A318050 -id:A318050 - cp's OEIS frontend

A173466 a(n) is the number of prime knots with n crossings such that the empirical unknotting numbers cannot be decided minimals using their signatures.

0, 0, 0, 1, 0, 2, 2, 13, 27, 114, 370, 1614

Offset: 1

Jonathan Vos Post, Nov 22 2010

From Franck Maminirina Ramaharo, Aug 14 2018: (Start)
Prime knots considered in the sequence are those satisfying (1/2)*abs(sigma(K)) < u(K), where sigma(K) denotes the signature of the knot K and u(K) the unknotting number.
Complement of A318052. (End)

			From _Franck Maminirina Ramaharo_, Aug 14 2018: (Start)
Let K denote a prime knot in Alexander-Briggs notation, s(K) = (1/2)*abs(sigma(K)) and u(K) = unknotting number of K. The following table gives some of the first prime knots with the property s(K) != u(K).
==============================================================
|  K   | 4_1 | 6_1 | 6_3 | 7_4 | 7_7 | 8_1 | 8_3 | 8_4 | 8_6 |
-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+
| s(K) |  0  |  0  |  0  |  1  |  0  |  0  |  0  |  1  |  1  |
-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+
| u(K) |  1  |  1  |  1  |  2  |  1  |  1  |  2  |  2  |  2  |
==============================================================
(End)

Peter R. Cromwell, Knots and Links, Cambridge University Press, November 15, 2004, pp. 151-154.
W. B. R. Lickorish, An introduction to Knot Theory, Springer, 1997, Table 8.1, p. 85.

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, A172184, A172293, A172441, A172444, A318050, A318051, A318052.

a(n) = A002863(n) - A318052(n). - Franck Maminirina Ramaharo, Aug 14 2018

Edited, new name, and corrected by Franck Maminirina Ramaharo, Aug 14 2018

A318051 Irregular triangle read by rows: T(n,k) is the number of prime knots with n crossings whose signatures are k in absolute value.

Original entry on oeis.org

0, 0, 1, 1, 0, 0, 1, 0, 1, 2, 0, 1, 1, 0, 3, 0, 2, 0, 1, 9, 0, 8, 0, 3, 0, 1, 11, 0, 21, 0, 12, 0, 4, 0, 1, 54, 0, 68, 0, 32, 0, 1, 0, 1, 148, 228, 0, 124, 0, 44, 7, 0, 1, 619, 0, 900, 0, 461, 0, 162, 0, 34

Offset: 3

Franck Maminirina Ramaharo, Aug 14 2018

The signature of a knot is a classical lower bound for the unknotting number of knots. If sigma(K) and u(K) denote the signature and the unknotting number of the knot K, respectively, then 0 <= (1/2)*abs(sigma(K)) <= u(K). If one can empirically find an unknotting number u*(K) = (1/2)*abs(sigma(K)), then it is its exact value.

Row n is a partition of A002863(n).

			Triangle begins:
n\k|   0   1   2   3   4   5   6   7   8   9  10
---+--------------------------------------------
3  |   0   0   1
4  |   1
5  |   0   0   1   0   1
6  |   2   0   1
7  |   1   0   3   0   2   0   1
8  |   9   0   8   0   3   0   1
9  |  11   0  21   0  12   0   4   0   1
10 |  54   0  68   0  32   0  10   0   1
11 | 148   0 228   0 124   0  44   0   7   0   1
12 | 619   0 900   0 461   0 162   0  34

P. R. Cromwell, Knots and Links, Cambridge University Press, 2004, pp. 151-154.
W. B. R. Lickorish, An introduction to Knot Theory, Springer, 1997, Table 8.1, p. 85.

J. C. Cha and C. Livingston, KnotInfo: Table of Knot Invariants
J. C. Cha and C. Livingston, Signature
K. Murasugi, On a certain numerical invariant of link types, Trans. Am. Math. Soc. Vol. 117 (1965), 387-422.
A. Stoimenow, Table of the signature
Eric Weisstein's World of Mathematics, Knot Signature
Wikipedia, Signature of a knot
Index entries for sequences related to knots

Cf. A002863, A172293, A172184, A172441, A172444, A172486, A173466, A318050, 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

A173466 a(n) is the number of prime knots with n crossings such that the empirical unknotting numbers cannot be decided minimals using their signatures.

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A318051 Irregular triangle read by rows: T(n,k) is the number of prime knots with n crossings whose signatures are k in absolute value.

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs