Milnor’s invariants and self $C_{k}$-equivalence
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- by Thomas Fleming and Akira Yasuhara PDF
- Proc. Amer. Math. Soc. 137 (2009), 761-770 Request permission
Abstract:
It has long been known that a Milnor invariant with no repeated index is an invariant of link homotopy. We show that Milnor’s invariants with repeated indices are invariants not only of isotopy, but also of self $C_{k}$-equivalence. Here self $C_{k}$-equivalence is a natural generalization of link homotopy based on certain degree $k$ clasper surgeries, which provides a filtration of link homotopy classes.References
- Dror Bar-Natan, Vassiliev homotopy string link invariants, J. Knot Theory Ramifications 4 (1995), no. 1, 13–32. MR 1321289, DOI 10.1142/S021821659500003X
- Tim D. Cochran, Derivatives of links: Milnor’s concordance invariants and Massey’s products, Mem. Amer. Math. Soc. 84 (1990), no. 427, x+73. MR 1042041, DOI 10.1090/memo/0427
- Nathan Habegger and Xiao-Song Lin, The classification of links up to link-homotopy, J. Amer. Math. Soc. 3 (1990), no. 2, 389–419. MR 1026062, DOI 10.1090/S0894-0347-1990-1026062-0
- Nathan Habegger and Gregor Masbaum, The Kontsevich integral and Milnor’s invariants, Topology 39 (2000), no. 6, 1253–1289. MR 1783857, DOI 10.1016/S0040-9383(99)00041-5
- Kazuo Habiro, Claspers and finite type invariants of links, Geom. Topol. 4 (2000), 1–83. MR 1735632, DOI 10.2140/gt.2000.4.1
- Xiao-Song Lin, Power series expansions and invariants of links, Geometric topology (Athens, GA, 1993) AMS/IP Stud. Adv. Math., vol. 2, Amer. Math. Soc., Providence, RI, 1997, pp. 184–202. MR 1470727, DOI 10.1090/amsip/002.1/10
- Jean-Baptiste Meilhan and Akira Yasuhara, On $C_n$-moves for links, preprint.
- John Milnor, Link groups, Ann. of Math. (2) 59 (1954), 177–195. MR 71020, DOI 10.2307/1969685
- John Milnor, Isotopy of links, Algebraic geometry and topology. A symposium in honor of S. Lefschetz, Princeton University Press, Princeton, N.J., 1957, pp. 280–306. MR 0092150
- Haruko Aida Miyazawa and Akira Yasuhara, Classification of $n$-component Brunnian links up to $C_n$-move, Topology Appl. 153 (2006), no. 11, 1643–1650. MR 2227018, DOI 10.1016/j.topol.2005.06.001
- Yasutaka Nakanishi, Delta link homotopy for two component links, Proceedings of the First Joint Japan-Mexico Meeting in Topology (Morelia, 1999), 2002, pp. 169–182. MR 1903689, DOI 10.1016/S0166-8641(01)00116-X
- Yasutaka Nakanishi and Yoshiyuki Ohyama, Delta link homotopy for two component links. II, J. Knot Theory Ramifications 11 (2002), no. 3, 353–362. Knots 2000 Korea, Vol. 1 (Yongpyong). MR 1905690, DOI 10.1142/S0218216502001664
- Yasutaka Nakanishi and Yoshiyuki Ohyama, Delta link homotopy for two component links. III, J. Math. Soc. Japan 55 (2003), no. 3, 641–654. MR 1978214, DOI 10.2969/jmsj/1191418994
- Tetsuo Shibuya, Self $\Delta$-equivalence of ribbon links, Osaka J. Math. 33 (1996), no. 3, 751–760. MR 1424684
- Tetsuo Shibuya and Akira Yasuhara, Boundary links are self delta-equivalent to trivial links, Math. Proc. Cambridge Philos. Soc. 143 (2007), no. 2, 449–458. MR 2364661, DOI 10.1017/S0305004107000254
- Tetsuo Shibuya and Akira Yasuhara, Self $C_k$-move, quasi self $C_k$-move and the Conway potential function for links, J. Knot Theory Ramifications 13 (2004), no. 7, 877–893. MR 2101233, DOI 10.1142/S0218216504003500
- Kouki Taniyama and Akira Yasuhara, Band description of knots and Vassiliev invariants, Math. Proc. Cambridge Philos. Soc. 133 (2002), no. 2, 325–343. MR 1912405, DOI 10.1017/S0305004102006138
Additional Information
- Thomas Fleming
- Affiliation: Department of Mathematics, University of California San Diego, 9500 Gilman Drive, La Jolla, California 92093-0112
- Email: tfleming@math.ucsd.edu
- Akira Yasuhara
- Affiliation: Department of Mathematics, Tokyo Gakugei University, Koganei-shi, Tokyo 184-8501, Japan
- MR Author ID: 320076
- Email: yasuhara@u-gakugei.ac.jp
- Received by editor(s): December 4, 2006
- Received by editor(s) in revised form: February 4, 2008
- Published electronically: August 28, 2008
- Additional Notes: The first author was supported by a Post-Doctoral Fellowship for Foreign Researchers ($\#$PE05003) from the Japan Society for the Promotion of Science.
The second author is partially supported by a Grant-in-Aid for Scientific Research (C) ($\#$18540071) of the Japan Society for the Promotion of Science. - Communicated by: Daniel Ruberman
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 137 (2009), 761-770
- MSC (2000): Primary 57M25
- DOI: https://doi.org/10.1090/S0002-9939-08-09521-X
- MathSciNet review: 2448599