Toroidal Spin Networks: Towards a Generalization of the Decomposition Theorem

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.