The fuzzy sphere ⋆-product and spin networks

We analyze the expansion of the fuzzy sphere noncommutative product in powers of the noncommutativity parameter. To analyze this expansion we develop a graphical technique that uses spin networks. This technique is potentially interesting in its own right as introducing spin networks of Penrose into noncommutative geometry. Our analysis leads to a clarification of the link between the fuzzy sphere noncommutative product and the usual deformation quantization of the sphere in terms of the ⋆-product.