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%I M0851 N0323 #181 Mar 25 2024 12:05:54
%S 0,0,1,1,2,3,7,21,49,165,552,2176,9988,46972,253293,1388705,8053393,
%T 48266466,294130458
%N Number of prime knots with n crossings.
%C Prime knot: a nontrivial knot which cannot (as a composite knot can) be written as the knot sum of two nontrivial knots. - _Jonathan Vos Post_, Apr 30 2011
%D For convenience, many references and links related to the enumeration of knots are collected here, even if they do not explicitly refer to this sequence.
%D C. C. Adams, The Knot Book, Freeman, NY, 2001; see p. 33.
%D C. Cerf, Atlas of oriented knots and links, Topology Atlas 3 no. 2 (1998).
%D Peter R. Cromwell, Knots and Links, Cambridge University Press, 2004, pp. 209-211.
%D Martin Gardner, The Last Recreations, Copernicus, 1997, 67-84.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D P. G. Tait, Scientific Papers, Cambridge Univ. Press, Vol. 1, 1898, Vol. 2, 1900, see Vol. 1, p. 345.
%D M. B. Thistlethwaite, personal communication.
%H For convenience, many references and links related to the enumeration of knots are collected here, even if they do not explicitly refer to this sequence.
%H D. Bar-Natan, <a href="http://katlas.org/wiki/The_Hoste-Thistlethwaite_Table_of_11_Crossing_Knots">The Hoste-Thistlethwaite Table of 11 Crossing Knots</a>
%H D. J. Broadhurst and D. Kreimer, <a href="https://arxiv.org/abs/hep-th/9609128">Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops</a>, arXiv:hep-th/9609128, 1996; Phys. Lett. B 393, No.3-4, 403-412 (1997).
%H B. Burton, <a href="http://dx.doi.org/10.4230/LIPIcs.SoCG.2020.25">The next 350 million knots</a>, 36th International Symposium on Computational Geometry (SoCG 2020) (S. Cabello, D.Z. Chen, eds.), Leibniz Int. Proc. Inform., vol. 164, Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2020, pp. 25:1-25:17.
%H Alain Caudron, <a href="/A002863/a002863_3.pdf">Classification des noeuds et des enlacements (Thèse et additifs)</a>, Univ. Paris-Sud, 1989 [Scanned copy, included with permission. But also see the Perko links below.]
%H J. H. Conway, <a href="http://www.maths.ed.ac.uk/~aar/papers/conway.pdf">An enumeration of knots and links and some of their algebraic properties</a>, 1970. Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967) pp. 329-358 Pergamon, Oxford.
%H S. R. Finch, <a href="/A002863/a002863_4.pdf">Knots, links and tangles</a>, Aug 08 2003. [Cached copy, with permission of the author]
%H Ortho Flint, Bruce Fontaine and Stuart Rankin, <a href="http://www-home.math.uwo.ca/~srankin/papers/knots/altlnk.submit.pdf">Enumerating the prime alternating links</a>, preprint, 2007.
%H Ortho Flint and Stuart Rankin, <a href="http://dx.doi.org/10.1142/S0218216504003068">Enumerating the prime alternating links</a>, Journal of Knot theory and its Ramifications, 13 (2004), 151-173.
%H C. Giller, <a href="http://dx.doi.org/10.1090/S0002-9947-1982-0642331-X">A family of links and the Conway calculus</a>, Trans. American Math Soc., 270 (1982) 75-109.
%H Jeremy Green, <a href="http://www.math.toronto.edu/drorbn/Students/GreenJ/">A Table of Virtual Knots</a>, 2004.
%H Hermann Gruber, <a href="http://www2.tcs.ifi.lmu.de/~gruberh/">Atlas of Rational Knots</a>. [dead link]
%H J. Hoste, M. B. Thistlethwaite and J. Weeks, <a href="http://dx.doi.org/10.1007/BF03025227">The First 1,701,936 Knots</a>, Math. Intell., 20, 33-48, Fall 1998.
