Linear motion planning with controlled collisions and pure planar braids

We compute the Lusternik-Schnirelmann category (LS-cat) and higher topological complexity ($TC_s$, $s\geq2$) of "no-$k$-equal" configuration space Conf$_k(\mathbb{R},n)$. This yields (with $k=3$) LS-cat Khovanov's group PP$_n$ pure planar braids on $n$ strands, which is an $\mathbb{R}$-analogue Artin's classical braid strands. Our methods can be used to describe optimal motion planners for provided small.