Linear motion planning with controlled collisions and pure planar braids

We compute the Lusternik–Schnirelmann category (LS-cat) and higher topological complexity ($\operatorname{TC}_s, s \geqslant 2$) of the “no-$k$-equal” configuration space $\operatorname{Conf}^{(k)} (\mathbb{R}, n)$. With $k = 3$, this yields the LS-cat and the higher topological complexity of Khovanov’s group $\operatorname{PP}_n$ 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 $\operatorname{PP}_n$ provided $n$ is small.

Keywords

motion planning, higher topological complexity, sectional category, configuration space, controlled collision, pure planar braid

2010 Mathematics Subject Classification

55M30, 55R80, 55S40, 68T40

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The second and third authors were supported by a Conacyt scholarship and a Conacyt Postdoctoral Fellowship, respectively.

Received 14 March 2020

Received revised 28 June 2020

Accepted 7 July 2020

Published 4 November 2020