Knot homotopy in subspaces of the 3-sphere

Yuya Koda and Makoto Ozawa

Vol. 282 (2016), No. 2, 389–414

DOI: 10.2140/pjm.2016.282.389

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 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 $S^3$ 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.

Keywords

knots, homotopies, transient, persistent, submanifolds of the 3-sphere

Mathematical Subject Classification 2010

Primary: 57M25

Secondary: 57M15, 57N10, 57Q35

Milestones

Received: 30 March 2015

Revised: 11 September 2015

Accepted: 7 October 2015

Published: 3 March 2016