Concept# Arithmetic topology

Summary

Arithmetic topology is an area of mathematics that is a combination of algebraic number theory and topology. It establishes an analogy between number fields and closed, orientable 3-manifolds.
Analogies
The following are some of the analogies used by mathematicians between number fields and 3-manifolds:
#A number field corresponds to a closed, orientable 3-manifold
#Ideals in the ring of integers correspond to links, and prime ideals correspond to knots.
#The field Q of rational numbers corresponds to the 3-sphere.
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). Therefore these primes have been called a "proper Borromean triple modulo 2" or "mod 2 Borromean primes".
History
In the 1960s topological in

Official source

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