## 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, 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 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.

Original language | English |
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Pages (from-to) | 389-414 |

Number of pages | 26 |

Journal | Pacific Journal of Mathematics |

Volume | 282 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2016 Jun 1 |

Externally published | Yes |

## Keywords

- Homotopies
- Knots
- Persistent
- Submanifolds of the 3-sphere
- Transient

## ASJC Scopus subject areas

- Mathematics(all)