\section{Introduction} \label{intro}

Let $R$ be a commutative ring, and fix two scalars $[2]_s,[2]_t \in R$.
The \defnemph{two-colored Temperley--Lieb algebra} $2\mathrm{TL}_R(\prescript{}{s}{n}):=2\mathrm{TL}_R(\prescript{}{s}{n};[2]_s,[2]_t)$ is the $R$-algebra with generators 
$e_i$ for $1 \leq i \leq n-1$,
subject to the relations
\begin{align}
e_i^2& =-[2]_s & & \text{$i$ odd,} \label{eq:oddquadratic} \\
e_i^2& =-[2]_t & & \text{$i$ even,} \label{eq:evenquadratic} \\
e_i e_j & =e_j e_i & & \text{for $|i-j|>1$,} \\
e_i e_{i\pm 1} e_i & =e_i & & \label{eq:braidreln}
\end{align}
The algebra $2\mathrm{TL}_R(\prescript{}{t}{n})$ is defined identically, except that the parity conditions on the relations \eqref{eq:oddquadratic} and \eqref{eq:evenquadratic} are swapped. 
These algebras form a generalization of the ordinary Temperley--Lieb algebra, which occurs as a special case when $[2]_s=[2]_t$. 
The relations immediately imply that there is an $R$-basis of $2\mathrm{TL}_R(\prescript{}{s}{n})$ consisting of monomials in the generators $e_i$.

We call a non-zero idempotent $\mathrm{JW}_R(\prescript{}{s}{n}) \in 2\mathrm{TL}_R(\prescript{}{s}{n})$ (and similarly for $\prescript{}{t}{n}$) a \defnemph{two-colored Jones--Wenzl projector} if $e_i\mathrm{JW}_R(\prescript{}{s}{n})=0$ for all $1 \leq i \leq n-1$ and the coefficient of $1$ in $\mathrm{JW}_R(\prescript{}{s}{n})$ is $1$. 
Such idempotents (if they exist) are unique. 


The behavior of $2\mathrm{TL}_R(\prescript{}{s}{n})$ is controlled by certain elements $[n]_s,[n]_t \in R$ for $n \in \mathbb{Z}$ called the \defnemph{two-colored quantum numbers}. 
These elements (defined in \eqref{eq:twocolqnum} below) are bivariate polynomials in $[2]_s$ and $[2]_t$ which are analogous to ordinary quantum numbers.
For an integer $0 \leq k \leq n$ the \defnemph{two-colored quantum binomial coefficient}
\begin{equation*}
\qbinom{n}{k}_{s}=\frac{[n]_{s}!}{[k]_{s}![n-k]_{s}!}=\frac{[n]_{s}[n-1]_{s}\dotsm [n-k+1]_{s}}{[k]_{s}[k-1]_{s} \dotsm [1]_{s}}
\end{equation*}
can also be shown to be an element of $R$.
Our first main result is the two-colored analogue of the well-known existence theorem for ordinary Jones--Wenzl projectors.

\begin{thmletter} \label{existence}
The two-colored Jones--Wenzl projector $\mathrm{JW}_R(\prescript{}{s}{n})$ exists if and only if $\qbinom{n}{k}_{s}$ is invertible in $R$ for each integer $0 \leq k \leq n$.
\end{thmletter}


The terminology for two-colored Temperley--Lieb algebras comes from their presentation as diagram algebras.
We associate the labels $s$ and $t$ with the colors red and blue, respectively, writing ${\color{red} s}$ and ${\color{blue} t}$ for emphasis.
A \defnemph{two-colored Temperley Lieb diagram} is a Temperley--Lieb diagram with the planar regions between strands colored with alternating colors. 
The algebra $2\mathrm{TL}_R(\prescript{}{{\color{red} s}}{n})$ is spanned by two-colored Temperley--Lieb diagrams with $n$ boundary points on the top and bottom whose leftmost region is colored red.
A blue circle inside a red region evaluates to $-[2]_{{\color{red} s}}$, while a red circle inside a blue region evaluates to $-[2]_{{\color{blue} t}}$. 
We draw the two-colored Jones--Wenzl projector as a rectangle labeled $\mathrm{JW}_R(\prescript{}{{\color{red} s}}{n})$:
\begin{equation*}
\begin{gathered}
        \begin{tikzpicture}[xscale=-.3,yscale=0.2]
          \begin{scope}
            \clip (-3,3.5) rectangle (3,-3.5);
            \draw[fill=dgrmred] (0,4) rectangle (4,-4);
            \draw[fill=dgrmblu] (1,4) rectangle (2,-4);
        
        
         \draw[fill=dgrmblu] (-4,4) rectangle (-2,-4);
         \draw[fill=white] (-2.5,2) rectangle (2.5,-2);
         \node at (0,0) {$\mathrm{JW}_R(\prescript{}{{\color{red} s}}{n})$};
         \node at (-1,2.7) {$\dots$};
         \node at (-1,-2.7) {$\dots$};
       \end{scope}
       \draw[dashed] (-3,3.5) to (3,3.5);
       \draw[dashed] (-3,-3.5) to (3,-3.5);
     \end{tikzpicture} \\
\text{$n$ odd}
\end{gathered} \qquad \qquad \qquad \qquad 
\begin{gathered}
        \begin{tikzpicture}[xscale=-.3,yscale=0.2]
          \begin{scope}
            \clip (-3,3.5) rectangle (3,-3.5);
            \draw[fill=dgrmred] (0,4) rectangle (4,-4);
            \draw[fill=dgrmblu] (1,4) rectangle (2,-4);
        
        
         \draw[fill=dgrmred] (-4,4) rectangle (-2,-4);
         \draw[fill=white] (-2.5,2) rectangle (2.5,-2);
         \node at (0,0) {$\mathrm{JW}_R(\prescript{}{{\color{red} s}}{n})$};
         \node at (-1,2.7) {$\dots$};
         \node at (-1,-2.7) {$\dots$};
       \end{scope}
       \draw[dashed] (-3,3.5) to (3,3.5);
       \draw[dashed] (-3,-3.5) to (3,-3.5);
     \end{tikzpicture} \\
\text{$n$ even}
\end{gathered}
\end{equation*}

