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

Linear motion planning with controlled collisions and pure planar braids