Again, it is easy to verify this gives a solution, shown in .

§ CONSTRUCTING EH3 VIA A Q-SYSTEM

We now show that $1\oplus \jw{6}\in \cE\cH_2$ can be endowed with the structure of a Q-system.
To do so, we prove a result similar to <cit.>.

A Q-system in a rigid $\rm C^*$-tensor category $\scrC$ is an algebra object $(A,\mu,i)$ which satisfies the following properties:

* ($\rm C^*$-Frobenius)
$(\id_A \otimes \mu)\circ (\mu^*\otimes \id_A) = \mu^*\circ \mu = (\mu\otimes \id_A)\circ (\id_A \otimes \mu^*)$,

* (special)
$\mu\circ \mu^*$ is a (non-zero) multiple of $\id_A$, and

* (standard)
$i^*\circ i = \sqrt{\dim_\scrC(A)} \id_{1_\scrC}$ and $\mu\circ \mu^* = \sqrt{\dim_\scrC(A)} \id_A$.

Suppose $\sigma$ is a symmetrically self-dual object in a unitary fusion category $\scrC$ with $\dim(\sigma)>1$.
If $T \in \Hom(\sigma\otimes \sigma \to \sigma)$ and $b>0$ such that

* (bigon)
\begin{tikzpicture}[baseline=-.6cm]
\node (T1) [circle, fill=white, thick, draw]  at (0,0) {$T$};
\node (T2) [circle, fill=white, thick, draw]  at (0,-1) {$T$};
\node [above=1pt] at (T1.west) {$\star$};
\node [below=1pt] at (T2.west) {$\star$};
\draw (T1) [in=45,out=-45] to (T2);
\draw (T1) [in=135,out=-135] to (T2);
\draw (T1) -- (0,1);
\draw (T2) -- (0,-2);
\node at (-.6,-.5) {\scriptsize{$\sigma$}};
\node at (.6,-.5) {\scriptsize{$\sigma$}};
\node at (0,1.2) {\scriptsize{$\sigma$}};
\node at (0,-2.2) {\scriptsize{$\sigma$}};
\end{tikzpicture}
\begin{tikzpicture}[baseline=-.1cm]
\draw (0,-1) -- (0,1);
\node at (0,-1.2) {\scriptsize{$\sigma$}};
\end{tikzpicture}

* (rotational invariance)
\begin{tikzpicture}[baseline=-.1cm]
\node (S) [circle, fill=white, thick, draw]  at (0,0) {$T$};
\node [below=1pt] at (S.west) {$\star$};
\draw (S) -- (0,1);
\draw (S) -- (-.6,-1);
\draw (S) -- (.6,-1);
\node at (-.6,-1.2) {\scriptsize{$\sigma$}};
\node at (.6,-1.2) {\scriptsize{$\sigma$}};
\node at (0,1.2) {\scriptsize{$\sigma$}};
\end{tikzpicture}
\begin{tikzpicture}[baseline=-.1cm]
\node (S) [circle, fill=white, thick, draw]  at (0,0) {$T$};
\node [below=1pt] at (S.east) {$\star$};
\draw (S) -- (0,1);
\draw (S) -- (-.6,-1);
\draw (S) -- (.6,-1);
\node at (-.6,-1.2) {\scriptsize{$\sigma$}};
\node at (.6,-1.2) {\scriptsize{$\sigma$}};
\node at (0,1.2) {\scriptsize{$\sigma$}};
\end{tikzpicture}

* (self-adjoint)
\begin{tikzpicture}[baseline=-.1cm, yscale=-1]
\node (S) [circle, fill=white, thick, draw]  at (0,0) {$T^*$};
\node [below=1pt] at (S.west) {$\star$};
\draw (S) -- (0,1);
\draw (S) -- (-.6,-1);
\draw (S) -- (.6,-1);
\node at (-.6,-1.2) {\scriptsize{$\sigma$}};
\node at (.6,-1.2) {\scriptsize{$\sigma$}};
\node at (0,1.2) {\scriptsize{$\sigma$}};
\end{tikzpicture}
\begin{tikzpicture}[baseline=-.1cm]
\node (S) [circle, fill=white, thick, draw]  at (0,0) {$T$};
\node [below=1pt] at (S.west) {$\star$};
\draw (S) -- (0,1);
\draw (S) -- (-.6,-1);
\draw (S) [out=-60,in=-90] to (1,-.2) -- (1,1);
\node at (-.6,-1.2) {\scriptsize{$\sigma$}};
\node at (1,1.2) {\scriptsize{$\sigma$}};
\node at (0,1.2) {\scriptsize{$\sigma$}};
\end{tikzpicture}
$, and

* (I=H)
\displaystyle
\begin{tikzpicture}[baseline=-.4cm]
\node (T1) [circle, fill=white, thick, draw]  at (0,.2) {$T$};
\node (T2) [circle, fill=white, thick, draw]  at (.8,-1) {$T$};
\node [below=1pt] at (T1.west) {$\star$};
\node [below=1pt] at (T2.west) {$\star$};
\draw (T1) -- (0,1);
\draw (T1) -- (-1.2,-1.6);
\draw (T1) -- (T2);
\draw (T2) -- (.4,-1.6);
\draw (T2) -- (1.2,-1.6);
\node at (-1.2,-1.8) {\scriptsize{$\sigma$}};
\node at (.4,-1.8) {\scriptsize{$\sigma$}};
\node at (1.2,-1.8) {\scriptsize{$\sigma$}};
\node at (.4,-.6) {\scriptsize{$\sigma$}};
\node at (0,1.2) {\scriptsize{$\sigma$}};
\end{tikzpicture}
\begin{tikzpicture}[baseline=-.4cm, xscale=-1]
\node (T1) [circle, fill=white, thick, draw]  at (0,.2) {$T$};
\node (T2) [circle, fill=white, thick, draw]  at (.8,-1) {$T$};
\node [below=1pt] at (T1.west) {$\star$};
\node [below=1pt] at (T2.west) {$\star$};
\draw (T1) -- (0,1);
\draw (T1) -- (-1.2,-1.6);
\draw (T1) -- (T2);
\draw (T2) -- (.4,-1.6);
\draw (T2) -- (1.2,-1.6);
\node at (-1.2,-1.8) {\scriptsize{$\sigma$}};
\node at (.4,-1.8) {\scriptsize{$\sigma$}};
\node at (1.2,-1.8) {\scriptsize{$\sigma$}};
\node at (.4,-.6) {\scriptsize{$\sigma$}};
\node at (0,1.2) {\scriptsize{$\sigma$}};
\end{tikzpicture}
\frac{b}{\dim(\sigma)-1}
\left(
\begin{tikzpicture}[baseline=-.1cm]
\draw (-.4,-1) -- (-.4,-.7) arc (180:0:.4cm) -- (.4,-1);
\draw (.8,-1) .. controls ++(90:1cm) and ++(270:1cm) .. (.2,1);
\node at (-.4,-1.2) {\scriptsize{$\sigma$}};
\node at (.4,-1.2) {\scriptsize{$\sigma$}};
\node at (.8,-1.2) {\scriptsize{$\sigma$}};
\node at (.2,1.2) {\scriptsize{$\sigma$}};
\end{tikzpicture}
\begin{tikzpicture}[baseline=-.1cm]
\draw (-.4,-1) -- (-.4,-.7) arc (180:0:.4cm) -- (.4,-1);
\draw (-.8,-1) .. controls ++(90:1cm) and ++(270:1cm) .. (-.2,1);
\node at (-.8,-1.2) {\scriptsize{$\sigma$}};
\node at (-.4,-1.2) {\scriptsize{$\sigma$}};
\node at (.4,-1.2) {\scriptsize{$\sigma$}};
\node at (-.2,1.2) {\scriptsize{$\sigma$}};
\end{tikzpicture}
\right)

then $1\oplus \sigma$ can be canonically endowed with the structure of a Q-system.
If moreover $\dim(\Hom_\scrC(1, \sigma)) =0$, then $1\oplus \sigma$ is irreducible.

If $\sigma\in\scrC$ is simple, then the existence of $b>0$ such that the bigon axiom holds for $T$ follows formally from the rotational invariance and self-adjoint axioms.

