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A002863
Number of prime knots with n crossings.
(Formerly M0851 N0323)
+10
46
0, 0, 1, 1, 2, 3, 7, 21, 49, 165, 552, 2176, 9988, 46972, 253293, 1388705, 8053393, 48266466, 294130458
(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.
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..19.
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.
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
FORMULA
a(n) = A051766(n) + A051769(n) + A051767(n) + A051768(n) + A052400(n). - Andrew Howroyd, Oct 15 2020
CROSSREFS
Cf. A002864, A086825.
Cf. A051766, A051767, A051768, A051769, A052400.
KEYWORD
nonn,hard,more,nice
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
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
STATUS
approved
A059739
Triangle T(n,k), n >= 1, giving number of prime unoriented alternating links with n crossings and k components.
+10
4
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,7
COMMENTS
A link is a not necessarily connected knot. Apart from the initial rows, the n-th row contains floor(n/2) terms.
REFERENCES
Ortho Flint, Bruce Fontaine and Stuart Rankin, The master array of a prime alternating link, preprint, 2007
LINKS
Stuart Rankin (srankin(AT)uwo.ca), Nov 05 2007, Table of n, a(n) for n = 0..133
S. 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.
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
EXAMPLE
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
...
CROSSREFS
First column gives numbers of knots, A002864. Second column gives A059741. Row sums give A049344.
KEYWORD
nonn,tabf,nice
AUTHOR
N. J. A. Sloane, Feb 10 2001
EXTENSIONS
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
STATUS
approved
A049344
Prime unoriented alternating links (not necessarily connected knots) with n crossings.
+10
3
0, 1, 1, 2, 3, 8, 14, 39, 96, 297, 915, 3308, 12417, 51347, 222595, 1016975, 4799520, 23301779, 115405815, 581071711, 2963793396, 15283327150, 79544488072, 417377448058
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
LINKS
Table of n, a(n) for n=1..24.
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. 626.
Bruce Fontaine, Knots/Links
Stavros Garoufalidis and Thao Vuong, Alternating knots, planar graphs, and q-series, The Ramanujan Journal 36.3 (2015): 501-527; arXiv:1304.1071 [math.GT], 2013. See (28).
CROSSREFS
Cf. A002864, A086771. Row sums of A059739.
KEYWORD
nonn,nice,hard,more
AUTHOR
N. J. A. Sloane, Feb 10 2001
EXTENSIONS
a(20)-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
STATUS
approved
A122495
Integers corresponding to rational knots in Conway's enumeration.
+10
2
1, 3, 22, 5, 32, 42, 312, 2112, 7, 52, 43, 322, 313, 2212, 21112, 62, 512, 44, 413, 4112, 332, 3212, 3113, 31112, 2312, 2222, 22112, 9, 72, 63, 54, 522, 513, 423, 4212, 4122, 41112, 342, 333, 3222, 3213, 31212, 31122, 311112, 2412, 2322, 23112, 22122, 21312
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
"Conway's motivation for studying tangles was to extend the [knot and link] catalogues.... here we shall concentrate on finding the first few rational links.
"The problem is reduced to listing sequences of integers and noting which sequences lead to isotopic links.
"The technique is so powerful that Conway claims to have verified the Tait-Little tables 'in an afternoon'.
"He then went on to list the 100-crossings knots and 10-crossing links.... A rational link (or its mirror image) has a regular continued fraction expansion in which all the integers are positive....
"We can discard all sequences that end in a 1 and that makes the regular sequence unique.... we do not need to keep both a sequence and its reverse.
"Applying these simple rules to the partitions of the first four integers, we see that we keep only the sequences shown in bold: 1, 2, 11, 3, 21, 12, 111, 4, 31, 22, 13, 211, 121, 112, 1111." [typographically, the bold subsequence is 1, 2, 3, 4, 22] "These sequences correspond to the trivial knot, the Hopf link, the trefoil, the (2,4) torus link and the figure 8 knot.
