
Suppose that F is an incompressible and pairwise incompressible
surface in a knot complement. We introduce topological graphs and
their moves (Rmove and Smove 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 "2sphere")
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.