Let $A=1\oplus \sigma$.
We need to construct maps $\mu: A\otimes A \to A$ and $i: 1\to A$ such that $(A,\mu,i)$ is a Q-system.
We define $\mu$ as follows:

\begin{array}{c|cc}
1 & 1 & \sigma
\\\hline
1 & \id_1 & 0
\\
\sigma & 0 &\ev_\sigma
\end{array}
\qquad\qquad
\begin{array}{c|cc}
\sigma & 1 & \sigma
\\\hline
1 & 0 & \id_\sigma
\\
\sigma & \id_\sigma & c T
\end{array}

We now solve for $c$ using associativity.
The only interesting diagram to check is when the boundaries are all $\sigma$, which simplifies to

\displaystyle
\begin{tikzpicture}[baseline=-.4cm]
\node (T1) [circle, fill=white, thick, draw]  at (0,.2) {$T$};
\node (T2) [circle, fill=white, thick, draw]  at (.8,-1) {$T$};
\node [below=1pt] at (T1.west) {$\star$};
\node [below=1pt] at (T2.west) {$\star$};
\draw (T1) -- (0,1);
\draw (T1) -- (-1.2,-1.6);
\draw (T1) -- (T2);
\draw (T2) -- (.4,-1.6);
\draw (T2) -- (1.2,-1.6);
\node at (-1.2,-1.8) {\scriptsize{$\sigma$}};
\node at (.4,-1.8) {\scriptsize{$\sigma$}};
\node at (1.2,-1.8) {\scriptsize{$\sigma$}};
\node at (.4,-.6) {\scriptsize{$\sigma$}};
\node at (0,1.2) {\scriptsize{$\sigma$}};
\end{tikzpicture}
\begin{tikzpicture}[baseline=-.4cm, xscale=-1]
\node (T1) [circle, fill=white, thick, draw]  at (0,.2) {$T$};
\node (T2) [circle, fill=white, thick, draw]  at (.8,-1) {$T$};
\node [below=1pt] at (T1.west) {$\star$};
\node [below=1pt] at (T2.west) {$\star$};
\draw (T1) -- (0,1);
\draw (T1) -- (-1.2,-1.6);
\draw (T1) -- (T2);
\draw (T2) -- (.4,-1.6);
\draw (T2) -- (1.2,-1.6);
\node at (-1.2,-1.8) {\scriptsize{$\sigma$}};
\node at (.4,-1.8) {\scriptsize{$\sigma$}};
\node at (1.2,-1.8) {\scriptsize{$\sigma$}};
\node at (.4,-.6) {\scriptsize{$\sigma$}};
\node at (0,1.2) {\scriptsize{$\sigma$}};
\end{tikzpicture}
\frac{1}{c^2}
\left(
\begin{tikzpicture}[baseline=-.1cm]
\draw (-.4,-1) -- (-.4,-.7) arc (180:0:.4cm) -- (.4,-1);
\draw (.8,-1) .. controls ++(90:1cm) and ++(270:1cm) .. (.2,1);
\node at (-.4,-1.2) {\scriptsize{$\sigma$}};
\node at (.4,-1.2) {\scriptsize{$\sigma$}};
\node at (.8,-1.2) {\scriptsize{$\sigma$}};
\node at (.2,1.2) {\scriptsize{$\sigma$}};
\end{tikzpicture}
\begin{tikzpicture}[baseline=-.1cm]
\draw (-.4,-1) -- (-.4,-.7) arc (180:0:.4cm) -- (.4,-1);
\draw (-.8,-1) .. controls ++(90:1cm) and ++(270:1cm) .. (-.2,1);
\node at (-.8,-1.2) {\scriptsize{$\sigma$}};
\node at (-.4,-1.2) {\scriptsize{$\sigma$}};
\node at (.4,-1.2) {\scriptsize{$\sigma$}};
\node at (-.2,1.2) {\scriptsize{$\sigma$}};
\end{tikzpicture}
\right)

Hence we must have $c = \pm \left(\frac{\dim(\sigma)-1}{b}\right)^{1/2}$.
Notice that these two choices for $c$ give algebras $A_\pm$ which are isomorphic via $(\id_1,-\id_\sigma): A_{\pm}\to A_{\mp}$.
Without loss of generality, we take $A = A_+$.

We see that $i = (\id_1, 0) : 1_\scrC \to A= 1_\scrC \oplus \sigma$ is the unique unit such that $(A, \mu, i)$ is an algebra.
By construction, $\mu$ is rotationally invariant and self-adjoint (as in hypotheses of this propostion).
Together with the facts that $(A,\mu,i)$ is an algebra and $(A, \mu^*, i^*)$ is a coalgebra, this immediately implies the $\rm C^*$-Frobenius property.
It is now straightforward to check that $(A, \mu, i)$ is actually special.
Indeed, this follows automatically as $\Hom(1_\scrC, A)$ is one dimensional c.f. <cit.>.
Finally, another straightforward calculation shows that $i^*\circ i = \id_{1_\scrC}$ and

\mu\circ \mu^*= (1+\dim(\sigma)) \id_{1_\scrC} + (2+c^2b)\id_\sigma = \dim_\scrC(A) \id_A.

Hence normalizing by a factor of $\sqrt{\dim_\scrC(A)}$ yields a (standard) Q-system.

The final statement now follows since $\dim(\Hom_\scrC(1,1\oplus \sigma))=1$.

§.§ Constructing the Q-system

Below, we use the notational convention that a solid bar across a number of strings denotes the Jones-Wenzl idempotent on that number of strands.
For example,

\begin{tikzpicture}[baseline=-.1cm]
\draw (-.2,-.4) -- (-.2,.4);
\draw (.2,-.4) -- (.2,.4);
\draw[very thick] (-.4,0) -- (.4,0);
\node at (-.2,-.6) {\scriptsize{$3$}};
\node at (.2,-.6) {\scriptsize{$3$}};
\end{tikzpicture}
\begin{tikzpicture}[baseline=-.1cm]
\draw (0,-.6) -- (0,.6);
\filldraw[fill=white,thick] (-.4,-.3) rectangle (.4,.3);
\node at (0,0) {$\JW{6}$};
\node at (0,-.8) {\scriptsize{$6$}};
\node at (0,.8) {\scriptsize{$6$}};
\end{tikzpicture}
\,.

We define the following elements in $\Hom(\jw{6}\otimes \jw{6} \to \jw{6})$:

\begin{tikzpicture}[baseline=-.1cm]
\draw (-.4,-1) -- (-.4,-.7) arc (180:0:.4cm) -- (.4,-1);
\draw (-.8,-1) .. controls ++(90:1cm) and ++(270:1cm) .. (-.2,1);
\draw (.8,-1) .. controls ++(90:1cm) and ++(270:1cm) .. (.2,1);
\draw[very thick] (-1,-.7) -- (-.2,-.7);
\draw[very thick] (1,-.7) -- (.2,-.7);
\draw[very thick] (-.4,.7) -- (.4,.7);
\node at (-.8,-1.2) {\scriptsize{$3$}};
\node at (-.4,-1.2) {\scriptsize{$3$}};
\node at (.4,-1.2) {\scriptsize{$3$}};
\node at (.8,-1.2) {\scriptsize{$3$}};
\node at (.2,1.2) {\scriptsize{$3$}};
\node at (-.2,1.2) {\scriptsize{$3$}};
\end{tikzpicture}
\qquad\qquad
\begin{tikzpicture}[baseline=-.1cm]
\node (S) [circle, fill=white, thick, draw]  at (0,0) {$S$};
\node [below=1pt] at (S.west) {$\star$};
\draw (S) -- (0,1);
\draw (S) -- (-.4,-1);
\draw (S) -- (.4,-1);
\draw (.8,-1) .. controls ++(90:1cm) and ++(270:1cm) .. (.4,1);
\draw[very thick] (0,-.7) -- (1,-.7);
\draw[very thick] (-.2,.7) -- (.6,.7);
\node at (-.4,-1.2) {\scriptsize{$6$}};
\node at (.4,-1.2) {\scriptsize{$5$}};
\node at (.8,-1.2) {\scriptsize{$1$}};
\node at (0,1.2) {\scriptsize{$5$}};
\node at (.4,1.2) {\scriptsize{$1$}};
\end{tikzpicture}
\qquad\qquad
\begin{tikzpicture}[baseline=-.1cm, xscale=-1]
\node (S) [circle, fill=white, thick, draw]  at (0,0) {$S$};
\node [below=-4pt] at (S.south) {$\star$};
\draw (S) -- (0,1);
\draw (S) -- (-.4,-1);
\draw (S) -- (.4,-1);
\draw (.8,-1) .. controls ++(90:1cm) and ++(270:1cm) .. (.4,1);
\draw[very thick] (0,-.7) -- (1,-.7);
\draw[very thick] (-.2,.7) -- (.6,.7);
\node at (-.4,-1.2) {\scriptsize{$6$}};
\node at (.4,-1.2) {\scriptsize{$5$}};
\node at (.8,-1.2) {\scriptsize{$1$}};
\node at (0,1.2) {\scriptsize{$5$}};
\node at (.4,1.2) {\scriptsize{$1$}};
\end{tikzpicture}
\qquad\qquad
\begin{tikzpicture}[baseline=-.1cm]
\node (S) [circle, fill=white, thick, draw]  at (0,.2) {$S$};
\node [below=1pt] at (S.east) {$\star$};
\draw (S) -- (0,1);
\draw (S) -- (-.8,-1);
\draw (S) -- (.8,-1);
\draw (-.4,-1) -- (-.4,-.8) arc (180:0:.4cm) -- (.4,-1);
\draw[very thick] (.2,-.8) -- (1,-.8);
\draw[very thick] (-.2,-.8) -- (-1,-.8);
\node at (-.8,-1.2) {\scriptsize{$5$}};
\node at (.8,-1.2) {\scriptsize{$5$}};
\node at (0,1.2) {\scriptsize{$6$}};
\end{tikzpicture}

Recall the following formula for coefficients in the Jones-Wenzl idempotent $\jw{k}$:

\underset{\in f^{(k)}}{\coeff}
\left(
\begin{tikzpicture}[baseline = -.1cm, yscale=.5]
\draw (-.5,-.8)--(-.5,.8);
\draw (-.3,.8) arc (-180:0:.2cm);
\draw (-.3,-.8)--(.3,.8);
\draw (-.1,-.8) arc (180:0:.2cm);
\draw (.5,-.8)--(.5,.8);
\node at (-.7,0) {{\scriptsize{$a$}}};
\node at (-.2,0) {{\scriptsize{$b$}}};
\node at (.7,0) {{\scriptsize{$c$}}};
\end{tikzpicture}
\right)

The set $\{T_0,T_1,T_2,T_3\}$ is linearly independent and is thus a basis for $\Hom(f^{(6)}\otimes f^{(6)}\to f^{(6)})$.

