  We discuss an "extrinsic" property of knots in a 3-subspace of the 3-sphere
$S^3$ to characterize how the subspace is embedded in $S^3$. Specifically, we
show 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 $S^3$ is said
to be transient if it can be moved by a homotopy within the subspace to the
trivial knot in $S^3$. To show this, we discuss 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
knot, we define an integer-valued invariant of knots in $S^3$ 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.
