%% if you are submitting an initial manuscript then you should have submission as an option here
%% if you are submitting a revised manuscript then you should have revision as an option here
%% otherwise options taken by the article class will be accepted
\documentclass[finalversion]{FPSAC2021}
\articlenumber{54}
%% but DO NOT pass any options (or change anything else anywhere) which alters page size / layout / font size etc

%% note that the class file already loads {amsmath, amsthm, amssymb}

%\usepackage{lipsum}



% % ========================================AMS packages:
% \usepackage{amsmath,amssymb,amsthm,amsfonts}
% \usepackage{mathtools}%Mathematical tools to use with amsmath

% % ========================================standard packages:
% %\usepackage[english]{babel}%Multilingual support for Plain TeX or LaTeX
% \usepackage{comment}%Selectively include/exclude portions of text
% % \usepackage[hypertex]{hyperref}
% \usepackage[bookmarksdepth=4]{hyperref}%Extensive support for hypertext in LaTeX
\usepackage{bbm}%"Blackboard-style" cm fonts
\usepackage{mathrsfs}%Support for using RSFS fonts in maths (\mathcal fonts via \mathscr)

% ========================================graphics packages:
\usepackage{tikz,graphicx,color}
\usepackage{tikz-cd}%Create commutative diagrams with TikZ
\usepackage{tikz-3dplot}%Coordinate transformation styles for 3d plotting in TikZ
% % for more on tikz libs, see 
% % http://tex.stackexchange.com/questions/42611/list-of-available-tikz-libraries-with-a-short-introduction
\usetikzlibrary{calc}%to make complex coordinate caculations.
\usetikzlibrary{arrows}%Provides various new and costumizable arrow tips
\usetikzlibrary{shapes}%Shape library, used to define shapes other than rectangle, circle and co-ordinate
% \usetikzlibrary{snakes}%curved lines, and are used either between nodes or as a border to a shape , or as independent shapes. DOESN'T COMPILE
\usetikzlibrary{patterns}%This package "defines patterns for filling areas".
\usetikzlibrary{positioning}%a way of positioning one node in respect with several others
\usetikzlibrary{arrows.meta}
\usetikzlibrary{decorations.markings}
\usepackage{epstopdf}%Convert EPS to PDF using Ghostscript
\pdfpageattr{/Group <</S /Transparency /I true /CS /DeviceRGB>>}

% ========================================extra packages:
% \usepackage{enumerate}%Enumerate with redefinable labels. Precedes enumitem
% \usepackage{enumitem}%Control layout of itemize, enumerate, description. Supersedes enumerate
\usepackage[all]{xy}%Typesetting Graphs and Diagrams in TeX 
\usepackage{diagbox}%Table heads with diagonal lines
\usepackage{subfig}%Figures broken into subfigures
% \usepackage{chngcntr} %Change the resetting of counters\PG{doesn't compile for me}
\usepackage{arcs}%Draw arcs over and under text. \overarc and \underarc
\usepackage{xcolor}%Driver-independent color extensions for LaTeX and pdfLaTeX
\usepackage{tabu}
\usepackage{booktabs}% for better rules in the table

% ========================================not recommended:
% \usepackage[margin=1in]{geometry}

\usepackage{soul}
\usepackage{accents}
% \usepackage[capitalise]{cleveref}
%\usepackage[T1]{fontenc}

\usepackage{xspace}

\usetikzlibrary{knots}


% Typography
\usepackage{enumitem}
% \usepackage{enumerate}
\usepackage{letltxmacro}
\usepackage{thmtools,etoolbox}

\def\myarabic#1{\normalfont(\roman{#1})}
\newlist{theoremlist}{enumerate}{1}
\setlist[theoremlist]{label=\myarabic{theoremlisti},ref={\myarabic{theoremlisti}},itemindent=0pt,labelindent=0pt,
leftmargin=*,noitemsep}


\makeatletter
\renewcommand{\p@theoremlisti}{\perh@ps{\thetheorem}}
\protected\def\perh@ps#1#2{\textup{#1#2}}
\newcommand{\itemrefperh@ps}[2]{\textup{#2}}
\newcommand{\itemref}[1]{\begingroup\let\perh@ps\itemrefperh@ps\ref{#1}\endgroup}
\makeatother



\usepackage{nameref,hyperref}
\usepackage[capitalize]{cleveref}


%https://tex.stackexchange.com/questions/100510/omit-specific-subsection-from-table-of-contents
\makeatletter
\ifcsname phantomsection\endcsname
    \newcommand*{\qrr@gobblenexttocentry}[5]{}
\else
    \newcommand*{\qrr@gobblenexttocentry}[4]{}
\fi
\newcommand*{\addsubsection}{%
    \addtocontents{toc}{\protect\qrr@gobblenexttocentry}%
    \subsection}
\makeatother




\newtheorem{theorem}{Theorem}[section]
\newtheorem*{statement}{Theorem}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{conjecture}[theorem]{Conjecture}
\theoremstyle{definition}
\newtheorem{remark}[theorem]{Remark}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{question}[theorem]{Question}
\theoremstyle{definition}
\newtheorem{problem}[theorem]{Problem}
\theoremstyle{definition}
\newtheorem{example}[theorem]{Example}



\crefname{figure}{Figure}{Figures}

\addtotheorempostheadhook[theorem]{\crefalias{theoremlisti}{theorem}}
\addtotheorempostheadhook[lemma]{\crefalias{theoremlisti}{lemma}}
\addtotheorempostheadhook[proposition]{\crefalias{theoremlisti}{proposition}}
\addtotheorempostheadhook[corollary]{\crefalias{theoremlisti}{corollary}}


% ================mathcal symbols
\def\Acal{\mathcal{A}}\def\Bcal{\mathcal{B}}\def\Ccal{\mathcal{C}}\def\Dcal{\mathcal{D}}\def\Ecal{\mathcal{E}}\def\Fcal{\mathcal{F}}\def\Gcal{\mathcal{G}}\def\Hcal{\mathcal{H}}\def\Ical{\mathcal{I}}\def\Jcal{\mathcal{J}}\def\Kcal{\mathcal{K}}\def\Lcal{\mathcal{L}}\def\Mcal{\mathcal{M}}\def\Ncal{\mathcal{N}}\def\Ocal{\mathcal{O}}\def\Pcal{\mathcal{P}}\def\Qcal{\mathcal{Q}}\def\Rcal{\mathcal{R}}\def\Scal{\mathcal{S}}\def\Tcal{\mathcal{T}}\def\Ucal{\mathcal{U}}\def\Vcal{\mathcal{V}}\def\Wcal{\mathcal{W}}\def\Xcal{\mathcal{X}}\def\Ycal{\mathcal{Y}}\def\Zcal{\mathcal{Z}}

% ================mathbf symbols
\def\abf{\mathbf{a}}\def\bbf{\mathbf{b}}\def\cbf{\mathbf{c}}\def\dbf{\mathbf{d}}\def\ebf{\mathbf{e}}\def\fbf{\mathbf{f}}\def\gbf{\mathbf{g}}\def\hbf{\mathbf{h}}\def\ibf{\mathbf{i}}\def\jbf{\mathbf{j}}\def\kbf{\mathbf{k}}\def\lbf{\mathbf{l}}\def\mbf{\mathbf{m}}\def\nbf{\mathbf{n}}\def\obf{\mathbf{o}}\def\pbf{\mathbf{p}}\def\qbf{\mathbf{q}}\def\rbf{\mathbf{r}}\def\sbf{\mathbf{s}}\def\tbf{\mathbf{t}}\def\ubf{\mathbf{u}}\def\vbf{\mathbf{v}}\def\wbf{\mathbf{w}}\def\xbf{\mathbf{x}}\def\ybf{\mathbf{y}}\def\zbf{\mathbf{z}}

% ================mathfrak symbols
\def\afr{{\mathfrak{a}}}
\def\nfr{{\mathfrak{n}}}
\def\Sfr{{ \mathfrak{S}}}
% ...

% ================mathbb symbols
\def\one{{\mathbbm{1}}}
\def\C{\mathbb{C}}
\def\R{\mathbb{R}}
\def\N{\mathbb{N}}
\def\Z{\mathbb{Z}}

% ==============Hat, bar, tilde, par
\def\Ahat{\hat{A}} 
\def\ipar{{(i)}}
\def\kpar{{(k)}}
\newcommand\parr[1]{{({#1})}}
% ...


% ==============Math notation
\def\<{{\langle}}
\def\>{{\rangle}}
\def\e{{\epsilon}}
\def\la{{\lambda}}
% \def\l{{\lambda}}
% \def\m{{\mu}}
% \def\lm{{\la/\m}}
% \def\RP{{\R P}}
% \def\CP{{\C P}}
\def\multiset#1#2{\left(\!\left({#1\atopwithdelims..#2}\right)\!\right)}
\def\toi{{\xhookrightarrow{i}}}

% ==============Operators
\def\Id{\operatorname{Id}}
\def\Vol{\operatorname{Vol}}
\def\det{{ \operatorname{det}}}
\def\tr{{ \operatorname{tr}}}
\def\Ker{{ \operatorname{Ker}}}
\def\Im{{ \operatorname{Im}}}
\def\diag{{ \operatorname{diag}}}
\def\rank{{ \operatorname{rank}}}
\def\rk{{\operatorname{rk}}}
\def\cork{ \operatorname{cork}}
\def\codim{ \operatorname{codim}}
\def\Vert{{ \operatorname{Vert}}}
\def\op{{ \operatorname{op}}}
\def\type{{\operatorname{type}}}
\def\sh{{ \operatorname{sh}}}
\def\Conv{ \operatorname{Conv}}
\def\Span{ \operatorname{Span}}
\def\proj{ \operatorname{proj}}
\def\sgn{ \operatorname{sgn}}
\def\wt{\operatorname{wt}}
\def\inv{\operatorname{inv}}
\def\Int{\operatorname{int}}
\def\Cone{\operatorname{Cone}}

\def\RR{{\mathbb R}}
\def\CC{{\mathbb C}}
\def\RP{{\RR\mathbb P}}
\def\CP{{\CC\mathbb P}}

\newcommand\fixit[1]{\textcolor{green}{TL:#1}}

\newcommand\notn[1]{\textcolor{blue}{#1}}
\newcommand\pc[1]{\Pi_{#1}^{>0}}
\newcommand\pctnn[1]{\Pi_{#1}^{\geq0}}
\newcommand\pcvar[1]{\Pi_{#1}^\circ}
\def\Grlam{\Gr_\lambda(k,n)}
\def\id{{\operatorname{id}}}
% \def\str{\varphi}
\def\spheref{S_f^\epsilon}
%\def\str{\operatorname{str}}
\newcommand{\arxiv}[1]{\href{https://arxiv.org/abs/#1}{\textup{\texttt{arXiv:#1}}}}
\newcommand{\Le}{\textup{\protect\scalebox{-1}[1]{L}}}

% \def\PGQ#1{\textcolor{brown!50!green}{{\bf[PGQ: #1]}}}

% \newcommand{\TODO}[1]{\textcolor{brown}{\bf[TODO: #1]}}
% \newcommand{\TODOTL}[1]{\textcolor{brown!50!green}{\bf[TODO-TL: #1]}}

% \def\todo{\textcolor{brown}{\bf[TODO]}\xspace}
% \def\todoc{\textcolor{brown}{\bf[TODO:cite]}\xspace}
% \def\todoe{\textcolor{brown}{\bf[TODO:example]}\xspace}
% \def\todof{\textcolor{brown}{\bf[TODO:figure]}\xspace}


\newcommand{\smat}[1]{\left[\begin{smallmatrix}
      #1
    \end{smallmatrix}\right]}
\newcommand{\pmat}[1]{\begin{pmatrix}
#1
  \end{pmatrix}}


\def\pj{^{\parr J}}
\def\Uj{U\pj}
\def\Cj{C\pj}
\def\Cuj{C\pj_u}
\def\Phij{\Phi\pj}
\def\gj{g\pj}
% \let\uu\u
% \def\u{h}
\def\hj{h^{\parr J}}
\def\Lie{\operatorname{Lie}}

\def\bs{\backslash}
\let\ii\i
% \def\i{{\mathbf{i}}}
\def\j{{\mathbf{j}}}
\def\t{{\mathbf{t}}}

\def\d#1{\dot{#1}}
\def\ds{\d{s}}
\def\dw{\d{w}}
\def\du{\d{u}}
\def\dv{\d{v}}
\def\da{\d{a}}
\def\db{\d{b}}
\def\dc{\d{c}}
\def\dz{\d{z}}
\def\dr{\d{r}}
\def\dq{\d{q}}
\def\gfr{\mathfrak{g}}
\def\alphacheck{\alpha^\vee}
% \def\X{\accentset{\circ}{X}}
\def\Xcl{X}


% \def\Racc{\accentset{\circ}{R}}
\def\Racc{R^\circ}
\def\Rich_#1^#2{\Racc_{#1,#2}}
\def\Richcl_#1^#2{R_{#1,#2}}
% \def\RRich_#1^#2{\Racc_{#1,#2}^\R}%open Richardson over R
% \def\RRichcl_#1^#2{R_{#1,#2}^\R}%closed Richardson over R

\def\sf{\operatorname{sf}}
\def\Tsf{T_{\sf}}
\def\bt{{\mathbf{t}}}
\def\bw{{\mathbf{w}}}
\def\bv{{\mathbf{v}}}
\def\bu{{\mathbf{u}}}
\def\ba{{\mathbf{a}}}
\def\bb{{\mathbf{b}}}
\def\bc{{\mathbf{c}}}
\def\wnot{{\mathbf{w_0}}}
\def\woj{w_J}
\def\dwoj{\dw_J}


\def\pd#1{_{(#1)}}
\def\pu#1{^{(#1)}}
% \def\p#1{{(#1)}}

\def\bx{{\mathbf{x}}}
\def\by{{\mathbf{y}}}
\def\bi{{\mathbf{i}}}
\def\bj{{\mathbf{j}}}
\def\Hom{\operatorname{Hom}}

\def\xrasim{\xrightarrow{\sim}}

%NOTATION

\def\chr{\operatorname{char}}

\let \oldlabel \label
\let\oldref\ref
\let\oldcref\cref




\def\O{{\Ocal}}
\def\g{{\mathfrak{g}}}
\def\h{{\mathfrak{h}}}
\def\SB{{\operatorname{SB}}}
\def\B{{\mathbf{B}}}
\def\w{{\mathbf{w}}}
% \def\proj{{\operatorname{proj}}}
\def\mod{{\operatorname{mod}}}
\def\Spec{\operatorname{Spec}}
\def\End{\operatorname{End}}
\def\Hom{\operatorname{Hom}}
\def\Ext{\operatorname{Ext}}
\def\IHom{\mathcal{H}om}
\def\Semis{\operatorname{Semis}}
\def\Perv{\operatorname{Perv}}
\def\Tilt{\operatorname{Tilt}}
% \def\SBim{\operatorname{SBim}}
\def\SBim{{\operatorname{\mathbb{S}Bim}}}
% \def\Smod{\operatorname{Smod}}
\def\SMod{\operatorname{\mathbb{S}Mod}}
\def\IC{\mathcal{IC}}
\def\HH{H\!H}
\def\HHH{H\!H\!H}
% \def\HHH{H^\bul\!H\!\!H}
\def\HHHC{\HHH_\C}
\def\H{{\mathbb{H}}}
\def\V{{\mathbb{V}}}
\def\pt{{\rm pt}}
% \def\P{{\mathcal{P}}}
\def\mix{{\operatorname{mix}}}
% \def\k{{\mathbf{k}}}
\def\Tate{{\langle 1 \rangle}}
\def\vDelta{{\check{\Delta}}}
\def\vIC{{\check{\IC}}}
\def\vT{{\check{\Tcal}}}
\def\vnabla{{\check{\nabla}}}
\def\vB{{\check{B}}}
\def\vG{{\check{G}}}
\def\F{{\mathbb{F}}}
\def\Q{{\mathbb{Q}}}
\def\bF{\overline{\F}}
\def\bQ{\overline{\Q}}
\def\Mod{{\operatorname{Mod}}}
\def\Fr{{\rm Fr}}
\def\RHom{{\operatorname{RHom}}}
\def\RHomcal{{\mathcal{RH}om}}
\def\RHomunder{{\underline{\RHom}}}
\def\ext{{\operatorname{ext}}}
\def\Fl{{\operatorname{Fl}}}
\def\tDcal{{\tilde \Dcal}}
\def\tFcal{{\tilde \Fcal}}
\def\tGcal{{\tilde \Gcal}}
\def\tP{{\tilde P}}
\def\gr{{\rm gr}}
\def\dgMod{{\rm dgMod}}
\def\For{{\operatorname{For}}}
\def\tDelta{{\tilde \Delta}}
% \def\TL#1{\textcolor{red}{{#1}}}




