Constructing essential laminations in 2-bridge knot surgered 3-manifolds

Abstract

A nontrivial knot that can be drawn with only two relative maxima in the vertical direction is called a 2-bridge knot, and one that can be drawn on a torus is called a torus knot. Loosely speaking, a lamination in a manifold M is a foliation of M, except that it can have a nonempty open complement in M, and very loosely speaking, the lamination is essential if each leaf of it L is incompressible, i.e. inclusion of L into M induces an injective homomorphism from 1(L) into 1(M). Our main result is: Theorem 2. Every 3-manifold obtained by surgery on a nontorus 2-bridge knot admits essential laminations. Some immediate corollaries are that these manifolds are covered by R 3 , and have innite fundamental group. So Property P is true for non-torus 2-bridge knots, i.e. surgery on these knots never yields a homotopy 3-sphere, or a counterexample to the Poincare conjecture. We use general techniques which are not specic to 2-bridge knots to nd and explicitly construct these laminations. In trying to understand 3-manifolds with the hope of eventually classifying them, as with 2-manifolds, one approach that has turned out to be fruitful is to study objects of codimension one in them, more specically, incompressible surfaces, Reebless foliations, and essential laminations. For a 3-manifoldM containing an incompressible surface, with some extra hypotheses, Waldhausen proved: The universal cover of M is R 3 , the topology of M is determined by its fundamental group, and homotopic homeomorphisms of M are isotopic. Similar theorems for manifolds containing Reebless foliations were proven by Haefliger, Novikov, Palmeira, Rosenberg, and others. The essential lamination was developed in the 1980’s as a generalization of the incompressible surface and the Reebless foliation, which themselves qualify as essential laminations. In fact they are just extreme cases of essential laminations: At one end we have compact properly embedded surfaces,