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A318052 Number of prime knots with n crossings whose unknotting numbers are given by their signatures.

%I A318052 #18 Apr 06 2025 07:42:05
%S A318052 0,0,1,0,2,1,5,8,22,51,182,562
%N A318052 Number of prime knots with n crossings whose unknotting numbers are given by their signatures.
%C A318052 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.
%D A318052 P. R. Cromwell, Knots and Links, Cambridge University Press, 2004, pp. 151-154.
%H A318052 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 A318052 J. C. Cha and C. Livingston, <a href="https://knotinfo.math.indiana.edu/">KnotInfo: Table of Knot Invariants</a>.
%H A318052 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 A318052 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 A318052 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/KnotSignature.html">Knot Signature</a>.
%H A318052 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/UnknottingNumber.html">Unknotting Number</a>.
%H A318052 <a href="/index/K#knots">Index entries for sequences related to knots</a>
%e A318052 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)).
%e A318052 ==================================================================
%e A318052 |    K     | 3_1 | 5_1 | 5_2 | 6_2 | 7_1 | 7_2 | 7_5 | 7_6 | 8_2 |
%e A318052 -----------+-----+-----+-----+-----+-----+-----+-----+-----+-----+
%e A318052 | sigma(K) | -2  | -4  | -2  | -2  | -6  | -2  | -4  | -2  | -4  |
%e A318052 -----------+-----+-----+-----+-----+-----+-----+-----+-----+-----+
%e A318052 |   u(K)   |  1  |  2  |  1  |  1  |  3  |  1  |  2  |  1  |  2  |
%e A318052 ==================================================================
%Y A318052 Cf. A002863, A078477, A089797, A089891, A089892, A172293, A172184, A172441, A172444, A172486, A173466, A318050, A318051.
%K A318052 nonn,hard,more
%O A318052 1,5
%A A318052 _Franck Maminirina Ramaharo_, Aug 14 2018