Linear motion planning with controlled collisions and pure planar braids

Abstract

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.

On Lusternik-Schnirelmann category and topological complexity of no k-equal manifolds

“…Thus, Theorem 1.1 describes cat and TC invariants for manifolds that (as far as it is currently known) might fail to be rationally formal. For instance, while the work in [6] shows that M (3) 2 (n) is rationally formal if and only if n ≤ 6, our calculations show that the equality TC s M (3) 2 (n) = s n k holds for n ≤ 12 with the possible exception of n = 11.…”

“…In this paper we will only deal with the case k ≥ 3. Further, as M (k) d (n) = R d n for n < k and M (k) d (k) S d k−d −1 , we will restrict our attention to the combinatorially more interesting case k < n. The aim of this paper is to extend the work in [3] where the Lusternik-Schnirelmann category (cat), topological complexity (TC) and sequential topological complexity (TC s ) of M (k) d (n) is computed for d = 1. Here we address the more subtle problem for d ≥ 2.…”

“…H * (M (3) 2 ( 6)) rank 6)) 20 H 4 (M (3) 2 ( 6)) 45 H 5 (M (3) 2 ( 6)) 36 H 6 1 (M (3) 2 ( 6)) 10 H 6 2 (M (3) 2 ( 6)) 10 H 7 (M (3) 2 ( 6)) 10…”

The planar pure braid group is a diagram group

“…. , x n q, such that x i " x j " x k , for some 1 ď i ă j ă k ď n. The fundamental group Γ n of this space has been studied under several different names (for a short guide, see the introduction to [16]), and in mathematical contexts ranging from low-dimensional topology [14] to topological robotics [7]. We will follow [16] in calling Γ n the n-strand planar pure braid group.…”

Topological Exchange Statistics in One Dimension

“…We call these non-abelian groups the traid group T N [28] (aka doodle group [57], planar braid group [58], or twin group [59]), the fraid group F N , and the free Coxeter group W N (aka universal Coxeter group [60]). Like the braid group B N , the strand groups T N , F N and W N can be understood as resulting from eliminating generator relations of the symmetric group [28].…”