It establishes an analogy between number fields and closed, orientable 3-manifolds.
The following are some of the analogies used by mathematicians between number fields and 3-manifolds:[1] Expanding on the last two examples, there is an analogy between knots and prime numbers in which one considers "links" between primes.
The triple of primes (13, 61, 937) are "linked" modulo 2 (the Rédei symbol is −1) but are "pairwise unlinked" modulo 2 (the Legendre symbols are all 1).
[3] In the 1960s topological interpretations of class field theory were given by John Tate[4] based on Galois cohomology, and also by Michael Artin and Jean-Louis Verdier[5] based on Étale cohomology.
Then David Mumford (and independently Yuri Manin) came up with an analogy between prime ideals and knots[6] which was further explored by Barry Mazur.