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
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.
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
KEYWORD
nonn,hard,more
AUTHOR
Franck Maminirina Ramaharo, Aug 14 2018
STATUS
approved