Suppose $\sum_{j=0}^3 \lambda_j T_j = 0$.
We calculate the coefficient of the following diagrams in $\sum_{j=0}^3 \lambda_j T_j$:

\begin{tikzpicture}[baseline=-.1cm]
\node (S) [circle, fill=white, thick, draw]  at (0,0) {$S$};
\node [below=1pt] at (S.west) {$\star$};
\draw (S) -- (0,1);
\draw (S) -- (0,-1);
\draw (.4,-1) arc (180:0:.2cm);
\node at (0,-1.2) {\scriptsize{$10$}};
\node at (0,1.2) {\scriptsize{$6$}};
\node at (.4,-1.2) {\scriptsize{$1$}};
\node at (.8,-1.2) {\scriptsize{$1$}};
\end{tikzpicture}

By expanding the Jones Wenzls in $\sum_{j=0}^3 \lambda_j T_j$ and noticing that almost all positions of cups or caps put a cap on $S$ (which is uncappable), we calculate that the coefficient in $\sum_{j=0}^3 \lambda_j T_j$ of $C$ is given by

\lambda_1
\underset{\in f^{(6)}}{\coeff}\left(\,
\begin{tikzpicture}[baseline = -.1cm]
\draw[thick] (-.6,-.4)--(-.6,.4)--(.6,.4)--(.6,-.4)--(-.6,-.4);
\draw (0,.4) arc (-180:0:.2cm);
\draw (-.4,-.4)--(-.4,.4);
\draw (0,-.4) arc (180:0:.2cm);
\node at (-.2,0) {{\scriptsize{$4$}}};
\end{tikzpicture}
\,\right)
\lambda_3
\omega_S^3
\underset{\in f^{(6)}}{\coeff}\left(\,
\begin{tikzpicture}[baseline = -.1cm]
\draw[thick] (-.6,-.4)--(-.6,.4)--(.8,.4)--(.8,-.4)--(-.6,-.4);
\draw (-.2,-.4)--(.4,.4);
\draw (.2,-.4) arc (180:0:.2cm);
\draw (-.4,.4) arc (-180:0:.2cm);
\node at (.3,0) {{\scriptsize{$4$}}};
\end{tikzpicture}
\,\right)
\frac{[5]}{[6]}
+(-1)^3 \lambda_3 \frac{-1}{[6]}
\frac{\lambda_3-[5]\lambda_1}{[6]}.

This quantity must be zero as $\sum_{j=0}^3 \lambda_j T_j=0$.
Now considering the 6-click rotations of $\sum_{j=0}^3 \lambda_j T_j$ and $C$, we also see that

\frac{\lambda_1-[5]\lambda_2}{[6]}=0
\qquad\text{and}\qquad
\frac{\lambda_2-[5]\lambda_3}{[6]}=0.

This is only possible when $\lambda_1 = \lambda_2=\lambda_3 = 0$.
We now conclude that $\lambda_0=0$ as well.

Any rotationally invariant map $\jw{6}\otimes \jw{6}\to \jw{6}$ is of the form $\alpha T_0 + \beta (T_1+T_2+T_3)$.

This follows from the previous lemma together with $T_1, T_2, T_3$ being rotations of each other.

The element $T=\alpha  T_0 + \beta (T_1+T_2+T_3)$ is self-adjoint if and only if $\alpha$ is real and $\beta$ is purely imaginary.

Since $\omega_S^3 = (-1)^3=-1$,
denoting $\sigma = \jw{6}$,
we have

\begin{equation*}
\begin{tikzpicture}[baseline=-.1cm, yscale=-1]
\node (S) [circle, fill=white, thick, draw]  at (0,0) {$T_0^*$};
\node [below=1pt] at (S.west) {$\star$};
\draw (S) -- (0,1);
\draw (S) -- (-.6,-1);
\draw (S) -- (.6,-1);
\node at (-.6,-1.2) {\scriptsize{$\sigma$}};
\node at (.6,-1.2) {\scriptsize{$\sigma$}};
\node at (0,1.2) {\scriptsize{$\sigma$}};
\end{tikzpicture}
\begin{tikzpicture}[baseline=-.1cm]
%	\draw (0,0) arc (-180:0:.6cm) -- (1.2,1);
\node (S) [circle, fill=white, thick, draw]  at (0,0) {$T_0$};
\node [below=1pt] at (S.west) {$\star$};
\draw (S) -- (0,1);
\draw (S) -- (-.6,-1);
\draw (S) [out=-60,in=-90] to (1,-.2) -- (1,1);
\node at (-.6,-1.2) {\scriptsize{$\sigma$}};
\node at (1,1.2) {\scriptsize{$\sigma$}};
\node at (0,1.2) {\scriptsize{$\sigma$}};
\end{tikzpicture}
\qquad
\begin{tikzpicture}[baseline=-.1cm, yscale=-1]
\node (S) [circle, fill=white, thick, draw]  at (0,0) {$T_2^*$};
\node [below=1pt] at (S.west) {$\star$};
\draw (S) -- (0,1);
\draw (S) -- (-.6,-1);
\draw (S) -- (.6,-1);
\node at (-.6,-1.2) {\scriptsize{$\sigma$}};
\node at (.6,-1.2) {\scriptsize{$\sigma$}};
\node at (0,1.2) {\scriptsize{$\sigma$}};
\end{tikzpicture}
\node (S) [circle, fill=white, thick, draw]  at (0,0) {$T_2$};
\node [below=1pt] at (S.west) {$\star$};
\draw (S) -- (0,1);
\draw (S) -- (-.6,-1);
\draw (S) [out=-60,in=-90] to (1,-.2) -- (1,1);
\node at (-.6,-1.2) {\scriptsize{$\sigma$}};
\node at (1,1.2) {\scriptsize{$\sigma$}};
\node at (0,1.2) {\scriptsize{$\sigma$}};
\end{tikzpicture}
\qquad
\text{and}
\qquad
\begin{tikzpicture}[baseline=-.1cm, yscale=-1]
\node (S) [circle, fill=white, thick, draw]  at (0,0) {$T_1^*$};
\node [below=1pt] at (S.west) {$\star$};
\draw (S) -- (0,1);
\draw (S) -- (-.6,-1);
\draw (S) -- (.6,-1);
\node at (-.6,-1.2) {\scriptsize{$\sigma$}};
\node at (.6,-1.2) {\scriptsize{$\sigma$}};
\node at (0,1.2) {\scriptsize{$\sigma$}};
\end{tikzpicture}
\node (S) [circle, fill=white, thick, draw]  at (0,0) {$T_3$};
\node [below=1pt] at (S.west) {$\star$};
\draw (S) -- (0,1);
\draw (S) -- (-.6,-1);
\draw (S) [out=-60,in=-90] to (1,-.2) -- (1,1);
\node at (-.6,-1.2) {\scriptsize{$\sigma$}};
\node at (1,1.2) {\scriptsize{$\sigma$}};
\node at (0,1.2) {\scriptsize{$\sigma$}};
\end{tikzpicture}.\qedhere
\end{equation*}

The element $T=\alpha T_0 +  \beta (T_1+T_2+T_3)$ for $\alpha \in \mathbb R$  and $\beta \in i \mathbb R$ satisfies the bigon axiom if and only if at least one of $\alpha$ and $\beta$ are nonzero.

Since $T$ is self-adjoint, the bigon axiom is equivalent to $T^*T\in \End(\jw{6})=\bbC$ being non-zero and positive, which holds if $T\neq 0$.

§.§ Calculations in the graph planar algebra

We still need to find $\alpha \in \mathbb R$ and $\beta \in i \mathbb R$, not both zero, so that $T=\alpha T_0+\beta (T_1+T_2+T_3)$ satisfies the $I=H $ relation.

This calculation is performed in the lopsided graph planar algebra <cit.> of the $\cE\cH_2-\cE\cH_1$ subfactor principal graph.
Recall that in <cit.> we gave explicit formulas for the image of the generator $S$ of the Extended Haagerup subfactor planar algebra in the (spherical) graph planar algebra.
It is just a small normalization issue to obtain this element in the lopsided graph planar algebra.
This lopsided graph planar algebra is built out of vector spaces over the number field $\bbQ(\lambda)$, where $\lambda$ is the largest imaginary root of $\lambda^6 + 2\lambda^4 - 3\lambda^2 -5$.

We first prepare the elements $T_0, T_1, T_2, T_3$ by direct calculation in the graph planar algebra (first constructing the 6-strand Jones-Wenzl idempotent using the usual Wenzl recursion formula).

