Homotopy, Delta-equivalence and concordance for knots in the complement of a trivial link

TL;DR

This paper investigates how Milnor invariants can determine the equivalence of knots in the complement of trivial links under link-homotopy, Delta-equivalence, and concordance, extending classical results to more complex link complements.

Contribution

It extends Milnor's invariants to analyze knots in the complement of trivial links, providing new criteria for null-homotopy, Delta-equivalence, and concordance in these settings.

Findings

Milnor invariants determine null-homotopy of knots in trivial link complements.

A sufficient condition for Delta-equivalence to the trivial knot is established.

Conditions for knots in trivial knot complements to be Delta-equivalent and concordant are provided.

Abstract

Link-homotopy and self Delta-equivalence are equivalence relations on links. It was shown by J. Milnor (resp. the last author) that Milnor invariants determine whether or not a link is link-homotopic (resp. self Delta-equivalent) to a trivial link. We study link-homotopy and self Delta-equivalence on a certain component of a link with fixing the rest components, in other words, homotopy and Delta-equivalence of knots in the complement of a certain link. We show that Milnor invariants determine whether a knot in the complement of a trivial link is null-homotopic, and give a sufficient condition for such a knot to be Delta-equivalent to the trivial knot. We also give a sufficient condition for knots in the complements of the trivial knot to be equivalent up to Delta-equivalence and concordance.