Planar pure braids on six strands

Abstract

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.

The planar pure braid group is a diagram group

“…The latter fact implies that Γ n is biautomatic [17]. We note also that presentations of Γ n [15,16] and integral homology groups of Γ n [1] have been computed. Thus, after comparing the above lists of corollaries, it appears that the proof that Γ n is bi-orderable is our main new result.…”

“…We will use the description of planar braids that appears in [16]. Fix a natural number n. A planar braid is a collection of n smooth descending arcs in R 2 connecting n distinct points on one horizontal line with n distinct points directly beneath them on another horizontal line.…”

“…. , 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.…”

“…Several of the above corollaries also follow from the literature. The introduction of [16] clearly shows that, for each n, Γ n is a finite-index subgroup of a right-angled Coxeter group. Coxeter groups are well-known to be linear, so the linearity of Γ n follows directly.…”

Linear motion planning with controlled collisions and pure planar braids

“…A direct computation (verifiable using the computational algebraic system GAP) using the Reidemeister-Schreier process reveals a group isomorphism PP 6 ∼ = H 6 * F , where F is a free group of rank at least 45. Details of such a fact, as well as potential extensions for groups PP n with n ≥ 6, are the topic of the forthcoming paper [20]. Here we remark that, in any decomposition PP n ∼ = H n * F with F free, the cat and TC values of H n are forced to agree with those of PP n .…”