bubble in LQG, and the divergence for a large spin sitting on an internal triangular face is called a spike.

The transition amplitudes can be made well defined and explicitly independent of the triangulation by a deformation of the $SU(2)$ symmetry, using quantum $SU(2)_q$ with a deformation parameter $q$ defined as a complex root of unity $q=e^{\frac{2\pi i}{r}}$, with $r$ an integer[^4]. The transition amplitudes is given by $$W_{q}(j_{l})=\mathcal{N}_{q}\sum_{j_{f}}\prod_{f}(-1)^{j_{f}}d_{q}(j_{f})\prod_{v}(-1)^{j_{v}}\{6j\}_{q},\label{eq:transitionamplitudequantumgroup}$$ where we have now, as the quantum-deformed analogues of the objects that appear in the transition amplitude, the quantum dimension $d_{q}(j_{f})$ and the quantum Wigner 6j-symbol $\{6j\}_{q}=\left\{ \begin{array}{ccc}
j_{1} & j_{2} & j_{3}\\
j_{4} & j_{5} & j_{6}
\end{array}\right\} _{q}$. From the representation theory of quantum groups [@Biedenharn] we know that the spin quantum numbers of $SU(2)_{q}$ are constrained to $j\leq\frac{r-2}{2}$, working as a cut off for the flow of spin on the bubbles. The usual triangular inequalities are modified and supplemented by the conditions $2j_{1},2j_{2},2j_{3}\leq j_{1}+j_{2}+j_{3}\leq r-2$, which are the triangular inequalities of a triangle on a sphere with a radius determined by $r$. The geometry of $q$-deformed spin networks is therefore consistent with the geometry of constant curvature space, with the curvature determined by the specific root of unity.

