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.
Best Answer
Le and Murakami (HERE and HERE) discovered several previously unknown relations between multiple zeta values through the study of quantum invariants of knots. Further relations were later discovered through knot theory by Takamuki, and by Ihara and Takamuki.
These relations stem from the fact that the Kontsevich invariant extends to an invariant not of tangles, but of bracketed tangles, also known as q-tangles or non-associative tangles (alternatively, to knotted trivalent graphs, e.g. HERE). An "associator" relates different bracketings, and the relations satisfied by the associator induce the relations between the multiple zeta values, upon choosing a weight system.
Now that these relations have been discovered, the scaffolding can be removed and they can be derived directly from the defining relations of the associator as shown by Furusho (see there for further references). The associator is an algebraic object rather than a topological one, so now the number theorist "no longer need the knot theorists" as far as the known relations between multiple zeta values are concerned.