A086825 -id:A086825 - cp's OEIS frontend

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

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

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.

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

Cf. A002864, A086825.

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

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

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

A018240 Number of rational knots (or two-bridge knots) with n crossings (up to mirroring).

Original entry on oeis.org

1, 1, 2, 3, 7, 12, 24, 45, 91, 176, 352, 693, 1387, 2752, 5504, 10965, 21931, 43776, 87552, 174933, 349867, 699392, 1398784, 2796885, 5593771, 11186176, 22372352, 44741973, 89483947, 178962432, 357924864, 715838805, 1431677611, 2863333376, 5726666752

Offset: 3

Alexander Stoimenow (stoimeno(AT)math.toronto.edu)

			The a(7)=7 rational knots with 7 crossings are 7, 52, 43, 322, 313, 2212, 21112. All the rational knots are listed in A122495.

S. Jablan and R. Sazdanović, LinKnot: Knot Theory by Computer, World Scientific Press, 2007.

Vincenzo Librandi, Table of n, a(n) for n = 3..1000
C. Ernst and D. W. Sumners, The Growth of the Number of Prime Knots, Math. Proc. Cambridge Philos. Soc. 102, 303-315, 1987 (see Theorem 5, formulas for TK_n).
Taizo Kanenobu and Toshio Sumi, Polynomial Invariants of 2-Bridge Knots through 22 Crossings, Math. Comp. 60 (1993), 771-778, S17 (see Table 2).
P.-V. Koseleff, D. Pecker, Conway polynomials of two-bridge links, arXiv:1011.5992 [math.GT], 2010-2012 (only version 1 contains tables).
P.-V. Koseleff, D. Pecker, On Alexander-Conway polynomials of two-bridge links, Journal of Symbolic Computation 68 (2015), 215-229.
A. Stoimenow, Generating functions, Fibonacci numbers and rational knots, Journal of Algebra, 310 (2007), 491-525.
Index entries for sequences related to knots
Index entries for linear recurrences with constant coefficients, signature (-1,5,5,-2,-2,-8,-8).

Cf. A122495, A005418, A002863, A086825, A089790.

Cf. A018240 = number of rational knots, A005418 = number of rational knots and links, A001045 = Jacobsthal sequence (the difference between the number of rational links and knots), A090597 = rational links with n crossings, A329908, A336398.

LinearRecurrence[{-1, 5, 5, -2, -2, -8, -8}, {1, 1, 2, 3, 7, 12, 24}, 50] (* Harvey P. Dale, Sep 03 2013 *)
CoefficientList[Series[(1 - 2 x^2 - x^3 - x^4)/((1 - 2 x) (1 + x) (1 - 2 x^2) (1 + x^2)), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 07 2014 *)

Vec((1-2*x^2-x^3-x^4)*x^3/((1-2*x)*(1+x)*(1-2*x^2)*(1+x^2))+O(x^66)) \\ Joerg Arndt, Aug 07 2014

a(n) = - a(n-1) + 5*(a(n-2)+a(n-3)) - 2*(a(n-4)+a(n-5)) - 8*(a(n-6)+a(n-7)). [Originally contributed as a separate sequence entry by Thomas A. Gittings, Dec 11 2003; see Stoimenow, Corollary 5.1 for proof]

G.f.: (1-2*x^2-x^3-x^4)*x^3/((1-2*x)*(1+x)*(1-2*x^2)*(1+x^2)). - R. J. Mathar, Sep 08 2008

Edited by Andrey Zabolotskiy, Jun 18 2020

A051765 Number of prime satellite knots with n crossings.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 6, 10, 29, 86, 245

Offset: 1

Eric W. Weisstein

Weisstein says that Hoste et al. said that all satellite knots are prime, but actually they didn't say it about all satellite knots; moreover, the conventional definition of satellite knots implies that all composite knots are satellite. - Andrey Zabolotskiy, Nov 25 2021

Benjamin A. Burton, The next 350 million knots, 36th International Symposium on Computational Geometry (SoCG 2020), Leibniz Int. Proc. Inform., vol. 164, Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2020, pp. 25:1-25:17. See also knot tables in Supporting Data for Regina.
Jim Hoste, Morwen Thistlethwaite and Jeff Weeks, The First 1,701,936 Knots, Math. Intell., 20, 33-48, Fall 1998.
Andrei Malyutin, On the question of genericity of hyperbolic knots, arXiv preprint arXiv:1612.03368 [math.GT], 2016.
Eric Weisstein's World of Mathematics, Hyperbolic Knot
Eric Weisstein's World of Mathematics, Knot
Eric Weisstein's World of Mathematics, Satellite Knot

Cf. A002863, A051764, A052408, A086825.

a(17)-a(19) from Burton's data added by Andrey Zabolotskiy, Nov 25 2021

A086826 Number of nonsplittable links (prime or composite) with n crossings.

