Artin's braids, Braids for three space, and groups ${\Gamma }_n^4$ and $G_n^k$

Abstract

We construct a group ${\Gamma }_n^4$ corresponding to the motion of points in ${\mathbb{R}}^3$ from the point of view of Delaunay triangulations. We study homomorphisms from pure braids on $n$ strands to the product of copies of ${\Gamma }_n^4$. We will also study the group of pure braids in ${\mathbb{R}}^3$, which is described by a fundamental group of the restricted configuration space of ${\mathbb{R}}^3$, and define the group homomorphism from the group of pure braids in ${\mathbb{R}}^3$ to ${\Gamma }_n^4$. In the end of this paper we give some comments about relations between the restricted configuration space of ${\mathbb{R}}^3$ and triangulations of the 3-dimensional ball and Pachner moves.