In a world with bigger and faster computers, we would next calculate the 32 compositions

\displaystyle
\begin{tikzpicture}[baseline=-.4cm]
\node (T1) [circle, fill=white, thick, draw]  at (0,.2) {$T_i$};
\node (T2) [circle, fill=white, thick, draw]  at (.8,-1) {$T_j$};
\node [below=1pt] at (T1.west) {$\star$};
\node [below=1pt] at (T2.west) {$\star$};
\draw (T1) -- (0,1);
\draw (T1) -- (-1.2,-1.6);
\draw (T1) -- (T2);
\draw (T2) -- (.4,-1.6);
\draw (T2) -- (1.2,-1.6);
\node at (-1.2,-1.8) {\scriptsize{$\sigma$}};
\node at (.4,-1.8) {\scriptsize{$\sigma$}};
\node at (1.2,-1.8) {\scriptsize{$\sigma$}};
\node at (.4,-.6) {\scriptsize{$\sigma$}};
\node at (0,1.2) {\scriptsize{$\sigma$}};
\end{tikzpicture}
\qquad
\begin{tikzpicture}[baseline=-.4cm, xscale=-1]
\node (T1) [circle, fill=white, thick, draw]  at (0,.2) {$T_i$};
\node (T2) [circle, fill=white, thick, draw]  at (.8,-1) {$T_j$};
\node [below=1pt] at (T1.west) {$\star$};
\node [below=1pt] at (T2.west) {$\star$};
\draw (T1) -- (0,1);
\draw (T1) -- (-1.2,-1.6);
\draw (T1) -- (T2);
\draw (T2) -- (.4,-1.6);
\draw (T2) -- (1.2,-1.6);
\node at (-1.2,-1.8) {\scriptsize{$\sigma$}};
\node at (.4,-1.8) {\scriptsize{$\sigma$}};
\node at (1.2,-1.8) {\scriptsize{$\sigma$}};
\node at (.4,-.6) {\scriptsize{$\sigma$}};
\node at (0,1.2) {\scriptsize{$\sigma$}};
\end{tikzpicture}

for $i,j=0,1,2,3$ and $\sigma = \jw{6}$, as well as the diagrams

\begin{tikzpicture}[baseline=-.1cm]
\draw (-.4,-1) -- (-.4,-.7) arc (180:0:.4cm) -- (.4,-1);
\draw (.8,-1) .. controls ++(90:1cm) and ++(270:1cm) .. (.2,1);
\node at (-.4,-1.2) {\scriptsize{$\sigma$}};
\node at (.4,-1.2) {\scriptsize{$\sigma$}};
\node at (.8,-1.2) {\scriptsize{$\sigma$}};
\node at (.2,1.2) {\scriptsize{$\sigma$}};
\end{tikzpicture}
\qquad
\begin{tikzpicture}[baseline=-.1cm]
\draw (-.4,-1) -- (-.4,-.7) arc (180:0:.4cm) -- (.4,-1);
\draw (-.8,-1) .. controls ++(90:1cm) and ++(270:1cm) .. (-.2,1);
\node at (-.8,-1.2) {\scriptsize{$\sigma$}};
\node at (-.4,-1.2) {\scriptsize{$\sigma$}};
\node at (.4,-1.2) {\scriptsize{$\sigma$}};
\node at (-.2,1.2) {\scriptsize{$\sigma$}};
\end{tikzpicture}

These are each elements in the 24 boundary point space of the graph planar algebra, which is unfortunately rather large, at 50,996,510-dimensional.
Each of the elements would take several gigabytes to store, and in practice this approach runs into tedious memory management issues.
Rather than preparing the actual element $I-H$ on the left hand side of the $I=H$ relation, as a quadratic expression in $\alpha$ and $\beta$, we attempt to solve the $I=H$ equation component by component in the graph planar algebra.

Although in many components this equation is trivially satisfied, unsurprisingly we quickly obtain enough equations to solve for $\alpha$ and $\beta$, finding that $\beta = i$ and $\alpha$ is the largest root (all are real) of $5 \lambda^6 + 17\lambda^4 - 18\lambda^2 + 1$, approximately 0.8932.
By Corollaries <ref>, <ref>, and <ref>, there is a unique such $T \in \Hom(\sigma\otimes \sigma \to \sigma)$, which by Proposition <ref>, gives two isomorphic Q-system structures on $1\oplus \sigma$.

Finally, we still need to check that these values gives a solution to the $I=H$ equation.
To do this, we directly evaluate $T$ in the graph planar algebra with these values of $\alpha$ and $\beta$, and compute the two compositions appearing on the left hand side of the $I=H$ relation.
This is a lengthy computation (about an hour, but as it eventually checks relations in a $\sim 51 \times 10^6$ dimensional vector space over a number field, the unoptimised calculation requires at least 64gb of RAM), and eventually verifies that we have found a Q-system structure on $1 \oplus \sigma$.

§ POSSIBLE SKEIN THEORETIC APPROACHES TO EH3 AND EH4

For Asaeda–Haagerup it turned out that one of the fusion categories in its Morita equivalence class was an Izumi quadratic category, which illuminated why the Asaeda–Haagerup category exists.  We would love for $\cE\cH_3$ or $\cE\cH_4$ to fit into a nicer story in a similar way thereby explaining the currently mysterious Extended Haagerup categories.  However, we have not yet found any direct approaches to understanding these new fusion categories.  In particular, none of these categories have any invertible objects or additional symmetries that were hidden in the original Extended Haagerup categories.  There are two skein theoretic approaches to understanding $\cE\cH_4$ that are potentially promising.

First, we saw in the <ref> that there is a $(3,3)$-supertransitive quadrilateral which is non-commuting and cocommuting where the lower inclusions are the subfactor of index roughly $8.0$ from Equation (<ref>) and the upper inclusions are the subfactor of index roughly $7.0$ from Equation (<ref>).  One could try to construct this quadrilateral directly.  Following <cit.> and unpublished planar algebra approaches of Grossman and Snyder, one can try to give a skein theoretic construction of highly supertransitive quadrilaterals.  Unfortunately this approach remains unpublished because it has not yet succeeded for any important examples.  Nonetheless, it doesn't seem entirely hopeless.

Second, there's an intriguing pattern that $\cE\cH_3$ comes from a Q-system on $1 \oplus (X \otimes X)$ in $\cE\cH_2$ while $\cE\cH_4$ would come from a Q-system on $1 \oplus (Y \otimes Y)$.  So it would be natural to try to classify triples of a tensor category, a non-self-dual simple object $X$, and a Q-system structure on $1 \oplus (X \otimes X)$.  By self-duality of the Q-system we see that $X \otimes X \cong \overline{X} \otimes \overline{X}$, and there is a very nice skein theoretic category which is universal for tensor categories with an object $X$ satisfying $X \otimes X \cong \overline{X} \otimes \overline{X}$

Let $\zeta$ be a fourth root of unity, let $\cU_\zeta$ be the oriented planar algebra given by the following generators

\begin{tikzpicture}[baseline=-.1cm]
\filldraw (0,0) node [left] {$\star$} circle (1mm);
\draw[->] (0,0) --(.25,.25);
\draw (.25,.25) --(.5,.5);
\draw[->] (0,0) --(.25,.-.25);
\draw (.25,-.25) --(.5,-.5);
\draw[->] (0,0) --(-.25,.25);
\draw (-.25,.25) --(-.5,.5);
\draw[->] (0,0) --(-.25,.-.25);
\draw (-.25,-.25) --(-.5,-.5);
\end{tikzpicture}
\hspace{1in}
\begin{tikzpicture}[baseline=-.1cm]
\draw (0,0) --(.25,.25);
\draw[<-] (.25,.25) --(.5,.5);
\draw (0,0) --(.25,.-.25);
\draw[<-] (.25,-.25) --(.5,-.5);
\draw (0,0) --(-.25,.25);
\draw[<-] (-.25,.25) --(-.5,.5);
\draw (0,0) --(-.25,.-.25);
\draw[<-] (-.25,-.25) --(-.5,-.5);
\filldraw [fill=white] (0,0) node [left] {$\star$} circle (1mm);
\end{tikzpicture}

