The aim of Geometric Group Theory is to get insights on the structure of groups looking at their actions on suitable topological spaces. A significant type of these spaces are classifying spaces for families of subgroups. Classifying spaces for families have been widely studied in the case of the families of finite subgroups and virtually cyclic subgroups, due to their connections with the celebrated Baum-Connes Conjecture and Farrell-Jones Conjecture, respectively. But the definitions are stated for all families of subgroups. The purpose of this seminar is to convince that even off the main road of the two standard families of finite and virtually cyclic subgroups there is fascinating mathematics, that still waits to be explored. In particular, this talk is intended to be an accessible invitation to the investigation of "exotic" classifying spaces in the realm of knot theory. We show how to build a classifying space for a significant family of subgroups of a prime knot group and we investigate its cohomological properties. While playing a little with it, interesting analogies with algebraic number theory will pop out.