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

1,6

Comments

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)

References

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.

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

Formula

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

Example

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)

Crossrefs

Cf. A002863, A172184, A172293, A172441, A172444, A318050, A318051, A318052.

Sequence in context: A306738 A151352 A155915 * A151367 A368957 A057648

Adjacent sequences: A173463 A173464 A173465 * A173467 A173468 A173469

Keyword

nonn,more,hard

Author

Jonathan Vos Post, Nov 22 2010

Extensions

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

Status

approved