Knot homotopy in subspaces of the 3-sphere

TL;DR

This paper introduces the transient number as a new invariant for knots in the 3-sphere, characterizing how subspaces are embedded and relating group-theoretic properties to knot homotopy.

Contribution

It defines the transient number for knots in $S^3$, establishes a criterion for knots being transient based on the subspace's exterior, and explores the relationship between various knot invariants.

Findings

A knot is transient iff the exterior of the subspace is a union of handlebodies.

The transient number is a new integer-valued knot invariant.

Knots with unknotting or tunnel number one are a proper subset of those with transient number one.

Abstract

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.