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A002863 Number of prime knots with n crossings.
(Formerly M0851 N0323)
18
0, 0, 1, 1, 2, 3, 7, 21, 49, 165, 552, 2176, 9988, 46972, 253293, 1388705 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

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]

REFERENCES

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.

C. C. Adams, The Knot Book, Freeman, NY, 2001; see p. 33.

Caudron, Alain. "Classification des noeuds et des entrelacs." PhD diss., Paris 11, 1987.

C. Cerf, Atlas of oriented knots and links, Topology Atlas 3 no. 2 (1998).

Peter R. Cromwell, Knots and Links, Cambridge University Press, 2004, pp. 209-211.

Martin Gardner, The Last Recreations, Copernicus, 1997, 67-84.

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.

LINKS

Table of n, a(n) for n=1..16.

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

D. J. Broadhurst and D. Kreimer, Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops, Phys. Lett. B 393, No.3-4, 403-412 (1997).

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.

S. R. Finch, Knots, links and tangles

Ortho Flint, Bruce Fontaine and Stuart Rankin, Enumerating the prime alternating links, preprint, 2007.

Ortho Flint and Stuart Rankin, Enumerating the prime alternating links, Journal of Knot theory and its Ramifications, 13 (2004), 151-173.

C. Giller, A family of links and the Conway calculus, Trans. American Msth Soc., 270 (1982) 75-109.

Jeremy Green, A Table of Virtual Knots, 2004.

Hermann Gruber, Atlas of Rational Knots.

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

Jim Hoste, The Enumeration and Classification of Knots and Links, in Handbook of Knot Theory, William W. Menasco and Morwen B. Thistlethwaite, Editors, Elsevier, 2015.

Knot Atlas, The Knot Atlas. 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.

Knotilus web site, Knotilus

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

C. Livingston and J. C. Cha, Table of knot invariants.

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

K. A. Perko, Jr., Primality of certain knots, 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.

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

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

S. Rankin and O. Flint Knot theory web page.

Stuart Rankin and Ortho Smith, Enumerating the Prime Alternating Links, arXiv:math/0211451 [math.GT], 2002

Stuart Rankin, Ortho Smith and John Schermann, Enumerating the Prime Alternating Knots, Part I, arXiv:math/0211346 [math.GT], 2002.

Stuart Rankin, Ortho Smith and John Schermann, Enumerating the Prime Alternating Knots, Part II, arXiv:math/0211348 [math.GT], 2002.

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.

R. G. Scharein, Number of Prime Links

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

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.

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

Eric Weisstein's World of Mathematics, Knot.

Eric Weisstein's World of Mathematics, Prime Knot.

Eric Weisstein's World of Mathematics, Alternating Knot.

Eric Weisstein's World of Mathematics, Prime Link

Index entries for sequences related to knots

CROSSREFS

Cf. A002864, A086825.

Sequence in context: A080021 A032313 A032223 * A047693 A212265 A107108

Adjacent sequences:  A002860 A002861 A002862 * A002864 A002865 A002866

KEYWORD

nonn,hard,more,nice,changed

AUTHOR

N. J. A. Sloane

EXTENSIONS

This is stated incorrectly in CRC Standard Mathematical Tables and Formulae, 30th ed., first printing, 1996, p. 320.

Terms from Hoste et al. added by Eric W. Weisstein

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

STATUS

approved

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Last modified September 3 04:49 EDT 2015. Contains 261304 sequences.