Classification of string links up to $2n$-moves and link-homotopy

Miyazawa, Haruko A.; Wada, Kodai; Yasuhara, Akira

doi:10.5802/aif.3407

Classification of string links up to

2 n

-moves and link-homotopy
[Classification des enlacements d’intervalles à

2 n

-mouvements et homotopie près]

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)

Annales de l'Institut Fourier, Tome 71 (2021) no. 3, pp. 889-911.

Résumé
Abstract

Deux enlacements d’intervalles sont équivalents à $2 n$ -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 $2 n$ -mouvements près pour tout $n > 0$ , alors ils sont homotopes.

Two string links are equivalent up to $2 n$ -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 2 $n$ -moves for every $n > 0$ , then they are link-homotopic.

Reçu le : 2020-01-16
Accepté le : 2020-05-13
Première publication : 2021-06-07
Publié le : 2022-03-15

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

Affiliations des auteurs :

Miyazawa, Haruko A. ¹ ; Wada, Kodai ² ; Yasuhara, Akira ³

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     title = {Classification of string links up to $2n$-moves and link-homotopy},
     journal = {Annales de l'Institut Fourier},
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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/

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