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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.

%I A173466 #44 Apr 06 2025 07:40:20
%S A173466 0,0,0,1,0,2,2,13,27,114,370,1614
%N 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.
%C A173466 From _Franck Maminirina Ramaharo_, Aug 14 2018: (Start)
%C A173466 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.
%C A173466 Complement of A318052. (End)
%D A173466 Peter R. Cromwell, Knots and Links, Cambridge University Press, November 15, 2004, pp. 151-154.
%D A173466 W. B. R. Lickorish, An introduction to Knot Theory, Springer, 1997, Table 8.1, p. 85.
%H A173466 S. A. Bleiler, <a href="http://dx.doi.org/10.1017/S0305004100062381">A note on unknotting number</a>, Math. Proc. Camb. Phil. Soc. Vol. 96 (1984).
%H A173466 J. C. Cha and C. Livingston, <a href="https://knotinfo.math.indiana.edu/">KnotInfo: Table of Knot Invariants</a>.
%H A173466 T. Kanenobu and S. Matsumura, <a href="https://doi.org/10.1142/S021821651540012X">Lower bound of the unknotting number of prime knots with up to 12 crossings</a>, Journal of Knot Theory and Its Ramifications Vol. 24 (2015).
%H A173466 K. Murasugi, <a href="http://dx.doi.org/10.2307/1994215">On a certain numerical invariant of link types</a>, Trans. Am. Math. Soc. Vol. 117 (1965), 387-422.
%H A173466 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/KnotSignature.html">Knot Signature</a>.
%H A173466 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/UnknottingNumber.html">Unknotting Number</a>.
%H A173466 <a href="/index/K#knots">Index entries for sequences related to knots</a>
%F A173466 a(n) = A002863(n) - A318052(n). - _Franck Maminirina Ramaharo_, Aug 14 2018
%e A173466 From _Franck Maminirina Ramaharo_, Aug 14 2018: (Start)
%e A173466 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).
%e A173466 ==============================================================
%e A173466 |  K   | 4_1 | 6_1 | 6_3 | 7_4 | 7_7 | 8_1 | 8_3 | 8_4 | 8_6 |
%e A173466 -------+-----+-----+-----+-----+-----+-----+-----+-----+-----+
%e A173466 | s(K) |  0  |  0  |  0  |  1  |  0  |  0  |  0  |  1  |  1  |
%e A173466 -------+-----+-----+-----+-----+-----+-----+-----+-----+-----+
%e A173466 | u(K) |  1  |  1  |  1  |  2  |  1  |  1  |  2  |  2  |  2  |
%e A173466 ==============================================================
%e A173466 (End)
%Y A173466 Cf. A002863, A172184, A172293, A172441, A172444, A318050, A318051, A318052.
%K A173466 nonn,more,hard
%O A173466 1,6
%A A173466 _Jonathan Vos Post_, Nov 22 2010
%E A173466 Edited, new name, and corrected by _Franck Maminirina Ramaharo_, Aug 14 2018