A002863 Number of prime knots with n crossings.

0, 0, 1, 1, 2, 3, 7, 21, 49, 165, 552, 2176, 9988, 46972, 253293, 1388705, 8053393, 48266466, 294130458

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• hard
• more
• nice

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.
• 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

• 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, arXiv:hep-th/9609128, 1996; Phys. Lett. B 393, No.3-4, 403-412 (1997).
• B. Burton, The next 350 million knots, 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.
• Alain Caudron, Classification des noeuds et des enlacements (Thèse et additifs), Univ. Paris-Sud, 1989 [Scanned copy, included with permission. But also see the Perko links below.]
• 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, Aug 08 2003. [Cached copy, with permission of the author]
• 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 Math Soc., 270 (1982) 75-109.
• Jeremy Green, A Table of Virtual Knots, 2004.
• Hermann Gruber, Atlas of Rational Knots. [dead link]
• 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.
• S. Jablan, L. H. Kauffman, and P. Lopes, The delunification process and minimal diagrams, Topology Appl., 193 (2015), 270-289, #5531; see also, arXiv:1406.2378 [math.GT], 2014.
• 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 [dead link]
• 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 A. H. Moore, KnotInfo: Table of Knot Invariants.
• Andrei Malyutin, On the question of genericity of hyperbolic knots, arXiv preprint arXiv:1612.03368 [math.GT], 2016.
• K. A. Perko, Jr., Abstract for Talk, 1973
• K. A. Perko, Jr., On covering spaces of knots, Glasnik Mathematicki, Tom 9 (29) No. 1 (1974), 141-145. (Annotated scanned copy)
• K. A. Perko, Jr., On the classification of knots, Proc. Amer. Math. Soc., 45 (1974), 262-266. (Annotated scanned copy)
• K. A. Perko, Jr., Letters to N. J. A. Sloane 1974-1977
• 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., On ninth order knottiness, Preprint (N. D.)
• K. A. Perko, Jr., Caudron's 1979 Knot Table, 2015 [Included with permission. See next link for list of errors.]
• K. A. Perko, Jr., Errors in Alain Caudron's 1989 thesis
• K. A. Perko, Jr., Review of Jablan-Kauffman-Lopes (2015)
• Stuart Rankin, Knot Theory Preprints of Ortho Flint Smith and Stuart Rankin
• S. Rankin and O. Flint Knot theory web page.
• Stuart Rankin and Ortho Flint Smith, Enumerating the Prime Alternating Links, arXiv:math/0211451 [math.GT], 2002
• Stuart Rankin, Ortho Flint Smith and John Schermann, Enumerating the Prime Alternating Knots, Part I, arXiv:math/0211346 [math.GT], 2002.
• Stuart Rankin, Ortho Flint Smith and John Schermann, Enumerating the Prime Alternating Knots, Part II, arXiv:math/0211348 [math.GT], 2002.
• 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.
• 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.
• R. G. Scharein, Number of Prime Links
• Silvia Sconza and Arno Wildi, Knot-based Key Exchange protocol, Cryptology ePrint Archive (2024), Art. No. 2024/471. See Table 2, p. 15.
• N. J. A. Sloane, Illustration of initial terms
• P. G. Tait, The first seven orders of knottiness [Annotated scan of Plate VI]
• M. B. Thistlethwaite, Home Page
• M. B. Thistlethwaite, Numbers of alternating 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.
• Daniel Tubbenhauer and Victor Zhang, Big data comparison of quantum invariants, arXiv:2503.15810 [math.GT], 2025. See p. 4.
• S. D. Tyurina, Diagram invariants of knots and the Kontsevich integral, J. Math. Sci. 134 (2) (2006) pp. 2017-2071.
• 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
• R. G. Wilson, V, Letter to N. J. A. Sloane, Oct. 1993
• Index entries for sequences related to knots

Crossrefs

Cf. A002864, A086825.

Cf. A051766, A051767, A051768, A051769, A052400.

Formula

a(n) = A051766(n) + A051769(n) + A051767(n) + A051768(n) + A052400(n). - Andrew Howroyd, Oct 15 2020

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

a(17)-a(19) computed by Benjamin Burton, added by Alex Klotz, Jun 21 2021

a(17)-a(19) computed by Benjamin Burton corrected by Andrey Zabolotskiy, Nov 25 2021