Linear motion planning with controlled collisions and pure planar braids

We compute the Lusternik-Schnirelmann category (LS-cat) and the higher topological complexity ($T{C}_s$, $s\ge 2$) of the “no-$k$-equal” configuration space Conf${}_k(\mathbb{R},n)$. This yields (with $k=3$) the LS-cat and the higher topological complexity of Khovanov’s group PP${}_n$ of pure planar braids on $n$ strands, which is an $\mathbb{R}$-analogue of Artin’s classical pure braid group on $n$ strands. Our methods can be used to describe optimal motion planners for PP${}_n$ provided $n$ is small.