Algebraic & Geometric Topology

Abelian quotients of the string link monoid

Jean-Baptiste Meilhan and Akira Yasuhara

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The set S(n) of n–string links has a monoid structure, given by the stacking product. When considered up to concordance, S(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 S(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

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

Received: 7 May 2013
Revised: 31 October 2013
Accepted: 6 November 2013
First available in Project Euclid: 19 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 57M27: Invariants of knots and 3-manifolds
Secondary: 20F38: Other groups related to topology or analysis

string links $C_n$–moves concordance claspers Milnor invariants


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.

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