Knot homotopy in subspaces of the 3-sphere

We discuss an extrinsic property of knots in a 3-subspace of the 3-sphere S³ to characterize how the subspace is embedded in S³. Specifically, weshow that every knot in a subspace of the 3-sphere is transient if and only if the exterior of the subspace is a disjoint union of handlebodies, i.e., regular neighborhoods of embedded graphs, where a knot in a 3-subspace of S³ is said to be transient if it can be moved by a homotopy within the subspace to the trivial knot in S³. To show this, we discuss the relation between certain group-theoretic and homotopic properties of knots in a compact 3-manifold, which can be of independent interest. Further, using the notion of transient knots, we define an integer-valued invariant of knots in S³ that we call the transient number. We then show that the union of the sets of knots of unknotting number one and tunnel number one is a proper subset of the set of knots of transient number one.