#### Volume 14, issue 3 (2014)

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Abelian quotients of the string link monoid

### Jean-Baptiste Meilhan and Akira Yasuhara

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

The set $\mathsc{S}\phantom{\rule{0.3em}{0ex}}\mathsc{ℒ}\left(n\right)$ of $n$–string links has a monoid structure, given by the stacking product. When considered up to concordance, $\mathsc{S}\phantom{\rule{0.3em}{0ex}}\mathsc{ℒ}\left(n\right)$ 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 $\mathsc{S}\phantom{\rule{0.3em}{0ex}}\mathsc{ℒ}\left(n\right)$ with a group structure, namely the ${C}_{k}$–equivalence introduced by Habiro in connection with finite-type invariants theory, and the ${C}_{k}$–concordance, which is generated by ${C}_{k}$–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