A089892 -id:A089892 - cp's OEIS frontend

A078477 Number of rational knots with n crossings and unknotting number = 1 (chiral pairs counted only once).

1, 1, 1, 3, 3, 6, 7, 15, 15, 30, 31, 63, 63, 126, 127, 255, 255, 510, 511, 1023, 1023, 2046, 2047, 4095, 4095, 8190, 8191, 16383, 16383, 32766, 32767, 65535, 65535, 131070, 131071, 262143, 262143, 524286, 524287, 1048575, 1048575, 2097150, 2097151

Offset: 3

Ralf Stephan, Jan 03 2003

From Alexander Adamchuk, Nov 16 2009: (Start)
For n>1 a(2n+1) = 2^(n-1) - 1 = A000225(n-1).
For n>1 a(4n) = a(4n+1) - 1 = 2^(2n-1) - 2.
For n>0 a(4n+2) = a(4n+3) = 2^(2n) - 1. (End)

Andrey Zabolotskiy, Table of n, a(n) for n = 3..1000
A. Stoimenow, Generating Functions, Fibonacci Numbers and Rational Knots, Journal of Algebra, 310 (2007), 491-525; arXiv:math/0210174 [math.GT], 2002.
Index entries for linear recurrences with constant coefficients, signature (0,2,0,1,0,-2).

Cf. A000225, A018240, A089891, A089892, A089797, A051449, A214927, A051450, A078478.

CoefficientList[Series[(2 x^7 + 2 x^6 - x^5 + x^3 - x^2 + x + 1) / ((x-1) (x+1) (x^2+1) (2 x^2-1)), {x, 0, 50}], x] (* Vincenzo Librandi, May 17 2013 *)

Vec(x^3*(1+x-x^2+x^3-x^5+2*x^6+2*x^7)/((1-x)*(1+x)*(1+x^2)*(1-2*x^2)) + O(x^60)) \\ Colin Barker, Dec 26 2015

G.f.: x^3*(1+x-x^2+x^3-x^5+2*x^6+2*x^7) / ((1-x)*(1+x)*(1+x^2)*(1-2*x^2)).

a(n) = 2*a(n-2)+a(n-4)-2*a(n-6) for n>10. - Colin Barker, Dec 26 2015

A318050 Triangle read by rows: T(n,k) is the number of prime knots with n crossings whose unknotting numbers are k.

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 1, 0, 3, 0, 3, 3, 1, 0, 9, 11, 1, 0, 17, 22, 9, 1

Offset: 3

Franck Maminirina Ramaharo, Aug 14 2018

The unknotting number of a knot is the minimal number of crossing switches required to convert a knot into the unknot (0 crossing).
Row n is a partition of A002863(n).
Row 10 cannot yet be completed because the unknotting number of some knots are still unknown.

			Triangle begins:
n\k|  0   1   2   3   4
---+-------------------
3  |  0   1
4  |  0   1
5  |  0   1   1
6  |  0   3
7  |  0   3   3   1
8  |  0   9  11   1
9  |  0  17  22   9   1

P. R. Cromwell, Knots and Links, Cambridge University Press, 2004, pp. 151-154.

S. A. Bleiler, A note on unknotting number, Math. Proc. Camb. Phil. Soc. Vol. 96 (1984).
M. Borodzik and S. Friedl, The unknotting number and classical invariants, I, Algebraic and Geometric Topology Vol. 15 (2015), 85-135.
J. C. Cha and C. Livingston, KnotInfo: Table of Knot Invariants.
S. Jablan and L. Radovic, Unknotting numbers of alternating knot and link families, Publications de l'Institut Mathématiques, Nouvelle série, tome 95 (2014), 87-99.
K. Murasugi, On a certain numerical invariant of link types, Trans. Am. Math. Soc. Vol. 117 (1965), 387-422.
Eric Weisstein's World of Mathematics, Unknotting Number.
Index entries for sequences related to knots

Cf. A002863, A078477, A089797, A089891, A089892, A173466, A318051, A318052.

A318052 Number of prime knots with n crossings whose unknotting numbers are given by their signatures.

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 5, 8, 22, 51, 182, 562

Offset: 1

Franck Maminirina Ramaharo, Aug 14 2018

a(n) counts the prime knots with n crossings satisfying u(K) = (1/2)*abs(sigma(K)), where u(K) denote the unknotting numbers of the knot K, and sigma(K) its signature.

			Let K denote a prime knot in Alexander-Briggs notation, and let sigma(K) and u(K) denote the signature and the unknotting number of the knot K, respectively. The following table gives some of the first prime knots with the property u(K) = (1/2)*abs(sigma(K)).
==================================================================
|    K     | 3_1 | 5_1 | 5_2 | 6_2 | 7_1 | 7_2 | 7_5 | 7_6 | 8_2 |
-----------+-----+-----+-----+-----+-----+-----+-----+-----+-----+
| sigma(K) | -2  | -4  | -2  | -2  | -6  | -2  | -4  | -2  | -4  |
-----------+-----+-----+-----+-----+-----+-----+-----+-----+-----+
|   u(K)   |  1  |  2  |  1  |  1  |  3  |  1  |  2  |  1  |  2  |
==================================================================

P. R. Cromwell, Knots and Links, Cambridge University Press, 2004, pp. 151-154.

S. A. Bleiler, A note on unknotting number, Math. Proc. Camb. Phil. Soc. Vol. 96 (1984).
J. C. Cha and C. Livingston, KnotInfo: Table of Knot Invariants.
T. Kanenobu and S. Matsumura, Lower bound of the unknotting number of prime knots with up to 12 crossings, Journal of Knot Theory and Its Ramifications Vol. 24 (2015).
K. Murasugi, On a certain numerical invariant of link types, Trans. Am. Math. Soc. Vol.117 (1965), 387-422.
Eric Weisstein's World of Mathematics, Knot Signature.
Eric Weisstein's World of Mathematics, Unknotting Number.
Index entries for sequences related to knots

Cf. A002863, A078477, A089797, A089891, A089892, A172293, A172184, A172441, A172444, A172486, A173466, A318050, A318051.

cp's OEIS Frontend

A078477 Number of rational knots with n crossings and unknotting number = 1 (chiral pairs counted only once).

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A318050 Triangle read by rows: T(n,k) is the number of prime knots with n crossings whose unknotting numbers are k.

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A318052 Number of prime knots with n crossings whose unknotting numbers are given by their signatures.

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