\def\Mod{{\operatorname{Mod}}}
\def\Fr{{\rm Fr}}
\def\RHom{{\operatorname{RHom}}}
\def\RHomcal{{\mathcal{RH}om}}
\def\RHomunder{{\underline{\RHom}}}
\def\ext{{\operatorname{ext}}}
\def\Fl{{\operatorname{Fl}}}
\def\tDcal{{\tilde \Dcal}}
\def\tFcal{{\tilde \Fcal}}
\def\tGcal{{\tilde \Gcal}}
\def\tP{{\tilde P}}
\def\gr{{\rm gr}}
\def\dgMod{{\rm dgMod}}
\def\For{{\operatorname{For}}}
\def\tDelta{{\tilde \Delta}}
% \def\TL#1{\textcolor{red}{[TL: {#1}]}}
% \def\PG#1{\textcolor{blue}{[PG: {#1}]}}
% \def\PGQ#1{\textcolor{green!70!black}{[PGQ: {#1}]}}%question
% \def\PGQX#1{\textcolor{blue}{[{\bf PGQ}: {#1}]}}%question



\def\Pio{\Pi^\circ}
% \def\qq#1{q^{#1/2}}
% \def\tt#1{t^{#1/2}}
% \def\qt#1{(qt)^{#1/2}}
% \def\qt#1#2{\sqrt{q^{#1}t^{#2}}}
\def\qt#1#2{q^{\frac{#1}2}t^{\frac{#2}2}}



\def\vw{_v^w}
\def\MHT{\operatorname{MHT}\vw}
\def\MHTt{\tilde{\operatorname{MHT}}\vw}
\def\ncyc{\operatorname{ncyc}}
\def\VRM{\operatorname{VRM}\vw}
\def\VRMt{\tilde{\operatorname{VRM}}\vw}
\def\SHF{\operatorname{SHF}\vw}
\def\SHFt{\tilde{\operatorname{SHF}}\vw}
\def\SOE{\operatorname{SOE}\vw}
\def\SOEt{\tilde{\operatorname{SOE}}\vw}

\def\SP{{\mathcal{P}}\vw}
\def\SPt{{\tilde{\mathcal{P}}\vw}}

% \def\qtn{(q^{1/2}-t^{-1/2})^{n-1}}
\def\HOMFLY{\operatorname{HOMFLY}}

% \def\Ro{\accentset{\circ}{R}}
\def\Ro{R}

\def\Dscr{{\mathscr{D}}}
\def\DA{{\Dscr_1}}
\def\DB{{\Dscr_2}}
\def\DC{{\Dscr_3}}

\def\DBFL{D^b_{B}(\Fl)}
\def\PBFL{\Perv_{B}(\Fl)}
\def\supp{\operatorname{supp}}
\def\cosupp{\operatorname{cosupp}}
% \def\bt{\bullet}
\def\VD{\mathbb{D}}
\def\Fb{{\overline{\F}}}
\def\Qb{{\overline{\Q}}}
\def\SS{\mathscr{S\!S}}

\def\BS{\operatorname{BS}}

\def\Parity{\operatorname{Parity}}

\def\Flaff{\widetilde{\Fl}}
\def\Saff{\widetilde{S}_n}
\def\Saffk{\widetilde{S}_{k,n}}
\def\Saffz{\widetilde{S}_{0,n}}

% \def\Boundkn{\Saff(-k,n-k)}
\def\BND{\operatorname{Bound}}
% \def\BND{\mathcal{B}}
\def\Boundkn{\BND(k,n)}
\def\Boundln{\BND(\l,n)}

\numberwithin{equation}{section}
% \numberwithin{equation}{subsection}

\def\pito{\overset \pi \to}

\def\FLY{HOMFLY\xspace}

% \def\Prob{\bm{P}}
\def\Prob{\operatorname{Prob}}
\def\Zpart{Z}
\def\M{M}
\def\Space{\Xcal}
\def\Closure{\overline{\Space}}
\def\OG{\operatorname{OG}}
\def\OGtnn{\OG_{\ge0}}
\def\OGtp{\OG_{>0}}
\def\doublemap{\phi}
\newcommand\double[1]{\widetilde{#1}}
\newcommand\Matsymx[1]{\Mat^{\operatorname{sym}}_{#1}(\R,1)}
\def\Matsym{\Matsymx{n}}

\def\insideprod{q}

% \def\Matsym{\Mat^{{\operatorname{sym}}}_n(\R)}
% \newcommand\Matsymx[1]{\Mat^{\operatorname{sym}}_{#1}(\R)}
% \def\doublemap{\operatorname{X}}

% \def\P{P}

\def\avg{\operatorname{avg}}

\def\Flafftnn{(\Flaff_n)_{\geq0}}
\def\Richafftp_#1^#2{{\Rcal_{#1,#2}^{>0}}}

\def\Richaff_#1^#2{\accentset{\circ}{\mathcal{R}}_{#1,#2}}



\def\mdd{\;\middle|\;}




% \def\a{a}
% \def\b{b}
% \def\Cat{\operatorname{Cat}}
\def\Cat{C}
\def\Cab{\Cat_{a,b}}
\def\Cn{\Cat_n}
\def\Ca{\Cat_n}
% \def\Ca{\Cat_a}
% \def\Ca{\Cat_{a,a+1}}
\newcommand{\qbinom}{\genfrac{[}{]}{0pt}{}}
\def\qint#1{[#1]_q}
\def\qbin#1#2{\qbinom{#1}{#2}_q}
\def\area{\operatorname{area}}
\def\dinv{\operatorname{dinv}}
\def\Dyck{\operatorname{Dyck}}
% \def\Dyckab{\Dyck_{a,b}}
% \def\Dyckab{\Dyck(a,b)}
\def\Dyckxx(#1,#2){\Dyck_{#1,#2}}
\def\Dyckab{\Dyckxx(a,b)}


\def\Poinc{\mathcal{P}}

\def\Pot{\tilde\Poinc}

\def\df{d_f}
\def\Matokn{\Mat^\circ_{k,n}}
\def\Piokn{\Pio_{k,n}}



\def\Poincf{\Poinc(\Pio_f;}
\def\Poinckn{\Poinc(\Pio_{k,n};}
\def\Poincvw{\Poinc(\Rich_v^w;}
\def\Potf{\Poinc(\Pio_f/T;}
\def\Potkn{\Poinc(\Pio_{k,n}/T;}
\def\Potvw{\Poinc(\Rich_v^w/T;}
% \def\Poincf{\Poinc_f(}

% \def\Fvw{F(\beta_{v,w})}
% \def\Fvw{F_{v,w}}
\def\Fvw{\FR_{v,w}}



\def\dkn{d_{k,n}}

% \def\Tn{T_n}
\def\Tn{T}
\def\Skn{S_{k,n}}
\def\ncyc{\operatorname{ncyc}}
\def\Sknc{S^{\ncyc=1}_{k,n}}

\def\qtn{\left(q^{\frac12}+t^{\frac12}\right)^{n-1}}
% \def\qtn{(q^{\frac12}+t^{\frac12})^{n-1}}


\def\betah{\hat\beta}

\def\fkn{{f_{k,n}}}
% \def\Pit{\tilde \Pi^\circ}
\def\Pit_#1{\Pi^\circ_{#1}/T}
\def\rowspan{\operatorname{RowSpan}}

\def\Braid{\mathcal{B}}
\def\Xo{X^\circ}
\def\k{\mathbbm{k}}
\def\Flk{\Fl_n(\k)}
\def\uk{\underline{\k}}
\def\ukw{\uk_w}
% \def\Fb{\overline{\F}}
\def\Ext{\operatorname{Ext}}

\def\mix{{\operatorname{mix}}}
\def\DEM{\Delta^\mix}
\def\DM{D_B^\mix(\Fl)}
% \def\IC{\mathcal{IC}_w}
\def\BS{\operatorname{BS}}
\def\bi{{\mathbf{i}}}
% \def\HH{H\!\!H}
% \def\HHH{H\!\!H\!\!H}

% \def\HH{\operatorname{HH}^\bullet}
% \def\HH{\operatorname{HH}}
% \def\HHH{\operatorname{HHH}}
% \def\HHH{\operatorname{HHH}^{\bullet\bullet}}
\def\KR{{\operatorname{KR}}}
% \def\PK{\tilde{\mathcal{P}}^{\KR}}
% \def\PK{\mathcal{P}^{\KR}}
\def\FR{F^\bullet}


\def\top{{\operatorname{top}}}

\def\PKR{\Pcal_\KR}
\def\PKRtop{\Pcal_\KR^\top}
\def\PK{\PKRtop}



% \let\pag\paragraph
\def\pag#1{


{\bf #1.}\ }

\def\SL{\operatorname{SL}}


\def\Ical{\mathcal{I}}
\def\tauk{\tau_k}
\def\dtauk{\dtau_k}
\def\leqop{\leq^\op}
\def\Povar_#1{\accentset{\circ}{\Pi}_{#1}}
\def\Povarcl_#1{\Pi_{#1}}
\def\RPovar_#1{\accentset{\circ}{\Pi}^\R_{#1}}
\def\RPovarcl_#1{\Pi^\R_{#1}}
\def\Povtp_#1{\Pi_{#1}^{>0}}
\def\Povtnn_#1{\Pi_{#1}^{\geq0}}

\def\Boundkn{\operatorname{Bound}(k,n)}


\def\FlCn{\Fl_n(\C)}


% ==============Total positivity
\def\GL{\operatorname{GL}}
\def\Fl{\operatorname{Fl}}
\def\Mat{\operatorname{Mat}}
\def\Gr{\operatorname{Gr}}
\def\Grtnn{\Gr_{\ge 0}}
\def\Grtp{\Gr_{>0}}
\def\alt{\operatorname{alt}}
\def\Mattp{\operatorname{Mat}_{>0}}
\def\Mattnn{\operatorname{Mat}_{\ge0}}

\def\Cast{\C^\ast}

\def\KLR_#1^#2{R_{#1,#2}(q)}

\def\Sym{\operatorname{Sym}} 



\def\sclbx{1}


\def\op{1}

\def\gridop{0.4}
% \def\elbowfill{black!30}
\def\elbowfill{white}



\def\drawelbow#1#2{
\def\a{#1}
\def\b{#2}
%   \draw[line width=1pt,dotted,opacity=\gridop,fill=\elbowfill] (\a,\b) rectangle (1+\a,1+\b);
  \draw[line width=2pt,opacity=\op] (\a.5,\b) to[out=90,in=0] (\a,\b.5);
  \draw[line width=2pt,opacity=\op] (\a.5,1+\b) to[out=-90,in=180] (1+\a,\b.5);
}

\def\drawelbowfill#1#2#3{
\def\a{#1}
\def\b{#2}
%   \draw[line width=1pt,dotted,opacity=\gridop,fill=#3] (\a,\b) rectangle (1+\a,1+\b);
  \fill[opacity=\gridop,fill=#3] (\a,\b) rectangle (1+\a,1+\b);
  \draw[line width=2pt,opacity=\op] (\a.5,\b) to[out=90,in=0] (\a,\b.5);
  \draw[line width=2pt,opacity=\op] (\a.5,1+\b) to[out=-90,in=180] (1+\a,\b.5);
}

\def\drawcrossing#1#2{
\def\a{#1}
\def\b{#2}
%   \draw[line width=1pt,dotted,opacity=\gridop] (\a,\b) rectangle (1+\a,1+\b);
  \draw[line width=2pt,opacity=\op] (\a.5,\b) -- (\a.5,1+\b);
  \draw[line width=2pt,opacity=\op] (\a,\b.5) -- (1+\a,\b.5);
}



\def\godiagx#1#2#3{
\foreach \x/\y in {#1}{
\drawelbow{\x}{\y}
}
\foreach \x/\y in {#2}{
\drawcrossing{\x}{\y}
}
\foreach \x/\y in {#3}{
\drawcrossing{\x}{\y}
}
}



\def\drawtree#1{
\foreach \x/\y/\z/\bnd in {#1}{
  \node[black,fill=black,circle,scale=0.3] (A\x) at (\x.5,\z.7) {};
  \node[black,fill=black,circle,scale=0.3] (A\y) at (\y.5,\z.7) {};
  \draw[line width=1pt] (A\x) to[bend right=-\bnd] (A\y);
}
}



\def\godiag#1#2#3{
\scalebox{\sclbx}{
\begin{tikzpicture}[xscale=0.5,yscale=0.5]
\godiagx{#1}{#2}{#3}
\end{tikzpicture}
}
}



\def\godiagtree#1#2#3#4{
\scalebox{\sclbx}{
\begin{tikzpicture}[xscale=0.5,yscale=0.5]
\godiagx{#1}{#2}{#3}

\foreach \x/\y/\z/\bnd in {#4}{
  \node[black,fill=black,circle,scale=0.3] (A\x) at (\x.5,\z.4) {};
  \node[black,fill=black,circle,scale=0.3] (A\y) at (\y.5,\z.4) {};
  \draw[line width=1pt] (A\x) to[bend right=-\bnd] (A\y);
}
\end{tikzpicture}
}
}






% \def\delkn{\delta}
\def\delkn{\chi_{k,n}}
% \def\delkn{d_{k,n}}
\def\dkn{d_{k,n}}
\def\df{d_f}
\def\dvw{d_{v,w}}

\def\KSBim{\Kb(\SBim)}
\def\HOMP{P}
\def\unkn{%
\begin{tikzpicture}[baseline=(ZUZU.base),scale=0.2]\coordinate(ZUZU) at (0,-0.6);
\draw[line width=0.7pt] (0,0) circle (1cm);
  \end{tikzpicture}
}


% \def\unlk{\unkn\,\unkn}
\def\unlk{
\scalebox{0.2}{%
\begin{tikzpicture}[baseline=(ZUZU.base)]\coordinate(ZUZU) at (0,-0.6);
\begin{knot}[
    % draft mode=crossings ,
    clip width=6,
    % flip crossing={1},
    ]
    \strand [line width=3.5pt] (0,0) circle (1.0cm);
    \strand [line width=3.5pt] (1,0) circle (1.0cm);
\end{knot}
\end{tikzpicture} 
}
}

\def\hopflink{
\scalebox{0.2}{%
\begin{tikzpicture}[baseline=(ZUZU.base)]\coordinate(ZUZU) at (0,-0.6);
\begin{knot}[
    %draft mode=crossings ,
    clip width=6,
    flip crossing={1},
    ]
    \strand [line width=3.5pt] (0,0) circle (1.0cm);
    \strand [line width=3.5pt] (1,0) circle (1.0cm);
\end{knot}
\end{tikzpicture} 
}
}

%https://tex.stackexchange.com/questions/506666/change-path-color-at-specified-points-to-tricolor-trefoil-knot
\def\trefoil{\!
\scalebox{0.22}{%
\begin{tikzpicture}[baseline=(ZUZU.base)]\coordinate(ZUZU) at (0.5,-0.6);
\begin{knot}[
    % draft mode=crossings ,
    clip width=6,
    consider self intersections
    ]
\strand[line width=3.5pt] (90:1)
            \foreach \x in {1,2,3} {
                to [bend left=117,looseness=1.9] ({90+120*\x}:1)
            }
        ;

        \flipcrossings{1,3}
\end{knot}
\end{tikzpicture} 
}\!
}

\def\top{{\operatorname{top}}}
\def\Ptop_#1{P^\top_{#1}(q)}
\def\Ptopnoq_#1{P^\top_{#1}}

% \def\Symh{\Sym(\h^*)}
\def\Symh{\C[\h^*]}

% \def\O{\mathscr{O}}
\def\O{\Ocal}
\def\g{\mathfrak{g}}

% \def\mda(#1){\deg^+_a(#1)}
\def\mda(#1){\deg^\top_a(#1)}
\def\Pvw{\HOMP_{v,w}}

\def\Ptopvw{\HOMP^\top_{v,w}(q)}

\def\us{{\underline{s}}}
\def\uu{{\underline{u}}}
\def\uv{{\underline{v}}}
\def\uw{{\underline{w}}}
\def\BS_#1{B_{#1}}
% \def\BS_#1{\operatorname{BS}(#1)}

\def\bul{\bullet}
\def\und#1{\underline{#1}}
\def\BraidW{\Braid_W}
\def\BraidSn{\Braid_{S_n}}


\def\HX#1#2{H^{#1,(#2,#2)}}
\def\HTX#1#2{H_T^{#1,(#2,#2)}}
% \def\HHX#1#2{\HHH^{0,#1,(#2)}}
% \def\HHB#1#2{\HHH^{0,#1,(#2)}}
% \def\HHX#1#2_#3{\HHB{#1}{#2}(\FR_{#3})}

\def\HHB#1#2(#3){H^{#1,(#2)}(\HH^0(#3))}
\def\HHBC#1#2(#3){H^{#1,(#2)}(\HHC^0(#3))}
\def\HHX#1#2_#3{H^{#1,(#2)}(\HH^0(\FR_{#3}))}
\def\HHXC#1#2_#3{H^{#1,(#2)}(\HHC^0(\FR_{#3}))}
\def\HHXk#1#2_#3{H^{#1,(#2)}(\HHk^0(\FR_{#3}))}
% % \def\HHXo#1#2{\overline{\HHH}^{#1,(#2)}}
% \def\Fo{\overline{F}}
% \def\HHX#1#2_#3{\HHH^{#1,(#2)}(F_{#3})}
% \def\HHXo#1#2_#3{\HHH^{#1,(#2)}(\Fo_{#3})}
% \def\HHCX#1#2_#3{\HHH_\C^{#1,(#2)}(F_{#3})}
% \def\HHCXo#1#2_#3{\HHH_\C^{#1,(#2)}(\Fo_{#3})}
% \def\q{\mathbf{q}}
% \def\t{\mathbf{t}}
% \def\Cat{\operatorname{Cat}}
% \def\SMod{\operatorname{SMod}}
% % \def\lvw{\ell(w)-\ell(v)}
% \def\lvw{\ell_{v,w}}



