Simple knots in compact, orientable $3$-manifolds

Abstract:

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 $\partial M$ contains no $2$-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 $3$-manifold is completely determined by its set $\mathcal {K}(M)$ of knot groups, i.e, the set of groups ${\pi _1}(M - J)$ 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 ${S^3}$ if and only if every simple closed curve in $M$ shrinks to a point inside a connected sum of graph manifolds and $3$-cells.