)\in\Gamma_{m,n}.$

###### Proof.

Statement (1) is clear from the definitions.

For statement (2), observe that $\gamma^{\prime}$ is surjective since $\gamma$
is. To prove that $\gamma^{\prime}$ is injective, note that modulo $(-m,n)$,
every $(x,y)\in\mathbb{Z}^{2}$ is equivalent to a cell in
$\mathbb{Z}\times[n]$, and Lemma 3.5 tells us that $\gamma$ restricted to
$\mathbb{Z}\times[n]$ is injective. This proves statement (2), while
precomposing with the bijection (3.1) proves statement (3).

For statement (4), observe that

  * •

$(\gamma^{\prime})^{-1}(0)=(-1,n)$,

  * •

$(\gamma^{\prime})^{-1}(k+n)=(\gamma^{\prime})^{-1}(k)+(-1,0)$,

  * •

$(\gamma^{\prime})^{-1}(k+m)=(\gamma^{\prime})^{-1}(k)+(0,-1)$ (modulo
$(k,0)\sim(k-m,n)$).

These observations make it clear that $(\gamma^{\prime})^{-1}(\Gamma_{m,n})$
equals $\mathbb{Z}_{\geq 0}\times[n]$, which completes the proof of statement
(4). ∎

######  Definition 3.9.

When $(r,s)=(m,n)\in\mathbb{Z}^{2}_{>0}$ are integers, it is natural to
identify an $m,n$-Dyck path $\pi=(\lambda,\tau_{m,n})$ with the lattice path
$\tilde{\pi}$ from $(m,0)$ to $(0,n)$ following the boundary of $\lambda$
(hence the name ‘Dyck path’). To avoid confusion, we will call $\tilde{\pi}$
the _associated lattice path_ of $\pi$.

######  Remark 3.10.

Note that the associated lattice path depends on the choice of the integers
$(m,n).$ For instance, one has $\tau_{n,n+1}=\tau_{n+1,n}$ and
$\mathrm{D}(n,n+1)=\mathrm{D}(n+1,n),$ but the associated lattice paths are
slightly different.

###  3.2. The $\mathtt{dinv}$ statistic

The _dinv_ statistic was introduced by Mark Haiman in the early $2000$’s [35,
36], but didn’t appear in print until $2009$, at least not in full generality.

######  Definition 3.11 ([45]).

Given any partition $\lambda$, define $\mathtt{dinv}_{p}(\lambda)$ to be the
cardinality of the following set:

$\mathtt{dinv}_{p}(\lambda):=\Big{|}\left\\{\Box\in\lambda:\frac{\mathrm{leg}(\Box)}{\mathrm{arm}(\Box)+1}<p\leq\frac{\mathrm{leg}(\Box)+1}{\mathrm{arm}(\Box)}\right\\}\Big{|},$

where $\mathrm{leg}(\Box)$ and $\mathrm{arm}(\Box)$ denote the numbers of
boxes in $\lambda$ that lie strictly above and in the same column and strictly
right and in the same row as $\Box$, respectively (see Figure 7).

$\overbrace{\hskip 113.81102pt}$leg$\underbrace{\hskip 170.71652pt}$arm$c$
Figure 7. The arm and leg of a box $c$ in $\mathbb{Z}^{2}$ denoted in red and
blue, respectively. The orange and purple diagonal lines denote the lines with
slopes $\frac{\mathrm{leg}(c)}{\mathrm{arm}(c)+1}$ and
$\frac{\mathrm{leg}(c)+1}{\mathrm{arm}(c)}$ used to determine the contribution
of $c$ towards $\mathtt{dinv}$.

