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

Thomas Fleming; Tetsuo Shibuya; Tatsuya Tsukamoto; Akira Yasuhara

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arXiv · 0909.1434

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

Thomas Fleming·Tetsuo Shibuya·Tatsuya Tsukamoto·Akira Yasuhara

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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.

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Thomas Fleming, Tetsuo Shibuya, Tatsuya Tsukamoto, Akira Yasuhara. 2009-09-08. Homotopy, Delta-equivalence and concordance for knots in the complement of a trivial link. https://arxiv.org/abs/0909.1434

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