The Φ₃Invariant Of Signed Graphic Hyperplane Arrangements

The topological structure of the complement of a hyperplane arrangement in a complex vector space is one of the central topics in the theory of hyperplane arrangements. While the research on the φ3invariant is the most important. Falk gave a general formula to compute the φ3invariant and asked for a combinatorial interpretation of the φ3invariant.In this paper, we mainly researched the φ3invariant of signed graphic hyperplane arrangement. Compute the φ3invariant of signed complete graph which has less than5vertexs, find some properties. Then we obtain a conclusion on the φ3invariant of signed graphic hyperplane arrangement and prove it. At last, we give some examples to apply this conclusion on the classification of nonlinear polymer topologies in chemistry.We give a simple introduction on the background and progress of the theory of hyperplane arrangements in Chapter1, and we introduce some basic concepts of hyperplane arrangements in Chapter2. Chapter3is the core of this article. We introduce the basic concepts of signed graph firstly, then we research some signed complete graphs, at last we give the conclusion on:he φ3invariant of signed graphic hyperplane arrangement and prove it. And we compute some φ3invariant of polymers in Chapter4.