for the 4096 combinations of $j$’s in ${0,1/2,1,3/2}$, are all null except for a small set of combinations, which can be computed from eq.(\[eq:symbol\]) restricted to the $SU(2)$ case and takes on only seven values: $\{0,\pm1,\pm\varphi,\pm\sqrt{\varphi}\}$, where $\varphi=\frac{\sqrt{5}-1}{2}=\phi^{-1}$, with $\phi=\frac{\sqrt{5}+1}{2}=1.618...$, the golden ratio.

More generally, this kind of amplitude is independent of the triangulation, and gives a topological invariant [@Turaev92], which means that it does not depend on the triangulation inside the 3 manifold, but depends only on the edge values and on the topology of the triangulated 3 manifold (the number of punctures on the boundary 2-sphere). Exploiting the possibility to build invariants, by fixing the representations labels on the edges, we can consider networks of these topological amplitudes, which act as building blocks for decomposing a higher dimensional tensor product space, which we will motivate more in the next section. For example, we can consider amplitudes, functions of representations on boundary faces of edges for closed loops along the 3 dimensional spin foam $$W_{c}(j_{lc})=\mathcal{N}_{q}\sum_{j_{f}|j_{lc}}\prod_{f}(-1)^{j_{f}}d_{q}(j_{f})\prod_{v}W_{\triangle_{1}}^{5}(j_{lv}),\label{eq:transitionamplitudequantumgroup-5}$$ where $c$ is a closed loop across the 3 dimensional triangulation and $j_{lc}=j_{1},...,j_{n}$ are the representations on the boundary faces of the $n$ edges contained in $c$. The sum $\sum_{j_{f}|j_{lc}}$ is taken while maintaining 