and relations

\filldraw (0,0) node [left] {$\star$} circle (1mm);
\draw[->] (0,0) --(.25,.25);
\draw (.25,.25) --(.5,.5);
\draw[->] (0,0) --(.25,.-.25);
\draw (.25,-.25) --(.5,-.5);
\draw[->] (0,0) --(-.25,.25);
\draw (-.25,.25) --(-.5,.5);
\draw[->] (0,0) --(-.25,.-.25);
\draw (-.25,-.25) --(-.5,-.5);
\end{tikzpicture}
= \zeta \begin{tikzpicture}[baseline=-.1cm]
\filldraw (0,0) node [below] {$\star$} circle (1mm);
\draw[->] (0,0) --(.25,.25);
\draw (.25,.25) --(.5,.5);
\draw[->] (0,0) --(.25,.-.25);
\draw (.25,-.25) --(.5,-.5);
\draw[->] (0,0) --(-.25,.25);
\draw (-.25,.25) --(-.5,.5);
\draw[->] (0,0) --(-.25,.-.25);
\draw (-.25,-.25) --(-.5,-.5);
\end{tikzpicture}
\begin{tikzpicture}[baseline=-.1cm]
\draw (0,0) --(.25,.25);
\draw[<-] (.25,.25) --(.5,.5);
\draw (0,0) --(.25,.-.25);
\draw[<-] (.25,-.25) --(.5,-.5);
\draw (0,0) --(-.25,.25);
\draw[<-] (-.25,.25) --(-.5,.5);
\draw (0,0) --(-.25,.-.25);
\draw[<-] (-.25,-.25) --(-.5,-.5);
\filldraw [fill=white] (0,0) node [left] {$\star$} circle (1mm);
\end{tikzpicture}
= \zeta^{-1} \begin{tikzpicture}[baseline=-.1cm]
\draw (0,0) --(.25,.25);
\draw[<-] (.25,.25) --(.5,.5);
\draw (0,0) --(.25,.-.25);
\draw[<-] (.25,-.25) --(.5,-.5);
\draw (0,0) --(-.25,.25);
\draw[<-] (-.25,.25) --(-.5,.5);
\draw (0,0) --(-.25,.-.25);
\draw[<-] (-.25,-.25) --(-.5,-.5);
\filldraw [fill=white] (0,0) node [below] {$\star$} circle (1mm);
\end{tikzpicture}
\begin{tikzpicture}[baseline=.8cm]
\draw[mid>] (0,.5) to (-.5,0);
\draw[mid>] (0,.5) to (.5,0);
\draw[mid>] (0,.5) to [out=45, in=-45] (0,1.5);
\draw[mid>] (0,.5) to [out=135, in=-135] (0,1.5);
\draw[mid<] (0,1.5) to (-.5,2);
\draw[mid<] (0,1.5) to (.5,2);
\filldraw (0,.5) node [left] {$\star$} circle (1mm);
\filldraw[fill=white] (0,1.5) node [left] {$\star$} circle (1mm);
\end{tikzpicture}
\begin{tikzpicture}[baseline=.4cm]
\draw[mid>] (-.5, 1) to [out=-45, in = 45] (-.5,0);
\draw[mid>] (.5, 1) to [out=-135, in = 135] (.5,0);
\end{tikzpicture}
\begin{tikzpicture}[baseline=0cm]
\draw[->] (0,0) arc (0:360:.5);
\end{tikzpicture} = d.

Using that the diagrams are bipartite, an easy Euler characteristic argument shows that you can evaluate all closed diagrams in $\cU_\zeta$, and that it has finite dimensional box spaces.  The simple objects in $\cU_\zeta$ are the tensor products of two Jones-Wenzls
$\jw{i} \otimes \jw{j}$ where each Jones-Wenzl alternates up-down, with the first one starting with an up arrow, and the second one starting with the same up/down that the previous one ended with.  (In the degenerate case where $i=0$ the second Jones-Wenzl starts with a down arrow.)  The fusion graph for $\cU_\zeta$ is given by the following:

\begin{tikzpicture}[baseline=-1mm, scale=.7]
\node (00) at (0,2*0)  {\scriptsize{$\jw{0} \otimes \jw{0}$}};
\node (01) at (0,2*1)  {\scriptsize{$\jw{0} \otimes \jw{1}$}};
\node (02) at (0,2*2)  {\scriptsize{$\jw{0} \otimes \jw{2}$}};
\node (03) at (0,2*3) {\scriptsize{$\vdots$}};
\node (10) at (3*1,2*0)  {\scriptsize{$\jw{1} \otimes \jw{0}$}};
\node (11) at (3*1,2*1)  {\scriptsize{$\jw{1} \otimes \jw{1}$}};
\node (12) at (3*1,2*2)  {\scriptsize{$\jw{1} \otimes \jw{2}$}};
\node (13) at (3*1,2*3)  {\scriptsize{$\vdots$}};
\node (20) at (3*2,2*0)  {\scriptsize{$\jw{2} \otimes \jw{0}$}};
\node (21) at (3*2,2*1)  {\scriptsize{$\jw{2} \otimes \jw{1}$}};
\node (22) at (3*2,2*2)  {\scriptsize{$\jw{2} \otimes \jw{2}$}};
\node (23) at (3*2,2*3)  {\scriptsize{$\vdots$}};
\node (30) at (3*3,2*0)  {\scriptsize{$\hdots$}};
\node (31) at (3*3,2*1)  {\scriptsize{$\hdots$}};
\node (32) at (3*3,2*2)  {\scriptsize{$\hdots$}};
\node (33) at (3*3,2*3)  {\scriptsize{$\iddots$}};
\draw[<-] (00) -- (01);
\draw[->] (01) -- (02);
\draw[<-] (02) -- (03);
\draw[->] (10) -- (11);
\draw[<-] (11) -- (12);
\draw[->] (12) -- (13);
\draw[<-] (20) -- (21);
\draw[->] (21) -- (22);
\draw[<-] (22) -- (23);

\draw[->] (00) -- (10);
\draw[<-] (10) -- (20);
\draw[->] (20) -- (30);
\draw[<-] (01) -- (11);
\draw[->] (11) -- (21);
\draw[<-] (21) -- (31);
\draw[->] (02) -- (12);
\draw[<-] (12) -- (22);
\draw[->] (22) -- (32);
\end{tikzpicture}

$\cU_\zeta$ can be constructed algebraically as a $\mathbb{Z}/4\mathbb{Z}$ extension of $\mathrm{SO}(3)_q \boxtimes \mathrm{SO}(3)_q$ for $d=q+q^{-1}$. Such extensions can be classified following <cit.>.  The Brauer–Picard group of $\mathrm{SO}(3)_q$ is $\mathbb{Z}/2\mathbb{Z}$ where the nontrivial bimodule is the the category of representations of $\mathrm{SU}(2)_q$ with odd highest weight.  The Brauer–Picard group of $\mathrm{SO}(3)_q\boxtimes \mathrm{SO}(3)_q$ is richer because of the outer automorphism interchanging the two factors.
There are four module categories with the correct dual category (picking a parity for the highest weight in each factor), but each gives two bimodule categories using the outer automorphism.  So the whole group is the $8$-element dihedral group.  In particular, there's an element of order $4$ in the Brauer group which we use to build our map from $\mathbb{Z}/4\mathbb{Z}$.  One then checks that two obstructions vanish, the first because there are no non-trivial invertible objects or gradings, and the second because $H^4(\mathbb{Z}/4, \mathbb{C}^\times)$ vanishes because the group is cyclic.  Finally there's a choice of element of $H^3(\mathbb{Z}/4\mathbb{Z}, \mathbb{C}^\times)$, which corresponds exactly to $\zeta$.

We now consider what it would mean to in addition have a Q-system structure on $1 \oplus (X \otimes X)$.  First, since Q-systems are symmetrically self-dual we see that $\zeta = \pm 1$.  Second, having a Q-system lets us build a shaded planar algebra where the $2$-strand shaded Jones-Wenzl corresponds to $(X \otimes X)$.  That is we have an oriented strand type corresponding to $X$ and an shaded strand type corresponding to $1 \oplus (X \otimes X)$ as a module over itself.  This gives us the following additional generators

\begin{tikzpicture}[baseline=-.1cm]
\draw[->] (0,0) -- (.25,.25);
\draw (.25,.25) -- (.5,.5);
\draw[->] (0,0) -- (-.25,.25);
\draw (-.25,.25) -- (-.5,.5);
\filldraw[shaded] (-.5,-.5) -- (0,0) -- (.5,-.5);
\filldraw[shaded] (0,0) circle (1mm);
\end{tikzpicture}
\hspace{1in}
\begin{tikzpicture}[baseline=-.1cm]
\draw (0,0) -- (.25,.25);
\draw[<-] (.25,.25) -- (.5,.5);
\draw (0,0) -- (-.25,.25);
\draw[<-] (-.25,.25) -- (-.5,.5);
\filldraw[shaded] (-.5,-.5) -- (0,0) -- (.5,-.5);
\filldraw[shaded] (0,0) circle (1mm);
\end{tikzpicture}