\def\ksa[#1]{[#1]}%cohom shift in D1
\def\psa#1{\{#1\}}%polyn shift in D1. So y has deg 2 and y\psa{-1} has deg 0

\def\FR{F^\bul}

\def\Qlb{\bQ_\ell}
\def\etale{\'etale\xspace}
\def\Etale{\'Etale\xspace}
\def\Poincare{Poincar\'e\xspace}
\def\lvw{\ell_{v,w}}
\def\eps{\epsilon}

\def\da{\kappa}%\operatorname{d}^a}
\def\davw{{\da_{v,w}}}




\def\chib{{\chi(\beta)}}
\def\chibvw{{\chi(\bvw)}}
\def\cb{\ncyc(\beta)}
\def\cbvw{\ncyc(\bvw)}

\def\tt{t^{\frac12}}
\def\qq{q^{\frac12}}
\def\tti{t^{-\frac12}}
\def\qqi{q^{-\frac12}}
\def\bvw{\beta_{v,w}}
\def\bhvw{\betah_{v,w}}




\def\k{{\mathbbm{k}}}
\def\ku{{\underline{\k}}}
\def\ko{{\overline{\k}}}

\def\Fqb{\overline{\F}_q}
\def\Qlb{\overline{\Q}_{\ell}}

\def\Dop{{\operatorname{D}}}
\def\Kop{{\operatorname{K}}}
\def\bop{{\operatorname{b}}}
\def\mop{{\operatorname{m}}}
\def\cop{{\operatorname{c}}}
\def\mix{{\operatorname{mix}}}
\def\real{{\bf {\operatorname{real}}}}
\def\perf{{\operatorname{perf}}}

% \def\Dbm{\Dop^\bop_\mop}
% \def\Dbc{\Dop^\bop_\cop}
% \def\Dbb{\Dop^\bop_B}
\def\Dbc{\Dop^\bop_{(B)}}
\def\DbcH{\Dop^\bop_{(H)}}
% \def\Dbcd{\Dop^\bop_{(B^\vee)}}
\def\Db{\Dop^\bop}
\def\Dmix{\Dop^\mix}


\def\Fq{\F_q}
\def\Xq{X}
\def\ptq{\pt}
\def\Xqb{\Xq_{\Fqb}}
% \def\X{\textcolor{red}{\Xq}}

\def\Xqw{\Xw}
% \def\Xw{X^\circ_w}
% \def\Xv{X^\circ_v}
% \def\Xqw{{\accentset{\circ}{X}_{0,w}}}
\def\Xw{{\accentset{\circ}{X}_w}}
\def\Xv{{\accentset{\circ}{X}_v}}
\def\Xu{{\accentset{\circ}{X}_u}}


\def\Xwcl{X_w}
\def\Xvcl{X_v}

\def\Xvm{\accentset{\circ}{X}_v^-}

\def\DB{{\Db_B(\Xq,\k)}}
\def\DBH{{\Db_H(Y,\k)}}
\def\DC{{\Db_B(\Xqb,\k)}}
\def\DCH{{\Db_H(\Yqb,\k)}}


\def\Nabla{\nabla}


% \def\iw{i_w}
% \def\iv{i_v}
% \def\iu{i_u}



\def\proj{\pi}
% \def\proj{p}



\def\Rsemi_#1^#2{\Racc_{#1,\overline{#2}}}
% \def\projvw{\proj_{v,w}}
\def\projvw{\proj}

\def\ibar{\overline{i}}

\def\qmap{r}
\def\kuw{\ku_{\Xw}}
\def\kuv{\ku_{\Xv}}
\def\kuqw{\ku_{\Xqw}}
\def\kuqv{\ku_{\Xqv}}
\def\BA{\phi_v}
\def\amap{a}


\def\PKRC{\Pcal_{\KR;\C}^\top}
\def\PKRCA{\Pcal_{\KR;\C}}
\def\Res{\operatorname{Res}}
\def\End{\operatorname{End}}
\def\Hom{\operatorname{Hom}}
\def\RHom{\operatorname{RHom}}
\def\sRHom{R\mathcal{H}om}
\def\Ext{\operatorname{Ext}}
\def\Spec{\operatorname{Spec}}
\def\ext{\operatorname{ext}}
\def\Free{\operatorname{Free}}
\def\HHC{\HH_\C}
\def\HHk{\HH_\k}





\def\brr#1{{[#1]}}

\def\ExtDC{\Ext}

\def\For{\operatorname{For}}
% \def\DeltaC{\Delta^\C}
\def\DeltaC{\Delta^{(B)}}


\def\Kb{\Kop^\bop}
\def\VERD{\mathbb{D}}
\def\dv{\dot v}
\def\Bkn{\Bcal_{k,n}}
\def\Bknc{\Bcal^{\ncyc=1}_{k,n}}
\def\ft{{\tilde f}}

\def\taukn{\tilde\tau_{k,n}}
\def\wtl{\tilde w}
\def\ut{\tilde u}
\def\vt{\tilde v}
\def\Piogauge{\Pi^{\circ,\Delta=1}}
\def\RowSpan{\operatorname{RowSpan}}

\def\Tp{T'}
\def\PGL{\operatorname{PGL}}
\def\wt{r}


\def\strvw{N^\circ_{v,w}}
\def\strvwi{N^\circ_{v^{-1},w^{-1}}}

\def\Rgauge_#1^#2{R^{\circ,\Delta=1}_{#1,#2}}
\def\Ngauge{N^{\circ,\Delta=1}_{v,w}}



\def\drawbox(#1,#2){
\draw[black!50,dashed] (#1-0.5,#2-0.5) rectangle (#1+0.5,#2+0.5);
}


\def\drawgrid#1#2#3#4{
\foreach\i in {#1,...,#3}{
\foreach \j in {#2,...,#4}{
\drawbox(\i,\j)
}
}
}

\def\crossing{\scalebox{0.4}{\begin{tikzpicture}[baseline=(ZUZU.base)]
\coordinate(ZUZU) at (0,-0.5);\drawgrid{0}{0}{0}{0}\draw[line width=3pt, rounded corners=12] (0.00,-0.50)--(0.00,0.50);\draw[line width=3pt, rounded corners=12] (-0.50,0.00)--(0.50,0.00);
\end{tikzpicture}}\xspace} 
\def\elbow{\scalebox{0.4}{\begin{tikzpicture}[baseline=(ZUZU.base),xscale=-1]
\coordinate(ZUZU) at (0,-0.5);\drawgrid{0}{0}{0}{0}\draw[line width=3pt, rounded corners=12] (0.00,-0.50)--(0.00,0.00)--(0.50,0.00);\draw[line width=3pt, rounded corners=12] (-0.50,0.00)--(0.00,0.00)--(0.00,0.50);
\end{tikzpicture}}\xspace}
\def\fil{D}

\def\maxx{\operatorname{max}}
\def\Go{\operatorname{Deo}}
\def\Gom{\Go^{\maxx}}



\def\elb{\operatorname{elb}}
\def\xing{\operatorname{xing}}
\def\pl{\operatorname{plab}}
\def\betapl{\betah^{\pl}}

% \def\Gof{$\Go_f$}
% \def\Gofkn{$\Go_\fkn$}
% \def\GO{$\Go$}
% \def\Gof{Go$_f$}
% \def\Gofkn{Go$_\fkn$}
% \def\GO{Go}



\def\Hecke{\mathcal{H}}
\def\Htrace{\epsilon}
\def\Tor{\operatorname{Tor}}
\def\wkn{w_{k,n}}
\def\dimT{d}





%% define your title in the usual way
\title{Positroids, knots, and $q,t$-Catalan numbers}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author{Pavel Galashin\thanks{\href{mailto:galashin@math.ucla.edu}{galashin@math.ucla.edu}. P.G.\ was supported by an Alfred P. Sloan Research Fellowship and by the National Science Foundation under Grants No.~DMS-1954121 and No.~DMS-2046915.}\addressmark{1} \and Thomas Lam\thanks{\href{mailto:tfylam@umich.edu}{tfylam@umich.edu}. T.L.\ was supported by a von Neumann Fellowship from the Institute for Advanced Study and by grant DMS-1464693 from the National Science Foundation.}\addressmark{2}}

%% then use \addressmark to match authors to institutions here
\address{\addressmark{1}Department of Mathematics, University of California, Los Angeles \\ 
\addressmark{2}Department of Mathematics, University of Michigan, Ann Arbor}

%% put the date of submission here
\received{\today}

%% leave this blank until submitting a revised version
%\revised{}

%% put your English abstract here, or comment this out if you don't have one yet
%% please don't use custom commands in your abstract / resume, as these will be displayed online
%% likewise for citations -- please don't use \cite, and instead write out your citation as something like (author year)
\abstract{We relate the mixed Hodge structure on the cohomology of open positroid varieties (in particular, their Betti numbers over $\mathbb{C}$ and point counts over $\mathbb{F}_q$) to Khovanov--Rozansky homology of the associated links. We deduce that the mixed Hodge polynomials of top-dimensional open positroid varieties are given by rational $q,t$-Catalan numbers.  Via the curious Lefschetz property, this implies the $q,t$-symmetry and unimodality properties of rational $q,t$-Catalan numbers.  We show that the $q,t$-symmetry phenomenon is a manifestation of Koszul duality for category $\Ocal$, and discuss relations with %extensions of Verma modules, 
equivariant derived categories of flag varieties, and open Richardson varieties.}

%% put your French abstract here, or comment this out if you don't have one
% \resume{\lipsum[2]}

%% put your keywords here, or comment this out if you don't have them yet
\keywords{Positroid varieties, $q,t$-Catalan numbers, HOMFLY polynomial, Khovanov--Rozansky homology, mixed Hodge structure, equivariant cohomology, Koszul duality.}

%% you can include your bibliography however you want, but using an external .bib file is STRONGLY RECOMMENDED and will make the editor's life much easier
%% regardless of how you do it, please use numerical citations, ie. [xx, yy] in the text

%% this sample uses biblatex, which (among other things) takes care of URLs in a more flexible way than bibtex
%% but you can use bibtex if you want
\usepackage[backend=bibtex]{biblatex}
\addbibresource{54Galashin.bib}
%% note the \printbibliography command at the end of the file which goes with these biblatex commands

\def\figref#1(#2){\hyperref[#1]{Figure~\ref*{#1}(#2)}}
% \def\parag#1{\subsubsection*{#1}}
\def\parag#1{\paragraph{#1.}}

% ==============================DOCUMENT BEGINS
\begin{document}

\maketitle
%% note that you DO NOT have to put your abstract here -- it is generated by \maketitle and the \abstract and \resume commands above

\section{Introduction}
% \PG{I'm not sure if this is a good intro. Feel free to write a new one, or we can also omit it altogether.}
% \TL{Might mention that the Gaussian polynomial is the Poincare polynomial of the Grassmannian, whose mixed Hodge structure is ``pure".  In the case of positroids, we get a $q,t$-version because of the lack of purity.}

The \Poincare polynomial of the complex Grassmannian $\Gr(k,n)$ is well known to be given by the Gaussian polynomial $\qbin{n}k$. The number of points of $\Gr(k,n)$ over a finite field $\F_q$ is given by the same polynomial. The reason these two polynomials coincide is that the mixed Hodge structure on the cohomology of $\Gr(k,n)$ is pure. The situation is different when one considers the top-dimensional positroid variety $\Piokn\subset\Gr(k,n)$, introduced in~\cite{KLS} building on the results of~\cite{Pos}. The space $\Piokn$ consists of row spans of full rank $k\times n$ matrices whose cyclically consecutive maximal minors are all nonzero. It turns out that the \Poincare polynomial and the point count of $\Piokn$ are given by two different $q$-analogs of (rational) Catalan numbers: one of our main results is that when $\gcd(k,n)=1$, the \Poincare polynomial of $\Piokn$ is given by $\sum_{P\in\Dyck_{k,n-k}} q^{\area(P)}$ while the number of points of $\Piokn$ over $\F_q$ equals $\frac{1}{\qint{n}}\qbin{n}k$, up to a simple factor. The mixed Hodge structure on $H^\bul(\Piokn)$ is non-pure, and we show that its bigraded \Poincare polynomial $\Poinc(\Piokn;q,t)$ coincides with the rational $q,t$-Catalan number $\Cat_{k,n-k}(q,t)$ introduced in~\cite{GaHa,LoWa}. Our proof proceeds via relating both sides to Khovanov--Rozansky knot homology~\cite{KR1}. Our results apply more generally to arbitrary positroid and Richardson varieties.


% Connections between knot invariants and Macdonald theory have received an enormous amount of attention in recent years; see e.g.~\cite{GORS,GN,Haglund,Mellit_torus,ORS,EH,MeHog}. In particular, Khovanov--Rozansky homology of torus knots and links was computed in~\cite{Mellit_torus,EH,MeHog}. For torus knots, the answer is given precisely by the rational $q,t$-Catalan numbers.

% Our main results are described in detail in the subsequent sections. We start by highlighting some consequences of our approach from several points of view. 

% {\bf Total positivity.} 
% % Positroid (and, more generally, Richardson) varieties arise naturally in the study of total positivity, cluster algebras, and Poisson geometry~\cite{Lus2,Pos,BGY,KLS}. 
% We associate a link to each positroid (and, more generally, Richardson) variety which captures many of its fundamental properties. We identify the Khovanov--Rozansky homology of this link with the mixed Hodge structure on the cohomology of the positroid variety, which includes its Betti numbers and the point count over a finite field. For  top-dimensional positroid varieties, the answer is given by rational $q,t$-Catalan numbers.

% {\bf Knot theory.} We introduce a class of \emph{Richardson links} for which we give a geometric interpretation of the top $a$-degree coefficient of Khovanov--Rozansky homology (and of the \FLY polynomial~\cite{HOMFLY,PT}). When a Richardson link is a knot, we show that the associated $q,t$-polynomial is $q,t$-symmetric. We also study a variant of Khovanov--Rozansky homology (where \emph{Soergel bimodules}~\cite{SoeHC} are replaced with \emph{Soergel modules}~\cite{SoeGarben}) which potentially gives a new knot invariant that is easier to compute.

% {\bf Representation theory.} The above objects compute extensions $\Ext^\bul(M_v,M_w)$ between Verma modules in the principal block $\O_0$ of the Bernstein–Gelfand–Gelfand category~$\O$ (see e.g.~\cite{Humphreys}). We show that the $q,t$-symmetry property, when translated in this  language, is a consequence of Koszul duality~\cite{BGS,BY}.
% %For a particular choice of $v,w$, the answer is given  by the rational $q,t$-Catalan numbers.

% {\bf Combinatorics.} The rational $q,t$-Catalan numbers $\Cat_{k,n-k}(q,t)$ are ubiquitous in combinatorics, and we find a few more appearances of them in the contexts above: we show that $\Cat_{k,n-k}(q,t)$ computes the cohomology of the top-dimensional positroid variety $\Piokn$, as well as the extension group $\Ext^\bul(M_{\id},M_{\wkn})$ between a particular pair of Verma modules. This gives a geometric proof that $\Cat_{k,n-k}(q,t)$ is $q,t$-symmetric and unimodal. Another potentially important consequence is that our results produce a whole family of $q,t$-symmetric and unimodal polynomials, which includes $\Cat_{k,n-k}(q,t)$ as a special case. This may lead to new recurrence relations, such as the one in~\cite{MuSp}.


% % Some recent work has also been done relating positroid varieties to knots~\cite{FPST,STWZ}.

