The Problem Of Roots Of Jones Polynomial And The Properties Of Incompressible Surface

In this paper ,we discuss mainly the problem of roots of Jones polynomial and the properties of incompressible surface in almost alternating link exteriors.First, by using the properties of knots and the Jones polynomial, we research the relation between them, thereby involving in the projection diagram with n crossings, discuss the state S of the projection diagram L , through the properties of vertexgraph, Euler Formula, one studied the problem of the equal of the knots. Second, on the basic of part one, we used relationship between the Kauffman polynomial and the Jones polynomial, find a collection of knots which have rational root and the root is 0 . Finally we deal with the properties of the properties of incompressible surfaces in almost alternating link exteriors, that the properties of incompressible pairwise incompressible surfaces in almost alternating link complements, hence, on the one hand, one give the properties of the surface F intersects with2-spheres in S~3-L, on the other hand, we proved the topological graph is special simple, when the genus of the surface is 0.