中文摘要：

通过对非简单拓扑图的性质进行讨论,研究了纽结补中的不可压缩、分段不可压缩曲面的性质.通过拓扑图的特征数给出了这些曲面亏格的性质,同时对不可压缩、分段不可压缩曲面的分支数与非简单拓扑图的非最内环道的字表示间的关系进行了研究.设曲面SS^3-K是不可压缩、分段不可压缩曲面,S±^2是二维球面,给出了拓扑图每条闭曲线C字的表示w±（C）,这些字母有P,S,它们描述了曲面的分支数和bubbles的相交,证明了当S∩S±^2的图（即拓扑图）的分支数等于3时,若拓扑图是非简单的,那么非简单拓扑图有唯一的形式,并通过单位拓扑图给出它们的表示,从而得到拓扑图的特征数是2,进而曲面S的亏格等于0（这里S是纽结补中的本质曲面）.

英文摘要：

In this paper,we deal with incompressible pairwise incompressible surfaces in link complements by studying the properties of non-simple topological graph.We give the properties of genus by using the characteristics number.We discuss the relation between the boundary components number of essential surfaces and the words of loops in non-simple topological graph.Let SS^3-Kbe incompressible pairwise incompressible surface and let S±^2 be 2-sphere.Each component Cof S∩S±^2can be associated a cyclic word w±（C）in letter P（=puncture）and S（=saddle）,which records,in order,the intersections of C with Kand with the bubbles,respectively.We give the properties of non-simple topological graph by making use of definitions,theorems and properties of the topological graph,essential surfaces in link complements.One can know that the topological graph with three components has a unique form.We prove that the characteristic number of the topological graph is two and the genus of the essential surface equals zero if the topological graph is non-simple and the component number of non-simple topological graph S∩S2±is three.