Properties Of Non-simple Topological Graph

In this paper, it deals with incompressible pairwise incompressible surfaces in link complements by studying the properties of non-simple topological graph, here calling the surfaces based surfaces. While it discusses the relation between the boundary components number of based surfaces and the words of loops in non-simple topological graph. This paper characterize the properties of non-simple topological graph, the component number of which is not more than six, by making use of definitions, theorems and properties of the topological graph, based surfaces in alternating link complements and almost alternating link complements. It proves that the genus of the based surface equals zero if the component number of non-simple topological graph is less than five. Furthermore, it analyzes the relation between the boundary components number of based surfaces and the words of loops in non-simple topological graph. One can show that the word of each loop of topological graph in alternating link complements contains at most n/2S’s and in almost alternating link complements has at most (n/2+1) S’s, if the boundary components number of based surfaces in link complements equals n.