Properties Of Incompressible Boundary Incompressible Surfaces In Knot Complements

Let K be a link in S~3, we position K so that it lies on S~2, exceptnear crossings of L, if L has n crossings, then there exist n bubbles,each bubble is a boundary of solid three sphere, and gives propertiesof the corresponding crossing, meanwhile, we got two 2-sphere S_-~2, S_+~2,Suppose F is a incompressible boundary incompressible surface and instandard position, by discussion properties about F∩S_+~2(F∩S_-~2), wecan conclude the genus and other properties of the surface. In this pa-per, in virtue of twist-crossing number, we give an upper bound on theEuler characteristic of a kind of incompressible boundary incompress-ible surface in the exterior of alternating knot or almost alternatingknot. at the same time, we can also estimate the genus of the surface.