  We formalize the arithmetic topology, i.e. a relationship between knots and
primes. Namely, using the notion of a cluster C*-algebra we construct a functor
from the category of 3-dimensional manifolds M to a category of algebraic
number fields K, such that the prime ideals (ideals, resp.) in the ring of
integers of K correspond to knots (links, resp.) in M. It is proved that the
functor realizes all axioms of the arithmetic topology conjectured in the
1960's by Manin, Mazur and Mumford.
