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A123192
Triangle read by rows: row n gives the coefficients in the expansion of x^abs(3*n - 2)*p(n;x), where p(n;x) denotes the bracket polynomial for the (2,n)-torus knots.
3
-1, 0, 0, 0, -1, 0, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, -1, -1, 0, 0, 0, 1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, -1, 0, 0, 0, 1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, -1, -1, 0, 0, 0, 1, 0, 0, 0, -1, 0, 0, 0
OFFSET
0,1
COMMENTS
From Franck Maminirina Ramaharo, Aug 11 2018: (Start)
Using Kauffman's notation, the formal expression of the bracket polynomial for the (2,n)-torus knot is defined as follows:
K(n;A,B,d) = A*K(n-1;A,B,d) + B*(A + B*d)^(n - 1) with K(0;A,B,d) = d.
- The polynomial in this sequence is defined as p(n;x) = K(n;x,1/x,-x^2-x^(-2)), and verifies p(n;x) = x*p(n-1;x) + (-1)^(n - 1)*x^(-3*n + 2).
- The polynomial x*K(n;1,1,x) yields (x + 1)^n + x^2 - 1 which is the bracket evaluated at the shadow diagram of the (2,n)-torus knot, see A300453.
- The polynomial sqrt(x)*K(n;-1,sqrt(x),sqrt(x)) yields (x - 1)^n + (x - 1)*(-1)^n. This is the chromatic polynomial for the n-cycle graph which is the medial graph of the (2,n)-torus knot, see A137396.
The planar diagram of the (2,0)-torus knot is two non-intersecting circles.
(End)
REFERENCES
Louis H. Kauffman, Knots and Physics (Third Edition), World Scientific, 2001. See p. 38 and p. 353.
LINKS
Louis H. Kauffman, State models and the Jones polynomial, Topology Vol. 26 (1987), 395-407.
Franck Ramaharo, Note on sequences A123192, A137396 and A300453, arXiv:1911.04528 [math.CO], 2019.
Eric Weisstein's World of Mathematics, Bracket Polynomial.
Eric Weisstein's World of Mathematics, Torus Knot.
Wikipedia, Torus knot.
Wikipedia, Medial graph.
EXAMPLE
From Franck Maminirina Ramaharo, Aug 11 2018: (Start)
The bracket polynomial for some value of n:
p(0;x) = -x^2 - 1/x^2;
p(1;x) = -x^3;
p(2;x) = -x^4 - 1/x^4;
p(3;x) = -x^5 - 1/x^3 + 1/x^7;
p(4;x) = -x^6 - 1/x^2 + 1/x^6 - 1/x^10;
p(5;x) = -x^7 - 1/x + 1/x^5 - 1/x^9 + 1/x^13;
p(6;x) = -x^8 - 1 + 1/x^4 - 1/x^8 + 1/x^12 - 1/x^16;
p(7;x) = -x^9 - x + 1/x^3 - 1/x^7 + 1/x^11 - 1/x^15 + 1/x^19;
...
The triangle giving the coefficients in x^abs(3*n - 2)*p(n;x) begins:
-1, 0, 0, 0, -1
0, 0, 0, 0, -1
-1, 0, 0, 0, 0, 0, 0, 0, -1
1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, -1
-1, 0, 0, 0, 1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, -1
1, 0, 0, 0, -1, 0, 0, 0, 1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, -1
...
(End)
PROG
(Maxima)
K(n, A, B, d) := if n = 0 then d else A*K(n - 1, A, B, d) + B*(A + B*d)^(n - 1)$
p(n, x) := x^abs(3*n - 2)*K(n, x, 1/x, -x^(-2) - x^2)$
t(n, k) := ratcoef(p(n, x), x, k)$
T:[]$
for n:0 thru 10 do T:append(T, makelist(t(n, k), k, 0, max(4, 4*n)))$
T; /* Franck Maminirina Ramaharo, Aug 11 2018 */
CROSSREFS
KEYWORD
tabf,sign
AUTHOR
Roger L. Bagula, Oct 03 2006
EXTENSIONS
Partially edited by N. J. A. Sloane, May 22 2007
Edited, new name, and corrected by Franck Maminirina Ramaharo, Aug 11 2018
STATUS
approved