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 (R) of n labelled points in R no three of which coincide.
We compute the Lusternik-Schnirelmann category (LS-cat) and higher topological complexity (TC s , s 2) of the "nok-equal" configuration space Conf (k) (R, n). With k = 3, this yields the LS-cat and the higher topological complexity of Khovanov's group PP n of pure planar braids on n strands, which is an 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.The second and third authors were supported by a Conacyt scholarship and a Conacyt Postdoctoral Fellowship, respectively.
We construct an Alexander-type invariant for oriented doodles from a deformation of the Tits representation of the twin group and from the Chebyshev polynomials of the second kind. Like the Alexander polynomial, our invariant vanishes on unlinked doodles with more than one component. We also include values of our invariant on several doodles.
We construct an Alexander type invariant for oriented doodles from a deformation of the Tits representation of the twin group. Similar to the Alexander polynomial, our invariant vanishes on unlinked doodles with more than one component. We also include values of our invariant on several doodles.
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