φ₃ Invariant Of Linear Arrangements

In this paper, we mainly consider the invariantφ3 of linear arrangements. This thesis consists of two parts:the computation for invariantφ3 of the affine plane linear arrangements and the characteristic polynomials of a class of the linear arrangements.In the first part, the invariantφ3 of the linear arrangements with up to 6 lines in affine plane have been studied. Through the algorithm analysis, theoretical proof and programming calculation, we obtained the generic algorithms for the invariantφ3 and a classification of these arrangements according to the value ofφ3 Three lines planar configuration is divided into three categories, four lines planar configuration is divided into 5 categories, five lines planar configuration is divided into 8 categories, and six lines planar configuration is divided into 13 categories. Then we researched some nice examples with special properties, and provided a proof of the common formula for the invariantφ3.In the second part, based on the Whitney Theorem, the paper studied the characteristic polynomials for the three-dimensional arrangements. Furthermore, we considered the computation of characteristic polynomials for a special planar complete graph arrangement and obtain the computing formula.