INCOMPRESSIBLE SURFACES AND (1, 1)-KNOTS

Abstract

Let M be S3, S1× S2, or a lens space L(p, q), and let k be a (1, 1)-knot in M. We show that if there is a closed meridionally incompressible surface in the complement of k, then the surface and the knot can be put in a special position, namely, the surface is the boundary of a regular neighborhood of a toroidal graph, and the knot is level with respect to that graph. As an application we show that for any such M there exist tunnel number one knots which are not (1, 1)-knots.