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.

-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

Author

Roger L. Bagula, Oct 03 2006

Keywords

• tabf
• sign

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)

Examples

			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)
		

References

• Louis H. Kauffman, Knots and Physics (Third Edition), World Scientific, 2001. See p. 38 and p. 353.

Links

• Paul Corbitt, Torus Links and the Bracket Polynomial.
• 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.

Crossrefs

Cf. A029694, A051764, A137396, A300453.

Programs

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 */

Extensions

Partially edited by N. J. A. Sloane, May 22 2007

Edited, new name, and corrected by Franck Maminirina Ramaharo, Aug 11 2018