A051764 -id:A051764 - cp's OEIS frontend

A052408 Number of hyperbolic knots with n crossings (A002863 - A051764 - A051765).

Original entry on oeis.org

0, 0, 0, 1, 1, 3, 6, 20, 48, 164, 551, 2176, 9985, 46969, 253285, 1388694, 8053363, 48266380, 294130212

Offset: 1

Eric W. Weisstein

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.

Cf. A052407.

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

A052407 Number of nonhyperbolic prime knots with n crossings (A051764+A051765).

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 3, 3, 8, 11

Offset: 1

Eric W. Weisstein

J. Hoste, M. B. Thistlethwaite and J. Weeks, The First 1,701,936 Knots, Math. Intell., 20, 33-48, Fall 1998.
Eric Weisstein's World of Mathematics, Hyperbolic Knot.

Cf. A052408.

A123192 Triangle read by rows: row n gives the coefficients in the expansion of x^abs(3n - 2)p(n;x), where p(n;x) denotes the bracket polynomial for the (2,n)-torus knots.

Original entry on oeis.org

-1, 0, 0, 0, -1, 0, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, -1, -1, 0, 0, 0, 1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, -1, 0, 0, 0, 1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, -1, -1, 0, 0, 0, 1, 0, 0, 0, -1, 0, 0, 0

Offset: 0

Roger L. Bagula, Oct 03 2006

From Franck Maminirina Ramaharo, Aug 11 2018: (Start)
Using Kauffman's notation, the formal expression of the bracket polynomial for the (2,n)-torus knot is defined as follows:
K(n;A,B,d) = A*K(n-1;A,B,d) + B*(A + B*d)^(n - 1) with K(0;A,B,d) = d.
  - The polynomial in this sequence is defined as p(n;x) = K(n;x,1/x,-x^2-x^(-2)), and verifies p(n;x) = x*p(n-1;x) + (-1)^(n - 1)*x^(-3*n + 2).
  - The polynomial x*K(n;1,1,x) yields (x + 1)^n + x^2 - 1 which is the bracket evaluated at the shadow diagram of the (2,n)-torus knot, see A300453.
  - The polynomial sqrt(x)*K(n;-1,sqrt(x),sqrt(x)) yields (x - 1)^n + (x - 1)*(-1)^n. This is the chromatic polynomial for the n-cycle graph which is the medial graph of the (2,n)-torus knot, see A137396.
The planar diagram of the (2,0)-torus knot is two non-intersecting circles.
(End)

			From _Franck Maminirina Ramaharo_, Aug 11 2018: (Start)
The bracket polynomial for some value of n:
  p(0;x) = -x^2 - 1/x^2;
  p(1;x) = -x^3;
  p(2;x) = -x^4 - 1/x^4;
  p(3;x) = -x^5 - 1/x^3 + 1/x^7;
  p(4;x) = -x^6 - 1/x^2 + 1/x^6 - 1/x^10;
  p(5;x) = -x^7 - 1/x   + 1/x^5 - 1/x^9  + 1/x^13;
  p(6;x) = -x^8 - 1     + 1/x^4 - 1/x^8  + 1/x^12 - 1/x^16;
  p(7;x) = -x^9 - x     + 1/x^3 - 1/x^7  + 1/x^11 - 1/x^15 + 1/x^19;
  ...
The triangle giving the coefficients in x^abs(3*n - 2)*p(n;x) begins:
  -1, 0, 0, 0, -1
   0, 0, 0, 0, -1
  -1, 0, 0, 0,  0, 0, 0, 0, -1
   1, 0, 0, 0, -1, 0, 0, 0,  0, 0, 0, 0, -1
  -1, 0, 0, 0,  1, 0, 0, 0, -1, 0, 0, 0,  0, 0, 0, 0, -1
   1, 0, 0, 0, -1, 0, 0, 0,  1, 0, 0, 0, -1, 0, 0, 0,  0, 0, 0, 0, -1
  ...
(End)

Louis H. Kauffman, Knots and Physics (Third Edition), World Scientific, 2001. See p. 38 and p. 353.

Paul Corbitt, Torus Links and the Bracket Polynomial.
Louis H. Kauffman, State models and the Jones polynomial, Topology Vol. 26 (1987), 395-407.
Franck Ramaharo, Note on sequences A123192, A137396 and A300453, arXiv:1911.04528 [math.CO], 2019.
Eric Weisstein's World of Mathematics, Bracket Polynomial.
Eric Weisstein's World of Mathematics, Torus Knot.
Wikipedia, Torus knot.
Wikipedia, Medial graph.

Cf. A029694, A051764, A137396, A300453.

K(n, A, B, d) := if n = 0 then d else A*K(n - 1, A, B, d) + B*(A + B*d)^(n - 1)$
p(n, x) := x^abs(3*n - 2)*K(n, x, 1/x, -x^(-2) - x^2)$
t(n, k) := ratcoef(p(n, x), x, k)$
T:[]$
for n:0 thru 10 do T:append(T, makelist(t(n,k), k, 0, max(4, 4*n)))$
T; /* Franck Maminirina Ramaharo, Aug 11 2018 */

Partially edited by N. J. A. Sloane, May 22 2007

Edited, new name, and corrected by Franck Maminirina Ramaharo, Aug 11 2018

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

A379242 Minimum crossing number at which there are n torus knots.

Original entry on oeis.org

1, 3, 15, 63, 189, 432, 792, 1232, 1584, 2880, 4320, 5040, 6336, 7920, 12096, 15120, 19008, 22176, 30240, 33264, 43200, 47520, 44352, 65520, 75600, 108000, 90720, 120960, 168480, 131040, 151200, 181440, 252000, 196560, 221760, 237600, 362880, 403200, 302400

Offset: 0

Alex Klotz, Dec 18 2024

nonn

Minimum number that can be factored N different ways into p*(q-1) for coprime p and q with p>q. e.g. 63=63*(2-1)=9*(8-1)=21*(4-1); 63 is the smallest crossing number with three torus knots. Odd numbers will admit an alternating (p,2) torus knot with p crossings, all others with q>2 are non-alternating. Based on definition of torus knot and data from A051764.

			3 = 3*(2-1), 15 = 15*(2-1) = 5*(4-1), 63 = 63*(2-1) = 9*(8-1) = 21*(4-1).

Alexander R. Klotz and Caleb J. Anderson, Ropelength and writhe quantization of 12-crossing knots, arXiv:2305.17204 [math.GT], 2023; Experimental Mathematics (2024): 1-8.

First occurrence of each n in A051764.

More terms from Alois P. Heinz, Dec 29 2024

cp's OEIS Frontend

A052408 Number of hyperbolic knots with n crossings (A002863 - A051764 - A051765).

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A052407 Number of nonhyperbolic prime knots with n crossings (A051764+A051765).

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A123192 Triangle read by rows: row n gives the coefficients in the expansion of x^abs(3*n - 2)*p(n;x), where p(n;x) denotes the bracket polynomial for the (2,n)-torus knots.

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A051765 Number of prime satellite knots with n crossings.

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A379242 Minimum crossing number at which there are n torus knots.

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A123192 Triangle read by rows: row n gives the coefficients in the expansion of x^abs(3n - 2)p(n;x), where p(n;x) denotes the bracket polynomial for the (2,n)-torus knots.