Narrow Results

We show how an incompressible Seifert surface F for a knot K in S³ can be used to create an essential lamination ℒ_F in the complement of each of an infinite class of knots associated to F. This lamination is persistent for these knots; it remains essential under all non-trivial Dehn fillings of the knot complement. This implies a very strong form of Property P for each of these knots.

We show how essential laminations can be used to provide an improvement on (some of) the results of the 2π-Theorem; at most 20 Dehn fillings on a hyperbolic 3-manifold with boundary a torus T can yield a reducible manifold, finite π₁ manifold, or exceptional Seifert-fibered space. Recent work of Wu allows us to add toroidal manifolds to this list, as well.

We consider the parallelism of two strings in alternating tangles. We show that if there is a pair of parallel strings in an alternating tangle then its alternating diagrams satify certain conditions. As a corollary, for a knot admitting a decomposition into two alternating tangles with two or three strings, we prove that its non-trivial Dehn surgery yields a 3-manifold with an essential lamination. Hence such a knot has property P and satisfy the cabling conjecture.

We show how to build tangles T in a 3-ball with the property that any knot obtained by tangle sum with T has a persistent lamination in its exterior, and therefore has property P. The construction is based on an example of a persistent lamination in the exterior of the twist knot 6₁, due to Ulrich Oertel. We also show how the construction can be generalized to n-string tangles.