Simple knots in compact, orientable 3-manifolds

Abstract

A simple closed curve J J in the interior of a compact, orientable 3 3 -manifold M M is called a simple knot if the closure of the complement of a regular neighborhood of J J in M M is irreducible and boundary-irreducible and contains no non-boundary-parallel, properly embedded, incompressible annuli or tori. In this paper it is shown that every compact, orientable 3 3 -manifold M M such that ∂ M \partial M contains no 2 2 -spheres contains a simple knot (and thus, from work of Thurston, a knot whose complement is hyperbolic). This result is used to prove that such a 3 3 -manifold is completely determined by its set K ( M ) \mathcal {K}(M) of knot groups, i.e, the set of groups π 1 ( M − J ) {\pi _1}(M - J) as J J ranges over all the simple closed curves in M M . In addition, it is proven that a closed 3 3 -manifold M M is homeomorphic to S 3 {S^3} if and only if every simple closed curve in M M shrinks to a point inside a connected sum of graph manifolds and 3 3 -cells.