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
Examples
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)
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.
- Index entries for sequences related to knots
Formula
Extensions
Edited, new name, and corrected by Franck Maminirina Ramaharo, Aug 14 2018
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