Incompressible pairwise incompressible surfaces in link complements

The central subject of studying in this paper is incompressible pairwise incompressible surfaces in link complements. Let L be a non-split prime link and let F be an incompressible pairwise incompressible surface in S3 – L. We discuss the properties that the surface F intersects with 2-spheres in S3 – L. The intersection forms a topological graph consisting of a collection of circles and saddle-shaped discs. We introduce topological graphs and their moves (R-move and S2-move), 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 a graph satisfies some conditions. By these ways, we characterize the properties of incompressible pairwise incompressible surfaces in alternating (or almost alternating) link complements. We prove that the genus of the surface equals zero if the component number of (or ) is less than five and the graph is simple for alternating or almost alternating links. Furthermore, one can prove that the genus of the surface is zero if #(∂F) ≤ 8.