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

0,1

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

Table of n, a(n) for n=0..81.

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.

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

Cf. A029694, A051764, A137396, A300453.

Sequence in context: A276791 A111900 A173860 * A120524 A014177 A014129

Adjacent sequences: A123189 A123190 A123191 * A123193 A123194 A123195

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