% % \PG{I feel like we should write nothing here and just dive into \S1.1. Do you agree?}

% % \PG{I collected some bibliography: } 
% % \cite{ARs,AchRid,ARV,BT,AMRW1,AMRW2,Bei,BBD,BGS,BL,BY,EW,Rid,Rou,Soe90,Soe97,Soe00,Soe01,Wil_ICM,Wil2,WW08,WW11,WW17}

% % \section{Main results}
\label{sec:main-results}
\section{Positroid varieties and Catalan numbers}
\parag{Rational $q,t$-Catalan numbers}
Let $a$ and $b$ be coprime positive integers. The \emph{rational $q,t$-Catalan number} $\Cab(q,t)\in\N[q,t]$ was introduced by Loehr--Warrington~\cite{LoWa}, generalizing the work of Garsia--Haiman~\cite{GaHa}. It is defined as follows:
\begin{equation}\label{eq:qt_cat}
  \Cab(q,t):=\sum_{P\in\Dyckab} q^{\area(P)} t^{\dinv(P)},
\end{equation}
where $\Dyckab$ is the set of lattice paths $P$ inside a rectangle of height $a$ and width $b$ that stay above the diagonal, $\area(P)$ is the number of unit squares fully contained between $P$ and the diagonal, and $\dinv(P)$ is the number of pairs $(h,v)$ satisfying the following conditions: $h$ is a horizontal step of $P$, $v$ is a vertical step of $P$ that appears to the right of $h$, and there exists a line of slope $a/b$ (parallel to the diagonal) intersecting both $h$ and $v$.
% such that (i) $h$ is a horizontal step of $P$ and $v$ is a vertical step of $P$, (ii) $h$ appears before $v$, and (iii) there exists a line with slope $b/a$ (parallel to the diagonal) intersecting both $h$ and $v$. 
 For example, \cref{fig:Cab_ex} shows that
\begin{equation}\label{eq:Cat_5_3}
\Cat_{3,5}(q,t)=q^{4} + q^{3} t + q^{2} t^{2} + q^{2} t + q t^{3} +q t^{2} +  t^{4}.
\end{equation} 

\begin{figure}
\centering

\def\dgr{
\draw[line width=0.3pt,dashed] (0,0) grid (5,3);
\draw[line width=0.5pt,red] (0,0)--(5,3);
}
\def\dyckscl{0.4}
\def\resc#1{\scalebox{2.5}{#1}}
\def\plw{3pt}
\def\pcol{blue}
\scalebox{\dyckscl}{
\begin{tabular}{ccccccc}
\begin{tikzpicture}
\dgr
\draw[line width=\plw,draw=\pcol] (0,0)--(0,3)--(5,3);
\end{tikzpicture}
&
\begin{tikzpicture}
\dgr
\draw[line width=\plw,draw=\pcol] (0,0)--(0,2)--(1,2)--(1,3)--(5,3);
\end{tikzpicture}
&
\begin{tikzpicture}
\dgr
\draw[line width=\plw,draw=\pcol] (0,0)--(0,2)--(2,2)--(2,3)--(5,3);
\end{tikzpicture}
&
\begin{tikzpicture}
\dgr
\draw[line width=\plw,draw=\pcol] (0,0)--(0,1)--(1,1)--(1,3)--(5,3);
\end{tikzpicture}
&
\begin{tikzpicture}
\dgr
\draw[line width=\plw,draw=\pcol] (0,0)--(0,1)--(1,1)--(1,2)--(2,2)--(2,3)--(5,3);
\end{tikzpicture}
&
\begin{tikzpicture}
\dgr
\draw[line width=\plw,draw=\pcol] (0,0)--(0,2)--(3,2)--(3,3)--(5,3);
\end{tikzpicture}
&
\begin{tikzpicture}
\dgr
\draw[line width=\plw,draw=\pcol] (0,0)--(0,1)--(1,1)--(1,2)--(3,2)--(3,3)--(5,3);
\end{tikzpicture}\\

\resc{$q^4t^0$} &\resc{$q^3t^1$}&\resc{$q^2t^2$}&\resc{$q^2t^1$}&\resc{$q^1t^3$}&\resc{$q^1t^2$}&\resc{$q^0t^4$}

\end{tabular}
}
  \caption{\label{fig:Cab_ex} Computing the rational $q,t$-Catalan number $\Cat_{3,5}(q,t)$.}
\end{figure}




% \def\ft{F}
% \def\ft{\dot{f}}
% \def\ft{\tilde{f}}
% \def\ft{\overline{f}}
\parag{Positroid varieties in the Grassmannian}\label{sec:positr-vari-grassm}
The \emph{Grassmannian} $\Gr(k,n)$ is the space of linear $k$-dimensional subspaces of $\C^n$. Alternatively, it can be identified with the space of full rank $k\times n$ matrices modulo row operations. Building on Postnikov's cell decomposition~\cite{Pos} of its totally nonnegative part, Knutson--Lam--Speyer~\cite{KLS} 
constructed a stratification $  \Gr(k,n)=\bigsqcup_{f\in\Skn} \Pio_f$ %$\Gr(k,n)=\bigsqcup_{f} \Pio_f$
% \begin{equation}\label{eq:Gr_strat}
%   \Gr(k,n)=\bigsqcup_{f\in\Skn} \Pio_f
% \end{equation}
 of the Grassmannian into \emph{(open) positroid varieties}. Roughly speaking,\footnote{More precisely, positroid varieties are in bijection with \emph{decorated} permutations, where a decoration of $f$ is an arbitrary coloring of fixed points of $f$ into black and white colors. The actual set $\Skn$ consists of decorated permutations $f$ satisfying $\#\{1\leq i\leq n\mid f(i)<i\}+\#\{\text{black fixed points of $f$}\}=k$. The most interesting special case for us occurs when $f$ is a single cycle, where the decoration is trivial.} each positroid variety $\Pio_f$ corresponds to a permutation $f\in S_n$ such that $\#\{1\leq i\leq n\mid f(i)<i\}=k$; the set of such permutations is denoted by $\Skn$.  For each $f\in \Skn$, the space $\Pio_f$ is a smooth algebraic variety. Two basic questions one can ask about such a space are: what is the number of points in $\Pio_f(\F_q)$ over a finite field $\F_q$ with $q$ elements, and what is the cohomology of $\Pio_f$ considered as a variety over $\C$? 

% \begin{enumerate}
% \item What is the number of points in $\Pio_f(\F_q)$ over a finite field $\F_q$ with $q$ elements?
% \item What is the cohomology of $\Pio_f$ considered as a variety over $\C$?
% \end{enumerate}

% \section{Mixed Hodge structure}\label{sec:mixed-hodge-struct}
These two questions turn out to be closely related to each other through the work of Deligne~\cite{Del71} on \emph{mixed Hodge structures}, explored in the case of cluster varieties in~\cite{LS}.  Since the work of Scott~\cite{Scott}, positroid varieties have been expected to admit a natural \emph{cluster algebra} structure arising from Postnikov diagrams. We recently proved this conjecture building on the results of~\cite{Lec,MuSp,SSBW}.
% \PG{On the one hand, I like this approach because it amplifies the importance of~\cite{LS,GL}. On the other hand, the polynomial $\Poincf q,t)$ is already defined through~\cite{Del71} via Frobenius weights, so we don't really need any cluster algebras to state \cref{thm:master_kn}.}
\begin{theorem}[{\cite{GL}}]\label{thm:GL}
  The coordinate ring of each positroid variety $\Pio_f$ is isomorphic to the associated cluster algebra.
\end{theorem}
This result allows one to study $\Pio_f$ as a \emph{cluster variety}, in which case Deligne's mixed Hodge structure can be explored using the machinery developed by Lam--Speyer~\cite{LS}. The mixed Hodge structure endows the cohomology $H^\bul(\Pio_f)$ of $\Pio_f$ with a second grading, and the suitably renormalized \Poincare polynomial $\Poincf q,t)$ of this bigraded vector space answers both of the above questions simultaneously:
%  in the following way.
% In particular, both of the above questions can be answered simultaneously in the following way.
\begin{theorem}[\cite{LS,GL}]\label{thm:LS}
  For each $f\in\Skn$, the \emph{bigraded \Poincare polynomial} $\Poincf q,t)\in\N[q^{\frac12},t^{\frac12}]$ satisfies the following properties:
\begin{theoremlist}
\item\label{thm:LS:qt_symm} $q,t$-symmetry: $\Poincf q,t)=\Poincf t,q)$;
\item\label{thm:LS:qt_unim} $q,t$-unimodality: for each $d$, the coefficients of $\Poincf q,t)$ at $q^dt^0, q^{d-1}t^1,\dots,q^0t^d$ form a unimodal sequence;
\item\label{thm:LS:poincare}  $\Poincf q^2,1)$ equals the \Poincare polynomial of $\Pio_f$ (considered as a variety over $\C$);
% \item\label{thm:LS:point_cnt} $q^{\frac12\dim\Pio_f}\cdot \Poincf q,1/q)$ equals\footnote{More precisely, one needs to substitute $t^{\frac12}=-q^{-\frac12}$.} the point count $\#\Pio_f(\F_q)$.
\item\label{thm:LS:point_cnt} $q^{\frac12\dim\Pio_f}\cdot \Poincf q,t) |_{t^{\frac12}=-q^{-\frac12}}$ equals the point count $\#\Pio_f(\F_q)$.
\end{theoremlist}
\end{theorem}
% \noindent Parts~\itemref{thm:LS:qt_symm} and~\itemref{thm:LS:qt_unim} are consequences of the \emph{curious Lefschetz property}, formalized in~\cite{HR} and proven to hold for certain cluster varieties in~\cite{LS}; see \cref{ex:E8} below.

The positroid stratification contains a unique open stratum, the \emph{top-dimensional positroid variety} $\Piokn:=\Pio_{\fkn}$ corresponding to the permutation $\fkn\in\Skn$ sending $i\mapsto i+k$ modulo $n$ for all $1\leq i\leq n$. It is given by 
% The space $\Piokn\subset\Gr(k,n)$ consists of row spans of 
% % In this case, the definition~\eqref{eq:Pio_dfn} of $\Piokn$ simplifies considerably. Consider the following set of
%  $k\times n$ complex matrices all of whose cyclically consecutive maximal minors are nonzero:
\begin{equation*}%\label{eq:Piokn_dfn}
  \Piokn:=\{\rowspan(A)\mid A\in\Mat(k,n;\C): \Delta_{1,2,\dots,k}(A),\Delta_{2,3,\dots,k+1}(A),\dots,\Delta_{n,1,\dots,k-1}(A)\neq0\}.
\end{equation*}
Here $\Delta_I(A)$ is the maximal minor of $A$ with column set $I\subset\{1,2,\dots,n\}$, $|I|=k$. 
% In particular, all matrices in $\Matokn$ have rank $k$, and $\Piokn\subset\Gr(k,n)$ is just the set of row spans of matrices in $\Matokn$.
% Denote ${\Poinckn q,t):=\Poinc(\Pio_{\fkn};q,t)}$.
 We are ready to state (the most important special case of) our main result.
\begin{theorem}\label{thm:master_kn}
Assume that $\gcd(k,n)=1$. Then 
\begin{equation}\label{eq:Poinc=Cat}
  \Poinckn q,t)=\qtn \Cat_{k,n-k}(q,t).
\end{equation}
\end{theorem}
\noindent 
The equality \eqref{eq:Poinc=Cat} arises as a conjecture from the works \cite{STZ,STWZ} and we thank Vivek Shende for drawing our attention to the conjecture. We generalize \cref{thm:master_kn} to all positroid varieties in \cref{thm:main} below.


Our proof of \cref{thm:master_kn} involves a number of ingredients, including Khovanov--Rozansky knot homology and  equivariant perverse sheaves. The point count specialization ($t^{\frac12}=-q^{-\frac12}$) turns out to require less advanced machinery. Namely, let us denote $ \qint{n}:=1+q+\dots+q^{n-1}$, $\qint{n}!:=\qint{1}\qint{2}\cdots\qint{n}$, and $\qbin{n}k:=\frac{\qint{n}!}{\qint{k}!\qint{n-k}!}$. 
% \begin{equation*}%\label{eq:*}
%   \qint{n}:=1+q+\dots+q^{n-1},\quad \qint{n}!:=\qint{1}\qint{2}\cdots\qint{n},\quad\text{and}\quad \qbin{n}k:=\frac{\qint{n}!}{\qint{k}!\qint{n-k}!}.
% \end{equation*}
We give an elementary proof of the following special case of \cref{thm:master_kn}.% in \cref{sec:point-count-fly}.
% \TL{If we have not done so already, should remind reader the relation between Grassmannian and $q$-binomials}%\PG{I'm assuming you plan to say something in the intro}
\begin{proposition}\label{prop:point_count}
Assume that $\gcd(k,n)=1$. Then $\#\Piokn(\F_q)=(q-1)^{n-1}\cdot \frac{1}{\qint{n}}\qbin{n}k.$
% \[\#\Piokn(\F_q)=(q-1)^{n-1}\cdot \frac{1}{\qint{n}}\qbin{n}k.\]
 In other words, the probability that a uniformly random $k$-dimensional subspace of $(\F_q)^n$ 
 belongs to $\Piokn(\F_q)$ equals $  \frac{(q-1)^n}{q^n-1}.$
% \begin{equation*}%\label{eq:*}
%   \frac{(q-1)^n}{q^n-1}.
% \end{equation*}
\end{proposition}
\begin{remark}
Surprisingly, this probability $\frac{(q-1)^n}{q^n-1}$ does not depend on $k$. We do not have a combinatorial explanation for this phenomenon.
\end{remark}
 Our proof proceeds by associating a \emph{link} $\betah_f$ to each positroid variety $\Pio_f$ (\cref{sec:knots-assoc-positr}) and then comparing the point count $\#\Pio_f(\F_q)$ to the \FLY polynomial of $\betah_f$. The \FLY polynomial is categorified by Khovanov--Rozansky knot homology, and our proof of \cref{thm:master_kn} may be considered a ``categorification'' of \cref{prop:point_count}.

\begin{remark}
After discovering the proof of \cref{prop:point_count} via the \FLY polynomial, we found that it can also be deduced from the results of~\cite{STZ,STWZ}. Our proof is new and yields a generalization (\cref{thm:homfly}) of \cref{prop:point_count} to arbitrary open positroid varieties. 
% \TL{Removed ``does not follow" part.}
%, which does not follow from the results of~\cite{STZ,STWZ}. \PG{is this alright?}
\end{remark}


% \qbin{a+b}a,\quad\text{and}\quad \Cab(q,1)=\sum_{P\in\Dyckab} q^{\area(P)}.
% \end{equation}


\parag{Torus action}
The appearance of the extra factor $\qtn$ in~\eqref{eq:Poinc=Cat}, as well as the condition $\gcd(k,n)=1$ are neatly explained by the \emph{torus action} on $\Gr(k,n)$. Let $\Tn\cong(\Cast)^{n-1}$ be the quotient of the group of diagonal $n\times n$ matrices by the group of scalar matrices. The group $\Tn$ acts on $\Gr(k,n)$ by rescaling the columns of $k\times n$ matrices. This action leaves each positroid variety $\Pio_f$ invariant. We say that $\Tn$ acts \emph{freely} on $\Pio_f$ if all non-identity elements of $\Tn$ act on $\Pio_f$ without fixed points. It is straightforward to check that the action of $\Tn$ on $\Pio_f$ is free if and only if the permutation $f$ is a single cycle. 
% The following is straightforward to check directly from the definitions.%; see \cref{cor:Pio_T_action}.
% \begin{equation}\label{eq:free}
% \text{The action of $\Tn$ on $\Pio_f$ is free if and only if the permutation $f$ is a single cycle.}
% \end{equation}
Note that $\fkn$ is a single cycle if and only if $\gcd(k,n)=1$. We will see later in \cref{dfn:knot} that $f$ is a single cycle precisely when the associated link $\betah_f$ has a single component, i.e., is a knot.



Let $\ncyc(f)$ denote the number of cycles of $f\in S_n$, and let $\Sknc:=\{f\in\Skn\mid \ncyc(f)=1\}$.
% \begin{equation}\label{eq:Sknc}
% \Sknc:=\{f\in\Skn\mid \ncyc(f)=1\}.
% \end{equation}
% From now on, we restrict our attention only to permutations in $\Skn$ that have one cycle, and denote the resulting set by $\Sknc$. 
 For $f\in\Sknc$, the quotient $\Pio_f/\Tn$ is again a smooth cluster variety, and \cref{thm:LS} applies to it.  The associated  bigraded \Poincare polynomials are related  as $ \Poincf q,t)=\qtn \cdot \Potf q,t).$
% follows:
 % We denote the resulting bigraded \Poincare polynomial of $\Pit_f$ by $\Potf q,t)$. The two polynomials are related  as follows:
% \begin{equation}\label{eq:mod_T_qtn}
%   \Poincf q,t)=\qtn \cdot \Potf q,t).
% \end{equation}
 In particular, in the setting of  \cref{thm:master_kn}, we find
\begin{equation}\label{eq:Poinc=Cat2}
  \Potkn q,t)=\Cat_{k,n-k}(q,t).
\end{equation}
Combining this with \cref{thm:LS} has consequences for $q,t$-Catalan numbers and positroid varieties which can be stated in an elementary way. Let us denote $\dkn:=(k-1)(n-k-1)=\dim(\Pit_{k,n}).$
% \begin{equation*}%\label{eq:*}
%   \dkn:=(k-1)(n-k-1)=\dim(\Pit_{k,n}).
% \end{equation*}
\begin{corollary}\label{cor:master_kn}
Assume that $\gcd(k,n)=1$. We have:% the rational $q,t$-Catalan numbers have the following properties.
\begin{theoremlist}
\item\label{cor:master:qt_symm} $q,t$-symmetry: $\Cat_{k,n-k}(q,t)=\Cat_{k,n-k}(t,q)$;
\item\label{cor:master:qt_unim} $q,t$-unimodality: for each $d$, the coefficients of $\Cat_{k,n-k}(q,t)$ at $q^dt^0, q^{d-1}t^1,\dots,q^0t^d$ form a unimodal sequence;
\item\label{cor:master:poincare}  the \Poincare polynomial of $\Pit_{k,n}$ is given by 
\begin{equation}\label{eq:Piokn_Poinc}
\sum_{d} q^{\frac d2} \dim H^{\dkn-d}(\Pit_{k,n})=\Cat_{k,n-k}(q,1)=\sum_{P\in\Dyckxx(k,n-k)} q^{\area(P)};
\end{equation}
\item\label{cor:master:point_cnt} the number of $\F_q$-points of $\Pit_{k,n}$ is given by
% \begin{equation}\label{eq:}
%  \#\Pit_{k,n}(\F_q)=q^{\frac{(k-1)(n-k-1)}2} \Cat_{k,n-k}(q,1/q)=\frac{1}{\qint{n}}\qbin{n}k=q^{\frac{(k-1)(n-k-1)}2} \sum_{P\in\Dyckxx(k,n-k)} q^{\area(P)-\dinv(P)}.
% \end{equation}
% \begin{equation}\label{eq:}
%   \#\Pit_{k,n}(\F_q)=q^{\delkn} \Cat_{k,n-k}(q,1/q)=\frac{1}{\qint{n}}\qbin{n}k=q^{\delkn} \sum_{P\in\Dyckxx(k,n-k)} q^{\area(P)-\dinv(P)},\quad\text{where}\quad \delkn:=\frac{(k-1)(n-k-1)}2.
% \end{equation}
% \begin{equation}\label{eq:Pit_kn_Fq}
%   \#(\Pit_{k,n})(\F_q)=\frac{1}{\qint{n}}\qbin{n}k=q^{\delkn}\cdot  \Cat_{k,n-k}(q,1/q)=q^{\delkn}\cdot  \sum_{P\in\Dyckxx(k,n-k)} q^{\area(P)-\dinv(P)},
% \end{equation}
\begin{equation}\label{eq:Pit_kn_Fq}
  \#(\Pit_{k,n})(\F_q)=\frac{1}{\qint{n}}\qbin{n}k=q^{\frac12\dkn}\cdot  \Cat_{k,n-k}(q,1/q).%=q^{\frac12\dkn}\cdot  \sum_{P\in\Dyckxx(k,n-k)} q^{\area(P)-\dinv(P)}.
\end{equation}
% where $\dkn:=(k-1)(n-k-1)=\dim(\Pit_{k,n})$.
% where $\delkn:=\frac{(k-1)(n-k-1)}2=\frac12\dim(\Pit_{k,n})$.
% $q^{\dim\Pio_f}\cdot \Poincf q,1/q)$ equals the point count $\#\Pio_f(\F_q)$.
\end{theoremlist}
\end{corollary}

