Simple Knots in Compact, Orientable 3-Manifolds

Robert Myers
1982 Transactions of the American Mathematical Society  
A simple closed curve J in the interior of a compact, orientable 3-manifold M is called a simple knot if the closure of the complement of a regular neighborhood of J in 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-manifold M such that 9AÍ contains no 2-spheres contains a simple knot (and thus, from work of Thurston, a knot whose complement is
more » ... ). This result is used to prove that such a 3-manifold is completely determined by its set %( M) of knot groups, i.e, the set of groups »,( M -J) as J ranges over all the simple closed curves in M. In addition, it is proven that a closed 3-manifold M is homeomorphic to S3 if and only if every simple closed curve in M shrinks to a point inside a connected sum of graph manifolds and 3-cells.
doi:10.2307/1999193 fatcat:zgo7vcdakzhmjpsl3vh5jr5m3y