Suppose both $\mathrm{JW}_R(\prescript{}{{\color{red} s}}{n})$ and $\mathrm{JW}_R(\prescript{}{{\color{blue} t}}{n})$ exist.
We say that $\mathrm{JW}_R(\prescript{}{{\color{red} s}}{n})$ is \defnemph{rotatable} if the (clockwise and counterclockwise) rotations of $\mathrm{JW}_R(\prescript{}{{\color{red} s}}{n})$ by one strand are equal to some scalar multiple of $\mathrm{JW}_R(\prescript{}{{\color{blue} t}}{n})$:
\begin{align*}
\begin{gathered}
  \begin{tikzpicture}[xscale=.5,yscale=.4]
    \begin{scope}
         \clip (-3,2) rectangle (3.5,-2);
    \draw[fill=dgrmblu] (-3.3,3) rectangle (1.4,-3);
    \draw[fill=dgrmred] (3.8,3) rectangle (1.4,-3);
             \draw[fill=dgrmred] (-.9,3) rectangle (0.5,-3);
             \draw[fill=white] (-.3,3) rectangle (0.8,-3);
             \draw[fill=dgrmblu] (2,3) to (2,-1)
         to[out=-90,in=180] (2.5,-1.5) to[out=0,in=-90] (3,-1) to
         (3,3) to (2,3);
         \draw[fill=dgrmred] (-1.5,-3) to (-1.5,1)
         to[out=90,in=0] (-2,1.5) to[out=180,in=90] (-2.5,1) to
         (-2.5,-3) to (-1.5,-3);
         \draw[fill=white] (-1.9,1) rectangle (2.4,-1);
         \node at (.25,0) {$\mathrm{JW}_R(\prescript{}{{\color{red} s}}{n})$};
         \node at (.3,1.4) {$\dots$}; 
         \node at (.3,-1.4) {$\dots$};
         \end{scope}
         \draw[dashed] (-3,2) to (3.5,2);
         \draw[dashed] (-3,-2) to (3.5,-2);
       \end{tikzpicture}
\end{gathered}
    & =
     \lambda
\begin{gathered}
  \begin{tikzpicture}[xscale=.5,yscale=.4]
    \begin{scope}
         \clip (-2.3,2) rectangle (2.8,-2);
    \draw[fill=dgrmblu] (-3.3,3) rectangle (.5,-3);
             \draw[fill=dgrmred] (-1.5,3) rectangle (-.9,-3);
    \draw[fill=dgrmred] (3.8,3) rectangle (.5,-3);
    \draw[fill=dgrmblu] (2,3) rectangle (1.4,-3);
             \draw[fill=white] (-.3,3) rectangle (.8,-3);
         \draw[fill=white] (-1.9,1) rectangle (2.4,-1);
         \node at (.25,0) {$\mathrm{JW}_R(\prescript{}{{\color{blue} t}}{n})$};
         \node at (.3,1.4) {$\dots$}; 
         \node at (.3,-1.4) {$\dots$};
         \end{scope}
         \draw[dashed] (-2.3,2) to (2.8,2);
         \draw[dashed] (-2.3,-2) to (2.8,-2);
       \end{tikzpicture}
\end{gathered}
=
\begin{gathered}
  \begin{tikzpicture}[xscale=.5,yscale=-.4]
    \begin{scope}
         \clip (-3,2) rectangle (3.5,-2);
    \draw[fill=dgrmblu] (-3.3,3) rectangle (1.4,-3);
    \draw[fill=dgrmred] (3.8,3) rectangle (1.4,-3);
             \draw[fill=dgrmred] (-.9,3) rectangle (0.5,-3);
             \draw[fill=white] (-.3,3) rectangle (0.8,-3);
             \draw[fill=dgrmblu] (2,3) to (2,-1)
         to[out=-90,in=180] (2.5,-1.5) to[out=0,in=-90] (3,-1) to
         (3,3) to (2,3);
         \draw[fill=dgrmred] (-1.5,-3) to (-1.5,1)
         to[out=90,in=0] (-2,1.5) to[out=180,in=90] (-2.5,1) to
         (-2.5,-3) to (-1.5,-3);
         \draw[fill=white] (-1.9,1) rectangle (2.4,-1);
         \node at (.25,0) {$\mathrm{JW}_R(\prescript{}{{\color{red} s}}{n})$};
         \node at (.3,1.4) {$\dots$}; 
         \node at (.3,-1.4) {$\dots$};
         \end{scope}
         \draw[dashed] (-3,2) to (3.5,2);
         \draw[dashed] (-3,-2) to (3.5,-2);
       \end{tikzpicture}
\end{gathered} & \text{($n$ odd)} \\
\begin{gathered}
  \begin{tikzpicture}[xscale=.5,yscale=.4]
    \begin{scope}
         \clip (-3,2) rectangle (3.5,-2);
    \draw[fill=dgrmblu] (-3.3,3) rectangle (3.8,-3);
             \draw[fill=dgrmred] (-.9,3) rectangle (1.4,-3);
             \draw[fill=white] (-.3,3) rectangle (.8,-3);
             \draw[fill=dgrmred] (2,3) to (2,-1)
         to[out=-90,in=180] (2.5,-1.5) to[out=0,in=-90] (3,-1) to
         (3,3) to (2,3);
         \draw[fill=dgrmred] (-1.5,-3) to (-1.5,1)
         to[out=90,in=0] (-2,1.5) to[out=180,in=90] (-2.5,1) to
         (-2.5,-3) to (-1.5,-3);
         \draw[fill=white] (-1.9,1) rectangle (2.4,-1);
         \node at (.25,0) {$\mathrm{JW}_R(\prescript{}{{\color{red} s}}{n})$};
         \node at (.3,1.4) {$\dots$}; 
         \node at (.3,-1.4) {$\dots$};
         \end{scope}
         \draw[dashed] (-3,2) to (3.5,2);
         \draw[dashed] (-3,-2) to (3.5,-2);
       \end{tikzpicture}
\end{gathered}
    & =
     \lambda
\begin{gathered}
  \begin{tikzpicture}[xscale=.5,yscale=.4]
    \begin{scope}
         \clip (-2.3,2) rectangle (2.8,-2);
    \draw[fill=dgrmblu] (-3.3,3) rectangle (3.8,-3);
             \draw[fill=dgrmred] (-1.5,3) rectangle (2,-3);
             \draw[fill=dgrmblu] (-.9,3) rectangle (1.4,-3);
             \draw[fill=white] (-.3,3) rectangle (.8,-3);
         \draw[fill=white] (-1.9,1) rectangle (2.4,-1);
         \node at (.25,0) {$\mathrm{JW}_R(\prescript{}{{\color{blue} t}}{n})$};
         \node at (.3,1.4) {$\dots$}; 
         \node at (.3,-1.4) {$\dots$};
         \end{scope}
         \draw[dashed] (-2.3,2) to (2.8,2);
         \draw[dashed] (-2.3,-2) to (2.8,-2);
       \end{tikzpicture}
\end{gathered}
=
\begin{gathered}
  \begin{tikzpicture}[xscale=.5,yscale=-0.4]
    \begin{scope}
         \clip (-3,2) rectangle (3.5,-2);
    \draw[fill=dgrmblu] (-3.3,3) rectangle (3.8,-3);
             \draw[fill=dgrmred] (-.9,3) rectangle (1.4,-3);
             \draw[fill=white] (-.3,3) rectangle (.8,-3);
             \draw[fill=dgrmred] (2,3) to (2,-1)
         to[out=-90,in=180] (2.5,-1.5) to[out=0,in=-90] (3,-1) to
         (3,3) to (2,3);
         \draw[fill=dgrmred] (-1.5,-3) to (-1.5,1)
         to[out=90,in=0] (-2,1.5) to[out=180,in=90] (-2.5,1) to
         (-2.5,-3) to (-1.5,-3);
         \draw[fill=white] (-1.9,1) rectangle (2.4,-1);
         \node at (.25,0) {$\mathrm{JW}_R(\prescript{}{{\color{red} s}}{n})$};
         \node at (.3,1.4) {$\dots$}; 
         \node at (.3,-1.4) {$\dots$};
         \end{scope}
         \draw[dashed] (-3,2) to (3.5,2);
         \draw[dashed] (-3,-2) to (3.5,-2);
       \end{tikzpicture}
\end{gathered} & \text{($n$ even)}
\end{align*}
Our second main result gives a combined condition for the existence and rotatability of two-colored Jones--Wenzl projectors.

\begin{thmletter} \label{existsrotates}
The two-colored Jones--Wenzl projectors $\mathrm{JW}_R(\prescript{}{{\color{red} s}}{n})$ and $\mathrm{JW}_R(\prescript{}{{\color{blue} t}}{n})$ exist and are rotatable if and only if $\qbinom{n+1}{k}_{{\color{red} s}}=\qbinom{n+1}{k}_{{\color{blue} t}}=0$ in $R$ for all integers $1 \leq k \leq n$.
\end{thmletter}

In the course of proving Theorems~\ref{existence} and \ref{existsrotates} we generalize several well-known results to the two-color setting which may be of independent interest. 
These including computations of the greatest common divisor and least common multiple of (two-colored) quantum binomial coefficients (\cref{qbinomideal} and \cref{qbinominvideal}) and the genericness of coefficients of $\mathrm{JW}_R(\prescript{}{{\color{red} s}}{n})$ (\cref{genericcoefs}) over arbitrary rings.

\subsection*{Soergel bimodules}

Two-colored Jones--Wenzl projectors lie at the heart of the Elias--Williamson construction of the diagrammatic Hecke category \cite{ew-soergelcalc}. 
Recently Abe has shown that there is a ``bimodule-theoretic'' category (a modification of the category of classical Soergel bimodules) which is equivalent to the diagrammatic Hecke category under certain assumptions \cite{abe-bimodhecke,abe-homBS}.
An important consequence of \cref{existsrotates} (which we discuss in the final section) is that these assumptions essentially always hold. 

\begin{corletter} \label{abeequalsew}
The diagrammatic Hecke category is equivalent to Abe's category of Soergel bimodules. 
\end{corletter}

More precisely, we use \cref{existsrotates} to give an algebraic condition on the base ring for determining when the diagrammatic Hecke category is well defined (\cref{realizcorrected}), completely correcting an error in \cite{ew-soergelcalc} (first identified and partially corrected in \cite{ew-localizedcalc}).
This algebraic condition is precisely Abe's \cite[Assumption~1.1]{abe-homBS}, so \cref{abeequalsew} follows from Abe’s results \cite[Theorem~3.9]{abe-homBS} and \cite[Theorem~5.6]{abe-bimodhecke}. 
We find it noteworthy that our correction gives the best possible equivalence result for two seemingly distinct categorifications of the Hecke algebra.


\subsection*{Acknowledgments}

We thank the Royal Commission for the Exhibition of 1851 and EPSRC (EP/V00090X/1) for financial support.


\section{Preliminaries}
\label{prelim}


Let $A=\mathbb{Z}[x_{{\color{red} s}},x_{{\color{blue} t}}]$ be the integral polynomial ring in two variables. 
The \defnemph{two-colored quantum numbers} are defined as follows. 
First set $[1]_{{\color{red} s}}=[1]_{{\color{blue} t}}=1$, $[2]_{{\color{red} s}}=x_{{\color{red} s}}$, and $[2]_{{\color{blue} t}}=x_{{\color{blue} t}}$ in $A$. 
For $n>1$ we inductively define
\begin{align}
[n+1]_{{\color{red} s}}& =[2]_{{\color{red} s}} [n]_{{\color{blue} t}} - [n-1]_{{\color{red} s}} \text{,} & [n+1]_{{\color{blue} t}} & =[2]_{{\color{blue} t}} [n]_{{\color{red} s}} - [n-1]_{{\color{blue} t}} \text{.} \label{eq:twocolqnum}
\end{align}
These formulas can be rearranged to inductively define $[n]_{{\color{red} s}}$ and $[n]_{{\color{blue} t}}$ for $n \leq 0$. 
For a commutative $A$-algebra $R$, we also define two-colored quantum numbers in $R$ to be the specializations of two-colored quantum numbers in $A$, which we will write in the same way.

These polynomials are bivariate extensions of the usual (one-colored) quantum numbers, which can be recovered as follows.
Let $\overline{A}=A/(x_{{\color{red} s}}-x_{{\color{blue} t}}) \cong \mathbb{Z}[x]$, where $x$ is the image of $x_{{\color{red} s}}$ or $x_{{\color{blue} t}}$.
Then the one-colored quantum number $[n]$ is the image of $[n]_{{\color{red} s}}$ or $[n]_{{\color{blue} t}}$ in $\overline{A}$.
When $n$ is odd, $[n]$ is an even polynomial, so we can formally evaluate $[n]$ at $x=\sqrt{x_{{\color{red} s}} x_{{\color{blue} t}}}$ to obtain an element of $A$.
When $n$ is even, $[n]/[2]$ is an even polynomial, which we can similarly formally evaluate at $x=\sqrt{x_{{\color{red} s}} x_{{\color{blue} t}}}$.
In both cases, it is easy to show by induction that
\begin{equation} \label{eq:colordependence}
\begin{aligned}
[n]_{{\color{red} s}}& =[n](\sqrt{x_{{\color{red} s}} x_{{\color{blue} t}}})=[n]_{{\color{blue} t}} & & \text{if $n$ is odd,} \\
\frac{[n]_{{\color{red} s}}}{[2]_{{\color{red} s}}}& =\left(\frac{[n]}{[2]}\right)(\sqrt{x_{{\color{red} s}} x_{{\color{blue} t}}})=\frac{[n]_{{\color{blue} t}}}{[2]_{{\color{blue} t}}} & & \text{if $n$ is even}
\end{aligned}
\end{equation}
in $A$.
In other words, two-colored quantum numbers are essentially the same as ordinary quantum numbers up to a factor of $[2]_{{\color{red} s}}$ and $[2]_{{\color{blue} t}}$ depending on color.

