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%I #40 Feb 02 2023 05:07:36
%S -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,
%T -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,
%U 0,0,0,0,0,0,0,-1,-1,0,0,0,1,0,0,0,-1,0,0,0
%N 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.
%C From _Franck Maminirina Ramaharo_, Aug 11 2018: (Start)
%C Using Kauffman's notation, the formal expression of the bracket polynomial for the (2,n)-torus knot is defined as follows:
%C 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.
%C - 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).
%C - 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.
%C - 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.
%C The planar diagram of the (2,0)-torus knot is two non-intersecting circles.
%C (End)
%D Louis H. Kauffman, Knots and Physics (Third Edition), World Scientific, 2001. See p. 38 and p. 353.
%H Paul Corbitt, <a href="http://educ.jmu.edu/~taalmala/OJUPKT/">Torus Links and the Bracket Polynomial</a>.
%H Louis H. Kauffman, <a href="https://doi.org/10.1016/0040-9383(87)90009-7">State models and the Jones polynomial</a>, Topology Vol. 26 (1987), 395-407.
%H Franck Ramaharo, <a href="https://arxiv.org/abs/1911.04528">Note on sequences A123192, A137396 and A300453</a>, arXiv:1911.04528 [math.CO], 2019.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BracketPolynomial.html">Bracket Polynomial</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TorusKnot.html">Torus Knot</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Torus_knot">Torus knot</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Knot_(mathematics)#Medial_graph">Medial graph</a>.
%e From _Franck Maminirina Ramaharo_, Aug 11 2018: (Start)
%e The bracket polynomial for some value of n:
%e p(0;x) = -x^2 - 1/x^2;
%e p(1;x) = -x^3;
%e p(2;x) = -x^4 - 1/x^4;
%e p(3;x) = -x^5 - 1/x^3 + 1/x^7;
%e p(4;x) = -x^6 - 1/x^2 + 1/x^6 - 1/x^10;
%e p(5;x) = -x^7 - 1/x + 1/x^5 - 1/x^9 + 1/x^13;
%e p(6;x) = -x^8 - 1 + 1/x^4 - 1/x^8 + 1/x^12 - 1/x^16;
%e p(7;x) = -x^9 - x + 1/x^3 - 1/x^7 + 1/x^11 - 1/x^15 + 1/x^19;
%e ...
%e The triangle giving the coefficients in x^abs(3*n - 2)*p(n;x) begins:
%e -1, 0, 0, 0, -1
%e 0, 0, 0, 0, -1
%e -1, 0, 0, 0, 0, 0, 0, 0, -1
%e 1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, -1
%e -1, 0, 0, 0, 1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, -1
%e 1, 0, 0, 0, -1, 0, 0, 0, 1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, -1
%e ...
%e (End)
%o (Maxima)
%o K(n, A, B, d) := if n = 0 then d else A*K(n - 1, A, B, d) + B*(A + B*d)^(n - 1)$
%o p(n, x) := x^abs(3*n - 2)*K(n, x, 1/x, -x^(-2) - x^2)$
%o t(n, k) := ratcoef(p(n, x), x, k)$
%o T:[]$
%o for n:0 thru 10 do T:append(T, makelist(t(n,k), k, 0, max(4, 4*n)))$
%o T; /* _Franck Maminirina Ramaharo_, Aug 11 2018 */
%Y Cf. A029694, A051764, A137396, A300453.
%K tabf,sign
%O 0,1
%A _Roger L. Bagula_, Oct 03 2006
%E Partially edited by _N. J. A. Sloane_, May 22 2007
%E Edited, new name, and corrected by _Franck Maminirina Ramaharo_, Aug 11 2018
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