Abelian quotients of the string link monoid

Abstract

The set $\mathcal{S}\mathcal{ℒ}\left(n\right)$ of $n$–string links has a monoid structure, given by the stacking product. When considered up to concordance, $\mathcal{S}\mathcal{ℒ}\left(n\right)$ 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 $\mathcal{S}\mathcal{ℒ}\left(n\right)$ 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.