It is self-evident that the automorphism of $A$ which exchanges $x_{{\color{red} s}}$ and $x_{{\color{blue} t}}$ (``color swap'') also exchanges $[n]_{{\color{red} s}}$ and $[n]_{{\color{blue} t}}$ for all $n$.
For this reason, we will generally write statements only for $[n]_{{\color{red} s}}$ and leave it to the reader to formulate color-swapped analogues.
Similarly we have $2\mathrm{TL}(\prescript{}{{\color{red} s}}{n}; [2]_{{\color{red} s}},[2]_{{\color{blue} t}}) \cong 2\mathrm{TL}(\prescript{}{{\color{blue} t}}{n}; [2]_{{\color{blue} t}},[2]_{{\color{red} s}})$, and this isomorphism maps $\mathrm{JW}_R(\prescript{}{{\color{red} s}}{n})$ to $\mathrm{JW}_R(\prescript{}{{\color{blue} t}}{n})$ when they exist, so we will only state our results for $2\mathrm{TL}_R(\prescript{}{{\color{red} s}}{n})$ and $\mathrm{JW}_R(\prescript{}{{\color{red} s}}{n})$.

As mentioned in \cref{intro}, the two-colored Temperley--Lieb algebra $2\mathrm{TL}(\prescript{}{{\color{red} s}}{n})$ has a basis consisting of reduced monomials in the generators $e_i$ (i.e.~monomials whose length cannot be reduced using \eqref{eq:oddquadratic}--\eqref{eq:braidreln}). 
Each basis element corresponds to a unique two-colored Temperley--Lieb diagram on $n$ strands, whose leftmost region is colored red.
Given an element $f \in 2\mathrm{TL}_R(\prescript{}{{\color{red} s}}{n})$ and a two-colored Temperley--Lieb diagram $D$ we will write
\begin{equation*}
\coeff_{{} \in f} D
\end{equation*}
for the coefficient of $D$ when $f$ is written in the diagrammatic basis.

If $R$ is a commutative $A$-algebra for which $\mathrm{JW}_R(\prescript{}{{\color{red} s}}{n})$ exists for all $n$ (e.g.~$R=\Frac A$), then the coefficients of $\mathrm{JW}_{R}(\prescript{}{{\color{red} s}}{n})$ can be calculated inductively as follows.
Suppose $D$ is a two-colored Temperley--Lieb diagram in $2\mathrm{TL}_R(\prescript{}{{\color{red} s}}{(n+1)})$.
Let $\hat{D}$ be the diagram with $n+2$ bottom boundary points and $n$ top boundary points obtained by folding down the strand connected to the top right boundary point of $D$.
If there is a strand connecting the $i$th and $(i+1)$th bottom boundary points of $\hat{D}$, let $D_i$ denote the two-colored Temperley--Lieb diagram with $n$ strands so obtained by deleting this cap. 
For example, if 
\begin{equation*}
D=\begin{gathered}
        \begin{tikzpicture}[xscale=.4,yscale=.2]
          \begin{scope}
            \clip (-3,3.5) rectangle (3,-3.5);
            \draw[fill=dgrmred] (2,4) rectangle (-3,-4);
            \draw[fill=dgrmblu] (2,4) rectangle (3,-4);
        
          \draw[fill=dgrmblu] (-2,-4) to (-2,4) to (-1,3.5) to[out=-90,in=90] (1,-3.5) to (1,-4) to (0,-3.5) to[out=90,in=0] (-.5,-2.5) to[out=180,in=90] (-1,-3.5) to (-2,-4);
          \draw[fill=dgrmblu] (0,4) to (0,3.5) to[out=-90,in=180] (.5,2.5) to[out=0,in=-90] (1,3.5) to (0,4);          
       \end{scope}
       \draw[dashed] (-3,3.5) to (3,3.5);
       \draw[dashed] (-3,-3.5) to (3,-3.5);
     \end{tikzpicture}
\end{gathered}
\end{equation*}
then
\begin{equation*}
\hat{D}=\begin{gathered}
        \begin{tikzpicture}[xscale=.4,yscale=.2]
          \begin{scope}
            \clip (-3,3.5) rectangle (3.5,-3.5);
            \draw[fill=dgrmred] (3.5,4) rectangle (-3,-4);
            \draw[fill=dgrmblu] (2,-4) to (2,-3.5) to[out=90,in=180] (2.5,-2.5) to[out=0,in=90] (3,-3.5) to (2,-4);
        
          \draw[fill=dgrmblu] (-2,-4) to (-2,4) to (-1,3.5) to[out=-90,in=90] (1,-3.5) to (1,-4) to (0,-3.5) to[out=90,in=0] (-.5,-2.5) to[out=180,in=90] (-1,-3.5) to (-2,-4);
          \draw[fill=dgrmblu] (0,4) to (0,3.5) to[out=-90,in=180] (.5,2.5) to[out=0,in=-90] (1,3.5) to (0,4);          
       \end{scope}
       \draw[dashed] (-3,3.5) to (3.5,3.5);
       \draw[dashed] (-3,-3.5) to (3.5,-3.5);
     \end{tikzpicture} \\
\end{gathered}
\end{equation*}
and
\begin{equation*}
D_2=\begin{gathered}
        \begin{tikzpicture}[xscale=.4,yscale=.2]
          \begin{scope}
            \clip (-3,3.5) rectangle (2,-3.5);
            \draw[fill=dgrmred] (3.5,4) rectangle (-3,-4);
            \draw[fill=dgrmblu] (-2,4) rectangle (-1,-4);
        
          \draw[fill=dgrmblu] (0,-4) to (0,-3.5) to[out=90,in=-180] (.5,-2.5) to[out=0,in=90] (1,-3.5) to (0,-4);          
          \draw[fill=dgrmblu] (0,4) to (0,3.5) to[out=-90,in=180] (.5,2.5) to[out=0,in=-90] (1,3.5) to (0,4);          
       \end{scope}
       \draw[dashed] (-3,3.5) to (2,3.5);
       \draw[dashed] (-3,-3.5) to (2,-3.5);
     \end{tikzpicture} \\
\end{gathered} \text{,} \qquad \qquad \qquad
D_5=\begin{gathered}
        \begin{tikzpicture}[xscale=.4,yscale=.2]
          \begin{scope}
            \clip (-3,3.5) rectangle (2,-3.5);
            \draw[fill=dgrmred] (3.5,4) rectangle (-3,-4);
        
          \draw[fill=dgrmblu] (-2,-4) to (-2,4) to (-1,3.5) to[out=-90,in=90] (1,-3.5) to (1,-4) to (0,-3.5) to[out=90,in=0] (-.5,-2.5) to[out=180,in=90] (-1,-3.5) to (-2,-4);
          \draw[fill=dgrmblu] (0,4) to (0,3.5) to[out=-90,in=180] (.5,2.5) to[out=0,in=-90] (1,3.5) to (0,4);          
       \end{scope}
       \draw[dashed] (-3,3.5) to (2,3.5);
       \draw[dashed] (-3,-3.5) to (2,-3.5);
     \end{tikzpicture} \\
\end{gathered} \text{.}
\end{equation*}

\begin{thm} \label{JWcoef}
Suppose $\mathrm{JW}_R(\prescript{}{{\color{red} s}}{n})$ and $\mathrm{JW}_R(\prescript{}{{\color{red} s}}{(n+1)})$ both exist.
We have
\begin{equation*}
\coeff_{{} \in \mathrm{JW}_{R}(\prescript{}{{\color{red} s}}{(n+1)})} D=\sum_{i} \frac{[i]_{{\color{Plum} u}}}{[n+1]_{{\color{red} s}}}\coeff_{{} \in \mathrm{JW}_{R}(\prescript{}{{\color{red} s}}{n})} D_i \text{,}
\end{equation*}
where the sum is taken over all positions $i$ where $D_i$ is defined, and ${\color{Plum} u}$ is the color of the deleted cap.
\end{thm}

\begin{proof}
The argument in the one-color setting (see \cite[Proposition~4.1]{morrison-JW} or \cite[Corollary~3.7]{frenkelkhovanov}) follows essentially unchanged from \cite[(6.29)]{ew-localizedcalc}.
\end{proof}

We will carefully show later that this computation is ``generic'', i.e.~if $\mathrm{JW}_R(\prescript{}{{\color{red} s}}{n})$ exists, then its coefficients are specializations of the coefficients of $\mathrm{JW}_{\Frac A}(n)$.

The existence criterion in \cref{existence} is known to hold in the one-color setting, i.e.~when the images of $x_{{\color{red} s}}$ and $x_{{\color{blue} t}}$ in $R$ are equal. 
In these circumstances we write $\mathrm{TL}_R(n)$ and $\mathrm{JW}_R(n)$ for the one-color Temperley--Lieb algebra and Jones--Wenzl projector.

\begin{thm}[{\cite[Theorem~A.2]{el-univcox}}] \label{existence-onecol}
Suppose $R$ is a commutative $A$-algebra which factors through $\overline{A}$. 
Then $\mathrm{JW}_R(n)$ exists if and only if the one-color quantum binomial coefficients
\begin{equation*}
\qbinom{n}{k}=\frac{[n]!}{[k]![n-k]!}=\frac{[n][n-1]\dotsm [n-k+1]}{[k][k-1] \dotsm [1]}
\end{equation*}
are invertible in $R$ for all integers $0 \leq k \leq n$.
\end{thm}

In light of the ``generic'' nature of the coefficients of $\mathrm{JW}_R(\prescript{}{{\color{red} s}}{n})$, we can interpret \cref{existence-onecol} as description of the denominators of the coefficients of $\mathrm{JW}_{\Frac \overline{A}}(n)$.
Unfortunately, none of the known proofs of this result (most of which use connections to Lie theory in a crucial way) generalize easily to the two-colored setting.

