have $$\begin{aligned}
\left[h_{a},L_{b}^{\pm}\right] & =\pm\beta\delta_{ab}L_{a}^{\pm}\nonumber \\
\left[L_{a}^{+},L_{b}^{-}\right] & =2\beta\delta_{ab}h_{a},\label{eq:commutationJplusminos-1}\end{aligned}$$ where $\beta$ has the dimension of area. The usual Casimir invariant is $L_{a}^{i}L_{a}^{i}=L_{a}^{+}L_{a}^{-}+h_{a}\left(h_{a}+\beta\right)$, commuting with each element of the algebra, and has eigenvalues[^5] ${\beta^{2}j_{a}(j_{a}+1)}$, $j_{a}\in{0,\frac{1}{2},1,\frac{3}{2},2,...}$, which gives a quantization of area $A_{a}=\beta\sqrt{j_{a}(j_{a}+1)}$ for a triangular face or $A_{j_{an}}=\beta\sum_{n}\sqrt{j_{an}(j_{an}+1)}$ for an arbitrary surface punctured by N punctures $n_{1},...,N$. As discussed in the previous section, there are no restrictions on the flow of spin representations in this case, suggesting we should implement a more fundamental topological symmetry. So we apply the $SU(2)_q$ deformation at the fifth root of unity $q=e^{\frac{2\pi i}{r}}$, with $r=5$ (from now on let us call it $SU(2)^5_q$), which affects the Cartan generator in the second term and constitutes essentially a quantization postulate for geometry: $$\begin{aligned}
\left[h_{a},L_{b}^{\pm}\right] & =\pm\delta_{ab}L_{a}^{\pm}\nonumber \\
\left[L_{a}^{+},L_{b}^{-}\right] & =2\delta_{ab}\left[2h_{a}\right]_{q},\label{eq:commutationJplusminos-1-1}\end{aligned}$$ with the q number $\left[x\right]_{q}$ defined by $$\left[x\right]_{q}=\frac{q^{\frac{x}{2}}-q^{-\frac{x}{2}}}{q^{\frac{1}{2}}-q^{-\frac{1}{2}}}=\frac{sin\left(\frac{\pi}{5}x\right)}{sin\left(\frac{\pi}{5}\right)}\label{eq:qnumber}$$ or writing the Cartan generator, $\mathcal{J}_{a}=q^{h_{a}}$: $$\begin{aligned}
\mathcal{J}_{a}L_{b}^{\pm}\mathcal{J}_{a}^{-1} & =q^{\pm1}L_{b}^{\pm}\nonumber \\
\left[L_{a}^{+},L_{b}^{-}\right] & =2\delta_{ab}\frac{\mathcal{J}_{a}-\mathcal{J}_{a}^{-1}}{q-q^{-1}}\label{eq:hopfalgebra}\end{aligned}$$ For this deformation to be dimensionally consistent, we start with everything adimensional and there is no need a priori for $\beta$. There is a notion of scale incorporated in the deformation parameter, $r=\frac{l_{c}}{l_{p}}$, with $r$ an integer and the large scale $l_{c}$ given in units 