Abelian quotients of the string link monoid

Meilhan, Jean-Baptiste¹ ; Yasuhara, Akira²

¹Institut Fourier, Université Grenoble 1, 100 rue des Maths, BP 74, 38402 Saint Martin d’Hères, France
²Department of Mathematics, Tokyo Gakugei University, 4-1-1 Nukuikita-Machi, Koganei-shi, Tokyo 184-8501, Japan

Algebraic and Geometric Topology, Tome 14 (2014) no. 3, pp. 1461-1488

Résumé

The set Sℒ(n) of n–string links has a monoid structure, given by the stacking product. When considered up to concordance, Sℒ(n) becomes a group, which is known to be abelian only if n = 1. In this paper, we consider two families of equivalence relations which endow Sℒ(n) with a group structure, namely the Ck–equivalence introduced by Habiro in connection with finite-type invariants theory, and the Ck–concordance, which is generated by Ck–equivalence and concordance. We investigate under which condition these groups are abelian, and give applications to finite-type invariants.

Détail

DOI : 10.2140/agt.2014.14.1461

Classification : 57M25, 57M27, 20F38
Keywords: string links, $C_n$–moves, concordance, claspers, Milnor invariants

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