Invariants of the Colored Braid Groupoid

Abstract

In this paper, a braid is regarded as a dynamical system of points in the plane. The states of this dynamical system are given by Delaunay triangulations. This construction makes it possible to define an abstract groupoid $\stackrel{abc}{{\mathcal{G}}_{n+3}^4}$, which gives a representation of the colored braid groupoid $\text{ColB}(n)$. We define homomorphisms $f_{n+3}:\stackrel{abc}{{\mathcal{G}}_{n+3}^4}\to {\text{GL}}_{2n+1}(\mathbb{Q})$ and $f_{n+3}^{\prime }:\stackrel{abc}{{\mathcal{G}}_{n+3}^4}\to {\text{GL}}_{2n+1}(\mathbb{C})$, and describe an algorithm for computing the resulting invariants.

30 pages, 13 figures