On finding A-polynomials

2012-07-30T23:38:32Z

I have two questions to obtain the explicit forms of A-polynomials.

Takata used the mathematica pacage qMultisum.m to obtain the recursion relation of the colored Jones polynomials for twist knots. As Stavros Garoufalidis and Xinyu Sun pointed out in this paper, the simple use of the mathematica pacage qZeil.m, qMultisum.m does not give the recursion relation of minimal order. They made use of the method, so-called creative telescoping, to get the recursion relation of minimal order by using the certificat function.

How do you implement this method in Mathematica, say, to get the recursion relation for $5_2$ and $6_1$ knots as in p.4 of the paper?

Recently, Gukov, Sulkowski and Fuji conjecture that, in the limit, \begin{equation} q = e^{\hbar} \to 1 \,, \qquad a = \text{fixed} \,, \qquad t = \text{fixed} \,, \qquad x = q^n = \text{fixed} \end{equation} the $n$-colored superpolynomials $P_n (K;a,q,t)$ exhibit the following ``large color'' behavior: \begin{equation} P_n (K;a,q,t) \;\overset{{n \to \infty \atop \hbar \to 0}}{\sim}\; \exp\left( \frac{1}{\hbar} \int \log y \frac{dx}{x} \,+\, \ldots \right) \end{equation} where ellipsis stand for regular terms (as $\hbar \to 0$) and the leading term is given by the integral on the zero locus of the super-$A$-polynomial: \begin{equation} A^{\text{super}} (x,y;a,t) \; = \; 0 \ . \end{equation}

For example, the critical points of the leading term of colored superpolynomials of torus knots $T^{2,2p+1}$ are give by \begin{eqnarray} 1 \; &=& \; -\frac{t^{-2-2p}(x-z_0)z_0^{-1-2p}(-1+t^2z_0)(1+ at^3 xz_0)}{(-1+z_0)(atx+z_0)(-1 + t^2 x z_0)} \cr y(x,t,a)&=& \frac{a^p t^{2 + 2 p} (-1 + x) x^{1 + 2 p} (atx + z_0) (1 + a t^3 x z_0)}{(1 + a t^3 x) (x - z_0) (-1 + t^2 x z_0)} , \end{eqnarray} which is written in Eq.(2.35) and (2.36). By eliminating $z_0$, you will obtain the super-$A$-polynomials for torus knots $T^{2,2p+1}$. Off course, it should be doable in principle, but

$2$. how can it be implemented explicitly to obtain the super-$A$-polynomials as in Table 5 of this paper? In other words, how do you explicitly eliminate $z_0$ in such a way that you will obtain the super-$A$-polynomials?

I have the same problem to obtain the $Q$-deformed $A$-polynomials from Eq.(A.21) in this paper.

On finding A-polynomials - MathOverflow

On finding A-polynomials