  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.
