Fox–Neuwirth cells, quantum shuffle algebras, and the homology of type-B Artin groups

In this paper, we will develop a family of braid representations of Artin groups of type B from braided vector spaces, and identify the homology of these groups with these coefficients with the cohomology of a specific bimodule over a quantum shuffle algebra. As an application, we give a complete characterization of the homology of type-B Artin groups with coefficients in one-dimensional braid representations over a field of characteristic 0. We will also discuss two different approaches to this computation: the first method extends a computation of the homology of braid groups due to Ellenberg–Tran–Westerland by means of induced representation, while the second method involves constructing a cellular stratification for configuration spaces of the punctured complex plane.