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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:^[1]

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"^[2] or "mod 2 Borromean primes".^[3]

History

In the 1960s topological interpretations of class field theory were given by John Tate^[4] based on Galois cohomology, and also by Michael Artin and Jean-Louis Verdier^[5] based on Étale cohomology. Then David Mumford (and independently Yuri Manin) came up with an analogy between prime ideals and knots^[6] which was further explored by Barry Mazur.^[7]^[8] In the 1990s Reznikov^[9] and Kapranov^[10] began studying these analogies, coining the term arithmetic topology for this area of study.

Notes

^ Sikora, Adam S. "Analogies between group actions on 3-manifolds and number fields." Commentarii Mathematici Helvetici 78.4 (2003): 832-844.
^ Vogel, Denis (February 13, 2004), Massey products in the Galois cohomology of number fields, doi:10.11588/heidok.00004418, urn:nbn:de:bsz:16-opus-44188
^ Morishita, Masanori (April 22, 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.

External links

Mazur’s knotty dictionary

[1] Sikora, Adam S. "Analogies between group actions on 3-manifolds and number fields." Commentarii Mathematici Helvetici 78.4 (2003): 832-844.

[2] Vogel, Denis (February 13, 2004), Massey products in the Galois cohomology of number fields, doi:10.11588/heidok.00004418, urn:nbn:de:bsz:16-opus-44188

[3] Morishita, Masanori (April 22, 2009), Analogies between Knots and Primes, 3-Manifolds and Number Rings, arXiv:0904.3399, Bibcode:2009arXiv0904.3399M

[4] J. Tate, Duality theorems in Galois cohomology over number fields, (Proc. Intern. Cong. Stockholm, 1962, p. 288-295).

[5] M. Artin and J.-L. Verdier, Seminar on étale cohomology of number fields, Woods Hole Archived May 26, 2011, at the Wayback Machine, 1964.

[6] Who dreamed up the primes=knots analogy? Archived July 18, 2011, at the Wayback Machine, neverendingbooks, lieven le bruyn's blog, May 16, 2011,

[7] Remarks on the Alexander Polynomial, Barry Mazur, c.1964

[8] B. Mazur, Notes on ´etale cohomology of number fields, Ann. scient. ´Ec. Norm. Sup. 6 (1973), 521-552.

[9] A. Reznikov, Three-manifolds class field theory (Homology of coverings for a nonvirtually b1-positive manifold), Sel. math. New ser. 3, (1997), 361–399.

[10] M. Kapranov, Analogies between the Langlands correspondence and topological quantum field theory, Progress in Math., 131, Birkhäuser, (1995), 119–151.