Finally, we will give an alternative criterion for checking rotatability.
For $f \in 2\mathrm{TL}_R(\prescript{}{{\color{red} s}}{n})$ define the \defnemph{partial trace} of $f$ to be
\begin{align*}
\pTr(f)& =\begin{gathered}
\begin{tikzpicture}[xscale=.6,yscale=0.8]
        \begin{scope}
          \clip (-1.5,1) rectangle (3,-1);
  \draw[fill=dgrmblu] (1,2) rectangle (3.5,-2);
  \draw[fill=dgrmred] (1.5,.6) to[out=90,in=180] (2,.8)
  to[out=0,in=90] (2.5,0) to[out=-90,in=0] (2,-.8) to[out=180,in=-90]
  (1.5,-.6) to (1.5,.6);
  \draw[fill=dgrmred] (0,2) rectangle (-2,-2);
  \draw[fill=dgrmblu] (-1,2) rectangle (-.5,-2);
  \draw[fill=white] (-1.25,.6) rectangle (1.9,-.6);
  \node at (.25,0) {$f$};
  \node at (.55,.8) {$\dots$};
  \node at (.55,-.8) {$\dots$};
  \end{scope}
  \draw[dashed] (-1.5,1) to (3,1);
  \draw[dashed] (-1.5,-1) to (3,-1);
\end{tikzpicture}
\end{gathered} & & 
\text{($n$ odd)} \\
\pTr(f)& =\begin{gathered}
\begin{tikzpicture}[xscale=.6,yscale=0.8]
        \begin{scope}
          \clip (-1.5,1) rectangle (3,-1);
  \draw[fill=dgrmred] (1,2) rectangle (3.5,-2);
  \draw[fill=dgrmblu] (1.5,.6) to[out=90,in=180] (2,.8)
  to[out=0,in=90] (2.5,0) to[out=-90,in=0] (2,-.8) to[out=180,in=-90]
  (1.5,-.6) to (1.5,.6);
  \draw[fill=dgrmred] (0,2) rectangle (-2,-2);
  \draw[fill=dgrmblu] (-1,2) rectangle (-.5,-2);
  \draw[fill=white] (-1.25,.6) rectangle (1.9,-.6);
  \node at (.25,0) {$f$};
  \node at (.55,.8) {$\dots$};
  \node at (.55,-.8) {$\dots$};
  \end{scope}
  \draw[dashed] (-1.5,1) to (3,1);
  \draw[dashed] (-1.5,-1) to (3,-1);
\end{tikzpicture}
\end{gathered} & &
\text{($n$ even)}
\end{align*}
From the definition of the Jones--Wenzl projector, it is easy to see that $\mathrm{JW}_R(\prescript{}{{\color{red} s}}{n})$ is rotatable if and only if $\pTr(\mathrm{JW}_R(\prescript{}{{\color{red} s}}{n}))=0$.
Using entirely standard techniques (e.g.~\cite[\S 6.6]{ew-localizedcalc}), one can show that
\begin{equation}
\pTr(\mathrm{JW}_R(\prescript{}{{\color{red} s}}{n}))=-\frac{[n+1]_{{\color{red} s}}}{[n]_{{\color{red} s}}}\mathrm{JW}_R(\prescript{}{{\color{red} s}}{(n-1)}) \label{eq:genericpTr}
\end{equation}
when both $\mathrm{JW}_R(\prescript{}{{\color{red} s}}{n})$ and $\mathrm{JW}_R(\prescript{}{{\color{red} s}}{(n-1)})$ exist. 
This gives the following partial rotatability criterion.

\begin{prop} \label{genericrotatability}
Suppose both $\mathrm{JW}_R(\prescript{}{{\color{red} s}}{n})$ and $\mathrm{JW}_R(\prescript{}{{\color{red} s}}{(n-1)})$ exist. 
Then $\mathrm{JW}_R(\prescript{}{{\color{red} s}}{n})$ is rotatable if and only if $[n+1]_{{\color{red} s}}=0$.
\end{prop}

The key to proving the full rotatability criterion will be to interpret \eqref{eq:genericpTr} generically.


\section{Principal ideals}

In this section, we show that several ideals generated by certain two-colored quantum numbers and binomial coefficients are principal.
Recall that for ordinary quantum numbers, one can show that if $d|n$ then $[d]|[n]$. 
Using $\eqref{eq:colordependence}$ it immediately follows that $[d]_{{\color{red} s}}|[n]_{{\color{red} s}}$.

\begin{lem}[Quantum B\'{e}zout's identity] \label{qbezout}
Let $m,n \in \mathbb{N}$. 
There exist polynomials $a,b \in A$ such that 
\begin{equation*}
a[m]_{{\color{red} s}}+b[n]_{{\color{red} s}}=[\gcd(m,n)]_{{\color{red} s}} \text{.}
\end{equation*}
\end{lem}

\begin{proof}
Suppose without loss of generality that $m<n$. 
We will show that the ideal in $A$ generated by $[m]_{{\color{red} s}}$ and $[n]_{{\color{red} s}}$ contains $[n-m]_{{\color{red} s}}$.
If $m$ and $n$ are not both odd, then
\begin{align*}
\begin{split}
[n-1]_{{\color{blue} t}}[m]_{{\color{red} s}} - [m-1]_{{\color{blue} t}}[n]_{{\color{red} s}} & =
([m+n-2]_{{\color{red} s}}+[m+n-4]_{{\color{red} s}}+\dotsb+[-(n-m)+2]_{{\color{red} s}}) \\
& \quad -([m+n-2]_{{\color{red} s}}+[m+n-4]_{{\color{red} s}}+\dotsb+[n-m+2]_{{\color{red} s}})
\end{split} \\
& =[n-m]_{{\color{red} s}}+[n-m-2]_{{\color{red} s}}+\dotsb+[-(n-m)+2]_{{\color{red} s}} \\
& =[n-m]_{{\color{red} s}}
\end{align*}
by \cite[(6.5a)--(6.5c)]{ew-localizedcalc}. 
If $m$ and $n$ are both odd, a similar calculation yields 
\begin{equation*}
[n-1]_{{\color{red} s}}[m]_{{\color{red} s}} - [m-1]_{{\color{red} s}}[n]_{{\color{red} s}}=[n-m]_{{\color{red} s}} \text{.}
\end{equation*}
By repeating this step multiple times, we can run Euclid's algorithm, and the result follows.
\end{proof}

Next we introduce the cyclotomic parts of quantum numbers, which are roughly analogous to cyclotomic polynomials.
Recall that the one-color quantum numbers are renormalizations of Chebyshev polynomials of the second kind. 
More precisely, if we evaluate a quantum number $[n]$ at $x=2\cos \theta$, we obtain
\begin{equation*}
[n](2\cos \theta) = \frac{\sin n\theta}{\sin \theta} \text{.}
\end{equation*}
Since $[n]$ is a monic polynomial in $x$ of degree $n-1$ we conclude that 
\begin{equation*}
[n]=\prod_{k=1}^{n-1} \left(x-2\cos \frac{k\pi}{n}\right)
\end{equation*}
We define the \defnemph{cyclotomic part} of the one-color quantum number $[n]$ to be the polynomial
\begin{equation*}
\Theta_n = \prod_{\substack{1 \leq k < n\\ (k,n)=1}} \left(x-2\cos \frac{k\pi}{n}\right) \text{.}
\end{equation*}

\begin{lem} \label{cyclofacts}
Let $n \in \mathbb{N}$. 
We have
\begin{enumerate}[label={\rm (\roman*)}]
\item \label{item:degree} $\Theta_n \in \mathbb{Z}[x]$, and $\deg \Theta_n=\varphi(n)$ when $n>1$;

\item \label{item:prod} $[n]=\prod_{k|n} \Theta_n$;

\item \label{item:mobinv} $\Theta_n=\prod_{k|n} [k]^{\mu(n/k)}$, where $\mu:\mathbb{N} \rightarrow \{\pm 1\}$ is the M\"{o}bius function.
\end{enumerate}
Moreover, if $n>2$ then we also have $\Theta_n(x)=\Psi_n(x^2)$, where $\Psi_n \in \mathbb{Z}[x]$ is the minimal polynomial of $4\cos^2(\pi/n)$.
\end{lem}

\begin{proof}
Both \ref{item:degree} and \ref{item:prod} follow from the definition and basic properties of cyclotomic fields and algebraic integers.
Applying M\"obius inversion to \ref{item:prod} yields \ref{item:mobinv}. 
For the final claim, we observe that if $n>2$ then $\Theta_n$ is an even polynomial, so is of the form of $\Psi_n(x^2)$ for some $\Psi_n \in \mathbb{Z}[x]$ of degree $\varphi(n)/2$. 
By construction $4\cos^2(\pi/n)$ is a root of $\Psi_n$. 
Since
\begin{equation*}
4\cos^2 \frac{\pi}{n}=2\cos \frac{2\pi}{n}+2
\end{equation*}
and $\mathbb{Q}(2\cos(2\pi/n)+2)=\mathbb{Q}(\cos(2\pi/n))$ is a field extension of $\mathbb{Q}$ of degree $\varphi(n)/2$, $\Psi_n$ must be the minimal polynomial of $4\cos^2(\pi/n)$.
\end{proof}

\begin{defn}
For $n \in \mathbb{N}$, we define the \defnemph{cyclotomic part} of the two-colored quantum number $[n]_{{\color{red} s}}$ to be
\begin{equation*}
\Theta_{n,{\color{red} s}}=\begin{cases}
\Psi_n(x_{{\color{red} s}} x_{{\color{blue} t}}) & \text{if $n>2$,} \\
x_{{\color{red} s}} & \text{if $n=2$,} \\
1 & \text{if $n=1$.}
\end{cases}
\end{equation*}
\end{defn}

Using \eqref{eq:colordependence} and \cref{cyclofacts} we similarly obtain $[n]_{{\color{red} s}}=\prod_{k|n} \Theta_{n,{\color{red} s}}$ and $\Theta_{n,{\color{red} s}}=\prod_{k|n} [n]_{{\color{red} s}}^{\mu(n/k)}$.