"Continuing in this fashion, we find that for knots and links with up to seven crossings, the sequences for rational knots are: 3, 22, 5, 32, 42, 312, 2112, 7, 52, 43, 322, 313, 2212, 21112 and the sequences for rational 2-component links are 2, 4, 212, 6, 33, 222, 412, 232, 3112.... we see that a sequence represents an amphicheiral knot or link only if the sequence is palindromic (equal to its reverse) and of even length (n even).
"This shows that the only amphicheiral knots in the list are the figure-8 knot (sequence 22) and the knot 6_3 (sequence 2112); all of the links are cheiral...." [Cromwell]
The ordering among the terms with the same sum of digits (i.e., number of crossings) is the inverse lexicographical. Each term is actually an ordered set of positive integers, concatenated; as long as all integers are 1-digit, it's not a problem, but a(97) requires "digit" 11, so at that point the sequence becomes not fully well-defined. An irregular array of these numbers would be well-defined. - Andrey Zabolotskiy, May 22 2017
REFERENCES
Peter R. Cromwell, Knots and Links, Cambridge University Press, 2004, pp. 209-211.
LINKS
Table of n, a(n) for n=1..49.
Alain Caudron, Classification des noeuds et des enlancements (see p. 168).
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.
Index entries for sequences related to knots
EXAMPLE
a(1) = 1 because 1 corresponds to the trivial knot.
a(2) = 3 because 3 corresponds to the trefoil.
a(3) = 22 because 22 corresponds to the figure-8 knot.
MATHEMATICA
whereTangle[{n_}] := If[EvenQ[n], 1, 2];
whereTangle[{rest__, n_}] := Switch[whereTangle[{rest}], 1, 3, 2, Switch[whereTangle[{n}], 1, 2, 2, 1, 3, 3], 3, whereTangle[{n}]];
FromDigits /@ Prepend[Select[Flatten[Table[Reverse@SortBy[Flatten[Permutations /@ IntegerPartitions[n], 1], PadRight[#, n] &], {n, 10}], 1], OrderedQ[{Reverse[#], #}] && Last[#] != 1 && whereTangle[#] != 1 (*change to "==1" for rational 2-component links*) &], {1}]
(* Andrey Zabolotskiy, May 22 2017 *)
CROSSREFS
Cf. A002863, A002864, A173637, A018240, A078666.
KEYWORD
nonn
AUTHOR
Jonathan Vos Post, Sep 16 2006
EXTENSIONS
Sequence edited and more terms added by Andrey Zabolotskiy, May 22 2017
STATUS
approved
A047051
Prime alternating tangle types (of knots) with n crossings.
+10
1
1, 2, 4, 10, 29, 98, 372, 1538, 6755, 30996, 146982, 715120, 3552254, 17951322, 92045058, 477882876, 2508122859, 13289437362, 71010166670, 382291606570, 2072025828101, 11298920776704, 61954857579594, 341427364138880
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
G.f. is related to the classes of 2- and 3-connected planar maps with n edges. Further terms are known.
REFERENCES
C. Sundberg and M. Thistlethwaite, The rate of growth of the number of prime alternating links and tangles, Pacif. J. Math., 182, No 2 (1998), 329-358.
LINKS
Herman Jamke (hermanjamke(AT)fastmail.fm), May 05 2007, Table of n, a(n) for n = 1..50
S. R. Finch, Knots, links and tangles
S. R. Finch, Knots, links and tangles, Aug 08 2003. [Cached copy, with permission of the author]
C. Sundberg and M. Thistlethwaite, The rate of growth of the number of prime alternating links and tangles, Pacif. J. Math., 182, No 2 (1998), 329-358.