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+{{Use mdy dates|date=March 2023}}
-'''Arithmetic topology''' is an area of [[mathematics]] that is a combination of [[algebraic number theory]] and [[topology]]. It developed from a series of analogies between [[number field]]s and [[3-manifolds]]; [[primes]] and [[Knot (mathematics)|knots]] pointed out by [[Barry Mazur]] and by [[Yuri Manin]] in the 1960s. In the 1990s Reznikov<ref>A. Reznikov, Three-manifolds class field theory (Homology of coverings for a nonvirtually b1-positive manifold), Sel. math. New ser. 3, (1997), 361&ndash;399.</ref> and Kapranov<ref>M. Kapranov, Analogies between the Langlands correspondence and topological quantum field theory, Progress in Math., 131, Birkhäuser, (1995), 119–151.</ref> began studying these analogies and coined the term '''arithmetic topology'''.
+'''Arithmetic topology''' is an area of [[mathematics]] that is a combination of [[algebraic number theory]] and [[topology]]. It establishes an analogy between [[number field]]s and closed, orientable [[3-manifold]]s.
+==Analogies==
+The following are some of the analogies used by mathematicians between number fields and 3-manifolds:<ref>Sikora, Adam S. "Analogies between group actions on 3-manifolds and number fields." Commentarii Mathematici Helvetici 78.4 (2003): 832-844.</ref>
+#A number field corresponds to a closed, orientable 3-manifold
+#[[Ideal (ring theory)|Ideals]] in the ring of integers correspond to [[link (knot theory)|links]], and [[prime ideal]]s correspond to knots.
+#The field '''Q''' of [[rational number]]s corresponds to the [[3-sphere]].
+Expanding on the last two examples,  there is an analogy between [[knot (mathematics)|knots]] and [[prime number]]s in which one considers "links" between primes. The triple of primes {{nowrap|(13, 61, 937)}} are "linked" modulo 2 (the [[Rédei symbol]] is −1) but are "pairwise unlinked" modulo 2 (the [[Legendre symbol]]s are all 1). Therefore these primes have been called a "proper Borromean triple modulo 2"<ref>{{Citation |last=Vogel |first=Denis |date=February 13, 2004 |title=Massey products in the Galois cohomology of number fields |doi=10.11588/heidok.00004418 |url=http://www.ub.uni-heidelberg.de/archiv/4418 |id={{URN|nbn|de:bsz:16-opus-44188}}}}</ref> or "mod 2 Borromean primes".<ref>{{Citation |last=Morishita |first=Masanori |date=April 22, 2009 |title=Analogies between Knots and Primes, 3-Manifolds and Number Rings |arxiv=0904.3399|bibcode=2009arXiv0904.3399M }}</ref>
+==History==
+In the 1960s topological interpretations of [[class field theory]] were given by [[John Tate (mathematician)|John Tate]]<ref>J. Tate,  Duality theorems in Galois cohomology over number fields, (Proc. Intern. Cong. Stockholm, 1962, p. 288-295).</ref> based on [[Galois cohomology]], and also by [[Michael Artin]] and [[Jean-Louis Verdier]]<ref>M. Artin and J.-L. Verdier,  [http://www.jmilne.org/math/Documents/WoodsHole3.pdf Seminar on étale cohomology of number fields, Woods Hole] {{webarchive |url=https://web.archive.org/web/20110526230017/http://www.jmilne.org/math/Documents/WoodsHole3.pdf |date=May 26, 2011 }}, 1964.</ref> based on [[Étale cohomology]]. Then [[David Mumford]] (and independently [[Yuri Manin]]) came up with an analogy between [[prime ideals]] and [[Knot (mathematics)|knots]]<ref>[http://www.neverendingbooks.org/who-dreamed-up-the-primesknots-analogy Who dreamed up the primes=knots analogy?] {{webarchive |url=https://web.archive.org/web/20110718061649/http://www.neverendingbooks.org/index.php/who-dreamed-up-the-primesknots-analogy.html |date=July 18, 2011 }}, neverendingbooks, lieven le bruyn's blog, May 16, 2011,</ref> which was further explored by [[Barry Mazur]].<ref>[http://www.math.harvard.edu/~mazur/papers/alexander_polynomial.pdf Remarks on the Alexander Polynomial], Barry Mazur, c.1964</ref><ref>B. Mazur, [http://archive.numdam.org/ARCHIVE/ASENS/ASENS_1973_4_6_4/ASENS_1973_4_6_4_521_0/ASENS_1973_4_6_4_521_0.pdf Notes on ´etale cohomology of number fields], Ann. scient. ´Ec. Norm. Sup. 6 (1973), 521-552.</ref> In the 1990s Reznikov<ref>A. Reznikov, [https://doi.org/10.1007%2Fs000290050015 Three-manifolds class field theory (Homology of coverings for a nonvirtually b1-positive manifold)], Sel. math. New ser. 3, (1997), 361–399.</ref> and Kapranov<ref>M. Kapranov, [https://books.google.com/books?hl=en&lr=&id=TOPa9irmsGsC&oi=fnd&pg=PA119 Analogies between the Langlands correspondence and topological quantum field theory], Progress in Math., 131, Birkhäuser, (1995), 119–151.</ref> began studying these analogies, coining the term '''arithmetic topology''' for this area of study.
 ==See also==
 *[[Arithmetic geometry]]
 *[[Arithmetic dynamics]]
+*[[Topological quantum field theory]]
+*[[Langlands program]]
 ==Notes==
+<references/>
-{{reflist}}
 ==Further reading==
-*Masanori Morishita (2009), [http://arxiv.org/abs/0904.3399v1 Analogies Between Knots And Primes, 3-Manifolds And Number Rings]
+*Masanori Morishita (2011), [https://www.springer.com/mathematics/numbers/book/978-1-4471-2157-2 Knots and Primes], Springer, {{ISBN|978-1-4471-2157-2}}
-*Christopher Deninger (2002), [http://arxiv.org/abs/math/0204274v1 A note on arithmetic topology and dynamical systems]
-*Adam S. Sikora (2001), [http://arxiv.org/abs/math/0107210v2 Analogies between group actions on 3-manifolds and number fields]
+*Masanori Morishita (2009), [https://arxiv.org/abs/0904.3399v1 Analogies Between Knots And Primes, 3-Manifolds And Number Rings]
+*Christopher Deninger (2002), [https://arxiv.org/abs/math/0204274v1 A note on arithmetic topology and dynamical systems]
-*Curtis T. McMullen (2003), [http://www.math.harvard.edu/~ctm/home/text/papers/fermat/fermat.pdf From dynamics on surfaces to rational points on curves]
+*Adam S. Sikora (2001), [https://arxiv.org/abs/math/0107210v2 Analogies between group actions on 3-manifolds and number fields]
+*[[Curtis T. McMullen]] (2003), [http://www.math.harvard.edu/~ctm/home/text/papers/fermat/fermat.pdf From dynamics on surfaces to rational points on curves]
+*Chao Li and Charmaine Sia (2012), [https://www.math.columbia.edu/~chaoli/tutorial2012/knots-and-primes.pdf Knots and Primes]
 ==External links==
-*[http://www.neverendingbooks.org/index.php/mazurs-dictionary.html Mazur’s knotty dictionary]
+*[http://www.neverendingbooks.org/mazurs-dictionary Mazur’s knotty dictionary]
 {{Number theory-footer}}
+{{Areas of mathematics}}
+[[Category:Algebraic number theory]]
-{{numtheory-stub}}
-{{topology-stub}}
-[[Category:Number theory]]
 [[Category:3-manifolds]]
 [[Category:Knot theory]]

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