## Algebraic & Geometric Topology

### Abelian quotients of the string link monoid

#### 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 $Ck$–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.

#### Article information

Source
Algebr. Geom. Topol., Volume 14, Number 3 (2014), 1461-1488.

Dates
Revised: 31 October 2013
Accepted: 6 November 2013
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.agt/1513715911

Digital Object Identifier
doi:10.2140/agt.2014.14.1461

Mathematical Reviews number (MathSciNet)
MR3190601

Zentralblatt MATH identifier
1315.57013

#### Citation

Meilhan, Jean-Baptiste; Yasuhara, Akira. Abelian quotients of the string link monoid. Algebr. Geom. Topol. 14 (2014), no. 3, 1461--1488. doi:10.2140/agt.2014.14.1461. https://projecteuclid.org/euclid.agt/1513715911

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