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

Author

N. J. A. Sloane

Keywords

• nonn
• hard
• more
• nice

Comments

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).

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 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.

Links

• 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

Crossrefs

Cf. A002863, A049344. A diagonal of A059739.

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 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