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

%I A059739 #41 Jul 21 2025 09:18:56
%S A059739 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,
%T A059739 384,146,17,1,1288,1408,500,100,11,1,4878,5100,2074,341,23,1,19536,
%U A059739 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
%N A059739 Triangle T(n,k), n >= 1, giving number of prime unoriented alternating links with n crossings and k components.
%C A059739 A link is a not necessarily connected knot. Apart from the initial rows, the n-th row contains floor(n/2) terms.
%D A059739 Ortho Flint, Bruce Fontaine and Stuart Rankin, The master array of a prime alternating link, preprint, 2007
%H A059739 Stuart Rankin (srankin(AT)uwo.ca), Nov 05 2007, <a href="/A059739/b059739.txt">Table of n, a(n) for n = 0..133</a>
%H A059739 Steven R. Finch, <a href="/A002863/a002863_4.pdf">Knots, links and tangles</a>, August 8, 2003. [Cached copy, with permission of the author]
%H A059739 Ortho Flint, Bruce Fontaine and Stuart Rankin, <a href="https://web.archive.org/web/20150912132825/http://www-home.math.uwo.ca/~srankin/papers/knots/altlnk.submit.pdf">Enumerating the prime alternating links</a>, preprint, 2007.
%H A059739 Ortho Flint and Stuart Rankin, <a href="http://dx.doi.org/10.1142/S0218216504003068">Enumerating the prime alternating links</a>, Journal of Knot theory and its Ramifications, 13 (2004), 151-173.
%H A059739 Knotilus web site, <a href="http://knotilus.math.uwo.ca">Knotilus</a>. [dead link]
%H A059739 S. Rankin and O. Flint, <a href="https://web.archive.org/web/20141007011116/http://www.math.uwo.ca/~srankin/knots/knotprint.html">Knot theory</a> web page.
%H A059739 M. B. Thistlethwaite, <a href="https://web.math.utk.edu/~mthistle/">Home Page</a>.
%H A059739 M. B. Thistlethwaite, <a href="https://web.math.utk.edu/~mthistle/png/link_stats.png">Numbers of knots and links with up to 19 crossings</a>.
%H A059739 <a href="/index/K#knots">Index entries for sequences related to knots</a>
%e A059739 First few rows of irregular triangle:
%e A059739    0
%e A059739    0  1
%e A059739    1
%e A059739    1  1
%e A059739    2  1
%e A059739    3  3  2
%e A059739    7  6  1
%e A059739   18 14  6 1
%e A059739   41 42 12 1
%e A059739   ...
%Y A059739 First column gives numbers of knots, A002864. Second column gives A059741. Row sums give A049344.
%K A059739 nonn,tabf,nice
%O A059739 0,7
%A A059739 _N. J. A. Sloane_, Feb 10 2001
%E A059739 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
%E A059739 Trailing 0 in row for n=2 removed by _N. J. A. Sloane_, Nov 21 2007