Essential laminations and Dehn surgery on 2-bridge knots

Abstract

In this paper we show that any nontrivial Dehn surgery on a nontorus 2-bridge knot produces a manifold which is covered by R 3. In particular, this manifold is irreducible and has infinite fundamental group. (As a consequence, it is also clear that 2-bridge knots satisfy property P, although this was shown previously by Takahashi (1981) using more algebraic techniques.) The result is a consequence of showing that such a manifold is laminar, that is it contains an essential lamination. We accomplish this by constructing in the exterior of each nontorus 2-bridge knot an essential lamination which remains essential in all manifolds produced by nontrivial Dehn filling. We call an essential lamination with this property persistent. The examples of essential laminations produced in this way are of interest since they tend support the conjecture that “most” manifolds with infinite fundamental group contain essential laminations.