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Homotopy, \(\Delta \)-equivalence and concordance for knots in the complement of a trivial link. (English) Zbl 1232.57006
Summary: 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 while fixing the other 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.
MSC:
57M25 Knots and links in the \(3\)-sphere (MSC2010)
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