# Planar pure braids on six strands

## TL;DR

This paper characterizes the structure of the planar pure braid group on six strands, revealing it as a free product of a free group and multiple free abelian groups, expanding understanding of these configuration space groups.

## Contribution

It provides a detailed description of the planar pure braid group on six strands, showing it as a free product of a free group and free abelian groups, a new structural insight.

## Key findings

- The planar pure braid group on 6 strands is a free product of a free group and free abelian groups.
- It consists of 71 generators in the free part.
- It includes 20 copies of the free abelian group of rank two.

## Abstract

The group of planar (or flat) pure braids on $n$ strands, also known as the pure twin group, is the fundamental group of the configuration space $F_{n,3}(\mathbb{R})$ of $n$ labelled points in $\mathbb{R}$ no three of which coincide. The planar pure braid groups on 3, 4 and 5 strands are free. In this note we describe the planar pure braid group on 6 strands: it is a free product of the free group on 71 generators and 20 copies of the free abelian group of rank two.