Constructing essential laminations in 2-bridge knot surgered 3-manifolds
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 \(\pi_1 (L)\) into \(\pi_1 (M)\). Our main result is: Theorem 2. Every 3-manifold obtained by surgery on a non-torus 2-bridge knot admits essential laminations. Some immediate corollaries are that these manifolds are covered by \(\mathbb{R}^3\), and have infinite 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 Poincaré conjecture. We use general techniques which are not specific to 2-bridge knots to find and explicitly construct these laminations.
scientific article; zbMATH DE number 950211