More on Kashaev limits of the quantum $A$-polynomials

“Colored” knot polynomials satisfy difference equation w.r.t. the highest weights of the underlying representation – which in the case of symmetrically colored Jones are named “quantum $A$-polynomials”. In the double scaling quasiclassical (Kashaev) limit, when representation size $r\sim {\mathrm{\hslash }}^{-1}$, there are different phases – in one of them the classical action vanishes and in another one it is a deformation of hyperbolic volume (of a knot complement in $S^3$). This corresponds to a splitting of the non-homogeneous version of the quantum $A$-polynomial into two pieces, which we illustrate by more examples than just a figure-eight knot $4_1$ in the original paper. From the point of view of quasiclassics, hyperbolic volume is just an integration constant, which is not fully determined by the $A$-polynomial equation – and actually remains ambiguous in this formalism. As a byproduct, we expect that classical $A$-polynomial at $L=1$ becomes proportional to Alexander: $A^{\mathcal{K}}(1,M)\sim {\mathrm{\Delta }}^{\mathcal{K}}(M)$ – this seems true, but $A$ should be consistent with the polynomiality of {\it non-homogeneous quantum} $\mathcal{A}$-polynomial, what sometime implies that it is not minimal.