P. Zinn-Justin and J.-B. Zuber, Matrix integrals and the generation and counting of virtual tangles and links, p. 12.
P. Zinn-Justin, The General O(n) Quartic Matrix Model and its application to Counting Tangles and Links
MATHEMATICA
max = 24; Clear[a, eq, s]; gf = Sum[a[k]*x^k, {k, 0, max}]; a[0] = 0; a[1] = 1; a[2] = 2; coes = CoefficientList[(x^4 - 2*x^3 + x^2)*gf^5 + (8*x^4 - 14*x^3 + 8*x^2 - 2*x)*gf^4 + (25*x^4 - 16*x^3 - 14*x^2 + 8*x + 1)*gf^3 + (38*x^4 + 15*x^3 - 30*x^2 - x + 2)*gf^2 + (28*x^4 + 36*x^3 - 5*x^2 - 12*x + 1)*gf + 8*x^4 + 17*x^3 + 8*x^2 - x, x]; eq[n_] := eq[n] = If[n == 1, Thread[Drop[coes, 3] == 0], eq[n-1] /. s[n-1] // First]; s[n_] := s[n] = (Print["n = ", n]; Solve[eq[n][[n]], a[n+2]]); sol = Table[s[n], {n, 1, max-2}] // Flatten; Table[a[n], {n, 1, max}] /. sol (* Jean-François Alcover, Apr 15 2014 *)
CROSSREFS
Cf. A002864, A000139, A000287.
KEYWORD
easy,nice,nonn
AUTHOR
Valery A. Liskovets
EXTENSIONS
More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), May 05 2007
STATUS
approved
A059741
Prime unoriented alternating links with n crossings and 2 components.
+10
1
0, 1, 0, 1, 1, 3, 6, 14, 42, 121, 384, 1408, 5100, 21854, 92234, 427079, 2005800, 9716848, 48184018, 241210386, 1228973463, 6301831944, 32663182521, 170407462900
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,6
LINKS
Table of n, a(n) for n=1..24.
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.
Bruce Fontaine, Knots/Links
Stuart Rankin, Knot Theory Work of Ortho Flint and Stuart Rankin
M. B. Thistlethwaite, Home Page
M. B. Thistlethwaite, Numbers of knots and links with up to 19 crossings
CROSSREFS
Column 2 of A059739. Cf. A002864.
KEYWORD
nonn,more
AUTHOR
N. J. A. Sloane, Feb 10 2001
EXTENSIONS
More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), May 05 2007
a(23)-a(24) corrected using Bruce Fontaine's table by Andrey Zabolotskiy, Jun 08 2022
STATUS
approved
A173637
Conway notation for rational 2-component links.
+10
1
2, 4, 212, 6, 33, 222, 412, 3112, 232, 8, 53, 422, 323, 3122, 242, 21212, 211112, 612, 5112, 432, 414, 4113, 3312, 32112, 3132, 31113, 252, 22212, 221112
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
The ordering of the list is based on increasing crossing numbers and inverse lexicographical order for the terms with the same crossing number.
This is to links what A122495 is to knots.
All these links are chiral.
Each term is actually an ordered set of positive integers, concatenated; as long as all integers are 1-digit, it's not a problem, but a(30) requires "digit" 10, so at that point the sequence becomes not fully well-defined. An irregular array of these numbers would be well-defined.
Number of the terms of this sequence with crossing number k plus number of the terms of A122495 with crossing number k equals A005418(k-2). - Andrey Zabolotskiy, May 23 2017
REFERENCES
C. Cerf, Atlas of oriented knots and links, Topology Atlas 3 no.2 (1998).
Peter R. Cromwell, Knots and Links, Cambridge University Press, November 15, 2004, p.210.
LINKS
Table of n, a(n) for n=1..29.
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.
C. Giller, A family of links and the Conway calculus, Trans. American Math Soc., 270 (1982) 75-109.
Index entries for sequences related to knots
EXAMPLE
a(1) = 2 because 2 is the Conway notation for the Hopf link.
a(2) = 4 because 4 is the Conway notation for the (2,4) torus link.
CROSSREFS
Cf. A002863, A002864, A018240, A090597, A122495.
