  For a prime knot group, the classifying space for the family of the subgroups
generated by the meridians can be seen as an abstract analogue of the ambient
manifold in which the knot lives. An explicit model of this ambient classifying
space is constructed as a branched covering space of the 3-sphere branched over
the knot; more general branched covering spaces, obtained quotienting the
ambient classifying space by finite-index normal subgroups, are studied.
Various homological properties of said spaces are established, some of which
have parallels in algebraic number theory. In particular, prime knot groups are
shown to be Bieri-Eckmann, but not Poincar\'e, duality groups for Bredon
cohomology with respect to the family of the meridians.
