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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Abelian quotients of the string link monoid

Jean-Baptiste Meilhan and Akira Yasuhara

Algebraic & Geometric Topology 14 (2014) 1461–1488
Abstract

The set S(n) of n–string links has a monoid structure, given by the stacking product. When considered up to concordance, S(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 S(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.

Keywords
string links, $C_n$–moves, concordance, claspers, Milnor invariants
Mathematical Subject Classification 2010
Primary: 57M25, 57M27
Secondary: 20F38
References
Publication
Received: 7 May 2013
Revised: 31 October 2013
Accepted: 6 November 2013
Published: 7 April 2014
Authors
Jean-Baptiste Meilhan
Institut Fourier
Université Grenoble 1
100 rue des Maths, BP 74
38402 Saint Martin d’Hères
France
http://www-fourier.ujf-grenoble.fr/~meilhan/
Akira Yasuhara
Department of Mathematics
Tokyo Gakugei University
4-1-1 Nukuikita-Machi
Koganei-shi
Tokyo 184-8501
Japan
http://www.u-gakugei.ac.jp/~yasuhara/