A059739 Triangle T(n,k), n >= 1, giving number of prime unoriented alternating links with n crossings and k components.

0, 0, 1, 1, 1, 1, 2, 1, 3, 3, 2, 7, 6, 1, 18, 14, 6, 1, 41, 42, 12, 1, 123, 121, 43, 9, 1, 367, 384, 146, 17, 1, 1288, 1408, 500, 100, 11, 1, 4878, 5100, 2074, 341, 23, 1, 19536, 21854, 8206, 1556, 181, 13, 1, 85263, 92234, 37222, 7193, 653, 29, 1, 379799, 427079, 172678, 33216, 3885, 301, 16, 1, 1769979, 2005800, 829904, 173549, 19122, 1129, 36, 1, 8400285, 9716848, 4194015, 876173, 105539, 8428, 471, 19, 1, 40619385, 48184018, 21207695, 4749914, 599433, 43513, 1813, 43, 1

Offset

0,7

Comments

A link is a not necessarily connected knot. Apart from the initial rows, the n-th row contains floor(n/2) terms.

References

Ortho Flint, Bruce Fontaine and Stuart Rankin, The master array of a prime alternating link, preprint, 2007

Links

Stuart Rankin (srankin(AT)uwo.ca), Nov 05 2007, Table of n, a(n) for n = 0..133

S. R. Finch, Knots, links and tangles, August 8, 2003. [Cached copy, with permission of the author]

Ortho Flint, Bruce Fontaine and Stuart Rankin, Enumerating the prime alternating links, preprint, 2007.

Ortho Flint and Stuart Rankin, Enumerating the prime alternating links, Journal of Knot theory and its Ramifications, 13 (2004), 151-173.

Knotilus web site, Knotilus.

S. Rankin and O. Flint, Knot theory web page.

M. B. Thistlethwaite, Home Page.

M. B. Thistlethwaite, Numbers of knots and links with up to 19 crossings.

Index entries for sequences related to knots

Example

First few rows of irregular triangle:
   0
   0  1
   1
   1  1
   2  1
   3  3  2
   7  6  1
  18 14  6 1
  41 42 12 1
  ...

Crossrefs

First column gives numbers of knots, A002864. Second column gives A059741. Row sums give A049344.

Sequence in context: A144305 A138635 A128182 * A035566 A110783 A333224

Adjacent sequences: A059736 A059737 A059738 * A059740 A059741 A059742

Keyword

nonn,tabf,nice

Author

N. J. A. Sloane, Feb 10 2001

Extensions

Terms for the 20-, 21-, 22- and 23-crossing prime alternating links (see the b-file) added Nov 03 2007 by Stuart Rankin, Ortho Flint and Bruce Fontaine

Trailing 0 in row for n=2 removed by N. J. A. Sloane, Nov 21 2007

Status

approved