English

Abstract:
Suppose that F is an incompressible and pairwise incompressible surface in a knot complement. We introduce topological graphs and their moves (R-move and S-move etc.), and define the characteristic number of the topological graph for. The characteristic number is unchanged under the moves. In fact, the number is exactly the Euler Characteristic number of the surface when the graph satisfies some conditions. By these ways, we characterize the properties of incompressible and pairwise incompressible surfaces in a knot exterior. First, we prove that the genus of the surface equals zero if the components number(of the surface intersection with the "2-sphere") is less than five and the graph is simple for an alternating or almost alternating knot. Furthermore, one can prove that the genus of the surface is zero if the components number of the surface is less than nine.