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A318051
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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.
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3
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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, 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, 162, 0, 34
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OFFSET
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3,10
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COMMENTS
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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.
Row n is a partition of A002863(n).
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REFERENCES
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P. R. Cromwell, Knots and Links, Cambridge University Press, 2004, pp. 151-154.
W. B. R. Lickorish, An introduction to Knot Theory, Springer, 1997, Table 8.1, p. 85.
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LINKS
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EXAMPLE
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Triangle begins:
n\k| 0 1 2 3 4 5 6 7 8 9 10
---+--------------------------------------------
3 | 0 0 1
4 | 1
5 | 0 0 1 0 1
6 | 2 0 1
7 | 1 0 3 0 2 0 1
8 | 9 0 8 0 3 0 1
9 | 11 0 21 0 12 0 4 0 1
10 | 54 0 68 0 32 0 10 0 1
11 | 148 0 228 0 124 0 44 0 7 0 1
12 | 619 0 900 0 461 0 162 0 34
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CROSSREFS
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KEYWORD
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nonn,hard,more,tabf
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AUTHOR
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STATUS
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approved
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