Ncompressible Pairwise Incompressible Surfaces In The Complements Of Knot And Spatial Graph

The existence and classification of incompressible pairwise incompressible surface in the complements of knot and spatial graph have very active role in knot and spatial graph theory.This paper from the perspective of the above, by using the properties of incompressible pairwise incompressible surface in the complements of knot and spatial and the combination methods of 3-manifolds, by the genus of incompressible pairwise incompressible surface F in the complements of alternating knot, it gives a constructing method of incompressible pairwise incompressible surfaces with genus at most three, and singular the form of topological graph of incompressible pairwise incompressible surfaces with boundaries at most twelve and analyses the difference between them for genus at most three. Furthermore, it proves that incompressible pairwise incompressible surfaces in a class of alternating spatial graph complements are punctured spheres. In particular, this paper will apply the techniques of the classification of incompressible pairwise incompressible surfaces in the complements of alternating knot to the classification of incompressible pairwise incompressible surfaces in alternating spatial graph. The results obtained in this paper plays a very important role for the classification of knots and spatial graphs up to isotopy.