In this section we remind the reader what a planar algebra is, and recall a few facts about the simplest planar algebra, Temperley-Lieb.

\subsection{What is a planar algebra?}\label{subsec:pa}
A planar algebra is a gadget specifying how to combine elements in planar
ways, rather as a ``linear" algebra is a gadget in which one can combine elements, given a linear ordering.  For  example,
$$ f \circ g \circ h \text{ \quad versus \quad }\mathfig{0.2}{tangles/cubic-fgh}.$$
Planar algebras were introduced in \cite{math.QA/9909027} to study subfactors,
and have since found more general use.

In the simplest version, a planar algebra $\mathcal{P}$ associates a
vector space $\mathcal{P}_k$ to each natural number $k$ (thought of as a
disc in the plane with $k$ points on its boundary) and associates a linear map
$\mathcal{P}(T)  : \mathcal{P}_{k_1} \tensor \mathcal{P}_{k_2} \tensor
\cdots \tensor \mathcal{P}_{k_r} \to \mathcal{P}_{k_0}$ to each
`spaghetti and meatballs' or `planar tangle' diagram $T$ with internal
discs with $k_1, k_2, \ldots, k_r$ points and $k_0$ points on the
external disc.  For example,
$$\mathfig{0.2}{tangles/cubic}$$ gives a map from $V_7 \tensor V_5 \tensor V_5 \rightarrow V_7$. Such maps (the `planar operations') must satisfy certain
properties: radial spaghetti induces the identity map, and composition of the
maps $\mathcal{P}(T)$ is compatible with the obvious composition of
spaghetti and meatballs diagrams by gluing some inside another.  When we glue, we match up base points;
each disk's base point is specified by a bullet.

The reason for these bullets in the definition is that they allow us to keep track of pictures which are not rotationally invariant.  For example, in Definition \ref{def:pa} we have used the marked points to indicate the way the generator $S$ behaves under rotation. Nevertheless we use the following conventions to avoid always drawing a bullet.
Instead of using marked points we will often instead use a ``rectangular" picture in which some of the strings go up, some go down, but none come out of the sides of generators.  This leaves a gap on the left side of every picture and the convention is that the marked points always lie in this gap.  Further, if we neglect to draw a bounding disk, the reader should imagine a rectangle around the picture (and therefore put the marked point on the left).  For example, the following pictures represent the same element of a planar algebra: $S$ rotated by one `click'.

\begin{center}
\begin{tikzpicture}[baseline]
    \node (S) [circle,draw] {$S$};
    \draw (0,0) circle (1cm);
    \filldraw (S.157) circle (.5mm);
    \draw (S.180)-- (180:10mm);
    \draw (S.135)-- (135:10mm);
    \draw (S.90)-- (90:10mm);
    \draw (S.45)-- (45:10mm);
    \draw (S.0)-- (0:10mm);
    \draw (S.-45)-- (-45:10mm);
    \filldraw (112:1cm) circle (.5mm);
    \draw (S.-90)-- (-90:10mm);
    \draw (S.-135)-- (-135:10mm);
\end{tikzpicture}
\quad , \quad
\begin{tikzpicture}[baseline]
    \clip (-1.1,-1.4) rectangle (1.1,1.4);

    \draw[rounded corners=2mm] (-.35,0) -- ++(90:.7cm) -- ++(180:.5cm) -- ++ (-90:1.7cm) -- ++(0:1.6cm) -- ++(90:3cm);
    \draw (.35,0) -- ++(90:15mm);
    \draw (-.25,0) -- ++(90:15mm);
    \draw (-.15,0) -- ++(90:15mm);
    \draw (-.05,0) -- ++(90:15mm);
    \draw (.05,0) -- ++(90:15mm);
    \draw (.15,0) -- ++(90:15mm);
    \draw (.25,0) -- ++(90:15mm);
    
    \node (S) at (0,0) [rectangle, draw, fill=white] {$\;\; S \;\;$};
\end{tikzpicture}
\end{center}

There are some special planar tangles which
induce operations with familiar names. First, each even vector space $\mathcal{P}_{2k}$ becomes an associative algebra using the `multiplication' tangle:
$$\mathfig{0.2}{tangles/multiplication}$$
Second, there is an involution $\overline{\phantom{X}} : \mathcal{P}_{2k} \to \mathcal{P}_{2k}$ given by the `dualising' tangle:
$$\mathfig{0.2}{tangles/dualising}$$
Third, for each $k$ there is a trace $\text{tr} : \mathcal{P}_{2k} \rightarrow \mathcal{P}_0$:
$$\mathfig{0.2}{tangles/trace}$$
If $\mathcal{P}_0$ is one-dimensional, this map really is a trace, and we can use it (along with multiplication) to build a bilinear form on $P_{2k}$ in the usual way.

