Homotopy, Δ-equivalence and concordance for knots in the complement of a trivial link |
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Authors: | Thomas Fleming Akira Yasuhara |
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Institution: | a Department of Mathematics, University of California, San Diego, 9500 Gilman Dr., La Jolla, CA 92093-0112, United States b Department of Mathematics, Osaka Institute of Technology, Asahi, Osaka 535-8585, Japan c Department of Mathematics, Tokyo Gakugei University, Koganei, Tokyo 184-8501, Japan |
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Abstract: | Link-homotopy and self Δ-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 Δ-equivalent) to a trivial link. We study link-homotopy and self Δ-equivalence on a certain component of a link with fixing the other components, in other words, homotopy and Δ-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 Δ-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 Δ-equivalence and concordance. |
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Keywords: | Link-homotopy Self Δ-equivalence Milnor invariants |
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