and relations

\begin{tikzpicture}[baseline=.5cm]
\filldraw[shaded] (-.5,-.5) -- (0,0) -- (.5,-.5);
\draw (0,1) --(.25,1.25);
\draw[<-] (.25,1+.25) --(.5,1+.5);
\draw (0,1+0) --(-.25,1+.25);
\draw[<-] (-.25,1+.25) --(-.5,1+.5);
\draw[->] (0,.0)[out=135,in=270] to (-.3,.5);
\draw[->] (0,.0)[out=45,in=270] to (.3,.5);
\draw (-.3,.5)[out=90,in=225] to (0,1);
\draw (.3,.5)[out=90,in=315] to (0,1);
\filldraw [fill=white] (0,1) node [left] {$\star$} circle (1mm);
\filldraw[shaded] (0,0) circle (1mm);
\end{tikzpicture} =
\begin{tikzpicture}[baseline=-.1cm]
\draw (0,0) -- (.25,.25);
\draw[<-] (.25,.25) -- (.5,.5);
\draw (0,0) -- (-.25,.25);
\draw[<-] (-.25,.25) -- (-.5,.5);
\filldraw[shaded] (-.5,-.5) -- (0,0) -- (.5,-.5);
\filldraw[shaded] (0,0) circle (1mm);
\end{tikzpicture}
\begin{tikzpicture}[baseline=.5cm]
\filldraw[shaded] (-.5,-.5) -- (0,0) -- (.5,-.5);
\draw[->] (0,1) --(.25,1.25);
\draw (.25,1+.25) --(.5,1+.5);
\draw[->] (0,1+0) --(-.25,1+.25);
\draw (-.25,1+.25) --(-.5,1+.5);
\draw (0,.0)[out=135,in=270] to (-.3,.5);
\draw (0,.0)[out=45,in=270] to (.3,.5);
\draw[<-] (-.3,.5)[out=90,in=225] to (0,1);
\draw[<-] (.3,.5)[out=90,in=315] to (0,1);
\filldraw (0,1) node [left] {$\star$} circle (1mm);
\filldraw[shaded] (0,0) circle (1mm);
\end{tikzpicture} =
\begin{tikzpicture}[baseline=-.1cm]
\draw[->] (0,0) -- (.25,.25);
\draw (.25,.25) -- (.5,.5);
\draw[->] (0,0) -- (-.25,.25);
\draw (-.25,.25) -- (-.5,.5);
\filldraw[shaded] (-.5,-.5) -- (0,0) -- (.5,-.5);
\filldraw[shaded] (0,0) circle (1mm);
\end{tikzpicture}
\begin{tikzpicture}[baseline=.5cm]
\draw[->] (0,0) -- (.25,-.25);
\draw (.25,-.25) -- (.5,-.5);
\draw[->] (0,0) -- (-.25,-.25);
\draw (-.25,-.25) -- (-.5,-.5);
\draw[->] (0,1) --(.25,1.25);
\draw (.25,1+.25) --(.5,1+.5);
\draw[->] (0,1+0) --(-.25,1+.25);
\draw (-.25,1+.25) --(-.5,1+.5);
\filldraw[shaded] (0,0)[out=135,in=225] to (0,1);
\filldraw[shaded] (0,0)[out=45,in=315] to (0,1);
\filldraw[shaded] (0,1) circle (1mm);
\filldraw[shaded] (0,0) circle (1mm);
\end{tikzpicture}
\begin{tikzpicture}[baseline=-.1cm]
\draw[->] (0,0) --(.25,.25);
\draw (.25,.25) --(.5,.5);
\draw[->] (0,0) --(.25,.-.25);
\draw (.25,-.25) --(.5,-.5);
\draw[->] (0,0) --(-.25,.25);
\draw (-.25,.25) --(-.5,.5);
\draw[->] (0,0) --(-.25,.-.25);
\draw (-.25,-.25) --(-.5,-.5);
\filldraw (0,0) node [left] {$\star$} circle (1mm);
\end{tikzpicture}
\begin{tikzpicture}[baseline=.5cm]
\draw (0,0) -- (.25,-.25);
\draw[<-] (.25,-.25) -- (.5,-.5);
\draw (0,0) -- (-.25,-.25);
\draw[<-] (-.25,-.25) -- (-.5,-.5);
\draw (0,1) --(.25,1.25);
\draw[<-] (.25,1+.25) --(.5,1+.5);
\draw (0,1+0) --(-.25,1+.25);
\draw[<-] (-.25,1+.25) --(-.5,1+.5);
\filldraw[shaded] (0,0)[out=135,in=225] to (0,1);
\filldraw[shaded] (0,0)[out=45,in=315] to (0,1);
\filldraw[shaded] (0,1) circle (1mm);
\filldraw[shaded] (0,0) circle (1mm);
\end{tikzpicture}
\begin{tikzpicture}[baseline=-.1cm]
\draw (0,0) --(.25,.25);
\draw[<-] (.25,.25) --(.5,.5);
\draw (0,0) --(.25,.-.25);
\draw[<-] (.25,-.25) --(.5,-.5);
\draw (0,0) --(-.25,.25);
\draw[<-] (-.25,.25) --(-.5,.5);
\draw (0,0) --(-.25,.-.25);
\draw[<-] (-.25,-.25) --(-.5,-.5);
\filldraw [fill=white] (0,0) node [left] {$\star$} circle (1mm);
\end{tikzpicture}$$

\begin{tikzpicture}[baseline=.5cm]
\draw (0,0) -- (.25,-.25);
\draw[<-] (.25,-.25) -- (.5,-.5);
\draw (0,0) -- (-.25,-.25);
\draw[<-] (-.25,-.25) -- (-.5,-.5);
\draw[->] (0,1) --(.25,1.25);
\draw (.25,1+.25) --(.5,1+.5);
\draw[->] (0,1+0) --(-.25,1+.25);
\draw (-.25,1+.25) --(-.5,1+.5);
\filldraw[shaded] (0,0)[out=135,in=225] to (0,1);
\filldraw[shaded] (0,0)[out=45,in=315] to (0,1);
\filldraw[shaded] (0,1) circle (1mm);
\filldraw[shaded] (0,0) circle (1mm);
\end{tikzpicture}
\begin{tikzpicture}[baseline=-.1cm]
\draw[mid>](-.5,-.5) [out=45, in=315] to (-.5,.5);
\draw[mid>](.5,-.5) [out=135, in=225] to (.5,.5);
\end{tikzpicture}
\begin{tikzpicture}[baseline=.5cm]
\filldraw[shaded] (-.5,-.5) -- (0,0) -- (.5,-.5);
%	\draw (0,1) --(.25,1.25);
%	\draw[<-] (.25,1+.25) --(.5,1+.5);
%	\draw (0,1+0) --(-.25,1+.25);
%	\draw[<-] (-.25,1+.25) --(-.5,1+.5);
\draw[->] (0,.0)[out=135,in=270] to (-.3,.5);
\draw[->] (0,.0)[out=45,in=270] to (.3,.5);
\draw (-.3,.5)[out=90,in=225] to (0,1);
\draw (.3,.5)[out=90,in=315] to (0,1);
\filldraw[shaded] (-.5,1.5) -- (0,1) -- (.5,1.5);
\filldraw[shaded] (0,1) circle (1mm);
\filldraw[shaded] (0,0) circle (1mm);
\end{tikzpicture} =
\begin{tikzpicture}[baseline=-.1cm]
\filldraw[shaded] (-.5,-.75) -- (0,0) -- (.5,-.75);
\filldraw[shaded] (-.5,.75) -- (0,0) -- (.5,.75);
\filldraw[fill=white,thick] (-.4,-.3) rectangle (.4,.3);
\node at (0,0) {$\jw{2}$};
\end{tikzpicture}
\begin{tikzpicture}[baseline=-.1cm]
\draw[->] (0,0) -- (.25,.25);
\draw (.25,.25) -- (.5,.5);
\draw[->] (0,0) -- (-.25,.25);
\draw (-.25,.25) -- (-.5,.5);
\filldraw[shaded] (0,-.5) [out=180,in=225,looseness=2] to (0,0);
\filldraw[shaded] (0,-.5) [out=0,in=315,looseness=2] to (0,0);
\filldraw[shaded] (0,0) circle (1mm);
\end{tikzpicture}
= 0 =
\begin{tikzpicture}[baseline=-.1cm]
\draw (0,0) -- (.25,.25);
\draw[<-] (.25,.25) -- (.5,.5);
\draw (0,0) -- (-.25,.25);
\draw[<-] (-.25,.25) -- (-.5,.5);
\filldraw[shaded] (0,-.5) [out=180,in=225,looseness=2] to (0,0);
\filldraw[shaded] (0,-.5) [out=0,in=315,looseness=2] to (0,0);
\filldraw[shaded] (0,0) circle (1mm);
\end{tikzpicture}
\begin{tikzpicture}[baseline=-.1cm]
\filldraw[shaded] (0,0) arc (0:360:.5);
\end{tikzpicture} = \sqrt{d^2+1}.

In the case of $X \in \cE\cH_3$ the space $\Hom(X^{\otimes 3},\overline{X}^{\otimes 3})=\Hom(\overline{X} \oplus W, X \oplus W)$ is only $1$ dimensional, so unless the coefficient magically vanishes, we have a Yang-Baxter relation (in the sense of <cit.>) saying the following diagrams are proportional (and the same if you reverse all arrows).

\begin{tikzpicture}[baseline=-.1cm]
\draw[mid>] (30:1) to (60:.5);
\draw[mid>] (90:1) to (60:.5);
\draw[mid>] (150:1) to (180:.5);
\draw[mid>] (210:1) to (180:.5);
\draw[mid>] (270:1) to (300:.5);
\draw[mid>] (330:1) to (300:.5);
\filldraw[shaded] (60:.5) -- (180:.5) -- (300:.5) -- (60:.5);
\filldraw[shaded] (60:.5) circle (1mm);
\filldraw[shaded] (180:.5) circle (1mm);
\filldraw[shaded] (300:.5) circle (1mm);
\end{tikzpicture} \; \sim \;
\begin{tikzpicture}[baseline=-.1cm]
\draw[mid>] (30:1) to (0:.5);
\draw[mid>] (90:1) to (120:.5);
\draw[mid>] (150:1) to (120:.5);
\draw[mid>] (210:1) to (240:.5);
\draw[mid>] (270:1) to (240:.5);
\draw[mid>] (330:1) to (0:.5);
\filldraw[shaded] (0:.5) -- (120:.5) -- (240:.5) -- (0:.5);
\filldraw[shaded] (0:.5) circle (1mm);
\filldraw[shaded] (120:.5) circle (1mm);
\filldraw[shaded] (240:.5) circle (1mm);
\end{tikzpicture}