A subfactor planar algebra is the best kind of planar algebra; it has additional properties which make it a nice place to work.  First and foremost, $\mathcal{P}_0$ must be one-dimensional.  In particular, a closed circle is equal to a multiple of the empty diagram, and the square of this multiple is called the \emph{index} of the planar algebra.  Note that this implies that the zero-ary planar operations, namely the `vegetarian' diagrams without any meatballs, induce the Temperley-Lieb diagrams (see below, \S \ref{ssTL}) as elements of the subfactor planar algebra.
There is thus a map $\TL_{2k} \to \mathcal{P}_{2k}$, although it need be neither surjective nor injective.

Second, subfactor planar algebras have the property that only spaces for discs with an even number of boundary points are nonzero. Third, subfactor planar algebras must be spherical, that is, for each element $T \in \mathcal{P}_2$, we have an identity in $\mathcal{P}_0$:
\begin{equation*}
\mathfig{0.1}{spherical/1} = \mathfig{0.1}{spherical/3}.
\end{equation*}
Fourth, there must be an anti-linear \emph{adjoint} operation $*:\mathcal{P}_k \to \mathcal{P}_k$ such that the sesquilinear form given by $\left< x , y \right> = \tr{y^*x}$ is positive definite.  Further, $*$ on $\mathcal{P}$ should be compatible with the horizontal reflection operation $*$ on planar tangles.  In particular, this means that the adjoint operation on Temperley-Lieb is reflection in a horizontal line.

Finally note that we use ``star'' to indicate the \emph{adjoint}, and ``bar'' to indicate the \emph{dual}. We apologise
to confused readers for this notation.

One useful way to generalize the definition of a planar algebra is by introducing a `spaghetti label set' and a `region label set,'
and perhaps insist that only certain labels can appear next to each other. When talking about subfactor planar algebras, only two simple cases of this
are required: a `standard' subfactor planar algebra has just two region labels, shaded and unshaded, which must alternate across spaghetti, while an `unshaded' subfactor planar algebra has no interesting labels at all.

From this point onwards, we'll be using the unshaded variety of planar algebra, essentially just for simplicity of exposition. The reader can easily
reconstruct the \emph{shaded} version of everything we say; checkerboard shade the regions, ensuring that the marked point of an $S$ box is always in an unshaded region.
This necessitates replacing relation \ref{rotateS} in Definition \ref{def:pa}, so that instead the ``2 click'' rotation of the $S$ box is $-1$ times the original unrotated box.
The one point at which reintroducing the shading becomes subtle is when we discuss braidings in \S \ref{sec:braiding}.

\subsection{The Temperley-Lieb (planar) algebra} \label{ssTL}
We work over the field $\Complex(q)$ of rational functions in a formal variable $q$.
It is often notationally convenient to use quantum numbers.

\begin{defn}
The $n^{th}$ quantum number $\qi{n}$ is defined as $$\frac{q^n-q^{-n}}{q-q^{-1}} = q^{n-1} + q^{n-3} + \cdots + q^{-n+1}.$$
 \end{defn}

Now let's recall some facts about the Temperley-Lieb algebra.

\begin{defn}
A Temperley-Lieb picture is a non-crossing matching of $2n$ points around the boundary of a disc, with a chosen first point.
\end{defn}

In practice, Temperley-Lieb pictures are often drawn with the points on two lines, and the chosen first point is the one on the top left.

\begin{defn}
The vector space $\TL_{2n}$ has as a basis the Temperley-Lieb pictures for $2n$ points.  These assemble into a planar
algebra by gluing diagrams into planar tangles, and removing each
closed circle formed in exchange for a coefficient of $\qi{2} = q + q^{-1}$.
\end{defn}

