TY - JOUR
T1 - Homotopy, Δ-equivalence and concordance for knots in the complement of a trivial link
AU - Fleming, Thomas
AU - Shibuya, Tetsuo
AU - Tsukamoto, Tatsuya
AU - Yasuhara, Akira
N1 - Funding Information:
E-mail addresses: tfleming@math.ucsd.edu (T. Fleming), shibuya@ge.oit.ac.jp (T. Shibuya), tsukamoto@ge.oit.ac.jp (T. Tsukamoto), yasuhara@u-gakugei.ac.jp (A. Yasuhara). 1 The author is partially supported by the Sumitomo Foundation (090729). 2 The author is partially supported by a Grant-in-Aid for Scientific Research (C) (#20540065) of the Japan Society for the Promotion of Science.
PY - 2010/5/1
Y1 - 2010/5/1
N2 - 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.
AB - 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.
KW - Link-homotopy
KW - Milnor invariants
KW - Self Δ-equivalence
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U2 - 10.1016/j.topol.2010.02.014
DO - 10.1016/j.topol.2010.02.014
M3 - Article
AN - SCOPUS:77949487397
SN - 0166-8641
VL - 157
SP - 1215
EP - 1227
JO - Topology and its Applications
JF - Topology and its Applications
IS - 7
ER -