\begin{lem} \label{thetabezout}
Let $m,n \in \mathbb{N}$ such that $m \nmid n$ and $n \nmid m$.
There exist polynomials $a,b \in A$ such that 
\begin{equation*}
a\Theta_{m,{\color{red} s}}+b\Theta_{n,{\color{red} s}}=1 \text{.}
\end{equation*}
\end{lem}

\begin{proof}
Suppose without loss of generality that $m<n$, and let $d=\gcd(m,n)$.
By \cref{qbezout} there exist $a',b' \in A$ such that
\begin{equation*}
a'[m]_{{\color{red} s}}+b'[n]_{{\color{red} s}}=[d]_{{\color{red} s}} \text{.}
\end{equation*}
By assumption $d<m<n$, so we have
\begin{align*}
\frac{[m]_{{\color{red} s}}}{[d]_{{\color{red} s}}} & \in \Theta_{m,{\color{red} s}} A \\
\frac{[n]_{{\color{red} s}}}{[d]_{{\color{red} s}}} & \in \Theta_{n,{\color{red} s}} A
\end{align*}
and thus dividing by $[d]_{{\color{red} s}}$ we obtain
\begin{equation*}
a\Theta_{m,{\color{red} s}}+b\Theta_{n,{\color{red} s}}=1 \text{.} \qedhere
\end{equation*}
\end{proof}

\begin{prop} \label{thetaprincipal}
Let $m_1,m_2,\dotsc,m_k,n_1,n_2,\dotsc,n_l \in \mathbb{N}$ such that for all $i,j$ either $m_i=n_j$ or $m_i \nmid n_j$ and $n_j \nmid m_i$.
Then the ideal 
\begin{equation*}
(\Theta_{m_1,{\color{red} s}}\Theta_{m_2,{\color{red} s}}\dotsm \Theta_{m_k,{\color{red} s}}, \Theta_{n_1,{\color{red} s}}\Theta_{n_2,{\color{red} s}}\dotsm \Theta_{n_l,{\color{red} s}})
\end{equation*}
in $A$ is principal.
\end{prop}

\begin{proof}
Let $I$ be the ideal above.
We may assume without loss of generality that $m_i \neq n_j$ for all $i,j$, i.e.~the generators of $I$ are coprime in $A$.
For each $i,j$ we can apply \cref{thetabezout} to obtain $a_{i,j},b_{i,j} \in A$ such that $a_{i,j}\Theta_{m_i,{\color{red} s}}+b_{i,j}\Theta_{n_j,{\color{red} s}}=1$.
Taking the product over all $i$ and $j$ we obtain
\begin{equation*}
1=\prod_{i} \left(\prod_{j} (a_{i,j}\Theta_{m_i,{\color{red} s}} + b_{i,j}\Theta_{n_j,{\color{red} s}})\right) \in \prod_{i} (\Theta_{m_i,{\color{red} s}},\Theta_{n_1,{\color{red} s}}\Theta_{n_2,{\color{red} s}}\dotsm \Theta_{n_l,{\color{red} s}}) \subseteq I
\end{equation*}
so $I=(1)$ is principal.
\end{proof}


For $f \in A$, we define the cyclotomic valuation $\nu_{l,{\color{red} s}}(f)$ to be the exponent of the highest power dividing $f$.
This extends to $\Frac A$ in the obvious way, namely we define $\nu_{l,{\color{red} s}}(f/g)=\nu_{l,{\color{red} s}}(f)-\nu_{l,{\color{red} s}}(g)$ for $f,g \in A$.
If $f$ and $g$ are products of ${\color{red} s}$-colored cyclotomic parts then
\begin{equation*}
\frac{f}{g}=\prod_l \Theta_{l,{\color{red} s}}^{\nu_{l,{\color{red} s}}(f/g)} \text{.}
\end{equation*}

\begin{lem} \label{valbinom}
Let $n,k$ be non-negative integers.
For all integers $1 \leq l \leq n$ we have
\begin{equation*}
\nu_{l,{\color{red} s}} \qbinom{n}{k}_{{\color{red} s}}=\left\lfloor \frac{n}{l} \right\rfloor - \left\lfloor \frac{k}{l} \right\rfloor - \left\lfloor \frac{n-k}{l} \right\rfloor \text{.}
\end{equation*} 
In particular, $\nu_{l,{\color{red} s}} \qbinom{n}{k}_{{\color{red} s}} \in \{0,1\}$.
\end{lem}

\begin{proof}
Clearly
\begin{equation*}
\nu_{l,{\color{red} s}}[m]_{{\color{red} s}}=\begin{cases}
1 & \text{if $l|m$,} \\
0 & \text{otherwise,}
\end{cases}
\end{equation*}
so $\nu_l([m]_{{\color{red} s}}!)=\lfloor m/l \rfloor$ and the equation above follows.
To show the bound on the valuation, note that $m/l-1<\lfloor m/l\rfloor \leq m/l$, so
\begin{multline*}
-1=\left(\frac{n}{l}-1\right)-\frac{k}{l}-\frac{n-k}{l}<\left\lfloor \frac{n}{l} \right\rfloor - \left\lfloor \frac{k}{l} \right\rfloor - \left\lfloor \frac{n-k}{l} \right\rfloor \\
<\frac{n}{l}-\left(\frac{k}{l}-1\right)-\left(\frac{n-k}{l}-1\right)=2 \text{.}
\end{multline*}
\end{proof}

\begin{thm} \label{qbinomideal}
Let $n \in \mathbb{N}$. 
The ideal
\begin{equation*}
\left(\qbinom{n}{1}_{{\color{red} s}}, \qbinom{n}{2}_{{\color{red} s}}, \dotsc, \qbinom{n}{n-1}_{{\color{red} s}}\right)
\end{equation*}
in $A$ is principal, generated by $\Theta_{n,{\color{red} s}}$.
\end{thm}

\begin{proof}
We will prove the result by induction.
Let
\begin{equation*}
I_m=\left(\qbinom{n}{1}_{{\color{red} s}},\qbinom{n}{2}_{{\color{red} s}}, \dotsc, \qbinom{n}{m}_{{\color{red} s}}\right)
\end{equation*}
and write 
\begin{equation*}
[n]_{{\color{red} s}}^{>m}=\prod_{\substack{k>m\\ k|n}} \Theta_{k,{\color{red} s}} \text{.}
\end{equation*}
Suppose we have shown that $I_m$ is principal, generated by $[n]_{{\color{red} s}}^{>m}$.
We will show that $I_{m+1}$ is principal, generated by $[n]_{{\color{red} s}}^{>m+1}$.
It is enough to show that
\begin{equation}
\left(\qbinom{n}{m+1}_{{\color{red} s}},[n]_{{\color{red} s}}^{>m}\right)=([n]_{{\color{red} s}}^{>m+1}) \text{.} \label{eq:qbinomideal-induction}
\end{equation}

Clearly $[n]_{{\color{red} s}}^{>m+1}$ divides $[n]_{{\color{red} s}}^{>m}$.
If $k>m+1$ and $k|n$ it is easy to see that
\begin{equation*}
\left\lfloor\frac{n}{k}\right\rfloor - \left\lfloor\frac{m+1}{k}\right\rfloor - \left\lfloor\frac{n-(m+1)}{k}\right\rfloor=1 \text{,}
\end{equation*}
so $\Theta_k$ divides $\qbinom{n}{m+1}_{{\color{red} s}}$ exactly once, and thus $[n]_{{\color{red} s}}^{>m+1}$ divides $\qbinom{n}{m+1}_{{\color{red} s}}$.
If $m+1 \nmid n$ then $[n]_{{\color{red} s}}^{>m+1}=[n]_{{\color{red} s}}^{>m}$, \eqref{eq:qbinomideal-induction} follows trivially.

Otherwise suppose $m+1|n$. 
We claim that if $\Theta_{l,{\color{red} s}}$ divides $\qbinom{n}{m+1}/[n]_{{\color{red} s}}^{>m+1}$ we must have $l \nmid m+1$ and $m+1 \nmid l$.
This implies that 
\begin{equation*}
\left(\frac{\qbinom{n}{m+1}_{{\color{red} s}}}{[n]_{{\color{red} s}}^{>m+1}},\Theta_{m+1,{\color{red} s}}\right)=(1)
\end{equation*}
by \cref{thetaprincipal}, from which \eqref{eq:qbinomideal-induction} holds and the result follows.

To prove the claim, suppose $l|m+1$.
It is straightforward to check that
\begin{equation*}
\left\lfloor\frac{n}{l}\right\rfloor - \left\lfloor\frac{m+1}{l}\right\rfloor - \left\lfloor\frac{n-(m+1)}{l}\right\rfloor=0 \text{,}
\end{equation*}
so $\Theta_{l,{\color{red} s}}$ does not divide $\qbinom{n}{m+1}_{{\color{red} s}}$, let alone $\qbinom{n}{m+1}_{{\color{red} s}}/[n]_{{\color{red} s}}^{>m+1}$.

Similarly, suppose $m+1|l$, and take $0 \leq r<l$ such that $n-(m+1)=ql+r$.
If $\Theta_{l,{\color{red} s}}$ divides $\qbinom{n}{m+1}_{{\color{red} s}}$ then 
\begin{equation*}
\left\lfloor\frac{n}{l}\right\rfloor - \left\lfloor\frac{m+1}{l}\right\rfloor - \left\lfloor\frac{n-(m+1)}{l}\right\rfloor=1
\end{equation*}
and we must have $r+m+1 \geq l$. 
Now let $d=\gcd(l,n-(m+1))$.
As $m+1|n$ and $m+1|l$, we have $m+1|d$ and in particular $m+1 \leq d$. 
We also have $d|r$, so in particular $l-r \geq d$.
We combine these two equalities to obtain $r+m+1 \leq l$, with equality if and only if $l-r=d$ and $m+1=d$.
This immediately implies that $l|n$, so $\Theta_{l,{\color{red} s}}$ does not divide $\qbinom{n}{m+1}_{{\color{red} s}}/[n]_{{\color{red} s}}^{>m+1}$.
\end{proof}



\begin{thm} \label{qbinominvideal}
Let $n \in \mathbb{N}$. 
The fractional ideal of $A$ generated by
\begin{equation*}
\qbinom{n}{0}_{{\color{red} s}}^{-1}, \qbinom{n}{1}_{{\color{red} s}}^{-1}, \dotsc, \qbinom{n}{n}_{{\color{red} s}}^{-1}
\end{equation*}
is principal, generated by 
\begin{equation*}
\left(\prod_{\substack{1 \leq k \leq n\\ k \nmid n+1}} \Theta_{k,{\color{red} s}}\right)^{-1} \text{.}
\end{equation*}
\end{thm}

