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Milnor's invariants and self  -equivalence
Authors:
Thomas Fleming and Akira Yasuhara
Journal:
Proc. Amer. Math. Soc. 137 (2009), 761-770
MSC (2000):
Primary 57M25
Posted:
August 28, 2008
MathSciNet review:
2448599
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Additional Information
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  -equivalence. Here self  -equivalence is a natural generalization of link homotopy based on certain degree  clasper surgeries, which provides a filtration of link homotopy classes.
- 1.
Dror
Bar-Natan, Vassiliev homotopy string link invariants, J. Knot
Theory Ramifications 4 (1995), no. 1, 13–32. MR 1321289
(96b:57004), http://dx.doi.org/10.1142/S021821659500003X
- 2.
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
(91c:57005)
- 3.
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
(91e:57015), http://dx.doi.org/10.1090/S0894-0347-1990-1026062-0
- 4.
Nathan
Habegger and Gregor
Masbaum, The Kontsevich integral and Milnor’s
invariants, Topology 39 (2000), no. 6,
1253–1289. MR 1783857
(2002b:57011), http://dx.doi.org/10.1016/S0040-9383(99)00041-5
- 5.
Kazuo
Habiro, Claspers and finite type invariants of links, Geom.
Topol. 4 (2000), 1–83 (electronic). MR 1735632
(2001g:57020), http://dx.doi.org/10.2140/gt.2000.4.1
- 6.
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
(98i:57014)
- 7.
Jean-Baptiste Meilhan and Akira Yasuhara, On
 -moves for links, preprint.
- 8.
John
Milnor, Link groups, Ann. of Math. (2) 59
(1954), 177–195. MR 0071020
(17,70e)
- 9.
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
(19,1070c)
- 10.
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
(2007b:57013), http://dx.doi.org/10.1016/j.topol.2005.06.001
- 11.
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
(2003i:57015), http://dx.doi.org/10.1016/S0166-8641(01)00116-X
- 12.
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
(2003i:57014), http://dx.doi.org/10.1142/S0218216502001664
- 13.
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
(2004a:57009), http://dx.doi.org/10.2969/jmsj/1191418994
- 14.
Tetsuo
Shibuya, Self Δ-equivalence of ribbon links, Osaka J.
Math. 33 (1996), no. 3, 751–760. MR 1424684
(97k:57012)
- 15.
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
(2008j:57009), http://dx.doi.org/10.1017/S0305004107000254
- 16.
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
(2006a:57009), http://dx.doi.org/10.1142/S0218216504003500
- 17.
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
(2003m:57034), http://dx.doi.org/10.1017/S0305004102006138
- 1.
- Dror Bar-Natan, Vassiliev homotopy string link invariants, J. Knot Theory Ramifications 4 (1995), no. 1, 13-32. MR 1321289 (96b:57004)
- 2.
- Tim D. Cochran, Derivatives of links: Milnor's concordance invariants and Massey's products, Mem. Amer. Math. Soc. 84 (1990), no. 427. MR 1042041 (91c:57005)
- 3.
- 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 (91e:57015)
- 4.
- Nathan Habegger and Gregor Masbaum, The Kontsevich integral and Milnor's invariants, Topology 39 (2000), no. 6, 1253-1289. MR 1783857 (2002b:57011)
- 5.
- Kazuo Habiro, Claspers and finite type invariants of links, Geom. Topol. 4 (2000), 1-83 (electronic). MR 1735632 (2001g:57020)
- 6.
- 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 (98i:57014)
- 7.
- Jean-Baptiste Meilhan and Akira Yasuhara, On -moves for links, preprint.
- 8.
- John Milnor, Link groups, Ann. of Math. (2) 59 (1954), 177-195. MR 0071020 (17:70e)
- 9.
- -, Isotopy of links. Algebraic geometry and topology, A symposium in honor of S. Lefschetz, Princeton University Press, Princeton, NJ, 1957, pp. 280-306. MR 0092150 (19:1070c)
- 10.
- H. A. Miyazawa and Akira Yasuhara, Classification of
 -component Brunnian links up to  -move, Topology Appl. 153 (2006), 1643-1650. MR 2227018 (2007b:57013)
- 11.
- Yasutaka Nakanishi, Delta link homotopy for two component links, Topology Appl. 121 (2002), 169-182. MR 1903689 (2003i:57015)
- 12.
- 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 (2003i:57014)
- 13.
- -, Delta link homotopy for two component links. III, J. Math. Soc. Japan 55 (2003), no. 3, 641-654. MR 1978214 (2004a:57009)
- 14.
- Tetsuo Shibuya, Self
 -equivalence of ribbon links, Osaka J. Math. 33 (1996), no. 3, 751-760. MR 1424684 (97k:57012)
- 15.
- Tetsuo Shibuya and Akira Yasuhara, Boundary links are self delta-equivalent to trivial links, Math. Proc. Cambridge Philos. Soc. 143 (2007), 449-458. MR 2364661
- 16.
- -, Self
 -move, quasi self  -move and the Conway potential function for links, J. Knot Theory Ramifications 13 (2004), no. 7, 877-893. MR 2101233 (2006a:57009)
- 17.
- 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 (2003m:57034)
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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
Email:
yasuhara@u-gakugei.ac.jp
DOI:
http://dx.doi.org/10.1090/S0002-9939-08-09521-X
PII:
S 0002-9939(08)09521-X
Received by editor(s):
December 4, 2006
Received by editor(s) in revised form:
February 4, 2008
Posted:
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
Article copyright:
© Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
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