%I A002864 M0847 N0322
%S A002864 0,0,1,1,2,3,7,18,41,123,367,1288,4878,19536,85263,379799,1769979,8400285,
%T A002864 40619385,199631989,990623857,4976016485,25182878921
%N A002864 Number of alternating prime knots with n crossings.
%C A002864 Ortho 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 A002864 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 A002864 J. Hoste, M. B. Thistlethwaite and J. Weeks, The First 1,701,936 Knots, 
               Math. Intell., 20, 33-48, Fall 1998.
%D A002864 W. B. R. Lickorish and K. C. Millett, The new polynomial invariants of 
               knots and links. Math. Mag. 61 (1988), no. 1, 3-23.
%D A002864 K. A. Perko, Jr., On the classification of knots, Proc. Amer. Math. Soc., 
               45 (1974), 262-266.
%D A002864 Stuart Rankin, Ortho Smith and John Schermann, Enumerating the Prime 
               Alternating Knots, Part I, Journal of Knot Theory and its Ramifications, 
               13 (2004), 57-100.
%D A002864 Stuart Rankin, Ortho Smith and John Schermann, Enumerating the Prime 
               Alternating Knots, Part II, Journal of Knot Theory and its Ramifications, 
               13 (2004), 101-149.
%D A002864 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002864 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002864 P. G. Tait, Scientific Papers, Cambridge Univ. Press, Vol. 1, 1898, Vol. 
               2, 1900, see Vol. 1, p. 345.
%D A002864 M. B. Thistlethwaite, personal communication.
%D A002864 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 A002864 D. Bar-Natan, <a href="http://www.math.toronto.edu/~drorbn/KAtlas/Knots11/
               index.html">The Hoste-Thistlethwaite Table of 11 Crossing Knots</
               a>
%H A002864 S. R. Finch, <a href="http://algo.inria.fr/bsolve/">Knots, links and 
               tangles</a>
%H A002864 Stuart Rankin, <a href="http://www.math.uwo.ca/~srankin/knotprint.html">
               Knot Theory Preprints of Ortho Smith and Stuart Rankin</a>
%H A002864 N. J. A. Sloane, <a href="a2863.gif">Illustration of initial terms</a>
%H A002864 M. B. Thistlethwaite, <a href="http://www.math.utk.edu/~morwen/index.html">
               Home Page</a>
%H A002864 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 A002864 University of Western Ontario Student Beowulf Initiative, <a href="http:/
               /baldric.uwo.ca/article.php3?section=baldric&article=knots">Project: 
               Prime Knots</a>
%H A002864 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               AlternatingKnot.html">Link to a section of The World of Mathematics 
               (1).</a>
%H A002864 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Knot.html">Link to a section of The World of Mathematics (2).</a>
%Y A002864 Cf. A002863. A diagonal of A059739.
%Y A002864 Sequence in context: A102226 A058334 A131093 this_sequence A005248 A032102 
               A100388
%Y A002864 Adjacent sequences: A002861 A002862 A002863 this_sequence A002865 A002866 
               A002867
%K A002864 nonn,hard,nice
%O A002864 1,5
%A A002864 N. J. A. Sloane (njas(AT)research.att.com).
%E A002864 Terms from Hoste et al. added by Eric Weisstein (eric(AT)weisstein.com). 
               Further terms from M. B. Thistlethwaite, Feb 10, 2001.
%E A002864 a(20) found by Ortho Smith and Stuart Rankin (srankin(AT)uwo.ca), with 
               coding done by Peter De Vries, Jun 26, 2003
%E A002864 Ortho 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 A002864 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