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".
In the 1960s topological interpretations of class field theory were given by John Tate based on Galois cohomology, and also by Michael Artin and Jean-Louis Verdier based on Étale cohomology. Then David Mumford (and independently Yuri Manin) came up with an analogy between prime ideals and knots which was further explored by Barry Mazur. In the 1990s Reznikov and Kapranov began studying these analogies, coining the term arithmetic topology for this area of study.
- Sikora, Adam S. "Analogies between group actions on 3-manifolds and number fields." Commentarii Mathematici Helvetici 78.4 (2003): 832-844.
- Vogel, Denis (13 February 2004), Massey products in the Galois cohomology of number fields, urn:nbn:de:bsz:16-opus-44188
- Morishita, Masanori (22 April 2009), Analogies between Knots and Primes, 3-Manifolds and Number Rings, arXiv:0904.3399, Bibcode:2009arXiv0904.3399M
- J. Tate, Duality theorems in Galois cohomology over number fields, (Proc. Intern. Cong. Stockholm, 1962, p. 288-295).
- M. Artin and J.-L. Verdier, Seminar on étale cohomology of number fields, Woods Hole Archived May 26, 2011, at the Wayback Machine., 1964.
- Who dreamed up the primes=knots analogy? Archived July 18, 2011, at the Wayback Machine., neverendingbooks, lieven le bruyn's blog, may 16, 2011,
- Remarks on the Alexander Polynomial, Barry Mazur, c.1964
- B. Mazur, Notes on ´etale cohomology of number fields, Ann. scient. ´Ec. Norm. Sup. 6 (1973), 521-552.
- A. Reznikov, Three-manifolds class field theory (Homology of coverings for a nonvirtually b1-positive manifold), Sel. math. New ser. 3, (1997), 361–399.
- M. Kapranov, Analogies between the Langlands correspondence and topological quantum field theory, Progress in Math., 131, Birkhäuser, (1995), 119–151.
- Masanori Morishita (2011), Knots and Primes, Springer, ISBN 978-1-4471-2157-2
- Masanori Morishita (2009), Analogies Between Knots And Primes, 3-Manifolds And Number Rings
- Christopher Deninger (2002), A note on arithmetic topology and dynamical systems
- Adam S. Sikora (2001), Analogies between group actions on 3-manifolds and number fields
- Curtis T. McMullen (2003), From dynamics on surfaces to rational points on curves
- Chao Li and Charmaine Sia (2012), Knots and Primes