Knot Fill In The Polynomial Width Of The Incompressible Surface And Four Of The Bifurcation Diagram

This paper consists of two parts .The first part is devoted to discussion of the properties of the incompressible pairwise incompressible surfaces in alternating knot complements. Let F c S3 K be an incompressible surface and if F r~ has three or four components ,we not only give the concrete location of topological graph but also prove that the genus of F equals zero; given certain conditions, the genus of the incompressible surface whose topological graph has n+l components equals zero if the genus of the incompressible surface whose topological graph has n components equals zero. The second part is devoted to computing the polynomial span of 4- regular graph in R3 by using the method of the calculation of the polynomial span in almost alternating links.