A318050 Triangle read by rows: T(n,k) is the number of prime knots with n crossings whose unknotting numbers are k.

0, 1, 0, 1, 0, 1, 1, 0, 3, 0, 3, 3, 1, 0, 9, 11, 1, 0, 17, 22, 9, 1

Offset: 3

Author

Franck Maminirina Ramaharo, Aug 14 2018

Keywords

• nonn
• hard
• more
• tabf

Comments

The unknotting number of a knot is the minimal number of crossing switches required to convert a knot into the unknot (0 crossing).

Row n is a partition of A002863(n).

Row 10 cannot yet be completed because the unknotting number of some knots are still unknown.

Examples

			Triangle begins:
n\k|  0   1   2   3   4
---+-------------------
3  |  0   1
4  |  0   1
5  |  0   1   1
6  |  0   3
7  |  0   3   3   1
8  |  0   9  11   1
9  |  0  17  22   9   1
		

References

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

Links

• S. A. Bleiler, A note on unknotting number, Math. Proc. Camb. Phil. Soc. Vol. 96 (1984).
• M. Borodzik and S. Friedl, The unknotting number and classical invariants, I, Algebraic and Geometric Topology Vol. 15 (2015), 85-135.
• J. C. Cha and C. Livingston, KnotInfo: Table of Knot Invariants.
• S. Jablan and L. Radovic, Unknotting numbers of alternating knot and link families, Publications de l'Institut Mathématiques, Nouvelle série, tome 95 (2014), 87-99.
• K. Murasugi, On a certain numerical invariant of link types, Trans. Am. Math. Soc. Vol. 117 (1965), 387-422.
• Eric Weisstein's World of Mathematics, Unknotting Number.
• Index entries for sequences related to knots

Crossrefs

Cf. A002863, A078477, A089797, A089891, A089892, A173466, A318051, A318052.