Abelian quotients of the string link monoid

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 nD1. In this paper, we consider two families of equivalence relations which endow SL. (n) 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.