  Knot polynomials colored with symmetric representations of $SL_q(N)$ satisfy
difference equations as functions of representation parameter, which look like
quantization of classical ${\cal A}$-polynomials. However, they are quite
difficult to derive and investigate. Much simpler should be the equations for
coefficients of differential expansion nicknamed quantum ${\cal
C}$-polynomials. It turns out that, for each knot, one can actually derive two
difference equations of a finite order for these coefficients, those with
shifts in spin $n$ of the representation and in $A=q^N$. Thus, the ${\cal
C}$-polynomials are much richer and form an entire ring. We demonstrate this
with the examples of various defect zero knots, mostly discussing the entire
twist family.