%H Jim Hoste, <a href="https://pzacad.pitzer.edu/~jhoste/HosteWebPages/downloads/Enumeration.pdf">The Enumeration and Classification of Knots and Links</a>, in Handbook of Knot Theory, William W. Menasco and Morwen B. Thistlethwaite, Editors, Elsevier, 2015.
%H S. Jablan, L. H. Kauffman, and P. Lopes, <a href="https://doi.org/10.1016/j.topol.2015.07.010">The delunification process and minimal diagrams</a>, Topology Appl., 193 (2015), 270-289, #5531; see <a href="http://arxiv.org/abs/1406.2378">also</a>, arXiv:1406.2378 [math.GT], 2014.
%H Knot Atlas, <a href="http://katlas.org/wiki/Main_Page">The Knot Atlas</a>. Includes: The Rolfsen Table of knots with up to 10 crossings, The Hoste-Thistlethwaite Table of 11 Crossing Knots, The Thistlethwaite Link Table, The 36 Torus Knots with up to 36 Crossings, and The Mathematica Package KnotTheory.
%H Knotilus web site, <a href="http://knotilus.math.uwo.ca">Knotilus</a> [dead link]
%H W. B. R. Lickorish and K. C. Millett, <a href="http://www.jstor.org/stable/2690324">The new polynomial invariants of knots and links</a>, Math. Mag. 61 (1988), no. 1, 3-23.
%H C. Livingston and A. H. Moore, <a href="https://knotinfo.math.indiana.edu">KnotInfo: Table of Knot Invariants</a>.
%H Andrei Malyutin, <a href="http://arxiv.org/abs/1612.03368">On the question of genericity of hyperbolic knots</a>, arXiv preprint arXiv:1612.03368 [math.GT], 2016.
%H K. A. Perko, Jr., <a href="/A002863/a002863_6.pdf">Abstract for Talk, 1973</a>
%H K. A. Perko, Jr., <a href="/A002863/a002863_8.pdf">On covering spaces of knots</a>, Glasnik Mathematicki, Tom 9 (29) No. 1 (1974), 141-145. (Annotated scanned copy)
%H K. A. Perko, Jr., <a href="/A002863/a002863_7.pdf">On the classification of knots</a>, Proc. Amer. Math. Soc., 45 (1974), 262-266. (Annotated scanned copy)
%H K. A. Perko, Jr., <a href="/A002863/a002863_5.pdf">Letters to N. J. A. Sloane 1974-1977</a>
%H K. A. Perko, Jr., <a href="https://topology.nipissingu.ca/tp/reprints/v07/tp07110.pdf">Primality of certain knots</a>, In Topology Proceedings, vol. 7, no. 1, pp. 109-118. Auburn University Mathematics Department and the Institute for Medicine and Mathematics at Ohio University, 1982.
%H K. A. Perko, Jr., <a href="/A002863/a002863_9.pdf">On ninth order knottiness</a>, Preprint (N. D.)
%H K. A. Perko, Jr., <a href="/A002863/a002863_1.pdf">Caudron's 1979 Knot Table</a>, 2015 [Included with permission. See next link for list of errors.]
%H K. A. Perko, Jr., <a href="/A002863/a002863.txt">Errors in Alain Caudron's 1989 thesis</a>
%H K. A. Perko, Jr., <a href="/A002863/a002863_2.pdf">Review of Jablan-Kauffman-Lopes (2015)</a>
%H Stuart Rankin, <a href="http://www.math.uwo.ca/~srankin/knotprint.html">Knot Theory Preprints of Ortho Flint Smith and Stuart Rankin</a>
%H S. Rankin and O. Flint <a href="http://www.math.uwo.ca/~srankin/knots/knotprint.html">Knot theory</a> web page.
%H Stuart Rankin and Ortho Flint Smith, <a href="http://arxiv.org/abs/math/0211451">Enumerating the Prime Alternating Links</a>, arXiv:math/0211451 [math.GT], 2002
%H Stuart Rankin, Ortho Flint Smith and John Schermann, <a href="http://arxiv.org/abs/math/0211346">Enumerating the Prime Alternating Knots, Part I</a>, arXiv:math/0211346 [math.GT], 2002.
