A318051 Irregular triangle read by rows: T(n,k) is the number of prime knots with n crossings whose signatures are k in absolute value.

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

Offset: 3

Author

Franck Maminirina Ramaharo, Aug 14 2018

Keywords

• nonn
• hard
• more
• tabf

Comments

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).

Examples

			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
		

References

• 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.

Links

• J. C. Cha and C. Livingston, KnotInfo: Table of Knot Invariants
• J. C. Cha and C. Livingston, Signature
• K. Murasugi, On a certain numerical invariant of link types, Trans. Am. Math. Soc. Vol. 117 (1965), 387-422.
• A. Stoimenow, Table of the signature
• Eric Weisstein's World of Mathematics, Knot Signature
• Wikipedia, Signature of a knot
• Index entries for sequences related to knots

Crossrefs

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