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%I M0847 N0322 #59 Jun 10 2022 07:57:24
%S 0,0,1,1,2,3,7,18,41,123,367,1288,4878,19536,85263,379799,1769979,
%T 8400285,40619385,199631989,990623857,4976016485,25182878921,
%U 128564665125
%N Number of alternating prime knots with n crossings.
%C Ortho Flint Smith and Stuart Rankin, with coding by Peter de Vries, calculated a(21) = 990623857 on a Compaq ES 45 in just under 14 hours on Jul 01 2003 (Canada Day).
%D See A002863 for many other references and links.
%D J. H. Conway, An enumeration of knots and links and some of their algebraic properties. 1970. Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967) pp. 329-358 Pergamon, Oxford.
%D J. Hoste, M. B. Thistlethwaite and J. Weeks, The First 1,701,936 Knots, Math. Intell., 20, 33-48, Fall 1998.
%D Stuart Rankin, Ortho Flint Smith and John Schermann, Enumerating the Prime Alternating Knots, Part I, Journal of Knot Theory and its Ramifications, 13 (2004), 57-100.
%D Stuart Rankin, Ortho Flint Smith and John Schermann, Enumerating the Prime Alternating Knots, Part II, Journal of Knot Theory and its Ramifications, 13 (2004), 101-149.
%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.
%D M. B. Thistlethwaite, Knot tabulations and related topics. Aspects of topology, 1-76, London Math. Soc. Lecture Note Ser., 93, Cambridge Univ. Press, Cambridge-New York, 1985.
%H See A002863 for many other references and links.
%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 S. R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/">Knots, links and tangles</a> [dead link]
%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 Steven R. Finch, <a href="https://doi.org/10.1017/9781316997741">Mathematical Constants II</a>, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 627.
%H Bruce Fontaine, <a href="https://pi.math.cornell.edu/~bfontain/knots.html">Knots/Links</a>
%H Abdullah Khan, Alexei Lisitsa, Viktor Lopatkin, and Alexei Vernitski, <a href="https://arxiv.org/abs/2108.02873">Circle graphs (chord interlacement graphs) of Gauss diagrams: Descriptions of realizable Gauss diagrams, algorithms, enumeration</a>, arXiv:2108.02873 [math.GT], 2021.
%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 K. A. Perko, Jr., <a href="https://doi.org/10.1090/S0002-9939-1974-0353294-X">On the classification of knots</a>, Proc. Amer. Math. Soc., 45 (1974), 262-266.
%H K. A. Perko, Jr., <a href="/A002863/a002863_1.pdf">Caudron's 1979 Knot Table</a>, 2015 [Included with permission]
%H Stuart Rankin, <a href="http://www.math.uwo.ca/~srankin/knotprint.html">Knot Theory Preprints of Ortho Flint Smith and Stuart Rankin</a> [dead link]
%H N. J. A. Sloane, <a href="/A002863/a002863.gif">Illustration of initial terms</a>
%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 knots and links with up to 19 crossings</a>
%H University of Western Ontario Student Beowulf Initiative, <a href="http://baldric.uwo.ca/article.php3?section=baldric&article=knots">Project: Prime Knots</a> [dead link]
%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/Knot.html">Knot.</a>
%H <a href="/index/K#knots">Index entries for sequences related to knots</a>
%Y Cf. A002863, A049344. A diagonal of A059739.
%K nonn,hard,more,nice
%O 1,5
%A _N. J. A. Sloane_
%E Terms from Hoste et al. added by _Eric W. Weisstein_; further terms from M. B. Thistlethwaite, Feb 10 2001
%E a(20) found by Ortho Flint Smith and Stuart Rankin (srankin(AT)uwo.ca), with coding done by Peter De Vries, Jun 26 2003
%E Ortho Flint Smith and Stuart Rankin, with coding by Peter de Vries, calculated a(22) = 4976016485 on an Intel Xeon 2.8ghz in 41.5 hours on Jul 07 2003
%E Ortho Flint and Stuart Rankin, with coding by Peter de Vries, calculated a(23) = 25182878921 on a Compaq ES 45 in 228 hours, finishing on Mar 14 2004
%E a(24) from Bruce Fontaine's table (produced by him together with Stuart Rankin and Ortho Flint in 2007) added by _Andrey Zabolotskiy_, Jun 08 2022
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