\begin{proof}
We follow a similar strategy as in the proof of \cref{qbinomideal}.
Let $I_m$ denote the fractional ideal generated by
\begin{equation*}
\qbinom{n}{0}_{{\color{red} s}}^{-1},\qbinom{n}{1}_{{\color{red} s}}^{-1}, \dotsc, \qbinom{n}{m}_{{\color{red} s}}^{-1}
\end{equation*}
and let 
\begin{equation*}
g_m=\prod_{\substack{k|n-m+i \text{ for some } 1 \leq i \leq m\\ k\nmid n+1}} \Theta_{k,{\color{red} s}} \text{.}
\end{equation*}
Suppose we have shown that $I_m$ is principal, generated by $g_m^{-1}$.
We will show that $I_{m+1}$ is principal, generated by $g_{m+1}^{-1}$.
It is enough to show that the fractional ideal generated by
\begin{equation*}
\qbinom{n}{m+1}_{{\color{red} s}}^{-1},g_m^{-1}
\end{equation*}
is equal to the principal fractional ideal generated by $g_{m+1}^{-1}$.
This is equivalent to proving equality of the following (ordinary) ideals
\begin{equation}
\left(g_m,\qbinom{n}{m+1}_{{\color{red} s}}\right)=\left(\frac{\qbinom{n}{m+1}_{{\color{red} s}}}{h_m}\right) \label{eq:qbinominvideal-induction}
\end{equation}
of $A$, where 
\begin{equation*}
h_m=\frac{g_{m+1}}{g_m}=\prod_{\substack{k|n-m\\ k \nmid n-m+1, k \nmid n-m+2, \dotsc ,k \nmid n+1}} \Theta_{k,{\color{red} s}} \text{.}
\end{equation*}
We first check that the ideal on the right-hand side of \eqref{eq:qbinominvideal-induction} is an ordinary ideal. 
If $\Theta_{k,{\color{red} s}}$ divides $h_m$ (i.e.~if $k|n-m$ and $k \nmid n-m+i$ for all $1 \leq i \leq m+1$) then $k \nmid m+1$ and the fractional part of $(n-(m+1))/k$ is $(k-1)/k$.
This implies that
\begin{equation*}
\left\lfloor \frac{n}{k} \right\rfloor - \left\lfloor \frac{m+1}{k} \right\rfloor - \left\lfloor \frac{n-(m+1)}{k} \right\rfloor = 1
\end{equation*}
so $\Theta_{k,{\color{red} s}}$ also divides $\qbinom{n}{m+1}_{{\color{red} s}}$.

It is clear that $\qbinom{n}{m+1}_{{\color{red} s}}/h_m$ divides $\qbinom{n}{m+1}_{{\color{red} s}}$. 
Suppose $\Theta_{k,{\color{red} s}}$ divides $\qbinom{n}{m+1}_{{\color{red} s}}/h_m$. 
Since we can write
\begin{equation*}
\qbinom{n}{m+1}=\frac{[n]_{{\color{red} s}} [n-1]_{{\color{red} s}} \dotsm [n-m]_{{\color{red} s}}}{[m+1]_{{\color{red} s}} [m]_{{\color{red} s}} \dotsm [1]_{{\color{red} s}}} \text{,}
\end{equation*}
this implies that either $k\nmid n-m$ and $k|n-m+i$ for some $1 \leq i \leq m$, or $k|n-m$ and $k|n-m+i$ for some $1 \leq i \leq m+1$. 
In either case, it is easy to check that $k\nmid n+1$, for otherwise $n/k$ has fractional part $(k-1)/k$, so
\begin{equation*}
\left\lfloor \frac{n}{k} \right\rfloor - \left\lfloor \frac{m+1}{k} \right\rfloor - \left\lfloor \frac{n-(m+1)}{k} \right\rfloor = 0
\end{equation*}
and $\Theta_{k,{\color{red} s}}$ cannot divide $\qbinom{n}{m+1}_{{\color{red} s}}$. 
This shows that $\qbinom{n}{m+1}_{{\color{red} s}}/h_m$ divides $g_m$.

We will now show that
\begin{equation*}
\left(\frac{g_{m+1}}{\qbinom{n}{m+1}_{{\color{red} s}}},h_m\right)=(1)
\end{equation*}
using \cref{thetaprincipal}, from which \eqref{eq:qbinominvideal-induction} holds and the result follows. 
It is enough to show that for any $l,d>1$, we do not have $\Theta_{l,{\color{red} s}}|g_{m+1}/\qbinom{n}{m+1}_{{\color{red} s}}$ and $\Theta_{ld,{\color{red} s}}|h_m$, or $\Theta_{ld,{\color{red} s}}|g_{m+1}/\qbinom{n}{m+1}_{{\color{red} s}}$ and $\Theta_{l,{\color{red} s}}|h_m$.

Suppose first that $\Theta_{l,{\color{red} s}}|g_{m+1}/\qbinom{n}{m+1}_{{\color{red} s}}$ and $\Theta_{ld,{\color{red} s}}|h_m$. 
Then $l \nmid n+1$ and $ld|n-m$, so $l|n-m$ and $l \nmid m+1$. 
This shows that the fractional part of $(n-(m+1))/l$ is $(l-1)/l$ and the fractional part of $(m+1)/l$ is non-zero, so
\begin{equation*}
\left\lfloor \frac{n}{l} \right\rfloor - \left\lfloor \frac{m+1}{l} \right\rfloor - \left\lfloor \frac{n-(m+1)}{l} \right\rfloor = 1
\end{equation*}
which contradicts $\Theta_{l,{\color{red} s}}|g_{m+1}/\qbinom{n}{m+1}_{{\color{red} s}}$.

Similarly, suppose that $\Theta_{ld,{\color{red} s}}|g_{m+1}/\qbinom{n}{m+1}_{{\color{red} s}}$ and $\Theta_{l,{\color{red} s}}|h_m$. 
Then $l|n-m$ and $l \nmid n-m+i$ for $1 \leq i \leq m+1$, while $ld|n-m+i$ for some $0 \leq i \leq m$ and $ld \nmid n+1$.
The only way this can happen is if $ld|n-m$. 
This implies that $ld \nmid m+1$, and we similarly obtain
\begin{equation*}
\left\lfloor \frac{n}{ld} \right\rfloor - \left\lfloor \frac{m+1}{ld} \right\rfloor - \left\lfloor \frac{n-(m+1)}{ld} \right\rfloor = 1
\end{equation*}
which contradicts $\Theta_{ld,{\color{red} s}}|g_{m+1}/\qbinom{n}{m+1}_{{\color{red} s}}$.
\end{proof}

\section{Existence and rotatability}

Let $Q=\Frac A$ and $\overline{Q}=\Frac \overline{A}$.
Our goal in this section is to prove \cref{existence} by showing that the denominators of the coefficients of $\mathrm{JW}_Q(\prescript{}{{\color{red} s}}{n})$ divide 
\begin{equation*}
\prod_{\substack{1 \leq k \leq n\\ k \nmid n+1}} \Theta_{k,{\color{red} s}}
\end{equation*}
by comparing them with the coefficients of $\mathrm{JW}_{\overline{Q}}(n)$.
First, we prove the analogous statement for $\mathrm{JW}_{\overline{Q}}(n)$.

\begin{lem} \label{onecolvalbound}
Let $k \in \mathbb{N}$, and let $D$ be a one-colored Temperley--Lieb diagram in $\mathrm{TL}_{\overline{Q}}(n)$.
Then
\begin{equation*}
\nu_k\left(\coeff_{{}\in \mathrm{JW}_{\overline{Q}}(n)} D \right) \geq -1 \text{,}
\end{equation*}
with equality only if $1 \leq k \leq n$ and $k \nmid n+1$.
\end{lem}

\begin{proof}
We proceed by induction. 
Suppose the result holds for $n=m$, and let $D$ be a one-colored Temperley--Lieb diagram in $\mathrm{TL}_{\overline{Q}}(m+1)$.
By the one-color version of \cref{JWcoef} we have
\begin{equation}
\begin{split}
\nu_k\left(\coeff_{{}\in \mathrm{JW}_{\overline{Q}}(m+1)} D \right)& =\nu_k\left(\sum_{\{i\}} \frac{[i]}{[m+1]}\coeff_{{}\in \mathrm{JW}_{\overline{Q}}(m)} D_i\right) \\
& \geq \min_{\{i\}} \left(\nu_k\left(\frac{[i]}{[m+1]}\right)+\nu_k\left(\coeff_{{}\in \mathrm{JW}_{\overline{Q}}(m)} D_i\right)\right) \text{.} 
\end{split}\label{eq:onecolvalbound}
\end{equation}
If $k \nmid m+1$, then $\nu_k([i]/[m+1]) \geq 0$ for any $i$ and $\nu_k(\coeff_{{}\in \mathrm{JW}_{\overline{Q}}(m)} D_i) \geq -1$.
On the other hand, if $k|m+1$, then $\nu_k([i]/[m+1]) \geq -1$ while $\nu_k(\coeff_{{}\in \mathrm{JW}_{\overline{Q}}(m)} D_i) \geq 0$.
In either case, the sum of the two valuations is at least $-1$, so the right-hand side of \eqref{eq:onecolvalbound} is at least $-1$.
In the case of equality, we must have $1 \leq k \leq m+1$ and $k \nmid m+2$ by \cref{existence-onecol} and the one-color version of \cref{qbinominvideal}.
\end{proof}

