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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Self delta-equivalence for links whose Milnor's isotopy invariants vanish


Author: Akira Yasuhara
Journal: Trans. Amer. Math. Soc. 361 (2009), 4721-4749
MSC (2000): Primary 57M25, 57M27
Published electronically: March 19, 2009
MathSciNet review: 2506425
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Abstract: For an $ n$-component link, Milnor's isotopy invariants are defined for each multi-index $ I=i_1i_2...i_m~(i_j\in\{1,...,n\})$. Here $ m$ is called the length. Let $ r(I)$ denote the maximum number of times that any index appears in $ I$. It is known that Milnor invariants with $ r=1$, i.e., Milnor invariants for all multi-indices $ I$ with $ r(I)=1$, are link-homotopy invariant. N. Habegger and X. S. Lin showed that two string links are link-homotopic if and only if their Milnor invariants with $ r=1$ coincide. This gives us that a link in $ S^3$ is link-homotopic to a trivial link if and only if all Milnor invariants of the link with $ r=1$ vanish. Although Milnor invariants with $ r=2$ are not link-homotopy invariants, T. Fleming and the author showed that Milnor invariants with $ r\leq 2$ are self $ \Delta$-equivalence invariants. In this paper, we give a self $ \Delta$-equivalence classification of the set of $ n$-component links in $ S^3$ whose Milnor invariants with length $ \leq 2n-1$ and $ r\leq 2$ vanish. As a corollary, we have that a link is self $ \Delta$-equivalent to a trivial link if and only if all Milnor invariants of the link with $ r\leq 2$ vanish. This is a geometric characterization for links whose Milnor invariants with $ r\leq 2$ vanish. The chief ingredient in our proof is Habiro's clasper theory. We also give an alternate proof of a link-homotopy classification of string links by using clasper theory.


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Additional Information

Akira Yasuhara
Affiliation: Department of Mathematics, Tokyo Gakugei University, Koganeishi, Tokyo 184-8501, Japan
Email: yasuhara@u-gakugei.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9947-09-04840-5
PII: S 0002-9947(09)04840-5
Keywords: $\Delta $-move, self $\Delta $-move, $C_n$-move, link-homotopy, self $\Delta $-equivalence, Milnor invariant, string link, Brunnian link, clasper
Received by editor(s): July 17, 2007
Published electronically: March 19, 2009
Additional Notes: The author was partially supported by a Grant-in-Aid for Scientific Research (C) ($#$18540071) of the Japan Society for the Promotion of Science.
Dedicated: Dedicated to Professor Tetsuo Shibuya on his 60th birthday.
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.