OFFSET
0,8
COMMENTS
This is essentially the usual Pascal triangle A007318, horizontally shifted and with the first three columns altered.
Let P(n;x) = (x + 1)^n + x^2 - 1. Then P(n;x) = P(n-1;x) + x*(x + 1)^(n - 1), with P(0;x) = x^2.
Let a (2,n)-torus knot be projected on the plane. The resulting projection is a planar diagram with n double points. Then, T(n,k) gives the number of state diagrams having k components that are obtained by deleting each double point, then pasting the edges in one of the two ways as shown below.
\ / \___/ \ / \ /
(1) \/ ==> (2) \/ ==> | |
/\ ___ /\ | |
/ \ / \ / \ / \
See example for the case n = 2.
REFERENCES
Colin Adams, The Knot Book, W. H. Freeman and Company, 1994.
Louis H. Kauffman, Knots and Physics, World Scientific Publishers, 1991.
LINKS
Michael De Vlieger, Table of n, a(n) for n = 0..11327 (rows 0 <= n <= 150, flattened).
Agnijo Banerjee, Knot theory [Foil knot family].
Allison Henrich, Rebecca Hoberg, Slavik Jablan, Lee Johnson, Elizabeth Minten and Ljiljana Radovic, The Theory of Pseudoknots, arXiv preprint arXiv:1210.6934 [math.GT], 2012.
Abdullah Kopuzlu, Abdulgani Şahin and Tamer Ugur, On polynomials of K(2,n) torus knots, Applied Mathematical Sciences, Vol. 3 (2009), 2899-2910.
Franck Ramaharo, A generating polynomial for the two-bridge knot with Conway's notation C(n,r), arXiv:1902.08989 [math.CO], 2019.
Franck Ramaharo, Note on sequences A123192, A137396 and A300453, arXiv:1911.04528 [math.CO], 2019.
Franck Ramaharo, A bracket polynomial for 2-tangle shadows, arXiv:2002.06672 [math.CO], 2020.
Wikipedia, Torus knot.
Xinfei Li, Xin Liu and Yong-Chang Huang, Tackling tangledness of cosmic strings by knot polynomial topological invariants, arxiv preprint arXiv:1602.08804 [hep-th], 2016.
FORMULA
EXAMPLE
The triangle T(n,k) begins
n\k 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
0: 0 0 1
1: 0 1 1
2: 0 2 2
3: 0 3 4 1
4: 0 4 7 4 1
5: 0 5 11 10 5 1
6: 0 6 16 20 15 6 1
7: 0 7 22 35 35 21 7 1
8: 0 8 29 56 70 56 28 8 1
9: 0 9 37 84 126 126 84 36 9 1
10: 0 10 46 120 210 252 210 120 45 10 1
11: 0 11 56 165 330 462 462 330 165 55 11 1
12: 0 12 67 220 495 792 924 792 495 220 66 12 1
13: 0 13 79 286 715 1287 1716 1716 1287 715 286 78 13 1
14: 0 14 92 364 1001 2002 3003 3432 3003 2002 1001 364 91 14 1
...
The states of the (2,2)-torus knot (Hopf Link) are the last four diagrams:
____ ____
/ \/ \
/ /\ \
| | | |
| | | |
\ \/ /
\____/\____/
___ ____ __________
/ \ / \ / __ \
/ / \ \ / / \ \
| | | | | | | |
| | | | | | | |
\ \/ / \ \/ /
\_____/\_____/ \____/\____/
____ ____ ____ ____ ____________ __________
/ \ / \ / \ / \ / __ \ / __ \
/ / \ \ / / \ \ / / \ \ / / \ \
| | | | | | | | | | | | | | | |
| | | | | | | | | | | | | | | |
\ \ / / \ \__/ / \ \ / / \ \__/ /
\____/ \____/ \____________/ \____/ \____/ \__________/
There are 2 diagrams that consist of two components, and 2 diagrams that consist of one component.
MATHEMATICA
f[n_] := CoefficientList[ Expand[(x + 1)^n + x^2 - 1], x]; Array[f, 12, 0] // Flatten (* or *)
CoefficientList[ CoefficientList[ Series[(x^2 + y*x/(1 - y*(x + 1)))/(1 - y), {y, 0, 11}, {x, 0, 11}], y], x] // Flatten (* Robert G. Wilson v, Mar 08 2018 *)
PROG
(Maxima)
P(n, x) := (x + 1)^n + x^2 - 1$
T : []$
for i:0 thru 20 do
T : append(T, makelist(ratcoef(P(i, x), x, n), n, 0, max(2, i)))$
T;
(PARI) row(n) = Vecrev((x + 1)^n + x^2 - 1);
tabl(nn) = for (n=0, nn, print(row(n))); \\ Michel Marcus, Mar 12 2018
CROSSREFS
Row sums: A000079 (powers of 2).
Triangles related to the regular projection of some knots: A299989 (connected summed trefoils); A300184 (chain links); A300454 (twist knot).
KEYWORD
nonn,tabf
AUTHOR
Franck Maminirina Ramaharo, Mar 06 2018
STATUS
approved