Essential Laminations and Branched Surfaces in the Exteriors of Links

In the same paper, by using the result above, he proposes a procedure for determining whether a given manifold contains an essential lamination or not. However the procedure does not work, at present, since (1) there is not a practical algorithm for determining whether a given branched surface is essential or not, and (2) there does not exist an algorithm for determining whether a given branched surface fully carries a lamination or not. The purpose of this paper is to try to carry out the procedure to the exteriors of links given by diagrams, by using various techniques in knot and link theory, and 3dimensional topology. In fact, we give a definition of standard position (with respect to a diagram of a given link) for branched surfaces contained in the exterior of links in section 2, which is a natural generalization of standard position of closed incompressible surfaces defined by W. Menasco [M1]. In section 3, we apply the result of [B], to show that any essential lamination in a link exterior can be deformed into one carried by an essential branched surface in standard position with respect to a given diagram. In section 4, we study about branched surfaces in standard positions with respect to alternating diagrams, and give a sufficient condition for the branched surfaces to be incompressible and Reebless, and possess indecomposable exteriors (for the definitions of these terms, see section 2). In [O], U.Oertel studied some fundamental properties of affine laminations in 3-manifolds. In section 5, we give a necessary and sufficient condition for a given branched surface in standard position to fully carry affine laminations, by using