A049344 -id:A049344 - cp's OEIS frontend

A002864 Number of alternating prime knots with n crossings.

0, 0, 1, 1, 2, 3, 7, 18, 41, 123, 367, 1288, 4878, 19536, 85263, 379799, 1769979, 8400285, 40619385, 199631989, 990623857, 4976016485, 25182878921, 128564665125

Offset: 1

N. J. A. Sloane

Ortho Flint 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).

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 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.
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.

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 [dead link]
S. R. Finch, Knots, links and tangles, Aug 08 2003. [Cached copy, with permission of the author]
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 627.
Bruce Fontaine, Knots/Links
Abdullah Khan, Alexei Lisitsa, Viktor Lopatkin, and Alexei Vernitski, Circle graphs (chord interlacement graphs) of Gauss diagrams: Descriptions of realizable Gauss diagrams, algorithms, enumeration, arXiv:2108.02873 [math.GT], 2021.
W. B. R. Lickorish and K. C. Millett, The new polynomial invariants of knots and links, Math. Mag. 61 (1988), no. 1, 3-23.
Alexei Lisitsa and Alexei Vernitski, Counting graphs induced by Gauss diagrams and families of mutant alternating knots, Examples Counterex. (2024) Vol. 6, Art. No. 100162.
K. A. Perko, Jr., On the classification of knots, Proc. Amer. Math. Soc., 45 (1974), 262-266.
K. A. Perko, Jr., Caudron's 1979 Knot Table, 2015 [Included with permission]
Stuart Rankin, Knot Theory Preprints of Ortho Flint Smith and Stuart Rankin [dead link]
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 [dead link]
Eric Weisstein's World of Mathematics, Alternating Knot.
Eric Weisstein's World of Mathematics, Knot.
Index entries for sequences related to knots

Cf. A002863, A049344. A diagonal of A059739.

Terms from Hoste et al. added by Eric W. Weisstein; further terms from M. B. Thistlethwaite, Feb 10 2001
a(20) found by Ortho Flint Smith and Stuart Rankin (srankin(AT)uwo.ca), with coding done by Peter De Vries, Jun 26 2003
Ortho Flint 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
a(24) from Bruce Fontaine's table (produced by him together with Stuart Rankin and Ortho Flint in 2007) added by Andrey Zabolotskiy, Jun 08 2022

A086771 Prime links (not necessarily connected knots) with n crossings.

Original entry on oeis.org

0, 1, 1, 2, 3, 9, 16, 50, 132, 452, 1559

Offset: 1

Steven Finch, Aug 02 2003

D. J. A. Welsh, On the number of knots and links, Sets, Graphs and Numbers (Budapest, 1991) 713-718, Colloq. Math. Soc. Janos Bolyai, 60, North-Holland, Amsterdam, 1992.

D. Bar-Natan, The Hoste-Thistlethwaite Link Table.
S. R. Finch, Knots, links and tangles
S. R. Finch, Knots, links and tangles, Aug 08 2003. [Cached copy, with permission of the author]
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 627.
R. G. Scharein, Number of Prime Links.
Eric Weisstein's World of Mathematics, Prime Link

Cf. A002863, A048952, A048953, A049344, A086771.

A059739 Triangle T(n,k), n >= 1, giving number of prime unoriented alternating links with n crossings and k components.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 2, 1, 3, 3, 2, 7, 6, 1, 18, 14, 6, 1, 41, 42, 12, 1, 123, 121, 43, 9, 1, 367, 384, 146, 17, 1, 1288, 1408, 500, 100, 11, 1, 4878, 5100, 2074, 341, 23, 1, 19536, 21854, 8206, 1556, 181, 13, 1, 85263, 92234, 37222, 7193, 653, 29, 1, 379799, 427079, 172678, 33216, 3885, 301, 16, 1, 1769979, 2005800, 829904, 173549, 19122, 1129, 36, 1, 8400285, 9716848, 4194015, 876173, 105539, 8428, 471, 19, 1, 40619385, 48184018, 21207695, 4749914, 599433, 43513, 1813, 43, 1

Offset: 0

N. J. A. Sloane, Feb 10 2001

A link is a not necessarily connected knot. Apart from the initial rows, the n-th row contains floor(n/2) terms.

			First few rows of irregular triangle:
   0
   0  1
   1
   1  1
   2  1
   3  3  2
   7  6  1
  18 14  6 1
  41 42 12 1
  ...

Ortho Flint, Bruce Fontaine and Stuart Rankin, The master array of a prime alternating link, preprint, 2007

Stuart Rankin (srankin(AT)uwo.ca), Nov 05 2007, Table of n, a(n) for n = 0..133
Steven R. Finch, Knots, links and tangles, August 8, 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.
Knotilus web site, Knotilus. [dead link]
S. Rankin and O. Flint, Knot theory web page.
M. B. Thistlethwaite, Home Page.
M. B. Thistlethwaite, Numbers of knots and links with up to 19 crossings.
Index entries for sequences related to knots

First column gives numbers of knots, A002864. Second column gives A059741. Row sums give A049344.

Terms for the 20-, 21-, 22- and 23-crossing prime alternating links (see the b-file) added Nov 03 2007 by Stuart Rankin, Ortho Flint and Bruce Fontaine

Trailing 0 in row for n=2 removed by N. J. A. Sloane, Nov 21 2007

cp's OEIS Frontend

A002864 Number of alternating prime knots with n crossings.

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A086771 Prime links (not necessarily connected knots) with n crossings.

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A059739 Triangle T(n,k), n >= 1, giving number of prime unoriented alternating links with n crossings and k components.

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