We investigate the Galois group G_S(k)(p) of the maximal p-extension unramified outside a finite set S of primes of a number field in the case when not all primes dividing p are in S. We show that the cohomology of G_S(k)(p) is `often' isomorphic to the etale cohomology of the scheme Spec(O_k)-S; in particular, G_S(k)(p) is of cohomological dimension 2 then. In the case k = Q this can be made explicit by using the analogy between knots and primes. Defining linking numbers between primes, one constructs the linking diagram attached to a set S of prime numbers. This is a directed graph from which one can read off properties of G_S(Q)(p). In the proofs we use Labute's results on mild pro-p groups in an essential way.