There is a well-known analogy between 3-manifolds and number fields, with knots corresponding to prime ideals. Are there any results in number theory that have been proven using topology through this analogy? I would also be interested in any result in number theory that had been motivated by an analogous result in topology.

Edit: In light of Dan Petersen's comment, I realize that this is to some extent a historical question. After all, if knot theory had been very developed in 1700, it is possible that quadratic reciprocity would be one example. I have not read all of the Morishita book, but it seems to be mainly concerned with proving topological analogues to some famous theorems in number theory.

provenusing the prime-knot dictionary. It really is just an analogy; there is no precise correspondence that allows you to infer statements on one side from statements on the other side. $\endgroup$ – Dan Petersen Jul 24 '16 at 12:23