Arithmetic topology

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Arithmetic topology is a branch of mathematics that blends algebraic number theory with topology. It builds a bridge between number fields and closed, orientable 3-dimensional spaces (3-manifolds). In this view, primes act like knots and the relationships between primes act like links between knots.

One idea is that groups of primes can be “linked” in a way similar to the Borromean rings. For example, the primes 13, 61, and 937 are linked modulo 2 (a certain invariant called the Rédei symbol is -1) but each pair is unlinked modulo 2 (the Legendre symbols are all 1). Such a triple is called a proper Borromean triple modulo 2, or mod 2 Borromean primes.

In the 1960s, mathematicians gave topological interpretations of class field theory using cohomology theories (Galois cohomology and étale cohomology). Later, Mumford (and independently Yuri Manin) proposed a direct analogy between prime ideals and knots, which was developed further by Barry Mazur.

In the 1990s, Reznikov and Kapranov explored these connections more deeply and coined the term arithmetic topology for this area.