######  Definition 3.12.

Let $(r,s)\in\mathbb{R}_{>0}^{2}$ be a pair of positive real numbers. Then for
an $r,s$-Dyck path $\pi=(\lambda,\tau_{r,s})$, set

$\mathtt{dinv}_{s/r}(\pi):=\mathtt{dinv}_{s/r}(\lambda).$

Note that the statistic $\mathtt{dinv}_{s/r}$ depends not only on the
partition $\tau_{r,s}$ and the set $\mathrm{D}(r,s),$ but also on the slope
$s/r.$ For instance, one has $\tau_{n,n+1}=\tau_{n+1,n}$ and
$\mathrm{D}(n,n+1)=\mathrm{D}(n+1,n),$ but $\mathtt{dinv}_{n/(n+1)}$ and
$\mathtt{dinv}_{(n+1)/n}$ are two different statistics. However, when the
slope $s/r$ (or $n/m$) is understood from the context, we will skip the
subscript and use $\mathtt{dinv}$ instead of $\mathtt{dinv}_{s/r}$ (or
$\mathtt{dinv}_{n/m}$).

###  3.3. Invariant Subsets

Let $(m,n)\in\mathbb{Z}^{2}_{>0}$ be a pair of relatively prime integers. In
this case, it will be more convenient for us to use an alternative definition
of the $\mathtt{dinv}=\mathtt{dinv}_{n/m}$ statistic originally inspired by
the geometry of compactified Jacobians and developed in [24]. See also [47]
and [26] for a simpler proof of the equivalence of the two definitions of
$\mathtt{dinv}.$ To do so, we need some additional combinatorial tools. We
follow the notations from [27].

######  Remark 3.13.

Note that in [27], the more general case where $m$ and $n$ are not necessarily
relatively prime is considered. However, we only need the relatively prime
case in this paper, and therefore we simplify the notations accordingly.

######  Definition 3.14.

For each subset $\Delta\in\mathbb{Z}_{\geq 0}$, let
$\Delta+n:=\\{k\in\mathbb{Z}\>|\>k-n\in\Delta\\}$ and denote by
$\overline{\Delta}=\mathbb{Z}_{\geq 0}\setminus\Delta$, the set complement of
$\Delta$. We say $\Delta$ is an _$(m,n)$ -invariant set_ if
$\Delta+n\subset\Delta$, $\Delta+m\subset\Delta.$ We will denote the set of
all $(m,n)$-invariant subsets by $\mathrm{Inv}_{m,n}$ and call the elements of
$\overline{\Delta}$ the _gaps_ in $\Delta$. In the case that $0\in\Delta$, we
say $\Delta$ is _0-normalized_ and write $\mathrm{Inv}_{m,n}^{0}$ for the
collection of all such sets.

The sets of _$n$ -generators, $m$-generators_ and _co-generators_ of any
$\Delta\in\mathrm{Inv}_{m,n}$ are defined as follows:

$\displaystyle\mathtt{ngen}(\Delta)$
$\displaystyle:=\Delta\setminus(\Delta+n)=\\{k\in\Delta\>|\>k-n\not\in\Delta\\},$
$\displaystyle\mathtt{mgen}(\Delta)$
$\displaystyle:=\Delta\setminus(\Delta+m)=\\{k\in\Delta\>|\>k-m\not\in\Delta\\},$
$\displaystyle\mathtt{cogen}(\Delta)$
$\displaystyle:=\\{k\in\mathbb{Z}\setminus\Delta\>|\>k+n\in\Delta,k+m\in\Delta\\}$
$\displaystyle\mathtt{cogen}_{\geq 0}(\Delta)$
$\displaystyle:=\mathtt{cogen}(\Delta)\cap\mathbb{Z}_{\geq 0}.$

In particular, there is precisely one generator per equivalence class (mod $n$
or $m$), so that $|\mathtt{ngen}|=n$ and $|\mathtt{mgen}|=m$.

######  Definition 3.15.

Let $[k,k+L]=\\{k,k+1,\dots,k+L\\}\in\mathbb{Z}$. For any
$\Delta\in\mathrm{Inv}_{m,n}$, set:

$\displaystyle\mathtt{area}(\Delta)$ $\displaystyle:=|\overline{\Delta}|,$
$\displaystyle\mathtt{codinv}(\Delta)$
$\displaystyle:=\sum_{k\in\mathtt{ngen}(\Delta)}|[k,k+m-1]\cap\overline{\Delta}|=\sum_{k\in\mathtt{ngen}(\Delta)}|\\{k\leq
x<k+m\>|\>x\not\in\Delta\\}|,$

so that area counts the number of gaps of $\Delta$ and codinv the number of
gaps in length $m$ intervals whose starting value is an $n$-generator. We also
define:

