环面自旋网络：分解定理的推广
Toroidal Spin Networks: Towards a Generalization of the Decomposition Theorem
摘要：
本文回顾了Moussouris的自旋网络分解算法，并检查了自旋网络与其状态和相关的分解定理中所做的隐含假设。它被发现，在原来的形式的定理隐藏的重要性，在自旋网络中的顶点的方向，更重要的是，该算法的自旋网络的评估假设一个循环的减少图工作在定理的证明。它表明，这并不总是如此，这引起了怀疑的一般性定理。考虑到这个问题，该定理被重述以解释环形自旋网络，即可嵌入环面中的蜂窝网络。为此，最小的非平面自旋网络的检查和算法扩展到占环形自旋网络的非平面性。此外，还发现了三种类型的极小非平面自旋网络，其中两种是环面网络，第三种是胞腔网络，只嵌入在双环面中，没有圈。给出了两种环形自旋网络的一些例子，并找出了它们之间的关系。最后，对分解定理的问题和推广的可能性作了简要的讨论。
英文摘要：
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.
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