Proper knot theory in open 3-manifolds

Abstract

This paper introduces a theory of proper knots, i.e., smooth proper embeddings of R 1 {{\mathbf {R}}^1} into open 3 3 -manifolds. Proper knot theory is distinguished by the fact that proper isotopies of knots are not ambient in general. A uniqueness theorem for proper knots is proved in case the target manifold is the interior of a one-dimensional handlebody.