Now let $A'=A[x]/(x^2-x_{{\color{red} s}} x_{{\color{blue} t}})$.
We view $A'$ as both an $A$-algebra and an $\overline{A}$-algebra in the obvious way. 
Writing $Q'=\Frac A'$, we have an isomorphism
\begin{equation*}
\begin{aligned}
\mathrm{TL}_{Q'}(n)& \longrightarrow 2\mathrm{TL}_{Q'}(\prescript{}{{\color{red} s}}{n})\\
e_i & \longmapsto \begin{cases}
\frac{x}{x_{{\color{red} s}}} e_i & \text{$i$ odd,} \\
\frac{x}{x_{{\color{blue} t}}} e_i & \text{$i$ even,}
\end{cases}
\end{aligned}
\end{equation*}
which maps $\mathrm{JW}_{Q'}(n) \mapsto \mathrm{JW}_{Q'}(\prescript{}{{\color{red} s}}{n})$.
So for any two-colored Temperley--Lieb diagram $D$, we have
\begin{equation*}
\coeff_{{}\in \mathrm{JW}_{Q}(\prescript{}{{\color{red} s}}{n})} D=\coeff_{{}\in \mathrm{JW}_{Q'}(\prescript{}{{\color{red} s}}{n})} D=x^a x_{{\color{red} s}}^b x_{{\color{blue} t}}^c \coeff_{{}\in \mathrm{JW}_{Q'}(n)} \overline{D}=x^a x_{{\color{red} s}}^b x_{{\color{blue} t}}^c \coeff_{{}\in \mathrm{JW}_{\overline{Q}}(n)} \overline{D}
\end{equation*}
for some integers $a,b,c$, where $\overline{D}$ denotes the one-color diagram obtained from $D$ by forgetting the coloring. 
It follows that when $k>2$ we have
\begin{equation}
\nu_{k,{\color{red} s}}\left(\coeff_{{}\in \mathrm{JW}_{Q}(n)} D \right)=\nu_k\left(\coeff_{{}\in \mathrm{JW}_{\overline{Q}}(\prescript{}{{\color{red} s}}{n})} \overline{D} \right) \text{.} \label{eq:valequalitynottwo}
\end{equation}

\begin{lem} \label{twocolvalbound}
Let $k \in \mathbb{N}$, and let $D$ be a two-colored Temperley--Lieb diagram in $2\mathrm{TL}_{Q}(\prescript{}{{\color{red} s}}{n})$.
Then
\begin{equation*}
\nu_{k,{\color{Plum} u}}\left(\coeff_{{}\in \mathrm{JW}_{Q}(\prescript{}{{\color{red} s}}{n})} D \right) \geq -1 \text{,}
\end{equation*}
and if we have equality then $1 \leq k \leq n$ and $k \nmid n+1$, and ${\color{Plum} u}={\color{red} s}$ if $k=2$.-
\end{lem}

\begin{proof}
By \cref{onecolvalbound} and \eqref{eq:valequalitynottwo} we need only concern ourselves with the case where $k=2$. 
We proceed by induction as in the proof of \cref{onecolvalbound}.
Suppose the result holds for $n=m$, and let $D$ be a two-colored Temperley--Lieb diagram in $\mathrm{TL}_{Q}(m+1)$.
By \cref{JWcoef} we have
\begin{equation}
\begin{split}
\nu_{2,{\color{Plum} u}}\left(\coeff_{{}\in \mathrm{JW}_{Q}(m+1)} D \right)& =\nu_{2,{\color{Plum} u}}\left(\sum_{\{i\}} \frac{[i]_{\color{OliveGreen} v}}{[m+1]_{{\color{red} s}}}\coeff_{{}\in \mathrm{JW}_{Q}(m)} D_i\right) \\
& \geq \min_{\{i\}} \left(\nu_{2,{\color{Plum} u}}\left(\frac{[i]_{{\color{OliveGreen} v}}}{[m+1]_{{\color{red} s}}}\right)+\nu_{2,{\color{Plum} u}}\left(\coeff_{{}\in \mathrm{JW}_{Q}(m)} D_i\right)\right) \text{.} 
\end{split}\label{eq:twocolvalbound}
\end{equation}
If $m$ is even or ${\color{Plum} u}={\color{blue} t}$, then for all $i$
\begin{equation*}
\nu_{2,{\color{Plum} u}}\left(\frac{[i]_{{\color{OliveGreen} v}}}{[m+1]_{{\color{red} s}}}\right) \geq 0 \qquad \text{ and } \qquad \nu_{2,{\color{Plum} u}}\left(\coeff_{{}\in \mathrm{JW}_{Q}(m)} D_i\right) \geq -1 \text{.}
\end{equation*}
On the other hand, if $m$ is odd and ${\color{Plum} u}={\color{red} s}$, then for all $i$ 
\begin{equation*}
\nu_{2,{\color{Plum} u}}\left(\frac{[i]_{{\color{OliveGreen} v}}}{[m+1]_{{\color{red} s}}}\right) \geq -1 \qquad \text{ and } \qquad \nu_{2,{\color{red} s}}\left(\coeff_{{}\in \mathrm{JW}_{\overline{Q}}(m)} D_i\right) \geq 0 \text{.}
\end{equation*}
In either case, the sum of the two valuations is at least $-1$, so the right-hand side of \eqref{eq:twocolvalbound} is at least $-1$.

Now suppose the left-hand side of \eqref{eq:twocolvalbound} is $-1$ but $m$ is even.
Let $D'$ be the diagram in $2\mathrm{TL}(\prescript{}{{\color{red} s}}{m+1})$ obtained by reflecting $D$ about a vertical axis, and then recoloring to obtain a diagram in $2\mathrm{TL}(\prescript{}{{\color{red} s}}{n})$.
This reflection map extends to a twisted involution of $2\mathrm{TL}(n)$ mapping $e_i$ to $e_{n-i}$ and swapping $[2]_{{\color{red} s}}$ and $[2]_{{\color{blue} t}}$.
Thus $\nu_{k,{\color{blue} t}}(\coeff_{{} \in \mathrm{JW}_Q(m+1)} D')=-1$, which is a contradiction.
\end{proof}



\begin{lem} \label{qbinominvcoef}
Let $n,k$ be integers with $0 \leq k \leq n$.
There exists a two-colored diagram $D$ such that $\coeff_{{}\in \mathrm{JW}(\prescript{}{{\color{red} s}}{n})} D=\qbinom{n}{k}_{{\color{red} s}}^{-1}$.
\end{lem}

\begin{proof}
Take $D$ to be the diagram with $k$ nested caps on the bottom left, $k$ nested cups on the top right, and all other strands connected from bottom to top. 
For example, if $n=5$ and $k=2$ we set
\begin{equation*}
D=\begin{gathered}
        \begin{tikzpicture}[xscale=.4,yscale=.2]
          \begin{scope}
            \clip (-3,3.5) rectangle (3,-3.5);
            \draw[fill=dgrmred] (2.5,4) rectangle (-4,-4);
            \draw[fill=dgrmblu] (4,-4) to (2,-3.5) to[out=90,in=-90] (-2,3.5) to (-2,4) to (-1,3.5) to[out=-90,in=180] (0.5,1.5) to[out=0,in=-90] (2,3.5) to (4,4) to (4,-4);
        
          \draw[fill=dgrmblu] (-2,-4) to (-2,-3.5) to[out=90,in=180] (-.5,-1.5) to[out=0,in=90] (1,-3.5) to (1,-4) to (0,-3.5) to[out=90,in=0] (-.5,-2.5) to[out=180,in=90] (-1,-3.5) to (-2,-4);
          \draw[fill=dgrmblu] (0,4) to (0,3.5) to[out=-90,in=180] (.5,2.5) to[out=0,in=-90] (1,3.5) to (0,4);          
       \end{scope}
       \draw[dashed] (-3,3.5) to (3,3.5);
       \draw[dashed] (-3,-3.5) to (3,-3.5);
     \end{tikzpicture}
\end{gathered}
\end{equation*}
The result follows by \cref{JWcoef} and induction on $n$.
\end{proof}

\begin{thm} \label{genericcoefs}
For each two-colored Temperley--Lieb diagram $D$ there are coprime elements $f_D,g_D \in A$ such that if $\mathrm{JW}_R(\prescript{}{{\color{red} s}}{n})$ exists,
\begin{equation*}
\coeff_{{} \in \mathrm{JW}_R(\prescript{}{{\color{red} s}}{n})} D
\end{equation*}
is the specialization of $f_D/g_D$ in $R$. 
In particular, the existence of $\mathrm{JW}_R(n)$ implies that the specialization of $g_D$ in $R$ is invertible for all diagrams $D$. 
\end{thm}

\begin{rem}
We consider generic computation of the coefficients of one-colored Jones--Wenzl projectors (at least for subrings of $\mathbb{C}$) to be mathematical folklore, i.e.~a ``known'' result without a published proof. 
In \cite[Theorem~6.13]{ew-localizedcalc} Elias--Williamson carefully prove an analogous result under the assumption that $R$ is both an integral domain and a henselian local ring.
Our proof does not require any restrictions on $R$ but is essentially equivalent to \cref{existence}.
\end{rem}

\begin{proof}
Let
\begin{equation*}
T_R=\{f \in 2\mathrm{TL}_R(\prescript{}{{\color{red} s}}{n}) : e_i f=0 \text{ for all } 1 \leq i \leq n-1\} \text{.}
\end{equation*}
In other words, $T_R$ is the (right) annihilator of the generators $e_1,\dotsc,e_{n-1}$. 
One can show that $\mathrm{JW}_R(\prescript{}{{\color{red} s}}{n})$ exists if and only if there exists $f \in T_R$ for which $\coeff_{{} \in f} 1$ is invertible in $R$ (see e.g.~\cite[Exercise~9.25]{emtw} for the one-colored case). 
When this happens, $T_R=R\mathrm{JW}_R(\prescript{}{{\color{red} s}}{n})$.

Clearly $\mathrm{JW}_Q(\prescript{}{{\color{red} s}}{n})$ exists so $T_Q=Q\mathrm{JW}_Q(n)$. 
Thus $T_A$ is a free $A$-module of rank $1$, generated by $c\mathrm{JW}_Q(n) \in \mathrm{TL}_A(n)$, where $c$ is the least common multiple of the denominators of the coefficients of $\mathrm{JW}_Q(n)$.
\Cref{twocolvalbound} implies that $c$ divides 
\begin{equation*}
g_n=\prod_{\substack{1 \leq k \leq n\\ k \nmid n+1}} \Theta_{k,{\color{red} s}} \text{,}
\end{equation*}
while \cref{qbinominvcoef} and \cref{qbinominvideal} give $c=g_n$.
If $\mathrm{JW}_R(n)$ exists, we have $T_R=R\mathrm{JW}_R(n) \geq R \otimes_A g_n\mathrm{JW}_Q(n)$, and thus $g_n\mathrm{JW}_R(n)=1 \otimes_A g_n\mathrm{JW}_Q(n)$.
But the coefficients of $g_n\mathrm{JW}_Q(n)$ (which lie in $A$) generate $(1)$ as an ideal of $A$ (again by \cref{qbinominvideal}), which directly implies that $g_n$ is invertible in $R$, and $\mathrm{JW}_R(n)=g_n^{-1} \otimes_A g_n \mathrm{JW}_Q(n)$.
\end{proof}

\begin{proof}[Proof of \cref{existence}]
The invertibility of $\qbinom{n}{k}_{{\color{red} s}}$ is necessary for $\mathrm{JW}_R(\prescript{}{{\color{red} s}}{n})$ to exist by \cref{qbinominvcoef}. 
Sufficiency follows from \cref{qbinominvideal}, \cref{twocolvalbound}, and \cref{genericcoefs} (or more simply from the proof of \cref{genericcoefs}).
\end{proof}