\begin{remark}
 % extensively in e.g.~\cite{GaHag,HagBook,HagConj,GoMa1,GoMa2}
When $a=n$ and $b=n+1$, $\Cab(q,t)$ recovers the famous \emph{$q,t$-Catalan numbers} $\Ca(q,t)$ of Garsia and Haiman~\cite{GaHa}. The fact that $\Ca(q,t)$ is $q,t$-symmetric and $q,t$-unimodal follows from the results of Haiman~\cite{Haiman94,Haiman02}. For arbitrary $a,b$, the $q,t$-symmetry property follows from the celebrated recent proof of the rational shuffle conjecture~\cite{Mellit_rat}. To our knowledge, $q,t$-unimodality of $\Cat_{k,n-k}(q,t)$ is a new result.
\end{remark}


% \begin{table}\centering
% \begin{tabular}{|c|ccccccccc|}\hline
% $H^k$ & $H^0$ & $H^1$ & $H^2$ & $H^3$ & $H^4$ & $H^5$ & $H^6$ &$H^7$ & $H^8$ \\\hline
% $k-p=0$&  $1$ &  $0$ &  $1$ &  $0$ &  $1$ &  $0$ &  $1$ &  $0$ &  $1$ \\
% $k-p=1$&      &      &      &      &  $1$ &  $0$ &  $1$ & & \\\hline
% \end{tabular}
%   \caption{\label{tab:E8} The mixed Hodge table recording the dimensions of $H^{k,(p,p)}(\Pio_{3,8}/T)$ for the cluster algebra of type $E_8$; see \cite[Table~5]{LS}. The dimensions agree with the coefficients of $\Cat_{3,5}(q,t)$; see \cref{ex:E8}.}
% \end{table}

\begin{example}\label{ex:E8}
For $k=3$, $n=8$, the coordinate ring of $\Piokn/T$ is a cluster algebra of type $E_8$ (with no frozen variables). The associated \emph{mixed Hodge table} recording the dimensions of $H^{k,(p,p)}(\Pio_{3,8}/T)$ has the following form.
\begin{center}
\begin{tabular}{|c|ccccccccc|}\hline
$H^k$ & $H^0$ & $H^1$ & $H^2$ & $H^3$ & $H^4$ & $H^5$ & $H^6$ &$H^7$ & $H^8$ \\\hline
$k-p=0$&  $1$ &  $0$ &  $1$ &  $0$ &  $1$ &  $0$ &  $1$ &  $0$ &  $1$ \\
$k-p=1$&      &      &      &      &  $1$ &  $0$ &  $1$ & & \\\hline
\end{tabular}
\end{center}
See~\cite[Table~5]{LS}. The grading conventions are chosen so that the first row contributes $q^{4} + q^{3} t + q^{2} t^{2} + q t^{3} +  t^{4}$ while the second row contributes $q^{2} t + q t^{2}$ to $\Poinckn q,t)$. Comparing the result with~\eqref{eq:Cat_5_3}, we find $\Potkn q,t)=\Cat_{k,n-k}(q,t)$.

The polynomial $\Cat_{3,5}(q,t)$ given in~\eqref{eq:Cat_5_3} is indeed $q,t$-symmetric and $q,t$-unimodal: fixing the total degree of $q$ and $t$, it splits into polynomials $q^{4} + q^{3} t + q^{2} t^{2} + q t^{3} +  t^{4}$ and $q^{2} t + q t^{2}$. We also have $\Cat_{3,5}(q,1)=q^4+q^3+2q^2+2q+1$; the coefficient of $q^{d/2}$ is equal to $\dim H^{\dkn-d}(\Piokn/T)$ for each $d$.
 % (these coefficients are column sums in \cref{tab:E8}). %which is the sum of the numbers in the column of \cref{tab:E8} labeled by $H^{2d}$.
 % This agrees with~\eqref{eq:Piokn_Poinc}.
\end{example}



\section{Links associated to positroid varieties}\label{sec:knots-assoc-positr}


In order to explain how knot theory comes into play, we need a way to represent $f\in \Skn$ in a slightly different form. Let us say that a permutation $w\in S_n$ is \emph{$k$-Grassmannian} if $w^{-1}(1)<w^{-1}(2)<\dots<w^{-1}(k)$ and $w^{-1}(k+1)<\dots<w^{-1}(n)$. We denote by $\leq$ the (strong) Bruhat order on $S_n$. The following result is well known.
% For $v,w\in S_n$, we denote by $v\leq w$ the Bruhat order on $S_n$.

\begin{proposition}[\cite{KLS}] \label{prop:v_w}
For every $f\in\Skn$, there exists a unique\footnote{If $f$ has fixed points, the pair $(v,w)$ must be compatible with the decoration of $f$.} pair of permutations $v, w\in S_n$ such that $v\leq w$, $w$ is $k$-Grassmannian, and $f=wv^{-1}$.
\end{proposition}
\noindent For example, when $f=\fkn$, we have $w=f$ and $v=\id$. The dimension of $\Pio_f$ equals $\lvw:=\ell(w)-\ell(v)$, where $\ell(w)$ is the number of inversions of $w$. 
% In the situation of \cref{prop:v_w}, we denote $f_{v,w}:=f$ and $(v_f,w_f):=(v,w)$.

The group $S_n$ is generated by simple transpositions $s_i=(i,i+1)$ for $1\leq i\leq n-1$. Similarly, let $\Braid_n$ be the \emph{braid group} on $n$ strands, generated by $\sigma_1,\dots,\sigma_{n-1}$ with relations $\sigma_i\sigma_{i+1}\sigma_i=\sigma_{i+1}\sigma_i\sigma_{i+1}$ and $\sigma_i\sigma_j=\sigma_j\sigma_i$ for $|i-j|>1$. Connecting the corresponding endpoints of a braid $\beta$ gives rise to a \emph{link} called the \emph{closure} $\betah$ of $\beta$; see \cref{fig:braids}.% If the link $\betah$ has one connected component, it is called a \emph{knot}.

 % ($1\leq i\leq n-1$) satisfying braid and commutation relations. In addition, each $s_i$ is an involution: $s_i^2=\id$. Removing this last relation gives rise to the \emph{braid group} $\Braid_n$ on $n$ strands. Thus $\Braid_n$ is generated by $\sigma_1,\dots,\sigma_{n-1}$ with relations $\sigma_i\sigma_{i+1}\sigma_i=\sigma_{i+1}\sigma_i\sigma_{i+1}$ ($1\leq i\leq n-2$) and $\sigma_i\sigma_j=\sigma_j\sigma_i$ ($|i-j|\geq2$). One can draw an element $\beta\in \Braid_n$ as in \cref{fig:braids}, and then connecting the corresponding endpoints gives rise to a link called the \emph{closure} $\betah$ of $\beta$. A \emph{link} is a union of several circles embedded in $\R^3$. These circles are called \emph{components} of a link, and a link with only one component is called a \emph{knot}.



\begin{figure}


\def\rc{5}
\def\lw{2pt}

\def\whdiam(#1,#2){
\fill[white] (#1-\whlw,#2)--(#1,#2-\whlw)--(#1+\whlw,#2)--(#1,#2+\whlw)--cycle;
}
\def\whdiamr(#1,#2){
\whdiam(#1,#2)
\draw[line width=\lw] (#1-\whlww,#2-\whlww)--(#1+\whlww,#2+\whlww);
}
\def\whdiaml(#1,#2){
\whdiam(#1,#2)
\draw[line width=\lw] (#1+\whlww,#2-\whlww)--(#1-\whlww,#2+\whlww);
}

\def\whlw{0.5}
\def\whlww{0.25}
\def\yscl{0.7}
\def\xscl{1}
\def\sclbx{0.5}

\def\hmar{0.5}
\def\vmar{0.5}

\def\hmarr{0.8}
\def\vmarr{0.7}

\def\hmarrr{1.1}
\def\vmarrr{0.9}

\def\hmarrrr{1.4}
\def\vmarrrr{1.1}

\def\recmar{0.3}
\def\ndscl{1.4}
\centering

\setlength{\tabcolsep}{6pt}
\scalebox{0.98}{
\makebox[1.0\textwidth]{
\begin{tabular}{ccc}

\def\mar{0.5}

\scalebox{\sclbx}{
\begin{tikzpicture}[yscale=1, baseline=(ZUZU.base)]
\coordinate(ZUZU) at (0,0);
\begin{knot}[
    % draft mode=crossings ,
    clip width=6,
    ]
\strand[rounded corners=\rc, line width=\lw] (-\mar,3)--(0,3)--(1,3)--(2,2)--(3,2)--(4,1)--(7,1)--(8,0)--(8+\mar,0);
\strand[rounded corners=\rc, line width=\lw] (-\mar,2)--(0,2)--(1,1)--(2,1)--(3,0)--(7,0)--(8,1)--(8+\mar,1);
\strand[rounded corners=\rc, line width=\lw] (-\mar,1)--(0,1)--(2,3)--(6,3)--(7,2)--(8,2)--(8+\mar,2);
\strand[rounded corners=\rc, line width=\lw] (-\mar,0)--(2,0)--(4,2)--(6,2)--(7,3)--(8+\mar,3);
\flipcrossings{1,6}
\end{knot}
\draw[line width=1pt,dashed,red!50] (0,-\recmar) rectangle (4,3+\recmar);
\draw[line width=1pt,dashed,blue!50] (6,-\recmar) rectangle (8,3+\recmar);
\node[scale=1.4,anchor=south,red](A) at (2,3+\recmar) {$\beta(w)=\sigma_2\sigma_1\sigma_3\sigma_2$};
\node[scale=1.4,anchor=south,blue](A) at (7,3+\recmar) {$\beta(v)^{-1}=\sigma_1^{-1}\sigma_3^{-1}$};
\node[scale=\ndscl,anchor=east] (A) at (-\mar,0) {$4$};
\node[scale=\ndscl,anchor=east] (A) at (-\mar,1) {$3$};
\node[scale=\ndscl,anchor=east] (A) at (-\mar,2) {$2$};
\node[scale=\ndscl,anchor=east] (A) at (-\mar,3) {$1$};

\node[scale=\ndscl,anchor=west] (A) at (8+\mar,0) {$4$};
\node[scale=\ndscl,anchor=west] (A) at (8+\mar,1) {$3$};
\node[scale=\ndscl,anchor=west] (A) at (8+\mar,2) {$2$};
\node[scale=\ndscl,anchor=west] (A) at (8+\mar,3) {$1$};
\end{tikzpicture}
}


&

\def\mar{0.2}

\scalebox{\sclbx}{
\begin{tikzpicture}[yscale=1, baseline=(ZUZU.base)]
\coordinate(ZUZU) at (0,0);
\begin{knot}[
    % draft mode=crossings ,
    clip width=6,
    ]
\strand[rounded corners=\rc, line width=\lw] (-\mar,3)--(0,3)--(1,3)--(2,2)--(3,2)--(4,1)--(7,1)--(8,0)--(8+\mar,0);
\strand[rounded corners=\rc, line width=\lw] (-\mar,2)--(0,2)--(1,1)--(2,1)--(3,0)--(7,0)--(8,1)--(8+\mar,1);
\strand[rounded corners=\rc, line width=\lw] (-\mar,1)--(0,1)--(2,3)--(6,3)--(7,2)--(8,2)--(8+\mar,2);
\strand[rounded corners=\rc, line width=\lw] (-\mar,0)--(2,0)--(4,2)--(6,2)--(7,3)--(8+\mar,3);
\flipcrossings{1,6}
\end{knot}
\draw[rounded corners=\rc, line width=\lw] 
(8+\mar,3)--(8+\hmar,3)--(8+\hmar,3+\vmar)--(-\hmar,3+\vmar)--(-\hmar,3)--(-\mar,3);
\draw[rounded corners=\rc, line width=\lw] (8+\mar,2)--(8+\hmarr,2)--(8+\hmarr,3+\vmarr)--(-\hmarr,3+\vmarr)--(-\hmarr,2)--(-\mar,2);
\draw[rounded corners=\rc, line width=\lw] (8+\mar,1)--(8+\hmarrr,1)--(8+\hmarrr,3+\vmarrr)--(-\hmarrr,3+\vmarrr)--(-\hmarrr,1)--(-\mar,1);
\draw[rounded corners=\rc, line width=\lw] (8+\mar,0)--(8+\hmarrrr,0)--(8+\hmarrrr,3+\vmarrrr)--(-\hmarrrr,3+\vmarrrr)--(-\hmarrrr,0)--(-\mar,0);
\end{tikzpicture}
}



&
\def\hbr{5}
\def\lw{3pt}
\def\lww{10pt}
\def\mar{2}
\def\ndscl{2}
\scalebox{0.35}{
\begin{tikzpicture}[baseline=(ZUZU.base),yscale=1,xscale=1]
\coordinate(ZUZU) at (0,0);
\begin{knot}[
    % draft mode=crossings ,
    clip width=5,
    ]
  \strand[line width=\lw,rounded corners=\rc] (-\mar,0)--(2,0)--(7,5)--(7+\mar,5);
  \strand[line width=\lw,rounded corners=\rc] (-\mar,1)--(1,1)--(6,6)--(7+\mar,6);
  \strand[line width=\lw,rounded corners=\rc] (-\mar,2)--(0,2)--(5,7)--(7+\mar,7);
  \strand[line width=\lw,rounded corners=\rc] (-\mar,3)--(0,3)--(3,0)--(7+\mar,0);
  \strand[line width=\lw,rounded corners=\rc] (-\mar,4)--(1,4)--(4,1)--(7+\mar,1);
  \strand[line width=\lw,rounded corners=\rc] (-\mar,5)--(2,5)--(5,2)--(7+\mar,2);
  \strand[line width=\lw,rounded corners=\rc] (-\mar,6)--(3,6)--(6,3)--(7+\mar,3);
  \strand[line width=\lw,rounded corners=\rc] (-\mar,7)--(4,7)--(7,4)--(7+\mar,4);
\foreach[count=\j]\i in {0,1,...,7}{
\node[scale=\ndscl,anchor=east] (A) at (-\mar,7-\i) {$\j$};
\node[scale=\ndscl,anchor=west] (A) at (7+\mar,7-\i) {$\j$};
}
% (5,-\mar)--(5,0)--(0,\hbr)--(0,\hbr+\mar);
%   \draw[line width=\lw,rounded corners=\rc] (6,-\mar)--(6,0)--(1,\hbr)--(1,\hbr+\mar);
%   \draw[line width=\lw,rounded corners=\rc] (7,-\mar)--(7,0)--(2,\hbr)--(2,\hbr+\mar);
\end{knot}
% \fatline(0,3)
% \fatline(1,4)
% \fatline(2,5)
% \fatline(3,6)
% \fatline(4,7)

  % \draw[line width=\lww,rounded corners=\rc,white] (0,-\mar)--(0,0)--(3,\hbr)--(3,\hbr+\mar);
  % \draw[line width=\lw,rounded corners=\rc] (0,-\mar)--(0,0)--(3,\hbr)--(3,\hbr+\mar);
\end{tikzpicture}
}

\\
braid $\beta_f=\beta(w)\cdot \beta(v)^{-1}$ & closure $\betah_f$ of $\beta_f$ & \quad$\beta_{\fkn}$ for $k=3$, $n=8$

\end{tabular}
}
}

  \caption{\label{fig:braids} Braids and links associated to positroid varieties.}
% Left: a braid $\beta$. Middle: the closure $\betah$ of $\beta$. Right: the braid $\beta_{\fkn}}
\end{figure}


% \emph{knot} (more precisely, a \emph{link}, if it has several connected components) called the \emph{braid closure} $\betah$ of $\beta$. 

