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

Abstract

Link-homotopy and self ∆-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 ∆-equivalent) to a trivial link. We study link-homotopy and self ∆-equivalence on a certain component of a link with fixing the rest components, in other words, homotopy and ∆-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 ∆-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 ∆-equivalence and concordance.