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
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.
FORMULA
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
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