For each element $u\in S_n$, let $\beta(u)$ denote the corresponding braid, obtained by choosing a reduced word $u=s_{i_1}s_{i_2}\cdots s_{i_{\ell(u)}}$ for $u$ and then replacing each $s_i$ with $\sigma_i$. 
% (where $\ell(u)$ is minimal possible) 
%We note that $\beta(u)^{-1}\neq \beta(u^{-1})$, since the former is a product of $\sigma_i^{-1}$-s while the latter is a product of $\sigma_i$-s. 
\begin{definition}\label{dfn:knot}
For $f\in\Skn$ and $v\leq w\in S_n$ as in \cref{prop:v_w}, let ${\beta_f:=\beta(w)\cdot \beta(v)^{-1}}$. 
% \begin{equation}\label{eq:beta_dfn}
% \beta_f:=\beta(w)\cdot \beta(v)^{-1}.
% \end{equation}
 We refer to the closure $\betah_f$ as \emph{the link associated to $f$}. See \cref{fig:braids} for an example.
\end{definition}
Observe that $\betah_f$ is a \emph{knot} (i.e., has one connected component) if and only if $f\in\Sknc$. We note that two other (more complicated) ways of assigning a Legendrian or a transverse link to a positroid variety have appeared recently in~\cite{STWZ,FPST}. 
% \PG{FPST links commented out} 
% We show in \cref{sec:FPST} that their links are isotopic to our link $\betah_f$.  Our presentation of $\betah_f$ as the closure of an explicit braid allows us to study its polynomial invariants and their categorifications in a more controlled way.


% \section{\FLY polynomial}\label{sec:HOMFLY}
The \emph{\FLY polynomial} $\HOMP(L)=\HOMP(L;a,z)$ of an (oriented) link $L$ is defined by a skein relation $a \HOMP(L_+) - a^{-1}\HOMP(L_-)=z \HOMP(L_0)$ and $\HOMP(\unkn)=1,$ 
% \begin{equation}\label{eq:HOMFLY_dfn}
% % a\HOMP(L_+)
%   a \HOMP(L_+) - a^{-1}\HOMP(L_-)=z \HOMP(L_0)\quad\text{and}\quad \HOMP(\unkn)=1,
% \end{equation}
 where $\unkn$ denotes the unknot and  $L_+$, $L_-$, $L_0$ are three links whose planar diagrams locally differ as  follows: 

\vspace{-0.1in}

\def\lw{2pt}
\def\lww{6pt}
\def\drcirc{
\draw[line width=0.3pt, dashed] (0,0) circle (1cm);
}
\def\scl{0.4}

\def\DA{30}
\begin{center}

\begin{tabular}{ccc}
\scalebox{\scl}{
\begin{tikzpicture}
\drcirc
% \draw (0:1)--(90:1);
\draw[line width=\lw,->,>=latex] (-90+\DA:1)--(90+\DA:1);
\draw[line width=\lww,white] (-90-\DA:1)--(90-\DA:1);
\draw[line width=\lw,->,>=latex] (-90-\DA:1)--(90-\DA:1);
\end{tikzpicture}
}
&


\scalebox{\scl}{
\begin{tikzpicture}
\drcirc
% \draw (0:1)--(90:1);
\draw[line width=\lw,->,>=latex] (-90-\DA:1)--(90-\DA:1);
\draw[line width=\lww,white] (-90+\DA:1)--(90+\DA:1);
\draw[line width=\lw,->,>=latex] (-90+\DA:1)--(90+\DA:1);
\end{tikzpicture}
}
&

\def\rc{10}
\scalebox{\scl}{
\begin{tikzpicture}
\drcirc
% \draw (0:1)--(90:1);
\draw[line width=\lw,->,>=latex,rounded corners=\rc] (-90-\DA:1)--(0,0)--(90+\DA:1);
\draw[line width=\lw,->,>=latex,rounded corners=\rc] (-90+\DA:1)--(0,0)--(90-\DA:1);
% \draw[line width=\lww,white] (-90+\DA:1)--(90+\DA:1);
% \draw[line width=\lw,->,>=latex] (-90+\DA:1)--(90+\DA:1);
\end{tikzpicture}
}
\\
$L_+$ & $L_-$ & $L_0$



\end{tabular}
\end{center}
\vspace{-0.1in}

\begin{example}\label{ex:S_2_homfly}
For $n=2$, we may take $L_+$ to be the closure of $\sigma_1$, in which case $L_-$ is the closure of $\sigma_1^{-1}$ and $L_0=\unlk$ is 
% $\hat \sigma_1=L_+$ and $\hat{\sigma_1^{-1}}=L_-$ are unknots while $\hat\id=L_0$ is
 the $2$-component unlink. Applying the skein relation, we find $\HOMP(L_0)=\frac{a-a^{-1}}z$.
\end{example} 

% \begin{theorem}\label{thm:homfly}
%   For all $f\in \Sknc$, $\#(\Pit_f)(\F_q)$ is obtained from the top $a$-degree term of $\HOMP(\betah_f;a,z)$ by substituting $a:=q^{-\frac12}$ and $z:=q^{\frac12}-q^{-\frac12}$. \TODO{state this for all $f\in \Skn$}
% % \begin{equation*}%\label{eq:*}
% %   \#\Pio_f(\F_q)=
% % \end{equation*}
% \end{theorem}


Surprisingly, the \FLY polynomial computes the number of $\F_q$-points of \emph{any} positroid variety.

\begin{theorem}\label{thm:homfly}
  For all $f\in \Skn$, let $\Ptop_f$ be obtained from the top $a$-degree term of $\HOMP(\betah_f;a,z)$ by substituting $a:=q^{-\frac12}$ and $z:=q^{\frac12}-q^{-\frac12}$. Then $  \#\Pio_f(\F_q)=(q-1)^{n-1}\cdot \Ptop_f.$
% \begin{equation}\label{eq:homfly_f}
%   \#\Pio_f(\F_q)=(q-1)^{n-1}\cdot \Ptop_f.
% \end{equation}
\end{theorem}


\begin{remark}\label{rmk:torus_knot}
When $\gcd(k,n)=1$, we have $\fkn\in\Sknc$, and the associated knot $\betah_{\fkn}$ is the \emph{$(k,n-k)$-torus knot}; see \figref{fig:braids}(right). The value of $\HOMP(\betah_{\fkn};a,z)$ was computed in~\cite{Jones}, and its relationship with Catalan numbers was clarified in~\cite{Gorsky_cat}. Thus \cref{prop:point_count} follows from \cref{thm:homfly} as a direct corollary. 
\end{remark}


\begin{example}
For $k=3$, $n=8$, one calculates (for instance, using  Sage\footnote{\url{https://doc.sagemath.org/html/en/reference/knots/sage/knots/link.html}}) that the top $a$-degree term of $\HOMP(\betah_{\fkn};a,z)$ equals $\frac{z^{8} + 8 \, z^{6} + 21 \, z^{4} + 21 \, z^{2} + 7}{a^{8}}$. 
% \[\HOMP(\betah_{\fkn};a,z)=\frac{z^{8} + 8 \, z^{6} + 21 \, z^{4} + 21 \, z^{2} + 7}{a^{8}} - \frac{z^{6} + 7 \, z^{4} + 14 \, z^{2} + 8}{a^{10}} + \frac{z^{2} + 2}{a^{12}}.\] 
Substituting $a:=q^{-\frac12}$ and $z:=q^{\frac12}-q^{-\frac12}$, we get 
\[\Ptop_f=q^{8} + q^{6} + q^{5} + q^{4} + q^{3} + q^{2} + 1=q^4\cdot \Cat_{3,5}(q,1/q).\]
This agrees with~\eqref{eq:Pit_kn_Fq} and \cref{thm:homfly}.%~\eqref{eq:homfly_f}.
% in agreement with \cref{thm:homfly} and \cref{prop:point_count}.
% ~\eqref{eq:cat_spec}, \cref{conj:master_kn}, and part (iv) of \cref{thm:LS}.
% \todo
% $k,n=3,7$, 
% $\#\Pit(\F_q)=q^6 + q^4 + q^3 + q^2 + 1$
% $\HOMP(K;a,z)=\frac{z^6 + 6z^4 + 10z^2 + 5}{a^6} - \frac{z^4 + 5z^2 + 5}{a^8} +\frac1{a^{10}}$
% So the top-degree-in-a term is
% $\frac{z^6 + 6z^4 + 10z^2 + 5}{a^6}$
% Plug in $z=q^{\frac12}-q^{-\frac12}$, $a=q^{-\frac12}$ and voila:
% $q^6 + q^4 + q^3 + q^2 + 1.$
\end{example}



\parag{Links associated to Richardson varieties}\label{sec:richardson-varieties}
By \cref{prop:v_w}, positroid varieties correspond to pairs ${v\leq w}$ of permutations such that $w$ is $k$-Grassmannian. There is a more general class of \emph{(open) Richardson varieties} $\Rich_v^w$, indexed by all pairs $v\leq w\in S_n$, and the majority of the above story generalizes to this setting. The varieties $\Rich_v^w$ form a stratification of the \emph{complete flag variety} $\FlCn$.  For a permutation $f\in\Skn$ corresponding to a pair $v\leq w\in S_n$ via \cref{prop:v_w}, the projection map $\FlCn\to\Gr(k,n)$ restricts to an isomorphism $\Pio_f\cong \Rich_v^w$. Thus  positroid varieties are special cases of Richardson varieties. 


Now, let $G$ be a complex semisimple algebraic group of adjoint type, and choose a pair $B,B_-\subset G$ of opposite Borel subgroups.  Let $T:=B\cap B_-$ be the maximal torus and $W:=N_G(T)/T$ the associated Weyl group. For the case $G=\SL_n(\C)$, we have $W=S_n$, the subgroups $B,B_-\subset G$ consist of upper and lower triangular matrices, and $T\cong (\Cast)^{n-1}$ is the group of diagonal matrices modulo scalar matrices.\footnote{Note that $G$ is assumed to be of \emph{adjoint type}, thus in type $A$ we should have $G = \PGL_n(\C)$.  However, we choose to work with $G=\SL_n(\C)$ for simplicity.} We have Bruhat decompositions $G=\bigsqcup_{w\in W} BwB=\bigsqcup_{v\in W} B_-vB$, and the intersection $BwB\cap B_-vB$ is nonempty if and only if $v\leq w$ in the Bruhat order on $W$. For $v\leq w$, we denote by $\Rich_v^w:=(BwB\cap B_-vB)/B$ an \emph{open Richardson variety} inside the \emph{complete flag variety} $G/B$. For $G=\SL_n(\C)$, $\Rich_v^w$ is the subset of $G/B=\FlCn$ obtained by  specifying the dimensions of the intersections of a given flag with a given coordinate flag and its opposite coordinate flag.


In the case $G=\SL_n(\C)$, one can similarly associate a braid $\bvw:=\beta(w)\cdot \beta(v)^{-1}$ to any pair $v\leq w$ and consider its closure $\bhvw$. We refer to the links of the form $\bhvw$ as \emph{Richardson links}. The point count $\#\Rich_v^w(\F_q)$ is given by the \emph{Kazhdan--Lusztig $R$-polynomial}~\cite{KL1}, and both the statement and the proof of \cref{thm:homfly} generalize to this setting.
% Most of the arguments in the previous subsection generalize to this setting as well.
 % However, in order to apply the results of~\cite{LS}, one would need an analog of \cref{thm:GL}, that is, a cluster algebra structure on $\Rich_v^w$. Leclerc~\cite{Lec} constructed a cluster algebra $\Acal_{v,w}$ for each $v\leq w$ and gave an injection $\Acal_{v,w}\hookrightarrow \C[\Rich_v^w]$. He conjectured that this map is an isomorphism. We aim to return to this question in future work.



\section{Main results}
All of the above results are actually special cases of a single statement which applies to arbitrary Richardson varieties. This includes all positroid varieties $\Pio_f$ for $f\in\Skn$, where $\ncyc(f)$ can be arbitrary. As a warm up, we start with the non-equivariant version.

\parag{Ordinary cohomology}
Let $\h:=\Lie(T)$ be the Cartan subalgebra of $\Lie(G)$ corresponding to $T$, and denote $R := \Symh$. For $G=\SL_n(\C)$, $R=\C[y_1,\dots,y_{n-1}]$ is the polynomial ring. Since $W$ is a Coxeter group, we can consider the category $\SBim$ of \emph{Soergel bimodules}. Each element $B\in\SBim$ is a graded $R$-bimodule, and we will be interested in its $R$-invariants, which by definition form the \emph{zeroth Hochschild cohomology $\HH^0(B)$ of $B$}. 
% : $\HH^0(B)$ consists of all $b\in B$ such that $rb=br$ for all $r\in R$. 
% \begin{equation}\label{eq:HH0}
% \HH^0(B):=\{b\in B\mid rb=br\text{ for all $r\in R$}\}.
% \end{equation}
% Thus $\HH^0(B)$ is a graded $R$-module.
% \begin{equation}\label{eq:HHC}
%   \HHC^0(B):=\HH^0(B)\otimes_R\C,
% \end{equation}
 Denote $\HHC^0(B):=\HH^0(B)\otimes_R\C,$ where $\C=R/(\h^\ast)$ is an $R$-module on which $\h^\ast$ acts by $0$. 
% For example, the quotient map $R\to R\otimes_R\C$ sends a polynomial to its constant term.
 While the functor $\HH^0$ involves Soergel bimodules, the functor $\HHC^0$ involves \emph{Soergel modules} instead.


 To any element $w\in W$, Rouquier~\cite{Rou} associates a cochain complex $\FR(w)$ of Soergel bimodules. He also associates another complex $\FR(w)^{-1}$ such that their tensor product $\FR(w)\otimes_R \FR(w)^{-1}$ is homotopic to the identity.
%  complex $\FR(\id):=(\dots\to0\to R\to 0\to\dots)$.
% , where $R$ is placed in cohomological degree $0$.
 For a braid $\bvw=\beta(w)\cdot \beta(v)^{-1}$, we set $\Fvw:=\FR(w)\otimes_R \FR(v)^{-1}$. Applying the functor $\HHC^0$ to each term of this complex yields a complex $\HHC^0(\Fvw)$ of graded $R$-modules. Taking its cohomology $\HHHC^0(\Fvw):=H^\bul(\HHC^0(\Fvw))$, 
% \begin{equation}\label{eq:HHH0}
% \HHHC^0(\Fvw):=H^\bul(\HHC^0(\Fvw)),
% \end{equation}
 we get a bigraded $R$-module. 
% Explicitly, we have  
% \begin{equation*}%\label{eq:*}
%   \HHHC^0(\Fvw)=\bigoplus_{k,p\in\Z} \HHXC kp_{v,w},
% \end{equation*}
 We denote by $\HHXC kp_{v,w}$ the polynomial degree $2p$ part of $H^{k}(\HHC^0(\Fvw))$. By convention, the elements of $\h^\ast\subset R$ are assumed to have polynomial degree $2$. On the other hand, 
% the bigrading on $H^\bul(\Rich_v^w)$ is denoted as
% \begin{equation}\label{eq:H_bigr}
%     H^\bul(\Rich_v^w)=\bigoplus_{k,p\in\Z} H^{k,(p,p)}(\Rich_v^w),
% \end{equation}
 let us denote by $H^{k,(p,p)}(\Rich_v^w)$ the $(p,p)$ part of the mixed Hodge structure on $H^k(\Rich_v^w)$. See \cref{ex:E8}.% for an example.