KEYWORD
nonn
AUTHOR
Jonathan Vos Post, Nov 23 2010
EXTENSIONS
Sequence edited and more terms added by Andrey Zabolotskiy, May 23 2017
STATUS
approved
A343358
Number of connected graphs with n vertices which are realizable (in the sense of realizability of Gauss diagrams).
+10
1
1, 1, 2, 3, 7, 18, 41, 123, 361, 1257, 4573
(list; graph; refs; listen; history; text; internal format)
OFFSET
3,3
COMMENTS
Consider a closed planar curve which crosses itself n times. Build a graph in which crossings are vertices, and two crossings c, d are not connected [connected] if respectively it is [is not] possible to travel along the curve from c to c without passing through d. A graph which can be produced in this way is called realizable. A classical related concept is that of a Gauss diagram (of a closed planar curve); realizable graphs are exactly the circle graphs of realizable Gauss diagrams.
The entries are produced by our code, and the entry for n=11 is corroborated by Section 4 in Bishler et al. which lists 6 pairs of alternating mutant knots of size 11. The entries for n=12, 13 are similarly corroborated by Stoimenow's data.
REFERENCES
L. Bishler et al. "Distinguishing mutant knots." Journal of Geometry and Physics 159 (2021): 103928.
LINKS
Table of n, a(n) for n=3..13.
L. Bishler, et al., Distinguishing mutant knots, arXiv:2007.12532 [hep-th], 2021.
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.
Alexei Lisitsa, Abdullah Khan, and Alexei Vernitski, An experimental approach to Gauss diagram realizability, 28th British Comb. Conf., Durham Univ. (UK, 2021), p. 107.
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.
A. Stoimenow, Knot data tables.
CROSSREFS
Cf. A002864, which starts with 1, 1, 2, 3, 7, 18, 41, 123, 367. This is because an alternating prime knot with 10 or fewer crossings is uniquely defined by the graph of the corresponding closed planar curve. Only starting from n=11 some alternating knots which share the same graph but are distinct knots (called "mutant knots") start appearing.
Cf. A264759, which starts with 1, 1, 2, 3, 10; there is a mismatch starting from size 7. Indeed, starting from n=7 there are some planar curves which share the same graph but have distinct Gauss diagrams.
KEYWORD
nonn,hard,more
AUTHOR
Alexei Vernitski, Apr 12 2021
STATUS
approved
A051763
Number of nonalternating prime knots with n crossings.
+10
0
0, 0, 0, 0, 0, 0, 0, 3, 8, 42, 185, 888, 5110, 27436, 168030, 1008906, 6283414, 39866181, 253511073
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,8
REFERENCES
See A002863 for many other references and links.
LINKS
Table of n, a(n) for n=1..19.
See A002863 for many other references and links.
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.
K. A. Perko, Jr., Caudron's 1979 Knot Table, 2015 [Included with permission]
Eric Weisstein's World of Mathematics, Alternating Knot.
Eric Weisstein's World of Mathematics, Knot.
CROSSREFS
Cf. A002863, A002864.
KEYWORD
nonn,nice,more
AUTHOR
Eric W. Weisstein
EXTENSIONS
Name clarified and a(17)-a(19) added from Burton's data by Andrey Zabolotskiy, Nov 25 2021
STATUS
approved
A116584
Number of prime knots with <= n crossings.
+10
0
0, 0, 1, 2, 4, 7, 14, 35, 84, 249, 801, 2977, 12965, 59937, 313230, 1701935
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
LINKS
Table of n, a(n) for n=1..16.
M. B. Thistlethwaite, Numbers of knots and links with up to 19 crossings. "These numbers have not been thoroughly checked."
CROSSREFS
Partial sums of A002863.
Cf. A002863, A002864, A086825.
KEYWORD
hard,nonn
AUTHOR
Jonathan Vos Post, May 09 2006
STATUS
approved
page 1

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