Incompressible Surfaces in Almost Alternating Link Exteriors

In this paper, we discuss mainly the properties of incompressible pairwise incomprcssiblc surfaccs in ahnost altcrnating link complcmcnts. Lct L bc a almost link and lct F be an incompressible pairwise incompressible surface in S^3 - L. First, we give the properties that the surface F intersects with 2-spheres in S^3- L. The intersection consisting of a collection of circles and saddle-shaped discs is called a topological graph. One can compute the Euler Characteristic number of the surface by calculating the characteristic number of the graph. Next, we prove that if the graph is special simple, then the genus of the surface is zero.