% \footnote{Equivalently, $H^{k,(p,p)}(\Rich_v^w)$ has the same dimension as the subspace of the \etale cohomology $H^k(\Rich_v^w(\Fb_q),\Qlb)$ where the Frobenius automorphism acts with an eigenvalue of norm $q^p$.}


% \begin{table}
%   \setlength{\tabcolsep}{1.5pt}
% \renewcommand{\arraystretch}{1.2}
% \makebox[1.0\textwidth]{
% \scalebox{0.85}{
% \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}\hline
% % Title & $n$ & $v$ & $w$ & $\lvw$ & $\bhvw$ & $Y:=\Rich_v^w$ & $H^{k,(p,p)}(Y)$ & $H^{k,(p,p)}_{T,c}(Y)$ & $\HHX kp_{v,w}$ & $\HHXC kp_{v,w}$\\
% Title & $n$ & $v$ & $w$ & $\lvw$ & $\bhvw$ & $\Rich_v^w$ & $H^{k,(p,p)}(\Rich_v^w)$ & $H^{k,(p,p)}_{T,c}(\Rich_v^w)$ & $\HHXC kp_{v,w}$ & $\HHX kp_{v,w}$\\\hline\hline
% % Name & $n$ & $v$ & $w$ & $\lvw$ & $\bhvw$ & $Y:=\Rich_v^w$ & $H^{k,(p,p)}(Y)$ & $H^{k,(p,p)}_{T,c}(Y)$ & $H^{k,(p)}\HH^0$ & $H^{k,(p)}\HHC^0$\\\hline\hline
% % Title & $n$ & $v$ & $w$ & $\lvw$ & $\bhvw$ & $\Rich_v^w$ & $H^{k,(p,p)}(\Rich_v^w)$ & $H^{k,(p,p)}_{T,c}(\Rich_v^w)$ & $H^{k,(p)}\HH^0$ & $H^{k,(p)}\HHC^0$\\\hline\hline
% % Name & $n$ & $v$ & $w$ & $\lvw$ & $\bhvw$ & $\Rich_v^w$ & $H^{k,(p,p)}$ & $H^{k,(p,p)}_{T,c}$ & $H^{k,(p)}\HH^0$ & $H^{k,(p)}\HHC^0$\\\hline\hline
% Unknot-I & $1$ & $\id$ & $\id$ & $0$ & $\unkn$ & $\pt$ & $k=0,\,p=0$ & $k=0,\,p=0$ & $k=0,\,p=0$ & $k=0,\,p=0$\\\hline
% Unknot-II & $2$ & $\id$ & $s_1$ & $1$ & $\unkn$ & $\Pio_{1,2}$ & \begin{tabular}{c} $k=0,\,p=0$\\ $k=1,\,p=1$\end{tabular} & $k=1,\,p=0$ & \begin{tabular}{c}$k=0,\,p=0$\\ $k=-1,\,p=1$\end{tabular} & $k=0,\,p=0$\\\hline
% $2$-cpt. unlink & $2$ & $s_1$ & $s_1$ & $0$ & $\unlk$ & $\pt$ & $k=0,\,p=0$ & $k=2p,\,p\in\Z_{\geq0}$ & $k=0,\,p=0$ & $k=0,\,p\in\Z_{\geq0}$\\\hline
% % \begin{tabular}{c}Hopf\\ link\end{tabular} 
% Hopf link
% & $4$ & $\id$ & $f_{2,4}$ & $4$ & $\hopflink$ & $\Pio_{2,4}$ & too big & \begin{tabular}{c} $k=4,\,p=0$\\ $k=2+2p,\,p\in\Z_{\geq2}$\end{tabular}  & too big & \begin{tabular}{c} $k=0,\,p=0$\\ $k=-2,\,p\in\Z_{\geq2}$\end{tabular}\\\hline
% % \begin{tabular}{c}Trefoil\\ knot\end{tabular}
% Trefoil knot
%  & $5$ & $\id$ & $f_{2,5}$ & $6$ & $\trefoil$ & $\Pio_{2,5}$ & too big & \begin{tabular}{c} $k=6,\,p=0$\\ $k=8,\,p=2$\end{tabular}  & too big & \begin{tabular}{c} $k=0,\,p=0$\\ $k=-2,\,p=2$\end{tabular}\\\hline
% \end{tabular}
% }
% }
%   \caption{\label{tab:examples} Some examples illustrating \cref{thm:ordinary,thm:main}. The last four columns list all values of $k,p$, for which the corresponding bigraded component is nonzero (in each case, it is $1$-dimensional). We have $f_{2,4}=s_2s_1s_3s_2$ and $f_{2,5}=s_3s_2s_1s_4s_3s_2$.}
% \end{table}

% \noindent The following is our main result in the case of the ordinary cohomology of $\Rich_v^w$.
\begin{theorem}\label{thm:ordinary}
 For all $v\leq w\in W$ and $k,p\in\Z$, we have
\begin{equation}\label{eq:ordinary}
\dim  H^{k,(p,p)}(\Rich_v^w) =\dim \HHXC {-k}p_{v,w}.
% H^\bul(\Rich_v^w)\cong \HHHC^0(\Fvw).
\end{equation}
\end{theorem}
% \noindent See \cref{tab:examples} for examples.
% \begin{theorem}\label{thm:ordinary}
% Let $f\in\Skn$ and $v\leq w\in S_n$ be as in \cref{prop:v_w}. Then 
% \begin{equation}\label{eq:ordinary}
%   \dim  H^{k,(p,p)}(\Pio_f) =\dim \HHXC {-k}p_{v,w} \qquad\text{for all $k,p\in\Z$.}
% % H^\bul(\Rich_v^w)\cong \HHHC^0(\Fvw).
% \end{equation}
% \end{theorem}



% This result relies, among other things, on the curious Lefschetz property given in \cref{cor:master:qt_symm}, which so far has only been proven for positroid varieties. For arbitrary Richardson varieties, the gradings are matched in a more involved way. Recall that we have set $\lvw=\ell(w)-\ell(v)=\dim\Rich_v^w$.

% \begin{theorem}\label{thm:ordinary_Rich}
%  For all $v\leq w\in W$ and $k,p\in\Z$, we have
% \begin{equation}\label{eq:ordinary_Rich}
% % \dim  H^{k,(p,p)}(\Rich_v^w) =\dim \HHXC {\lvw-k-2p}{\lvw-p}_{v,w}.
% % \dim  H^{\lvw+k-2p,(\lvw-p,\lvw-p)}(\Rich_v^w) =\dim \HHXC {-k}{p}_{v,w}.
% \dim  H^{k,(p,p)}(\Rich_v^w) =\dim \HHXC {k-\lvw+2p}{\lvw-p}_{v,w}.
% % H^\bul(\Rich_v^w)\cong \HHHC^0(\Fvw).
% \end{equation}
% \end{theorem}




\parag{Equivariant cohomology}%\label{sec:main-result-equivariant}
The spaces $\HHH^0(\Fvw)$ and $\HHHC^0(\Fvw)$ are closely related. By \cref{thm:ordinary}, $\HHHC^0(\Fvw)$ yields the cohomology of $\Rich_v^w$. It turns out that $\HHH^0(\Fvw)$ yields the \emph{torus-equivariant} cohomology of $\Rich_v^w$. 


The algebraic torus $T$ acts on each Richardson variety $\Rich_v^w$, and thus we can consider its \emph{$T$-equivariant cohomology with compact support}, denoted $H^\bul_{T,c}(\Rich_v^w)$. It is equipped with an action of the ring $H^\bul_{T,c}(\pt)\cong R$. Similarly to the positroid case, $H^\bul_{T,c}(\Rich_v^w)$ admits a second grading via the mixed Hodge structure and is therefore a bigraded $R$-module. 
% We are ready to state the equivariant version of our main result.

\begin{theorem}\label{thm:main}
 For all $v\leq w\in W$, we have an isomorphism of bigraded $R$-modules
\begin{equation}\label{eq:main}
H^\bul_{T,c}(\Rich_v^w)\cong \HHH^0(\Fvw).
\end{equation}
It restricts to a vector space isomorphism $H^{\lvw+2p+k,(p,p)}_{T,c}(\Rich_v^w)\cong \HHX {k}p_{v,w}$ for each $k,p\in\Z$, 
% \begin{equation}\label{eq:grading:intro}
%   H^{\lvw+2p+k,(p,p)}_{T,c}(\Rich_v^w)\cong \HHX {k}p_{v,w},
%   % H^{\ell(w)-\ell(v)+2p+k,(p,p)}_{T,c}(\Rich_v^w)\cong \HHX {k}p_{v,w}.
% \end{equation}
where $\lvw=\ell(w)-\ell(v)=\dim\Rich_v^w$.
\end{theorem}
% \noindent See \cref{tab:examples} for examples.

% Unlike in the case of \cref{thm:ordinary}, the bigradings are matched in a more complicated fashion. Denote $\lvw:=\ell(w)-\ell(v)=\dim\Rich_v^w$. Then the isomorphism~\eqref{eq:main} restricts to a vector space isomorphism 
% \begin{equation}\label{eq:grading:intro}
%   H^{\lvw+2p+k,(p,p)}_{T,c}(\Rich_v^w)\cong \HHX {k}p_{v,w} \quad\text{for each $k,p\in\Z$.}
% \end{equation}
 % We show in~\todo that the left (resp., right) hand side of~\eqref{eq:main} is a direct sum of the left (resp., right) hand sides of~\eqref{eq:grading:intro} over $k,p\in\Z$.
% See~\todo for the precise details on how the two bigradings match each other.



% \scalebox



% \begin{remark} %(cf.\ \cref{thm:homfly})
% The bigraded \Poincare polynomial of $\HHH^0(\Fvw)$, denoted $\PK(\bhvw;q,t)$, is the top $a$-degree coefficient of the celebrated \emph{Khovanov--Rozansky homology}~\cite{KR1,KR2,KhoSoe} of $\bhvw$. The dimensions of bigraded components $\HHX {k}p_{v,w}$, encoded in the polynomial $\PK(\bhvw;q,t)$, form a link invariant, however, the $R$-module structure of $\HHH^0(\Fvw)$ is not itself a link invariant. Nevertheless, $\HHHC^0(\Fvw)$ depends on this $R$-module structure and it can be used to form a new link invariant.\todo
% \end{remark}


% We now explain how \cref{thm:main} generalizes our above results. 
% \begin{remark}
% Let $G=\SL_n(\C)$. It is well known that forgetting one of the gradings in Khovanov--Rozansky homology recovers the \FLY polynomial of the link. The (bigraded) cohomology with compact support yields the point count by the Grothendieck--Lefschetz trace formula~\cite{Gro66}. Thus \cref{thm:main} specializes to \cref{thm:homfly}.
% \end{remark}
% \begin{remark}
% Let $G=\SL_n(\C)$ and choose $f\in\Sknc$, $v\leq w\in S_n$ as in \cref{prop:v_w}, thus we have $\Rich_v^w=\Pio_f$. Denote 
% \begin{equation*}%\label{eq:*}
%   \lvw:=\ell(w)-\ell(v)=\dim(\Rich_v^w)=\dim(\Pio_f) \quad\text{and}\quad\df:=\dim(\Pio_f/T)=\lvw-n+1.  
% \end{equation*}
%  Since $\ncyc(f)=1$, the $T$-action on $\Pio_f$ is free by~\eqref{eq:free}. In this case, we have
% \begin{equation*}%\label{eq:HTc_vs_Hc}
% H^{k+n-1,(p,p)}_{T,c}(\Pio_f)\cong H^{k,(p,p)}_c(\Pit_f)\cong H^{2\df-k,(\df-p,\df-p)}(\Pit_f)\cong H^{\df+2p-k,(p,p)}(\Pit_f),
% \end{equation*}
% where $H_c$ denotes cohomology with compact support, the first isomorphism is Lemma~\ref{lem:HTfree}, the second isomorphism is the \Poincare duality~\eqref{eq:Poincare}, and the third isomorphism is the curious Lefschetz property~\eqref{eq:Koszul}. 
% % \begin{equation}\label{eq:}
% % \df:=\dim(\Pit_f)=\ell(w)-\ell(v)-(n-1).
% % \end{equation}
%  % The isomorphisms in~\eqref{eq:HTc_vs_Hc} respect the bigradings, t
% Therefore~\eqref{eq:main} yields 
% \begin{equation}\label{eq:main_KR}
% \Potf q,t)=   \left(\qq\tt\right)^{\chi(\beta_f)} \PK(\betah_f;q,t),
% \end{equation}
% where $\chi(\beta_f):=\frac{\lvw-n+\ncyc(f)}2$; see~\eqref{eq:chi_dfn}. In the special case $f=\fkn$ and $\gcd(k,n)=1$, \cref{rmk:torus_knot} implies that $\betah_f$ is a torus knot, for which the right hand side of~\eqref{eq:main_KR} was shown by Mellit~\cite{Mellit_torus} to coincide with $\Cat_{k,n-k}(q,t)$. Thus, \cref{thm:master_kn} follows from \cref{thm:main}.
% \end{remark}



% \begin{remark}\label{rmk:new_lk_inv}
% Similarly to the case of $\HHH^0(\FR(\beta))$, one can study its suitably renormalized bigraded \Poincare polynomial $\PKRC(\beta;q,t)$; see~\eqref{eq:PKRC}. It is not hard to check that $\PKRC(\beta;q,t)$ is invariant under braid conjugation and positive Markov moves. However, both $\HHH^0(\FR(\beta))$ and $\HHHC^0(\FR(\beta))$ are \emph{not} invariant under the negative Markov moves, cf.\ \cref{ex:unknot_III}. We expect that after taking the full Hochschild cohomology into account, $\PKRC(\beta;q,t)$ becomes the top $a$-degree coefficient of a link invariant $\PKRCA(\betah;a,q,t)$. This will be addressed in a separate paper. The precise relationship between $\PKR(\betah;a,q,t)$ and $\PKRCA(\betah;a,q,t)$ --- as well as the geometric meaning its of lower $a$-degree coefficients --- remain mysterious to us. 
% As we show in \cref{sec:ordinary-cohomology}, when $\bhvw$ is a knot, we have
% \begin{equation}\label{eq:PKRtop=PKRC}
%   \PKRtop(\bhvw;q,t)=\PKRC(\bvw;q,t).
% \end{equation}
% We expect the same result to hold for arbitrary knots $\betah$; see \cref{conj:PKRtop=PKRC}. It is not clear what happens in the case $\ncyc(\betah)>1$.
% % Conjecturally~\todo, the same result holds for arbitrary knots $\betah$. It is not clear what happens in the case $\ncyc(\betah)>1$.
% \end{remark}
% % When $\ncyc(\betah)>1$, the relationship between the two polynomials is unclear.

% % \TODOTL{mention Tor result here and deduce \cref{thm:ordinary} from \cref{thm:main}}



\parag{Koszul duality and $q,t$-symmetry}
One can encode the dimensions of bigraded components of $\HHH^0(\FR(\beta))$, resp., $\HHHC^0(\FR(\beta))$ in a two-variable polynomial $\PK(\beta;q,t)$, resp., $\PKRC(\beta;q,t)$.\footnote{The polynomial $\PK(\beta;q,t)$ the top $a$-degree coefficient of \emph{Khovanov--Rozansky homology}~\cite{KR1} of $\betah$.}
% By the results of~\cite{LS,GL}, 
For any $f\in\Sknc$, the positroid variety $\Pit_f$ is a cluster variety~\cite{GL}, so the polynomial $\PK(\betah_f;q,t)$ satisfies the properties~\itemref{thm:LS:qt_symm}--\itemref{thm:LS:point_cnt} listed in \cref{thm:LS} by the results of~\cite{LS}. In particular, it is $q,t$-symmetric and $q,t$-unimodal. 
% For a general link $\betah$, the question of whether $\PK(\betah;q,t)$ is $q,t$-symmetric has been a major open problem\footnote{At the final stages of the preparation of this manuscript, we found that the $q,t$-symmetry of $\PK(\betah;q,t)$ for the case when $\betah$ is a knot has been established in a very recent preprint~\cite{ObRoz}.} going back to~\cite{DGR}.

Richardson varieties are not yet known to admit cluster structures (see~\cite{Lec}), in particular, it does not follow from \cref{thm:LS} that $\PK(\bhvw;q,t)$ is $q,t$-symmetric for arbitrary $v\leq w\in S_n$. We show that the $q,t$-symmetry phenomenon for such links is a manifestation of \emph{Koszul duality} for mixed perverse sheaves~\cite{BGS,BY}.

% Despite the above lack of cluster structure~\todo, one can still study the polynomial $\PK(\bhvw;q,t)$ defined through Khovanov--Rozansky homology. In fact, we have shown the following result, which is a manifestation of \emph{Koszul duality} for mixed perverse sheaves~\cite{AR2,BY}. 

\begin{theorem}[Koszul duality]\label{thm:Koszul_intro}
%  \ 
% \begin{itemize}
% \item   For any $v\leq w\in S_n$, we have $\PKRC(\bvw;q,t) =\PKRC(\bvw;t,q).$ 
% \item  If $\bhvw$ is a knot then $\PK(\bhvw;q,t) =\PK(\bhvw;t,q).$
% \end{itemize}
  For any $v\leq w\in S_n$, we have
\begin{equation*}%\label{eq:*}
  \PKRC(\bvw;q,t) =\PKRC(\bvw;t,q).
\end{equation*}
%  If $\bhvw$ is a knot then
% \begin{equation*}%\label{eq:*}
%   \PK(\bhvw;q,t) =\PK(\bhvw;t,q).
% \end{equation*}
%   For any $v\leq w\in S_n$, we have $\PKRC(\bvw;q,t) =\PKRC(\bvw;t,q).$ 
% % \begin{equation*}%\label{eq:*}
% %   \PKRC(\bvw;q,t) =\PKRC(\bvw;t,q).
% % \end{equation*}
%  If $\bhvw$ is a knot then $\PK(\bhvw;q,t) =\PK(\bhvw;t,q).$
% \begin{equation*}%\label{eq:*}
%   \PK(\bhvw;q,t) =\PK(\bhvw;t,q).
% \end{equation*}
\end{theorem}
\noindent If $\bhvw$ is a knot then it follows that $\PK(\bhvw;q,t) =\PK(\bhvw;t,q).$  This gives a new proof of the $q,t$-symmetry of $\Cat_{k,n-k}(q,t)$ for $\gcd(k,n)=1$.

\section{Catalan numbers associated to positroid varieties}
An important combinatorial consequence of our results is an embedding of rational $q,t$-Catalan numbers $\Cat_{k,n-k}(q,t)$ into a family of $q,t$-polynomials $\Poinc(\Pio_f/T;q,t)\in\N[\qq,\tt]$ indexed by permutations $f\in\Sknc$ (all of which are $q,t$-symmetric and $q,t$-unimodal).
 % The goal of this section is to give a combinatorial interpretation to a specialization of $\Poinc(\Pio_f/T;q,t)$.% related to the point count as in \cref{thm:LS:point_cnt}.
\begin{definition}
  For $f\in\Sknc$, define the \emph{$f$-Catalan number} $\Cat_f\in\Z$ as the $q=1$ specialization of the point count polynomial $\#(\Pio_f/T)(\F_q)$.
% \begin{equation*}%\label{eq:*}
%   \Cat_f:= \Poinc(\Pio_f/T;q,t) \big|_{\qq=1,\tt=-1}.
% \end{equation*}
% Alternatively, $\Cat_f$ is the $q=1$ specialization of the point count polynomial $\#(\Pio_f/T)(\F_q)$, and we also have $\Cat_f=\Ptopnoq_f(1)$, where the polynomial $\Ptop_f$ is defined in \cref{thm:homfly}.
\end{definition}
% \noindent 
 % In particular, $\Cat_{\fkn}=\Cat_{k,n-k}(1,1)=\#\Dyck_{k,(n-k)}$ is the usual rational Catalan number when $\gcd(k,n)=1$. 

Let us focus on the case $f=\fkn$ with $\gcd(k,n)=1$. We will show that $\Cat_{\fkn}=\Cat_{k,n-k}(1,1)=\#\Dyck_{k,(n-k)}$ counts certain pipe dreams inside a $k\times (n-k)$ rectangle. This interpretation extends to arbitrary $f\in\Skn$ in a straightforward fashion. 

\begin{definition}
Let $\gcd(k,n)=1$. A \emph{maximal $\fkn$-Deogram} (short for \emph{Deodhar diagram}) is a way of placing $n-1$ elbows in a $k\times(n-k)$ rectangle and filling the rest with crossings so that (i) the resulting permutation obtained by following the paths is the identity, and (ii) the following \emph{distinguished condition}~\cite{Deodhar} is satisfied: if any two paths have crossed an odd number of times, they cannot form an elbow.
 % See \cref{fig:Go_3_8} for an example.
% the distinguished condition in \cref{dfn:Go} is satisfied. 
\end{definition}
% \noindent An example of a filling satisfying condition (i) but violating condition (ii) is shown in \figref{fig:NC_alt}(right).



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% }
%   \caption{\label{fig:Go_3_8} The sets $\Gom_\fkn$ and $\Dyck_{k,n-k}$ have the same cardinality by~\eqref{eq:Cat=Go}. Compare with \cref{fig:Cab_ex}.}
% \end{figure}

Denote the set of maximal $\fkn$-Deograms by $\Gom_\fkn$. It follows by combining our results with~\cite{Deodhar} that $\Cat_\fkn$ equals the number of maximal $\fkn$-Deograms:
\begin{equation}\label{eq:Cat=Go}
  \Cat_\fkn=\#\Gom_\fkn.
\end{equation}
An analogous result holds for arbitrary $f\in\Skn$. It would be interesting to give a bijective proof of~\eqref{eq:Cat=Go}. For example, the maximal $f_{3,8}$-Deograms are shown below.

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		\end{tikzpicture}
			}
&
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		\draw[line width=2pt] (3.50,3.95)--(3.50,4.15);
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		\draw[line width=2pt] (4.50,3.95)--(4.50,4.15);
		\node[anchor=south,scale=0.50,inner sep=1pt] (A) at (4.50,4.15) {$$};
		\end{tikzpicture}
			}\end{tabular}
}
According to~\eqref{eq:Cat=Go}, these objects are in bijection with Dyck paths in \cref{fig:Cab_ex}.

\begin{problem}\label{prob:catalan}
Find a bijection between $\Gom_\fkn$ and $\Dyck_{k,(n-k)}$ for the case $\gcd(k,n)=1$.
% Give a bijective proof of~\eqref{eq:Cat=Go}. 
% That is, find a bijection between $\Gom_\fkn$ and $\Dyck_{k,(n-k)}$ for the case $\gcd(k,n)=1$.
\end{problem}
% \noindent For instance, \cref{fig:Cab_ex,fig:Go_3_8} both have $7$ objects in them, but it is unclear which objects correspond to which.


% \begin{figure}
% \centering
% \makebox[1.0\textwidth]{

% \begin{tabular}{c|c}


% \setlength{\tabcolsep}{3pt}
% \begin{tabular}{ccccc}
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% 					\draw[line width=2pt] (1.50,1.05)--(1.50,0.85);
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% 		\node[anchor=south,scale=0.50,inner sep=1pt] (A) at (0.50,4.15) {$$};
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% 		\draw[line width=2pt] (3.50,3.95)--(3.50,4.15);
% 		\node[anchor=south,scale=0.50,inner sep=1pt] (A) at (3.50,4.15) {$$};
% 		\drawtree{2/3/4/15,1/3/4/30,0/3/4/45}\end{tikzpicture}
% 			}
% &
% \scalebox{0.80}{
% 	\begin{tikzpicture}[xscale=0.5,yscale=0.5]
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% 				\draw[line width=2pt] (0.50,3.95)--(0.50,4.15);
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% 		\draw[line width=2pt] (1.50,3.95)--(1.50,4.15);
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% 		\draw[line width=2pt] (3.50,3.95)--(3.50,4.15);
% 		\node[anchor=south,scale=0.50,inner sep=1pt] (A) at (3.50,4.15) {$$};
% 		\drawtree{0/3/4/45,1/2/4/15,0/2/4/30}\end{tikzpicture}
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% \scalebox{0.80}{
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% 		\draw[line width=2pt] (2.50,3.95)--(2.50,4.15);
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% 		\draw[line width=2pt] (3.50,3.95)--(3.50,4.15);
% 		\node[anchor=south,scale=0.50,inner sep=1pt] (A) at (3.50,4.15) {$$};
% 		\drawtree{2/3/4/15,0/3/4/45,0/1/4/15}\end{tikzpicture}
% 			}
% &
% \scalebox{0.80}{
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% 	\godiagx{0/3,3/3,0/2,2/2,0/1,1/1}{3/2,2/1,3/1}{1/3,2/3,1/2}
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% 				\draw[line width=2pt] (0.50,3.95)--(0.50,4.15);
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% 		\draw[line width=2pt] (2.50,3.95)--(2.50,4.15);
% 		\node[anchor=south,scale=0.50,inner sep=1pt] (A) at (2.50,4.15) {$$};
% 		\draw[line width=2pt] (3.50,3.95)--(3.50,4.15);
% 		\node[anchor=south,scale=0.50,inner sep=1pt] (A) at (3.50,4.15) {$$};
% 		\drawtree{0/3/4/45,0/2/4/30,0/1/4/15}\end{tikzpicture}
% 			}\end{tabular}

% &

% \scalebox{0.7}{
% 	\begin{tikzpicture}[xscale=0.5,yscale=0.5,baseline=(ZUZU.base)]
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% 					\node[anchor=north,scale=0.50,inner sep=1pt] (A) at (0.50,0.85) {$1$};
% 					\draw[line width=2pt] (1.50,1.05)--(1.50,0.85);
% 					\node[anchor=north,scale=0.50,inner sep=1pt] (A) at (1.50,0.85) {$2$};
% 					\draw[line width=2pt] (2.50,1.05)--(2.50,0.85);
% 					\node[anchor=north,scale=0.50,inner sep=1pt] (A) at (2.50,0.85) {$3$};
% 					\draw[line width=2pt] (3.50,1.05)--(3.50,0.85);
% 					\node[anchor=north,scale=0.50,inner sep=1pt] (A) at (3.50,0.85) {$4$};
% 					\draw[line width=2pt] (4.50,1.05)--(4.50,0.85);
% 					\node[anchor=north,scale=0.50,inner sep=1pt] (A) at (4.50,0.85) {$5$};
% 					\draw[line width=2pt] (5.50,1.05)--(5.50,0.85);
% 					\node[anchor=north,scale=0.50,inner sep=1pt] (A) at (5.50,0.85) {$6$};
% 					\draw[line width=2pt] (6.50,1.05)--(6.50,0.85);
% 					\node[anchor=north,scale=0.50,inner sep=1pt] (A) at (6.50,0.85) {$7$};
% 					\draw[line width=2pt] (6.95,1.50)--(7.15,1.50);
% 				\node[anchor=west,scale=0.50,inner sep=1pt] (A) at (7.15,1.50) {$8$};
% 				\draw[line width=2pt] (0.05,1.50)--(-0.15,1.50);
% 				\node[anchor=east,scale=0.50,inner sep=1pt] (A) at (-0.15,1.50) {$1$};
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% 				\node[anchor=west,scale=0.50,inner sep=1pt] (A) at (7.15,2.50) {$9$};
% 				\draw[line width=2pt] (0.05,2.50)--(-0.15,2.50);
% 				\node[anchor=east,scale=0.50,inner sep=1pt] (A) at (-0.15,2.50) {$2$};
% 				\draw[line width=1.5pt,black!50] (6.95,3.00)--(7.05,3.00);\draw[line width=1.5pt,black!50] (7.00,2.95)--(7.00,4.05);\draw[line width=2pt] (6.95,3.50)--(7.15,3.50);
% 				\node[anchor=west,scale=0.50,inner sep=1pt] (A) at (7.15,3.50) {$10$};
% 				\draw[line width=2pt] (0.05,3.50)--(-0.15,3.50);
% 				\node[anchor=east,scale=0.50,inner sep=1pt] (A) at (-0.15,3.50) {$3$};
% 				\draw[line width=1.5pt,black!50] (6.95,4.00)--(7.05,4.00);\draw[line width=1.5pt,black!50] (7.00,3.95)--(7.00,5.05);\draw[line width=2pt] (6.95,4.50)--(7.15,4.50);
% 				\node[anchor=west,scale=0.50,inner sep=1pt] (A) at (7.15,4.50) {$11$};
% 				\draw[line width=2pt] (0.05,4.50)--(-0.15,4.50);
% 				\node[anchor=east,scale=0.50,inner sep=1pt] (A) at (-0.15,4.50) {$4$};
% 				\draw[line width=1.5pt,black!50] (6.95,5.00)--(7.05,5.00);\draw[line width=1.5pt,black!50] (7.00,4.95)--(7.00,6.05);\draw[line width=2pt] (6.95,5.50)--(7.15,5.50);
% 				\node[anchor=west,scale=0.50,inner sep=1pt] (A) at (7.15,5.50) {$12$};
% 				\draw[line width=2pt] (0.05,5.50)--(-0.15,5.50);
% 				\node[anchor=east,scale=0.50,inner sep=1pt] (A) at (-0.15,5.50) {$5$};
% 				\draw[line width=2pt] (0.50,5.95)--(0.50,6.15);
% 		\node[anchor=south,scale=0.50,inner sep=1pt] (A) at (0.50,6.15) {$6$};
% 		\draw[line width=2pt] (1.50,5.95)--(1.50,6.15);
% 		\node[anchor=south,scale=0.50,inner sep=1pt] (A) at (1.50,6.15) {$7$};
% 		\draw[line width=2pt] (2.50,5.95)--(2.50,6.15);
% 		\node[anchor=south,scale=0.50,inner sep=1pt] (A) at (2.50,6.15) {$8$};
% 		\draw[line width=2pt] (3.50,5.95)--(3.50,6.15);
% 		\node[anchor=south,scale=0.50,inner sep=1pt] (A) at (3.50,6.15) {$9$};
% 		\draw[line width=2pt] (4.50,5.95)--(4.50,6.15);
% 		\node[anchor=south,scale=0.50,inner sep=1pt] (A) at (4.50,6.15) {$10$};
% 		\draw[line width=2pt] (5.50,5.95)--(5.50,6.15);
% 		\node[anchor=south,scale=0.50,inner sep=1pt] (A) at (5.50,6.15) {$11$};
% 		\draw[line width=2pt] (6.50,5.95)--(6.50,6.15);
% 		\node[anchor=south,scale=0.50,inner sep=1pt] (A) at (6.50,6.15) {$12$};
% 		\drawelbowfill{3}{3}{red!40}\end{tikzpicture}
% 			}

% \\

% & \textcolor{red}{not} distinguished
% \end{tabular}
% }
%   \caption{\label{fig:NC_alt} Left: for $n=2k+1$, maximal $\fkn$-Deograms are in bijection with non-crossing alternating trees. Right: a non-distinguished filling.}
% \end{figure}



% \begin{remark}\label{rmk:NC_alt}
For the case $n=2k+1$ of the standard Catalan numbers, the maximal $\fkn$-Deograms are easily seen (exercise) to be in bijection with \emph{non-crossing alternating trees} on $n+1$ vertices. 
% Explicitly, given $\fil\in\Gom_\fkn$, we assign a vertex to each of the $n+1$ columns of $\fil$. One can show that every row of $\fil$ must contain exactly two elbows, and connecting the two corresponding vertices by an edge for each of the $n$ rows, one obtains a non-crossing alternating tree.
 % The case $k=3$, $n=7$ is illustrated in \cref{fig:NC_alt}.
% \end{remark}
% \begin{remark}
  A recursive proof of~\eqref{eq:Cat=Go} for the case $n=dk\pm1$ ($d\geq 2$) was found by David Speyer. 
% It appears that when $n=dk\pm1$, the distinguished condition is automatically satisfied for any maximal $\fkn$-Deogram, however, this is not the case for instance when $k=5$ and $n=12$; see \cref{fig:Go_ex}(right).
 We were able to find a recursive proof of~\eqref{eq:Cat=Go} for arbitrary $k,n$. The problem of finding a bijective proof remains open.
% This and some other enumerative consequences of our results will appear in a separate paper.
% \end{remark}



% \TL{Remark: the proof can be extended to $q,t$-symmetry for other links.}
% \TODO{Other type knot invariants}


% \section{Extensions of Verma modules}\label{sec:Verma_intro}
% The above theorems have representation-theoretic applications to \emph{Verma modules}, which are certain infinite-dimensional modules over the Lie algebra $\g$ of $G$. Consider the principal block $\O_0$ of the Bernstein–Gelfand–Gelfand category~$\O$, and let $M_w$ be the Verma module with highest weight $w(\rho)-\rho$, where $\rho$ is half the sum of positive roots of the root system of $\g$. We also denote by $L_w$ the corresponding simple module. The graded dimensions of $\Ext^\bul(M_v,L_w)$ famously coincide with the coefficients of the \emph{Kazhdan--Lusztig $P$-polynomials} $P_{v,w}(q)$; see e.g.~\cite[Theorem~3.11.4]{BGS}. On the other hand, computing extension groups $\Ext^\bul(M_v,M_w)$ is an important open problem.

% A \emph{graded} version of $\O_0$ was introduced by Beilinson--Ginzburg--Soergel~\cite{BGS}. They constructed the (essentially unique) graded lifts of Verma modules $M_w$ (see also~\cite{Stroppel}), thus endowing the space $\Ext^\bul(M_v,M_w)$ with a second grading:
% \begin{equation*}%\label{eq:*}
%   \Ext^\bul(M_v,M_w)=\bigoplus_{k,\wt\in\Z} \Ext^{k,(\wt/2)}(M_v,M_w).
% \end{equation*}
% These $\Ext$-groups can be related to the cohomology of open Richardson varieties using the \emph{localization theorem} of~\cite{BB,BK}.
% % \TL{Replaced by this.}
% %\TL{How about the following sentence?  The $\Ext$-group $\Ext^\bul(M_v,M_w)$ can be directly related to the $\Ext$-groups $\Ext^\bul(\DeltaC_v,\DeltaC_w)$ by  the \emph{localization theorem} of~\cite{BB,BK}.} \PG{I like it but we would have to define $\DeltaC_v$ for that. Not sure how to proceed here.}
% % (A closely related result is the \emph{localization theorem} of~\cite{BB,BK}.) 
% For the case of Kazhdan--Lusztig polynomials, the groups $\Ext^\bul(M_v,L_w)$ are ``pure": the two gradings agree.  On the other hand, the bigrading on $\Ext^\bul(M_v,M_w)$ is quite non-trivial. As a corollary to our approach, we obtain the following result.
% \begin{theorem}\label{thm:Verma}
%   For all $v\leq w\in W$ and $k,\wt\in\Z$, we have an isomorphism
% \begin{equation*}%\label{eq:*}
%   \Ext^{k,((\wt-\lvw)/2)}(M_v,M_w)\cong H^{\wt-k,(\wt/2,\wt/2)}(\Rich_v^w)
% \end{equation*}
% of vector spaces. (In particular, both sides are zero for odd $\wt$.)
% \end{theorem}


\begin{remark}
For a class of \emph{repetition-free permutations}, a combinatorial interpretation of the numbers $\Cat_f$ in terms of Dyck paths avoiding a convex shape was recently given in~\cite{cat_combin}. We refer to~\cite[Section~7]{cat_combin} for relations between their $q,t$-analogs and the results of~\cite{BHMPS,GHSR,GNR,OR_Cox}.
\end{remark}


\acknowledgements{We are grateful to David Speyer for his contributions to this project at the early stages. We also thank Pramod Achar, Sergey Fomin, Alexei Oblomkov, and Vivek Shende for stimulating discussions.}

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