$\displaystyle\delta(m,n)$ $\displaystyle:=\frac{(m-1)(n-1)}{2},$
$\displaystyle\mathtt{coarea}(\Delta)$
$\displaystyle:=\delta(m,n)-\mathtt{area}(\Delta),$
$\displaystyle\mathtt{dinv}(\Delta)$
$\displaystyle:=\delta(m,n)-\mathtt{codinv}(\Delta).$

Note that the definition of $\delta(m,n)$ above agrees with Definition 2.14.
An $\mathtt{area}$ and $\mathtt{dinv}$ preserving bijection between
$\mathrm{D}(m,n)$ and $\mathrm{Inv}_{m,n}^{0}$ was introduced in [24]:

######  Definition 3.16 ([24]).

Let $\pi=(\lambda,\tau_{m,n})$ be a $m,n$-Dyck path, then the set of _gaps_ of
$\pi$ is given by

$\mathrm{Gaps}(\pi):=\\{\gamma_{m,n}(\Box)|\Box\in\tau_{m,n}\setminus\lambda\\}.$

The bijection $\mathcal{A}_{m,n}:\mathrm{D}(m,n)\to\mathrm{Inv}_{m,n}^{0}$ is
then defined by

$\mathcal{A}_{m,n}(\pi):=\mathbb{Z}_{\geq 0}\setminus\mathrm{Gaps}(\pi).$

Notice that $\mathcal{A}_{m,n}$ depends on the choice of the relatively prime
integers $(m,n),$ not only on the partition $\tau_{m,n}$ and the set
$\mathrm{D}(m,n).$ Nonetheless, whenever $(m,n)$ are clear from the context,
we will skip the subscript and write $\mathcal{A}=\mathcal{A}_{m,n}.$

######  Definition 3.17.

For a partition $\lambda$, a box $\Box\not\in\lambda$ is called _addable_ if
$\lambda\cup\Box$ is a partition diagram. Equivalently, $\Box$ is addable if
both its south and west boundaries intersect the boundary of $\lambda.$ Thus,
a box is addable for a Dyck path $\pi=(\lambda,\tau)$ if it is addable for
$\lambda$.

Let $\tilde{\pi}$ be the associated lattice path of
$\pi=(\lambda,\tau_{m,n})\in\mathrm{D}(m,n).$ Then under this bijection, we
have the following correspondences:

$\displaystyle\\{\gamma(x,y)\;|\;\text{ the top boundary of the box
}(x,y)\text{ is a horizontal step of
}\tilde{\pi}\\}=\mathtt{mgen}(\mathcal{A}(\pi)),$
$\displaystyle\\{\gamma(x,y)\;|\;\text{ the right boundary of the box
}(x,y)\text{ is the a vertical step of
}\tilde{\pi}\\}=\mathtt{ngen}(\mathcal{A}(\pi)),$
$\displaystyle\\{\gamma(x,y)\ \;|\;(x,y)\text{ is an addable box of
}\lambda\\}=\mathtt{cogen}(\mathcal{A}(\pi)).$

See Figure 8 for an example.

12840951-362-2-63-1-5-90-4-8-12 Figure 8. The Dyck path
$\pi=((1),(2,1))\in\mathrm{D}(4,3)$ with the path $\tilde{\pi}$ denoted in
red. The set $\mathrm{Gaps}(\pi)=\\{1,2\\}$ obtained from $(2,1)\setminus(1)$
is highlighted in yellow and coincide in this case with the addable boxes, or
co-generators, of $\pi$. The boxes corresponding to the $4$-generators of
$\mathcal{A}(\pi)=\mathbb{Z}_{\geq 0}\setminus\mathrm{Gaps}(\pi)$ are shaded
green, the boxes corresponding to the $3$-generators are shaded blue. Note
that $5$ and $0$ are both $3$ and $4$-generators.

######  Theorem 3.18 ([24]).

The bijection $\mathcal{A}:\mathrm{D}(m,n)\to\mathrm{Inv}^{0}_{m,n}$ is
$\mathtt{area}$ and $\mathtt{dinv}$ preserving: