Classification of string links up to 2n-moves and link-homotopy
[Classification des enlacements d’intervalles à 2n-mouvements et homotopie près]
Annales de l'Institut Fourier, Tome 71 (2021) no. 3, pp. 889-911.

Deux enlacements d’intervalles sont équivalents à 2n-mouvements et homotopie près si et seulement si leurs invariants d’homotopie de Milnor sont congrus modulo n. De plus, l’ensemble des classes d’équivalence forme un groupe fini engendré par des éléments d’ordre n. Cette classification implique que si deux enlacements d’intervalles sont équivalents à 2n-mouvements près pour tout n>0, alors ils sont homotopes.

Two string links are equivalent up to 2n-moves and link-homotopy if and only if their all Milnor link-homotopy invariants are congruent modulo n. Moreover, the set of the equivalence classes forms a finite group generated by elements of order n. The classification induces that if two string links are equivalent up to 2n-moves for every n>0, then they are link-homotopic.

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DOI : 10.5802/aif.3407
Classification : 57K10
Keywords: Milnor invariant, link, string link, $2n$-move, link-homotopy, Fox’s congruence class, clasper
Mot clés : Invariants de Milnor, entrelacs, enlacements d’intervalles, $2n$-mouvements, homotopie, classes de congruence de Fox, claspers
Miyazawa, Haruko A. 1 ; Wada, Kodai 2 ; Yasuhara, Akira 3

1 Institute for Mathematics and Computer Science Tsuda University 2-1-1 Tsuda-machi, Kodaira Tokyo 187-8577 (Japan)
2 Department of Mathematics Kobe University 1-1 Rokkodai-cho, Nada-ku Kobe 657-8501 (Japan)
3 Faculty of Commerce Waseda University 1-6-1 Nishi-Waseda, Shinjuku-ku Tokyo 169-8050 (Japan)
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Miyazawa, Haruko A.; Wada, Kodai; Yasuhara, Akira. Classification of string links up to $2n$-moves and link-homotopy. Annales de l'Institut Fourier, Tome 71 (2021) no. 3, pp. 889-911. doi : 10.5802/aif.3407. http://www.numdam.org/articles/10.5802/aif.3407/

[1] Audoux, Benjamin; Bellingeri, Paolo; Meilhan, Jean-Baptiste; Wagner, Emmanuel Homotopy classification of ribbon tubes and welded string links, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 17 (2017) no. 2, pp. 713-761 | MR | Zbl

[2] Audoux, Benjamin; Meilhan, Jean-Baptiste; Wagner, Emmanuel On codimension two embeddings up to link-homotopy, J. Topol., Volume 10 (2017) no. 1, pp. 1107-1123 | DOI | MR | Zbl

[3] Casson, Andrew J. Link cobordism and Milnor’s invariant, Bull. Lond. Math. Soc., Volume 7 (1975), pp. 39-40 | DOI | MR | Zbl

[4] Chen, Kuo-Tsai Commutator calculus and link invariants, Proc. Am. Math. Soc., Volume 3 (1952), pp. 44-55 | DOI | MR | Zbl

[5] Conant, James; Schneiderman, Rob; Teichner, Peter Higher-order intersections in low-dimensional topology, Proc. Natl. Acad. Sci. USA, Volume 108 (2011) no. 20, pp. 8131-8138 | DOI | MR | Zbl

[6] Conant, James; Schneiderman, Rob; Teichner, Peter Whitney tower concordance of classical links, Geom. Topol., Volume 16 (2012) no. 3, pp. 1419-1479 | DOI | MR | Zbl

[7] Conant, James; Schneiderman, Rob; Teichner, Peter Milnor invariants and twisted Whitney towers, J. Topol., Volume 7 (2014) no. 1, pp. 187-224 | DOI | MR | Zbl

[8] Dąbkowski, Mieczysław K.; Przytycki, Józef H. Burnside obstructions to the Montesinos–Nakanishi 3-move conjecture, Geom. Topol., Volume 6 (2002), pp. 355-360 | DOI | MR | Zbl

[9] Dąbkowski, Mieczysław K.; Przytycki, Józef H. Unexpected connections between Burnside groups and knot theory, Proc. Natl. Acad. Sci. USA, Volume 101 (2004) no. 50, pp. 17357-17360 | DOI | MR | Zbl

