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Irregular triangle read by rows: T(n,k) is the number of prime knots with n crossings whose signatures are k in absolute value.
3

%I #26 Feb 16 2025 08:33:56

%S 0,0,1,1,0,0,1,0,1,2,0,1,1,0,3,0,2,0,1,9,0,8,0,3,0,1,11,0,21,0,12,0,4,

%T 0,1,54,0,68,0,32,0,1,0,1,148,228,0,124,0,44,7,0,1,619,0,900,0,461,0,

%U 162,0,34

%N Irregular triangle read by rows: T(n,k) is the number of prime knots with n crossings whose signatures are k in absolute value.

%C The signature of a knot is a classical lower bound for the unknotting number of knots. If sigma(K) and u(K) denote the signature and the unknotting number of the knot K, respectively, then 0 <= (1/2)*abs(sigma(K)) <= u(K). If one can empirically find an unknotting number u*(K) = (1/2)*abs(sigma(K)), then it is its exact value.

%C Row n is a partition of A002863(n).

%D P. R. Cromwell, Knots and Links, Cambridge University Press, 2004, pp. 151-154.

%D W. B. R. Lickorish, An introduction to Knot Theory, Springer, 1997, Table 8.1, p. 85.

%H J. C. Cha and C. Livingston, <a href="http://www.indiana.edu/~knotinfo">KnotInfo: Table of Knot Invariants</a>

%H J. C. Cha and C. Livingston, <a href="https://www.indiana.edu/~knotinfo/descriptions/signature.html">Signature</a>

%H 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 A. Stoimenow, <a href="http://stoimenov.net/stoimeno/homepage/ptab/sig10.html">Table of the signature</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/KnotSignature.html">Knot Signature</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Signature_of_a_knot">Signature of a knot</a>

%H <a href="/index/K#knots">Index entries for sequences related to knots</a>

%e Triangle begins:

%e n\k| 0 1 2 3 4 5 6 7 8 9 10

%e ---+--------------------------------------------

%e 3 | 0 0 1

%e 4 | 1

%e 5 | 0 0 1 0 1

%e 6 | 2 0 1

%e 7 | 1 0 3 0 2 0 1

%e 8 | 9 0 8 0 3 0 1

%e 9 | 11 0 21 0 12 0 4 0 1

%e 10 | 54 0 68 0 32 0 10 0 1

%e 11 | 148 0 228 0 124 0 44 0 7 0 1

%e 12 | 619 0 900 0 461 0 162 0 34

%Y Cf. A002863, A172293, A172184, A172441, A172444, A172486, A173466, A318050, A318052.

%K nonn,hard,more,tabf,changed

%O 3,10

%A _Franck Maminirina Ramaharo_, Aug 14 2018