  A powerful way to study groups is via their actions on suitable spaces.
Classifying spaces for families of subgroups are a type of these spaces,
obtained by imposing some strict conditions on the fixed-point sets. We show
how in the framework of knot theory the notion of classifying spaces for
families pops out naturally and we provide explicit models for them. The study
of their topological properties hints at the existence of arcane connections
between knot theory and algebraic number theory. Trying to track these
connections, we discovered a behavioural analogy between the fundamental group
of our classifying spaces and the ideal class group of an algebraic number
field, a 9-term exact sequence for knots mimicking Poitou-Tate exact sequence
for algebraic number fields and, in some cases, a second analogy between the
first (co)homology group of the classifying spaces and the first Shafarevich
group of a number field extension.