Temperley-Lieb is a subfactor planar algebra (with the adjoint operation being horizontal reflection) except that the sesquilinear form need not be positive definite (see \S \ref{sec:special-TL}).  Some important elements of the Temperley-Lieb algebra are

\begin{itemize}
\item The identity (so-called because it's the identity for the multiplication given by vertical stacking):
$$\id =\inputtikz{Identity};$$
\item The Jones projections in $\TL_{2n}$:
$$e_i=\inputtikz{JonesProjection}, \, i \in \{ 1, \ldots, n-1\};$$
\item
The Jones-Wenzl projection \cite{MR873400} $\JW{n}$ in $\TL_{2n}$:  The unique projection
with the property
$$\JW{n} e_i = e_i \JW{n} =0, \, \text{for each $i \in \{1, \ldots, n-1 \}$};$$
\item The crossing in $\TL_4$:
\begin{equation*}
\inputtikz{ResolvedCrossing}.
\end{equation*}
\end{itemize}

Recall that the crossing satisfies Reidemeister relations $2$ and $3$, but not Reidemeister $1$.  Instead the positive twist factor is $i q^{3/2}$.

\begin{fact}
Here are some useful identities involving the Jones-Wenzl projections:
\begin{enumerate}
\item
{ \bf Wenzl's relation:}
\begin{equation} \label{WenzlsRelation} \inputtikz{WenzlRelation} \end{equation}

and $\tr{\JW{m}} = \qi{m+1}$.

\item
{\bf Partial trace relation:} \begin{equation}\label{eq:PartialTraceRelation}\inputtikz{PartialTraceRelation}\end{equation}

\end{enumerate}
\end{fact}

\subsection{Temperley-Lieb when $f^{(4n-3)} = 0$}
\label{sec:special-TL}%
At any `special value' $q=e^{\frac{i \pi}{k+2}}$ (equivalently $\delta = q+q^{-1} = 2 \cos(\frac{\pi}{k+2})$), the
Temperley-Lieb planar algebra is degenerate, with radical generated by
the Jones-Wenzl projection $f^{(k+1)}$. We therefore pass to a
quotient, by imposing the relation $f^{(k+1)} = 0$. In the physics literature $k$ would be called the level.  We're interested in the case $k=4n-4$, so
$q=e^{i \pi/(4n-2)}$ and $\delta=2 \cos{ \frac{\pi}{4n-2}}$.  For this value of $q$, $\qi{m}=\qi{4n-2-m}$.



We record several facts about this quotient of Temperley-Lieb which we'll need later.
(In the following diagrams, we're just drawing $3$ or $4$ parallel strands where we really mean $4n-5$ or $4n-4$ respectively; make sure you read the labels of the boxes.)

\begin{lem} Strands cabled by $f^{(4n-4)}$ can be reconnected.
\label{lem:2f}
$$\mathfig{0.3}{consistency/2f} = \rotatemathfig{0.3}{90}{consistency/2f}$$
\end{lem}
\begin{rem}
Any relation in Temperley-Lieb also holds if superimposed on top of, or
behind, another Temperley-Lieb diagram; this is just the statement that
Temperley-Lieb is braided. We'll need to use all these variations of the
identity in the above lemma later.
\end{rem}

\begin{lem}
\label{lem:f-twist}
\begin{align*}
\rotatemathfig{0.15}{-90}{consistency/f-twist-1} & =
\rotatemathfig{0.15}{-90}{consistency/f-twist-2}
     =
     %q^{\frac {k} {2} +1} i^{k}
     q^{2n-1} \rotatemathfig{0.12}{-90}{consistency/f-no-twist}
     =
     %i^{k+1}
     i \rotatemathfig{0.12}{-90}{consistency/f-no-twist}
\end{align*}
(the twisted strand here indicates just a single strand, while the 3 parallel strands actually represent 4n-5 strands)
and as an easy consequence
$$\rotatemathfig{0.25}{-90}{consistency/f-twist-full} =
%(-1)^{\frac {k(k+1)}{2}}
\rotatemathfig{0.12}{-90}{consistency/f-no-twist}.$$
\end{lem}

The first two equalities hold in Temperley-Lieb at any value of $q$. The
third equality simply specialises to the relevant value. Note that the
crossings in the above lemma are all undercrossings for the single strand. Changing each
of these to an overcrossing for that strand, we have the same identities, with $q$ replaced by
$q^{-1}$, and $i$ replaced by $-i$.

\begin{lem} Overcrossings, undercrossings and the $2$-string identity cabled by $f^{(4n-4)}$ are all the same.
\label{lem:f-crossings}
$$\mathfig{0.3}{consistency/f-crossings-1} = \mathfig{0.3}{consistency/f-crossings-2} =
%(-1)^{\frac {k(k+1)}{2}}
\mathfig{0.3}{consistency/2f}$$
\end{lem}
