Summary Analogies Edit
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 Edit See also Edit Notes Edit
^ 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.
Further reading Edit
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 External links Edit
Mazur’s knotty dictionary