Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Knot homotopy in subspaces of the 3-sphere   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.

24
The categories of this paper are: dg-ga
8
The categories of this paper are: hep-ex
14
The categories of this paper are: math-ph
23
The categories of this paper are: solv-int
4
The categories of this paper are: cond-mat
11
The categories of this paper are: hep-lat
26
The categories of this paper are: bayes-an
21
The categories of this paper are: adap-org
35
The categories of this paper are: mtrl-th
31
The categories of this paper are: ao-sci
28
The categories of this paper are: alg-geom
10
The categories of this paper are: q-bio
17
The categories of this paper are: q-fin
3
The categories of this paper are: physics
9
The categories of this paper are: nlin
12
The categories of this paper are: nucl-th