Artin's braids, Braids for three space, and groups $\Gamma_{n}^{4}$ and $G_{n}^{k}$

TL;DR

This paper introduces a new group $\Gamma_{n}^{4}$ related to point motions in 3D space via Delaunay triangulations, explores homomorphisms from pure braids to this group, and discusses connections to 3D triangulations and Pachner moves.

Contribution

It constructs the group $\Gamma_{n}^{4}$, studies homomorphisms from pure braids to this group, and links configuration spaces with 3D triangulations and Pachner moves.

Findings

Defined the group $\Gamma_{n}^{4}$ for point motions in $\mathbb{R}^{3}$.

Established homomorphisms from pure braids to $\Gamma_{n}^{4}$.

Commented on relations between configuration spaces, triangulations, and Pachner moves.

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.