Pentagon equations, Delaunay triangulations and pure braid group invariant

TL;DR

This paper introduces a novel matrix representation of point motions on the plane using Delaunay triangulations and constructs a homomorphism from the pure braid group to a general linear group, revealing new algebraic structures.

Contribution

It presents a new matrix construction linked to Delaunay triangulations and defines a homomorphism from the pure braid group to a linear group, bridging geometry and algebra.

Findings

Matrix representation of point motions via Delaunay triangulations

Homomorphism from pure braid group to GL_{2n+1}(Q)

New algebraic insights into braid group invariants

Abstract

We construct $(2n+1)\times (2n+1)$ matrices corresponding to a motion of points on the plane from the point of view of Delaunay triangulations. We define a homomorphism from the pure braid group on ($n+3$) strands to the general linear group $\text{GL}_{2n+1}(\mathbb{Q})$.