A Note on Pure Braids and Link Concordance

TL;DR

This paper investigates the algebraic structure of string link concordance groups, showing that even after factoring out pure braid subgroups, these groups remain non-abelian, revealing complex underlying relationships.

Contribution

It proves that the quotient of each string link concordance group by its pure braid subgroup is non-abelian, extending previous understanding of these groups' structures.

Findings

String link concordance groups are non-abelian.

Quotients by pure braid subgroups are also non-abelian.

Supports the complexity of link concordance group structures.

Abstract

The knot concordance group can be contextualized as organizing problems about 3- and 4-dimensional spaces and the relationships between them. Every 3-manifold is surgery on some link, not necessarily a knot, and thus it is natural to ask about such a group for links. In 1988, Le Dimet constructed the string link concordance groups and in 1998, Habegger and Lin precisely characterized these groups as quotients of the link concordance sets using a group action. Notably, the knot concordance group is abelian while, for each $n$, the string link concordance group on $n$ strands is non-abelian as it contains the pure braid group on $n$ strands as a subgroup. In this work, we prove that even the quotient of each string link concordance group by its pure braid subgroup is non-abelian.