On classifying proper knots in open 3-manifolds

Abstract

We examine proper embeddings of the real line into open 3-manifolds and their proper isotopy classes, i.e., proper knots and their equivalence classes. In particular, for proper knots running between distinct ends of an open 3-manifold M, we give conditions on the structure of the ends of M under which proper homotopy implies proper isotopy. To prove this result, geometric techniques are employed which enable one to properly isotope a proper knot that is wild in the neighbourhood of an end to one that is tame.