Original entry on oeis.org

1, 0, 1, 1, 3, 4, 15, 24, 82

Offset: 0

Steven Finch, Aug 07 2003

nonn

A link L is splittable if we can embed a plane in R^3, disjoint from L, that separates one or more components of L from other components of L. Otherwise L is nonsplittable.

			a(5)=4 since we have 2 prime knots, as well as the Whitehead link; and the trefoil knot linked with a circle.
a(6)=15 since we have 3 prime knots, as well as 2 composite knots (the square & granny knots); 6 prime links; a chain of four circles simply-intertwined; four circles simply-intertwined in the shape of a "T"; three circles, two doubly-intertwined and two simply-intertwined; and the figure-eight knot linked with a circle.

S. R. Finch, Knots, links and tangles, August 8, 2003. [Cached copy, with permission of the author]
Stéphane Legendre, Table of nonsplittable links
Eric Weisstein's World of Mathematics, Splittable Link
Index entries for sequences related to knots

Cf. A086771, A086825.

a(7) and a(8) from Stéphane Legendre, Jan 06 2014

A283314 Total number of knots (prime or composite) with <= n crossings.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 10, 18, 44

Offset: 0

N. J. A. Sloane, Mar 08 2017, following a suggestion from Daniel Forgues

Partial sums of A086825.

Cf. A283315.

a(8) corrected from A086825 by Kyle Miller, Oct 14 2020

A277740 Knot diagrams with n crossings.

Original entry on oeis.org

1, 2, 6, 36, 276, 2936, 35872, 484088, 6967942, 105555336, 1664142836

Offset: 0

N. J. A. Sloane, Nov 07 2016

From Andrey Zabolotskiy, Dec 24 2018: (Start)
It follows from Definition 7 of the paper by Cantarella, Chapman & Mastin that every one of A008988(n) knot shadows contributes some diagrams to a(n). The number of diagrams contributed is shown near knot shadows in the illustration linked below (e.g., the knot shadow looking like an n-gon with petals contributes A000031(n) inequivalent diagrams, while non-symmetric knot shadows contribute 2^n diagrams each). Note that the orientation of knot shadows is not shown when it is not needed to distinguish two knot shadows, but all the knot shadows are oriented, which spoils the symmetry of many of them. (Even though the paper itself starts from unoriented knot shadows counted by A008989, eventually the orientation is assigned to the diagrams.)
The more symmetric a knot shadow is, the fewer inequivalent diagrams it contributes. The knot shadows and the diagrams generated by them are oriented but immersed into unoriented spheres. When determining equivalent diagrams, they can be rotated on the sphere and turned "inside out" but not reflected. Some planar drawings of knot shadows make them look less symmetric than they actually are (taking into account their immersion into the sphere).
Note that this equivalence relation distinguishes many more knot diagrams than the ambient isotopy of knots, cf. A086825. (End)

J. Cantarella, H. Chapman, M. Mastin, Knot Probabilities in Random Diagrams, arXiv preprint arXiv:1512.05749 [math.GT], 2015. See Tables I and III.
Andrey Zabolotskiy, Illustration of the knot shadows contributing to a(0)-a(4)

Cf. A000031, A008988, A008989, A086825, A277740.

a(0)-a(2) from Andrey Zabolotskiy, Dec 24 2018

cp's OEIS Frontend

A002863 Number of prime knots with n crossings.

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A018240 Number of rational knots (or two-bridge knots) with n crossings (up to mirroring).

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A051765 Number of prime satellite knots with n crossings.

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A086826 Number of nonsplittable links (prime or composite) with n crossings.

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A283314 Total number of knots (prime or composite) with <= n crossings.

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A277740 Knot diagrams with n crossings.

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A283315 Total number of nontrivial knots (prime or composite) with <= n crossings.

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A116584 Number of prime knots with <= n crossings.

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A375605 Number of composite knots with n crossings.

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