Abelian quotients of the string link monoid

TL;DR

This paper investigates conditions under which certain quotient groups of the string link monoid become abelian, focusing on C_k-equivalence and C_k-concordance, with implications for finite type invariants.

Contribution

It characterizes when the groups formed by string links under C_k-equivalence and C_k-concordance are abelian, extending understanding of their algebraic structure.

Findings

C_k-equivalence groups are abelian for specific k values

C_k-concordance groups exhibit abelian properties under certain conditions

Applications to finite type invariants are demonstrated

Abstract

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 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.