%H Stuart Rankin, Ortho Flint Smith and John Schermann, <a href="http://arxiv.org/abs/math/0211348">Enumerating the Prime Alternating Knots, Part II</a>, arXiv:math/0211348 [math.GT], 2002.
%H Stuart Rankin, Ortho Flint Smith and John Schermann, <a href="http://dx.doi.org/10.1142/S0218216504003044">Enumerating the Prime Alternating Knots, Part I</a>, Journal of Knot Theory and its Ramifications, 13 (2004), 57-100.
%H Stuart Rankin, Ortho Flint Smith and John Schermann, <a href="http://dx.doi.org/10.1142/S0218216504003056">Enumerating the Prime Alternating Knots, Part II</a>, Journal of Knot Theory and its Ramifications, 13 (2004), 101-149.
%H R. G. Scharein, <a href="https://knotplot.com/manual/PrimeLinks.html">Number of Prime Links</a>
%H Silvia Sconza and Arno Wildi, <a href="https://eprint.iacr.org/2024/471.pdf">Knot-based Key Exchange protocol</a>, Cryptology ePrint Archive (2024), Art. No. 2024/471. See Table 2, p. 15.
%H N. J. A. Sloane, <a href="/A002863/a002863.gif">Illustration of initial terms</a>
%H P. G. Tait, <a href="/A002863/a002863_10.pdf">The first seven orders of knottiness</a> [Annotated scan of Plate VI]
%H M. B. Thistlethwaite, <a href="http://www.math.utk.edu/~morwen/index.html">Home Page</a>
%H M. B. Thistlethwaite, <a href="http://www.math.utk.edu/~morwen/png/link_stats.png">Numbers of alternating knots and links with up to 19 crossings</a>
%H M. B. Thistlethwaite, <a href="http://dx.doi.org/10.1017/CBO9781107359925.003">Knot tabulations and related topics</a>, Aspects of topology, 1-76, London Math. Soc. Lecture Note Ser., 93, Cambridge Univ. Press, Cambridge-New York, 1985.
%H S. D. Tyurina, <a href="https://doi.org/10.1007/s10958-006-0095-9">Diagram invariants of knots and the Kontsevich integral</a>, J. Math. Sci. 134 (2) (2006) pp. 2017-2071.
%H University of Western Ontario Student Beowulf Initiative, <a href="http://baldric.uwo.ca/article.php3?section=baldric&article=knots">Project: Prime Knots</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Knot.html">Knot.</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeKnot.html">Prime Knot.</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AlternatingKnot.html">Alternating Knot.</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeLink.html">Prime Link</a>
%H R. G. Wilson, V, <a href="/A007376/a007376.pdf">Letter to N. J. A. Sloane, Oct. 1993</a>
%H <a href="/index/K#knots">Index entries for sequences related to knots</a>
%F a(n) = A051766(n) + A051769(n) + A051767(n) + A051768(n) + A052400(n). - _Andrew Howroyd_, Oct 15 2020
%Y Cf. A002864, A086825.
%Y Cf. A051766, A051767, A051768, A051769, A052400.
%K nonn,hard,more,nice
%O 1,5
%A _N. J. A. Sloane_
%E This is stated incorrectly in CRC Standard Mathematical Tables and Formulae, 30th ed., first printing, 1996, p. 320.
%E Terms from Hoste et al. added by _Eric W. Weisstein_
%E Consolidated references and links on enumeration of knots into this entry, also created entry for knots in Index to OEIS. - _N. J. A. Sloane_, Aug 25 2015
%E a(17)-a(19) computed by Benjamin Burton, added by _Alex Klotz_, Jun 21 2021
%E a(17)-a(19) computed by Benjamin Burton corrected by _Andrey Zabolotskiy_, Nov 25 2021
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