  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.