Similarly (unless $\alpha$ is miraculously zero) there is a Yang-Baxter relation of the following form:

\begin{tikzpicture}[baseline=-.1cm]
\filldraw[shaded] (30:1) -- (60:.5) -- (90:1);
\draw[mid>] (150:1) to (180:.5);
\draw[mid>] (330:1) to (300:.5);
\draw[mid>] (60:.5) -- (180:.5) ;
\draw[mid>] (60:.5) -- (300:.5) ;
\filldraw[shaded] (210:1) -- (180:.5) -- (300:.5) -- (270:1);
\filldraw[shaded] (60:.5) circle (1mm);
\filldraw[shaded] (180:.5) circle (1mm);
\filldraw[shaded] (300:.5) circle (1mm);
\end{tikzpicture}
= \alpha
\begin{tikzpicture}[baseline=-.1cm]
\filldraw[shaded] (30:1) to (0:.5) to (120:.5) to (90:1);
\draw[mid>] (150:1) to (120:.5);
\filldraw[shaded] (210:1) to (240:.5) to (270:1);
\draw[mid>] (330:1) to (0:.5);
\draw[mid>] (120:.5) -- (240:.5);
\draw[mid>] (0:.5) -- (240:.5);
\filldraw[shaded] (0:.5) circle (1mm);
\filldraw[shaded] (120:.5) circle (1mm);
\filldraw[shaded] (240:.5) circle (1mm);
\end{tikzpicture}
+ \beta
\begin{tikzpicture}[baseline=-.1cm]
\filldraw[shaded] (30:1) to [out=210,in=270,looseness=2] (90:1);
\draw[mid>] (150:1) to (0,0);
\draw[mid>] (330:1) to (0,0);
\filldraw[shaded] (210:1) to (0,0) to (270:1);
\filldraw[shaded] (0,0) circle (1mm);
\end{tikzpicture}
+ \gamma
\begin{tikzpicture}[baseline=-.1cm, rotate=180]
\filldraw[shaded] (30:1) to [out=210,in=270,looseness=2] (90:1);
\draw[mid>] (150:1) to (0,0);
\draw[mid>] (330:1) to (0,0);
\filldraw[shaded] (210:1) to (0,0) to (270:1);
\filldraw[shaded] (0,0) circle (1mm);
\end{tikzpicture}

Unfortunately, these relations do not suffice to simplify all closed diagrams following <cit.>, because several diagrams do not satisfy any Yang-Baxter relations.  Specifically the following two diagrams (together with their rotations, and switching the directions of all arrows) do not satisfy Yang-Baxter relations because there are no $4$-boxes with the appropriate shading/orientation on the other side.

\begin{tikzpicture}[baseline=-.1cm]
\draw[mid<] (30:1) to (60:.5);
\draw[mid<] (90:1) to (60:.5);
\draw[mid>] (150:1) to (180:.5);
\draw[mid>] (210:1) to (180:.5);
\draw[mid>] (270:1) to (300:.5);
\draw[mid>] (330:1) to (300:.5);
\filldraw[shaded] (60:.5) -- (180:.5) -- (300:.5) -- (60:.5);
\filldraw[shaded] (60:.5) circle (1mm);
\filldraw[shaded] (180:.5) circle (1mm);
\filldraw[shaded] (300:.5) circle (1mm);
\end{tikzpicture}
\; \text{ and } \;
\begin{tikzpicture}[baseline=-.1cm]
\draw[mid<] (30:1) to (60:.5);
\draw[mid<] (90:1) to (60:.5);
\draw[mid>] (150:1) to (180:.5);
\draw[mid>] (330:1) to (300:.5);
\draw[mid>] (60:.5) -- (180:.5) ;
\draw[mid>] (60:.5) -- (300:.5) ;
\filldraw[shaded] (210:1) -- (180:.5) -- (300:.5) -- (270:1);
\filldraw[fill=white] (60:.5) node [above left = -.15 and 0] {$\star$} circle (1mm);
\filldraw[shaded] (180:.5) circle (1mm);
\filldraw[shaded] (300:.5) circle (1mm);
\end{tikzpicture}

There are closed diagrams which cannot be easily simplified using on the specific Yang-Baxter relations that we have.  The simplest such diagram we found involves $16$-vertices.  Nonetheless this approach seems promising, even if it's beyond current skein theory techniques.

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Recently, as an interesting alternative, attosecond streaking was employed to inspect transport effects in dielectric nanoparticles, cf.  Seiffert [54]. The schematic setup of the experiment is shown in fig:Seiffert_NatPhys13_2017_1a. A beam of isolated $d = \SI{50}{nm}$ nanospheres was irradiated by isolated 250as XUV pulses (central photon energy $\approx \SI{28}{eV}$, peak intensities $\approx \SI{2e12}{W/cm^2}$) and synchronized 5fs NIR pulses (central wavelength $\approx\SI{720}{nm}$, peak intensities $\approx \SI{1e12}{W/cm^2}$) with adjustable delay $\Delta t$. Emitted electrons were recorded via single-shot VMI. The nanosphere density within the incident beam was sufficiently low, such that hits of multiple nanospheres within one laser shot were negligible. Further, only roughly every fifth laser shot hit a nanoparticle, resulting in VMI images including only contributions from the carrier gas and images containing signal from both the gas and a nanosphere. Typical single shot projected momentum images are shown in fig:Seiffert_NatPhys13_2017_1b,c for a gas only measurement and when including nanoparticles. Note that the data includes electrons stemming from a nanosphere as well as contributions from the carrier gas. However, due to the higher ionization energy the latter only contributes to the low momentum region (i.e. within the dashed circles). Efficient classification of laser shots hitting only gas or hitting a nanosphere was enabled by inspecting the number of detected photoelectron events and the average electron momentum in the laser propagation direction (cf. blue and red symbols in fig:Seiffert_NatPhys13_2017_1b,c, also termed 'shadowing' [106]). For nanoparticle hits, the former is higher and the latter is shifted towards negative values resulting from stronger photoionization at the front side of the sphere where the XUV field is absorbed (cf. left inset in fig:Seiffert_NatPhys13_2017_1a). Resulting averaged momentum distributions for gas and nanospheres are shown in fig:Seiffert_NatPhys13_2017_1d,e.

Attosecond streaking with dielectric nanospheres. a Schematic setup of the experiment. Synchronized attosecond XUV and few-cycle NIR pulses with adjustable delay $\Delta t$ were shot on a beam of isolated nanopspheres. Emitted electrons were recorded by a single-shot VMI spectrometer. b,c Typical single-shot VMI momentum projections recorded from a gas-only measurement (b) and when including silica nanospheres (c) for $\Delta t = \SI{0}{as}$. Dashed circles indicate the low momentum region ($< \SI{0.75}{a.u.}$), where the measured signal contains additional contributions from residual gas background. The grey arrow in (b) illustrates the x-component of the projected momentum of the i-th measured event. Blue and red dots indicate respective averaged projected momenta. d,e Respective averaged momentum distributions. Adapted from [54].

Streaking spectrograms for gas and nanospheres were extracted by inspecting the corresponding energies of electrons emitted along the laser polarization direction (obtained via integration over a 50 full opening angle) as function of the pulse delay, see Seiffert_NatPhys13_2017_2a,b. Both spectrograms revealed pronounced delay-dependent oscillations as well as a relative delay of the nanoparticle data with respect to the gas. To quantify the individual streaking delays, contour-lines (i.e. delay-dependent energies with equal signal strength) were determined and filtered to eliminate measurement-induced high frequency noise. As an example, two filtered contour lines are indicated by blue and red dots in Seiffert_NatPhys13_2017_2a,b. The streaking delay is extracted from each contour line via fitting a few-cycle waveform
$E_\text{fit}(\Delta t) = E + A\cos\left(\omega \left[\Delta t - \delta t\right]\right) e^{-\frac{1}{2}\frac{(\Delta t - t_0)^2)}{\tau^2}}$
with asymptotic energy $E$ and streaking delay $\delta t$ (solid curves). The resulting energy-dependent streaking delays $\delta t(E)$ extracted from several contour lines in the high-energy spectral regions ($20$–$\SI{30}{eV}$) of the gas and nanoparticle spectrograms (shown as blue and red symbols in Seiffert_NatPhys13_2017_2c) unveiled two prominent features. First, the streaking delays of both gas and nanospheres increased linearly with energy which could be attributed mainly to the chirp of the XUV pulses. Second, and most importantly, the and gas data exhibited an only slightly energy-dependent relative delay of around 100as.

Streaking spectrograms and extracted straking delays. a,b Attosecond streaking spectrograms obtained from angular integration of projected momentum maps around the laser polarization direction (cf. area indicated in fig:Seiffert_NatPhys13_2017_1d,e) for gas (a) and silica nanoparticles (b). To extract the streaking delays energy-dependent frequency-filtered isolines (blue and red dots) were fitted with few-cycle waveforms (blue and red curves) as described in the text. The fits carrier phases define the respective streaking delays $\delta t$. c Streaking delays extracted in the high energy range of the measured gas and nanoparticle streaking spectrograms (blue and red dots). Blue and black curves show delays predicted by gas simulations and nanoparticle simulations for the experimental parameters, respectively. The red curve represents the delay extracted from a mixed spectrogram. The dark gray shaded area reflects the maximal variation of the extracted streaking delay when performing the nanosphere simulations with charge-interaction enabled or anisotropic elastic collisions. d Ratio of nanoparticle to gas signal in dependence of energy extracted from the experiment (red symbols) and from the simulations (black curve). Adapted from [54].

