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

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

Offset

1,5

Comments

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.

References

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

Links

Table of n, a(n) for n=1..12.

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

Example

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  |
==================================================================

Crossrefs

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

Sequence in context: A167816 A316292 A222542 * A026078 A349635 A176665

Adjacent sequences: A318049 A318050 A318051 * A318053 A318054 A318055

Keyword

nonn,hard,more

Author

Franck Maminirina Ramaharo, Aug 14 2018

Status

approved