For $f \in Q$, say that $f$ exists in $R$ if there are $a,b \in A$ with $f=a/b$ and $b$ invertible in $R$.

\begin{lem}
Suppose $\mathrm{JW}_R(\prescript{}{{\color{red} s}}{n})$ exists. 
Then $\frac{[n+1]_{{\color{red} s}}}{[k]_{{\color{red} s}}}$ exists in $R$ for any integer $1 \leq k \leq n+1$.
\end{lem}

\begin{proof}
We have
\begin{equation*}
\frac{[n+1]_{{\color{red} s}}}{[k]_{{\color{red} s}}}=\frac{\prod_{l|n+1} \Theta_{l,{\color{red} s}}}{\prod_{l|k} \Theta_{l,{\color{red} s}}}=\frac{\prod_{\substack{l|n+1\\ l \nmid k}} \Theta_{l,{\color{red} s}}}{\prod_{\substack{l|k\\ l \nmid n+1}} \Theta_{l,{\color{red} s}}} \text{,}
\end{equation*}
and the denominator of the right-hand side divides
\begin{equation*}
\prod_{\substack{1 \leq l \leq n\\ l \nmid n+1}} \Theta_{l,{\color{red} s}}
\end{equation*}
which is invertible by \cref{existence} and \cref{qbinominvideal}.
\end{proof}

\begin{prop} \label{simple-rotatability}
Suppose the two-colored Jones--Wenzl projectors $\mathrm{JW}_R(\prescript{}{{\color{red} s}}{n})$ and $\mathrm{JW}_R(\prescript{}{{\color{blue} t}}{n})$ exist.
Then $\mathrm{JW}_R(\prescript{}{{\color{red} s}}{n})$ is rotatable if and only if $\frac{[n+1]_{{\color{red} s}}}{[k]_{{\color{red} s}}}=0$ for all integers $1 \leq k \leq n$.
\end{prop}

\begin{proof}
Calculating generically, we have
\begin{equation*}
\pTr(\mathrm{JW}_Q(\prescript{}{{\color{red} s}}{n}))=-\frac{[n+1]_{{\color{red} s}}}{[n]_{{\color{red} s}}}\mathrm{JW}_Q(\prescript{}{{\color{red} s}}{(n-1)})
\end{equation*}
by \eqref{eq:genericpTr}.
From the proof of \cref{genericcoefs} the coefficients of $\mathrm{JW}_Q(\prescript{}{{\color{red} s}}{(n-1)})$ can be written as sums of fractions of the form $a\qbinom{n-1}{k}_{{\color{red} s}}^{-1}$ for some $a \in A$ and some integer $0 \leq k \leq n-1$. 
Now observe that
\begin{equation*}
-\frac{[n+1]_{{\color{red} s}}}{[n]_{{\color{red} s}}}\frac{a}{\qbinom{n-1}{k}_{{\color{red} s}}}=-\frac{[n+1]_{{\color{red} s}}[k]_{{\color{red} s}}! a}{[n]_{{\color{red} s}}[n-1]_{{\color{red} s}} \dotsm [n-k]_{{\color{red} s}}}=-\frac{[n+1]_{{\color{red} s}}}{[k+1]_{{\color{red} s}}}\frac{a}{\qbinom{n}{k+1}_{{\color{red} s}}}
\end{equation*}
noting that since $\mathrm{JW}_R(\prescript{}{{\color{red} s}}{n})$ exists, $\qbinom{n}{k+1}_{{\color{red} s}}$ is invertible. 
Thus $\mathrm{JW}_R(\prescript{}{{\color{red} s}}{n})$ is rotatable if $\frac{[n+1]_{{\color{red} s}}}{[k+1]_{{\color{red} s}}}=0$.
Conversely, by \cref{qbinominvcoef} there is a a diagram whose coefficient in $\mathrm{JW}_Q(\prescript{}{{\color{red} s}}{n})$ is exactly $\qbinom{n-1}{k}_{{\color{red} s}}^{-1}$, so the above calculation shows that rotatability implies $\frac{[n+1]_{{\color{red} s}}}{[k+1]_{{\color{red} s}}}=0$.
\end{proof}

\begin{proof}[Proof of \cref{existsrotates}]
The condition on quantum binomial coefficients is the same as \cite[Assumption~1.1]{abe-homBS}. 
By \cite[Proposition~3.4]{abe-homBS} this implies that the quantum binomial coefficients $\qbinom{n}{k}_{{\color{red} s}}$ and $\qbinom{n}{k}_{{\color{blue} t}}$ are all invertible. 
Since
\begin{equation}
\qbinom{n+1}{k}_{{\color{red} s}}=\frac{[n+1]_{{\color{red} s}}}{[k]_{{\color{red} s}}}\qbinom{n}{k-1}_{{\color{red} s}} \label{eq:binomcoefincrement}
\end{equation}
and similarly for ${\color{blue} t}$, we conclude that $\frac{[n+1]_{{\color{red} s}}}{[k]_{{\color{red} s}}}=\frac{[n+1]_{{\color{blue} t}}}{[k]_{{\color{blue} t}}}=0$ for all integers $1 \leq k \leq n$. 
Conversely, if the two-colored Jones--Wenzl projectors exist and are rotatable, then \eqref{eq:binomcoefincrement} combined with \cref{existence} and \cref{simple-rotatability} show that $\qbinom{n+1}{k}_{{\color{red} s}}$ and $\qbinom{n+1}{k}_{{\color{blue} t}}$ vanish for all integers $1 \leq k \leq n$.
\end{proof}

\section{Applications to Soergel bimodules}

The diagrammatic Hecke category of Elias--Williamson is constructed from a reflection representation of a Coxeter group called a \defnemph{realization}.
For each finite parabolic dihedral subgroup they identify a corresponding two-colored Temperley--Lieb algebra, whose defining parameters depend on the realization \cite[\S 5.2]{ew-soergelcalc}.
In \cite[\S 5]{ew-localizedcalc} Elias--Williamson highlight some hidden assumptions about their realizations from \cite{ew-soergelcalc}.
Their most basic assumption (without which the diagrammatic Hecke category is not well defined) is that certain two-colored Jones--Wenzl projectors exist and are rotatable. 
For the benefit of future work we give a corrected definition of a realization (which ensures the existence and rotatability of these Jones--Wenzl projectors) below.

\begin{defn} \label{realizcorrected}
Let $\Bbbk$ be an integral domain.
A \defnemph{realization} of a Coxeter system $(W,S)$ over $\Bbbk$ consists of a free, finite rank $\Bbbk$-module $V$ along with subsets 
\begin{align*}
\{\alpha_s : s \in S\} & \subset V &  \{\alpha_s^\vee : s \in S\} \subset V^\ast=\Hom_\Bbbk(V,\Bbbk)
\end{align*}
 such that
\begin{enumerate}[label=(\roman*)]
\item $\langle \alpha_s^\vee,\alpha_s \rangle=2$ for all $s \in S$;

\item the assignment 
\begin{equation*}
s(\beta)=\beta-\langle \alpha_s^\vee, \beta \rangle \alpha_s
\end{equation*} 
for all $s \in S$ and $\beta \in V$ defines a representation of the Coxeter group $W$ on $V$; 

\item \label{item:abecond} for all $s,t \in S$ with $m_{st}<\infty$, we have
\begin{equation*}
\qbinom{m_{st}}{k}_{{\color{red} s}}(\langle \alpha_s^\vee,\alpha_t\rangle, \langle \alpha_t^\vee,\alpha_s\rangle)=\qbinom{m_{st}}{k}_{{\color{blue} t}}(\langle \alpha_s^\vee,\alpha_t\rangle, \langle \alpha_t^\vee,\alpha_s\rangle)=0
\end{equation*}
for all integers $1 \leq k \leq m_{st}-1$.
\end{enumerate}
\end{defn}

By \cref{existsrotates}, condition \ref{item:abecond} above is equivalent to the existence and rotatability of $\mathrm{JW}_{\Bbbk}(\prescript{}{{\color{red} s}}{(m_{st}-1)})$ and $\mathrm{JW}_{\Bbbk}(\prescript{}{{\color{blue} t}}{(m_{st}-1)})$ for $[2]_{{\color{red} s}}=\langle \alpha_s^\vee,\alpha_t\rangle$ and $[2]_{{\color{blue} t}}=\langle \alpha_t^\vee,\alpha_s\rangle$.
This condition is exactly Abe's assumption \cite[Assumption~1.1]{abe-homBS}, so \cref{abeequalsew} immediately follows by Abe's results \cite[Theorem~3.9]{abe-homBS} and \cite[Theorem~5.6]{abe-bimodhecke}.
It is also equivalent to 
\begin{equation} \label{eq:minpolycond}
\begin{aligned}
\Psi_{m_{st}}(\langle \alpha_s^\vee,\alpha_t \rangle \langle \alpha_t^\vee,\alpha_s \rangle)& =0 & &\text{if $m_{st}>2$,} \\
\langle \alpha_s^\vee,\alpha_t \rangle=\langle \alpha_t^\vee, \alpha_s^\vee \rangle& =0 & & \text{if $m_{st}=2$,}
\end{aligned}
\end{equation}
by \cref{qbinomideal}.

\begin{rem}
In \cite{ew-soergelcalc} Elias--Williamson incorrectly state that
\begin{equation} \label{eq:ewtechnicalcondition}
[m_{st}]_{{\color{red} s}}(\langle \alpha_s^\vee,\alpha_t\rangle, \langle \alpha_t^\vee,\alpha_s\rangle)=[m_{st}]_{{\color{blue} t}}(\langle \alpha_s^\vee,\alpha_t\rangle, \langle \alpha_t^\vee,\alpha_s\rangle)=0
\end{equation}
is enough to ensure the existence and rotatability of $\mathrm{JW}_\Bbbk(\prescript{}{{\color{red} s}}{(m_{st}-1)})$.
(This error was identified in \cite{ew-localizedcalc} but only partially resolved there.)
In the same paper Elias--Williamson also incorrectly state that \eqref{eq:ewtechnicalcondition} is equivalent to \eqref{eq:minpolycond}.
Amusingly, when these two statements are combined these errors accidentally cancel and the result is a correct statement!
\end{rem}

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