  In this paper Moussouris' algorithm for the decomposition of spin networks is
reviewed and the implicit assumptions made in the Decomposition Theorem
relating a spin network with its state sum are examined. It is found that the
theorem in the original form hides the importance of the orientation of the
vertices in the spin networks and, more important, that the algorithm for the
evaluation of spin networks assumes a cycle for the reduction of the graph to
work in the proof of the theorem. It is shown that this is not always the case
and this rises doubt about the generality of the theorem. Having this issue in
mind, the theorem is restated to account for toroidal spin networks, i.e.
networks cellular embeddable in the torus. For this, the minimal non-planar
spin network is examined and the algorithm is extended to account for the
non-planarity of toroidal spin networks. Furthermore, three types of minimal
non-planar spin networks are found, two of them are toroidal and the third is
cellular embeddable only in the double torus and without a cycle. Some examples
for both toroidal spin networks are given and the relation between these is
found. Finally, the issues and the possibility of generalizing the
Decomposition Theorem is shortly discussed.