The origin of the relative delay could be clarified by corresponding semi-classical simulations for the experimental parameters. As the XUV chirp could not be extracted accurately from the experiment it was determined by comparing streaking delays extracted from a simulation for the reference gas to the experiment. Best agreement was found for a chirp parameter of $\zeta = \SI{-7e-3}{fs^2}$, compare blue curve to blue symbols in Seiffert_NatPhys13_2017_2c. Considering this chirp for the nanosphere simulations resulted in an extracted streaking delay exceeding that from the experiment (compare black curve to red symbols). The discrepancy was attributed to the contributions from carrier gas in the nanoparticle signal which results in the measured nanoparticle streaking delay actually reflecting a mixture of the signals from nanosphere and gas. The energy-dependent ratio of nanoparticle and gas signal could be extracted from the experiment via background subtraction, see red symbols in Seiffert_NatPhys13_2017_2d. It was accounted for in the simulations by combining the individual streaking spectrograms for gas and nanoparticles in a mixed spectrogram $S_\text{mix}(E,\Delta t) = S_\text{SiO$_2$}(E,\Delta t) + \eta S_\text{gas}(E,\Delta t)$ by adjusting the parameter $\eta$ for best agreement with the experimental ratio (compare black curve to red symbols in Seiffert_NatPhys13_2017_2d). The excellent quantitative agreement of the streaking delay extracted from the mixed spectrogram with the experiment (compare red curve to red symbols in Seiffert_NatPhys13_2017_2c) support the assumption that the simulations include the relevant physics and can thus be utilized to identify the physical picture behind the essentially  relative delay $\delta t_\text{rel} = \delta t_\text{SiO$_2$} - \delta t_\text{ref} \approx \SI{150}{as}$ between the (pure) nanoparticle delay $\delta t_\text{SiO$_2$}$ and the delay of the gas reference $\delta t_\text{ref}$ (cf. black arrow and gray shaded area in Seiffert_NatPhys13_2017_2c).

Systematic analysis of the collisional streaking delay calculated via assuming unchirped XUV pulses and energy-independent scattering times to eliminate energy-dependencies of the streaking delays. a Streaking delays in dependence of elastic and inelastic scattering times (as indicated). b Streaking delays as function of the material’s attenuation factor $\alpha = 1/\varepsilon_\text{r}$, where $\varepsilon_\text{r}$ is the relative permittivity at the wavelength of the NIR field. Gray shaded areas in both panels visualize the variation of the streaking delay in dependence of the elastic scattering time. The gray rectangle and the black arrows in (b) indicate permittivities of typical dielectric materials. The dashed black line marks the permittivity of . Adapted from [54]. c Schematic illustration of attosecond streaking on the surfaces of transparent (top), dielectric (center) and metallic (bottom) targets. After ionization by the XUV pulse (blue) the electron is streaked by the near-field of the optical pulse (red). Adapted from [22].

A detailed analysis of the impacts of various contributions to the individual delays for the gas and nanospheres revealed that delays imposed by the XUV chirp as well as field propagation and retardation, and near-field inhomogeneities were negligible in the inspected scenario (for a detailed analysis see the supporting material of [54] or [22]). Hence, the relative delay was in good approximation solely governed by the material-specific collisional electron transport dynamics, i.e. $\delta t_\text{rel} = \delta t_\text{coll.}$ To systematically explore the effect of electron transport within the material and its significance for the collisional streaking delay simulations were performed with two central simplifications. First, the XUV pulses were considered to be unchirped in order to eliminate the energy-dependent tilt of the streaking delays. Second, electron scattering was modeled via fixed energy-independent scattering times for both elastic and inelastic collisions. These times were considered as $\tau_\text{el} = \SI{100}{as}$ and $\tau_\text{inel} = \SI{300}{as}$ to closely resemble the energy-dependent scattering times in the relevant energy range (cf. fig:theory_inelastic_scatteringd) and were each varied by factors of $1/3$ and $3$. The nine resulting energy-dependent streaking delays shown in fig:Seiffert_NatPhys13_2017_3a allowed three key conclusions. First, neglecting the XUV chirp and assuming fixed scattering times results in essentially energy-independent streaking delays. Second, the delays are strongly sensitive to the inelastic scattering time, where increasing the inelastic scattering time results in larger streaking delays (see three separated groups of curves indicated by the gray areas). Third, the sensitivity to the elastic scattering time is much weaker (compare colored curves within each gray group).

The observed strong sensitivity of the streaking delays to the inelastic scattering time and only weak dependence on the elastic scattering time raised the question if this behavior is general for solid-state attosecond streaking from arbitrary materials. A key difference between streaking at metallic and dielectric systems is the screening of the NIR field within the material and hence its effect on the internal streaking during the electron transport to the surface (cf. fig:Seiffert_NatPhys13_2017_3c). The impact of the internal streaking was inspected by repeating the simulations shown in fig:Seiffert_NatPhys13_2017_3a and additionally adjusting the strength of the internal field by varying the materials relative permittivity $\varepsilon_\text{r}$ for each run. As the streaking delays were energy-independent, each individual simulation for a set of parameters ($\tau_\text{el}$, $\tau_\text{inel}$ and $\varepsilon_\text{r}$) resulted in one (energy-averaged) value for the streaking delay. The results of this systematic analysis are shown in fig:Seiffert_NatPhys13_2017_3b in dependence on the field attenuation factor $\alpha = 1/\varepsilon_\text{r}$. In the two limiting cases $\alpha \rightarrow 0$ and $1$ the delays were found to be sensitive to both the inelastic and the elastic scattering time, as visualized by the broadening of the gray areas. The dependence on the latter, however, had opposite signs in the two limits. In the transparent case ($\alpha \rightarrow 1$, $\varepsilon_\text{r} \rightarrow 1$), the streaking delay decreased for larger elastic scattering times (compare blue to green curves at the top). In contrast, in the metal limit ($\alpha \rightarrow 0$, $\varepsilon_\text{r} \rightarrow \infty$) larger elastic scattering times resulted in larger streaking delays (compare blue to green curves at the bottom). The elastic collision effect reversed (and thus effectively vanishes) for permittivity values of typical dielectric materials (light gray shaded area). This nearly exclusive sensitivity of the streaking delays to the inelastic scattering time for dielectrics therefore enabled the retrieval of quantitative scattering times by matching the simulation results to experimental data. For the presented scenario and electron energies around 25eV the authors extracted an inelastic scattering time of $\tau_\text{inel}\approx \SI{370}{as}$ which corresponds to an inelastic mean free path of approximately 10Å in good agreement with values calculated from optical data by Tanuma [83] and obtained from Monte-Carlo simulations by Kuhr [81] (cf. fig:theory_inelastic_scatteringb).

§ CONCLUSIONS AND FUTURE PERSPECTIVES

The selected set of results from the first decade of strong-field physics with nanospheres has clearly demonstrated that these targets provide a suitable and experimentally as well as theoretically accessible model system to tackle various  fundamental questions in attosecond nanophysics. These range from confirming that many of the established pictures and tools from atomic strong-field physics are also applicable to more complex nanotargets (e.g. the backscattering picture of HATI), over the identification of qualitatively new aspects such as the impacts of charge interaction or field propagation to the development of new metrologies for the quantification of transport properties in dielectrics. For example, the at first glance seemingly spurious charge interaction effects also revealed surprising advantages like further enhancements of the energies of fast recollision electrons or the suppression of certain emission channels (e.g. direct photoemission) and even the realization of a material-independent photoemission response at high laser-intensities.

Although nanospheres will probably not be very suitable for future technical  applications themselves, it is expected that many of the fundamental physical insights first obtained utilizing these simple model systems are also of major importance for more applications-relevant systems and application fields like sub-cycle laser processing or Petahertz optoelectronics employing other nanostructures such as nanotips [29]. For example, only recently the impact of charge interaction on the photoemission, first observed with dielectric nanospheres, has also been demonstrated with metal nanotips [65].

The initial proof-of-principles demonstrated in some of the seminal publications reviewed in this work still leave open a long list of questions for future studies. For example, the attosecond streaking metrology, so far demonstrated for silica nanospheres, could be extended to higher photon energies or other materials for which quantification of inelastic mean-free paths is not accessible via other techniques. New milestones could also be reached by combining phase-dependent excitation of nanospheres via few-cycle fields with X-ray imaging techniques to track plasma dynamics in space and time simultaneously [63]. Another approach to gain further insights into strong-field ionization of solids could be to extend the excitation to light fields even more extreme than few-cycle pulses, for example by deploying sub single cycle fields [107]. Hence, the strong-field physics with nanospheres remains a fascinating platform for future scientific activity in the field of attosecond nanophysics.

§ ACKNOWLEDGMENTS

We gratefully acknowledge the invaluable contributions of our colleagues and collaborators on related original work that this article is based on.

§ DISCLOSURE STATEMENT

No potential conflict of interest was reported by the authors.

§ FUNDING

This work was supported by the German Research Foundation (DFG) under grant 652714; S.Z. was supported from the DFG via ZH582/1-1; M.F.K. acknowledges support from the Max Planck Society via the Max Planck Fellow program; We acknowledge support from the DFG within SPP 1391 Ultrafast Nanooptics, SPP 1840 QUTIF and SFB 652/3; T.F. acknowledges financial support from the DFG via Heisenberg Grant 398382624; We are grateful for the computing time provided by the North-German super-computing center HLRN (project mvp00011 and mvp00017)

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