Simple Topological Graph And Closed Surfaces In Almost Alternating Link Complements

The main content of this paper is to study the properties of incompressible and pairwise incompressible surfaces with simple topological graph in non-split link complements and closed surfaces in non-split almost alternating link complements.Firstly,we introduce the concepts of standard position of surface,incompressible surface,pairwise incompressible surface,topological graph,characteristic number of topological graph and connected sum of topological graph,and define a method of moving topological graph.Then,by studying the relationship between the Euler characteristic of the surface and the characteristic number of the surface's topological graph,and using the properties of the connected sum of the topological graph,we prove that if the topological graph of the incompressible and pairwise incompressible surface can be decomposed into the connected sum of several simple topological graphs,then the surface must be a punctured sphere.At the same time,we also study the relationship between the simple topological graph and the genus of the surface when there is at most one saddle in each bubble of the surface's topological graph.By moving the topological graph,we prove that if the topological graph corresponding to the incompressible and pairwise incompressible surface in non-split link complements satisfies that there is at most one saddle-shaped disk in each bubble and the topological graph cannot be decomposed into the connected sum of several simple topological graphs,the topological graph contains at least one basic local graph.And the sufficient and necessary condition for the surface's genus to be 0 in this case is that the topological graph can be decomposed into the connected sum of several simple topological graphs.Finally,we study the special position that closed incompressible and pairwise incompressible surfaces must satisfy in non-split almost alternating link complements.Each topological graph is corresponded to a dissection of square,and by using the properties of the dissection of square,we prove that when the number of crossings of the almost alternating projection of the non-split almost alternating link is less than 10,the closed incompressible surface in almost alternating link complement must contain a circle which is isotopic in link complement to a meridian of the link.At the same time,we prove that there are a finite number of the closed,incompressible and pairwise incompressible surfaces in the non-split almost alternating link complements,up to isotopy.