Incompressible surfaces and (1,1)-knots

Details Incompressible surfaces and (1,1)-knots Incompressible surfaces and (1,1)-knots

2002-01-15

arxiv.org/abs/math/0201121v2

Mathematics

Geometric Topology

11 pages, 4 figures. Revised version. To appear in JKTR

Scientific paper

Let M be $S^3$, $S^1\times S^2$, or a lens space L(p,q), and let k be a (1,1)-knot in M, i.e., a knot which is of 1-bridge with respect to a Heegaard torus. 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.

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