Menu
Home
People
Places
Arts
History
Plants & Animals
Science
Life & Culture
Technology
Reference.org
Arithmetic topology
open-in-new
<h2 id="analogies">Analogies</h2> <p>The following are some of the analogies used by mathematicians between number fields and 3-manifolds:<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> </p> <ol><li>A number field corresponds to a closed, orientable 3-manifold</li> <li><a href="/facts/Ideal_(ring_theory)/vCvytduX">Ideals</a> in the ring of integers correspond to <a href="/facts/Link_(knot_theory)/hoFDYmVj">links</a>, and <a href="/facts/Prime_ideal/pmrGf6DA">prime ideals</a> correspond to knots.</li> <li>The field Q of <a href="/facts/Rational_number/VJQmyY05">rational numbers</a> corresponds to the <a href="/facts/3-sphere/u67nImpY">3-sphere</a>.</li></ol> <p>Expanding on the last two examples, there is an analogy between <a href="/facts/Knot_(mathematics)/UMtKJ36m">knots</a> and <a href="/facts/Prime_number/L20FLkgn">prime numbers</a> 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 <a href="/facts/Legendre_symbol/MwHrJ7as">Legendre symbols</a> are all 1). Therefore these primes have been called a "proper Borromean triple modulo 2"<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> or "mod 2 Borromean primes".<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p> <h2 id="history">History</h2> <p>In the 1960s topological interpretations of <a href="/facts/Class_field_theory/4cvKbn6a">class field theory</a> were given by <a href="/facts/John_Tate_(mathematician)/nBntLCG0">John Tate</a><a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> based on <a href="/facts/Galois_cohomology/nzC1EVYT">Galois cohomology</a>, and also by <a href="/facts/Michael_Artin/Tfk0f6rK">Michael Artin</a> and <a href="/facts/Jean-Louis_Verdier/nR3cH3Lj">Jean-Louis Verdier</a><a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> based on <a href="/facts/%25C3%2589tale_cohomology/KXJ8a6Hw">Étale cohomology</a>. Then <a href="/facts/David_Mumford/WJoJ8K2u">David Mumford</a> (and independently <a href="/facts/Yuri_Manin/gDlvYGCq">Yuri Manin</a>) came up with an analogy between <a href="/facts/Prime_ideals/pmrGf6DA">prime ideals</a> and <a href="/facts/Knot_(mathematics)/UMtKJ36m">knots</a><a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> which was further explored by <a href="/facts/Barry_Mazur/4tGbAnCX">Barry Mazur</a>.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a><a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> In the 1990s Reznikov<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> and Kapranov<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> began studying these analogies, coining the term arithmetic topology for this area of study. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Arithmetic_geometry/Bs1mfNu4">Arithmetic geometry</a></li> <li><a href="/facts/Arithmetic_dynamics/11Ir59ko">Arithmetic dynamics</a></li> <li><a href="/facts/Topological_quantum_field_theory/zHjrp3b8">Topological quantum field theory</a></li> <li><a href="/facts/Langlands_program/koK9PhID">Langlands program</a></li></ul> <h2 id="notes">Notes</h2> <h2 id="further-reading">Further reading</h2> <ul><li>Masanori Morishita (2011), <a href="https://www.springer.com/mathematics/numbers/book/978-1-4471-2157-2">Knots and Primes</a>, Springer, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4471-2157-2</li> <li>Masanori Morishita (2009), <a href="https://arxiv.org/abs/0904.3399v1">Analogies Between Knots And Primes, 3-Manifolds And Number Rings</a></li> <li>Christopher Deninger (2002), <a href="https://arxiv.org/abs/math/0204274v1">A note on arithmetic topology and dynamical systems</a></li> <li>Adam S. Sikora (2001), <a href="https://arxiv.org/abs/math/0107210v2">Analogies between group actions on 3-manifolds and number fields</a></li> <li><a href="/facts/Curtis_T._McMullen/WKEXxuc1">Curtis T. McMullen</a> (2003), <a href="http://www.math.harvard.edu/~ctm/home/text/papers/fermat/fermat.pdf">From dynamics on surfaces to rational points on curves</a></li> <li>Chao Li and Charmaine Sia (2012), <a href="https://www.math.columbia.edu/~chaoli/tutorial2012/knots-and-primes.pdf">Knots and Primes</a></li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="http://www.neverendingbooks.org/mazurs-dictionary">Mazur’s knotty dictionary</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Sikora, Adam S. "Analogies between group actions on 3-manifolds and number fields." Commentarii Mathematici Helvetici 78.4 (2003): 832-844. <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>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 <a href="http://www.ub.uni-heidelberg.de/archiv/4418" target="_blank">http://www.ub.uni-heidelberg.de/archiv/4418</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Morishita, Masanori (April 22, 2009), Analogies between Knots and Primes, 3-Manifolds and Number Rings, arXiv:0904.3399, Bibcode:2009arXiv0904.3399M <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>J. Tate, Duality theorems in Galois cohomology over number fields, (Proc. Intern. Cong. Stockholm, 1962, p. 288-295). <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>M. Artin and J.-L. Verdier, Seminar on étale cohomology of number fields, Woods Hole Archived May 26, 2011, at the Wayback Machine, 1964. <a href="http://www.jmilne.org/math/Documents/WoodsHole3.pdf" target="_blank">http://www.jmilne.org/math/Documents/WoodsHole3.pdf</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Who dreamed up the primes=knots analogy? Archived July 18, 2011, at the Wayback Machine, neverendingbooks, lieven le bruyn's blog, May 16, 2011, <a href="http://www.neverendingbooks.org/who-dreamed-up-the-primesknots-analogy" target="_blank">http://www.neverendingbooks.org/who-dreamed-up-the-primesknots-analogy</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Remarks on the Alexander Polynomial, Barry Mazur, c.1964 <a href="http://www.math.harvard.edu/~mazur/papers/alexander_polynomial.pdf" target="_blank">http://www.math.harvard.edu/~mazur/papers/alexander_polynomial.pdf</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>B. Mazur, Notes on ´etale cohomology of number fields, Ann. scient. ´Ec. Norm. Sup. 6 (1973), 521-552. <a href="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" target="_blank">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</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>A. Reznikov, Three-manifolds class field theory (Homology of coverings for a nonvirtually b1-positive manifold), Sel. math. New ser. 3, (1997), 361–399. <a href="https://doi.org/10.1007%2Fs000290050015" target="_blank">https://doi.org/10.1007%2Fs000290050015</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>M. Kapranov, Analogies between the Langlands correspondence and topological quantum field theory, Progress in Math., 131, Birkhäuser, (1995), 119–151. <a href="https://books.google.com/books?hl=en&lr=&id=TOPa9irmsGsC&oi=fnd&pg=PA119" target="_blank">https://books.google.com/books?hl=en&lr=&id=TOPa9irmsGsC&oi=fnd&pg=PA119</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> </ol>