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A002864 Number of alternating prime knots with n crossings.
(Formerly M0847 N0322)
9
0, 0, 1, 1, 2, 3, 7, 18, 41, 123, 367, 1288, 4878, 19536, 85263, 379799, 1769979, 8400285, 40619385, 199631989, 990623857, 4976016485, 25182878921 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

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).

REFERENCES

See A002863 for many other references and links.

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.

J. Hoste, M. B. Thistlethwaite and J. Weeks, The First 1,701,936 Knots, Math. Intell., 20, 33-48, Fall 1998.

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.

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.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

P. G. Tait, Scientific Papers, Cambridge Univ. Press, Vol. 1, 1898, Vol. 2, 1900, see Vol. 1, p. 345.

M. B. Thistlethwaite, personal communication.

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.

LINKS

Table of n, a(n) for n=1..23.

See A002863 for many other references and links.

D. Bar-Natan, The Hoste-Thistlethwaite Table of 11 Crossing Knots

S. R. Finch, Knots, links and tangles

S. R. Finch, Knots, links and tangles, Aug 08 2003. [Cached copy, with permission of the author]

W. B. R. Lickorish and K. C. Millett, The new polynomial invariants of knots and links, Math. Mag. 61 (1988), no. 1, 3-23.

K. A. Perko, Jr., On the classification of knots, Proc. Amer. Math. Soc., 45 (1974), 262-266.

K. A. Perko, Jr., Cuadron's 1979 Knot Table, 2015 [Included with permission]

Stuart Rankin, Knot Theory Preprints of Ortho Smith and Stuart Rankin

N. J. A. Sloane, Illustration of initial terms

M. B. Thistlethwaite, Home Page

M. B. Thistlethwaite, Numbers of knots and links with up to 19 crossings

University of Western Ontario Student Beowulf Initiative, Project: Prime Knots

Eric Weisstein's World of Mathematics, Alternating Knot.

Eric Weisstein's World of Mathematics, Knot.

Index entries for sequences related to knots

CROSSREFS

Cf. A002863. A diagonal of A059739.

Sequence in context: A058334 A303090 A131093 * A005248 A032102 A100388

Adjacent sequences:  A002861 A002862 A002863 * A002865 A002866 A002867

KEYWORD

nonn,hard,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

Terms from Hoste et al. added by Eric W. Weisstein; further terms from M. B. Thistlethwaite, Feb 10 2001

a(20) found by Ortho Smith and Stuart Rankin (srankin(AT)uwo.ca), with coding done by Peter De Vries, Jun 26 2003

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

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

STATUS

approved

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Last modified November 18 19:59 EST 2019. Contains 329288 sequences. (Running on oeis4.)