[10] Fenn, Roger A. Techniques of geometric topology, London Mathematical Society Lecture Note Series, 57, Cambridge University Press, 1983 | MR | Zbl

[11] Fleming, Thomas; Yasuhara, Akira Milnor’s invariants and self C k -equivalence, Proc. Am. Math. Soc., Volume 137 (2009) no. 2, pp. 761-770 | DOI | MR | Zbl

[12] Fox, Ralph H. Congruence classes of knots, Osaka J. Math., Volume 10 (1958), pp. 37-41 | MR | Zbl

[13] Habegger, Nathan; Lin, Xiao-Song The classification of links up to link-homotopy, J. Am. Math. Soc., Volume 3 (1990) no. 2, pp. 389-419 | DOI | MR | Zbl

[14] Habiro, Kazuo Claspers and finite type invariants of links, Geom. Topol., Volume 4 (2000), pp. 1-83 | DOI | MR | Zbl

[15] Kinoshita, Shin’ichi On Wendt’s theorem of knots, Osaka J. Math., Volume 9 (1957), pp. 61-66 | MR | Zbl

[16] Kinoshita, Shin’ichi On the distribution of Alexander polynomials of alternating knots and links, Proc. Am. Math. Soc., Volume 79 (1980) no. 4, pp. 644-648 | DOI | MR | Zbl

[17] Lackenby, Marc Fox’s congruence classes and the quantum- SU (2) invariants of links in 3-manifolds, Comment. Math. Helv., Volume 71 (1996) no. 4, pp. 664-677 | DOI | MR | Zbl

[18] Levine, Jerome P. An approach to homotopy classification of links, Trans. Am. Math. Soc., Volume 306 (1988) no. 1, pp. 361-387 | DOI | MR | Zbl

[19] Meilhan, Jean-Baptiste; Yasuhara, Akira Characterization of finite type string link invariants of degree <5, Math. Proc. Camb. Philos. Soc., Volume 148 (2010) no. 3, pp. 439-472 | DOI | MR | Zbl

[20] Meilhan, Jean-Baptiste; Yasuhara, Akira Milnor invariants and the HOMFLYPT polynomial, Geom. Topol., Volume 16 (2012) no. 2, pp. 889-917 | DOI | MR | Zbl

[21] Meilhan, Jean-Baptiste; Yasuhara, Akira Arrow calculus for welded and classical links, Algebr. Geom. Topol., Volume 19 (2019) no. 1, pp. 397-456 | DOI | MR | Zbl

[22] Milnor, John W. Link groups, Ann. Math., Volume 59 (1954), pp. 177-195 | DOI | MR | Zbl

[23] Milnor, John W. Isotopy of links. Algebraic geometry and topology, A symposium in honor of S. Lefschetz, Princeton University Press, 1957, pp. 280-306 | Zbl

[24] Miyazawa, Haruko Aida; Wada, Kodai; Yasuhara, Akira Milnor invariants, 2n-moves and V n -moves for welded string links (2021) (to appear in Tokyo J. Math.)

[25] Nakanishi, Yasutaka On Fox’s congruence classes of knots. II, Osaka J. Math., Volume 27 (1990) no. 1, pp. 207-215 | MR | Zbl

[26] Nakanishi, Yasutaka; Suzuki, Shin’ichi On Fox’s congruence classes of knots, Osaka J. Math., Volume 24 (1987) no. 1, pp. 217-225 | MR | Zbl

[27] Przytycki, Józef H. t k moves on links, Braids (Santa Cruz, CA, 1986) (Contemporary Mathematics), Volume 78, American Mathematical Society, 1988, pp. 615-656 | MR | Zbl

[28] Stallings, John Homology and central series of groups, J. Algebra, Volume 2 (1965), pp. 170-181 | DOI | MR | Zbl

[29] Takabatake, Y.; Kuboyama, T.; Sakamoto, H. StringCMP: Faster calculation for Milnor invariant, 2013 (available at https://code.google.com/archive/p/stringcmp/)

[30] Yasuhara, Akira Classification of string links up to self delta-moves and concordance, Algebr. Geom. Topol., Volume 9 (2009) no. 1, pp. 265-275 | DOI | MR | Zbl

[31] Yasuhara, Akira Self delta-equivalence for links whose Milnor’s isotopy invariants vanish, Trans. Am. Math. Soc., Volume 361 (2009) no. 9, pp. 4721-4749 | DOI | MR | Zbl

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