Reference for birational equivalence of $A$-polynomial curve and character variety

Question

For $K$ a knot in $S^3$, the character variety $\mathfrak{X}_K$ parametrizes conjugacy classes of representations $\pi_1(S^3 \setminus K) \to \operatorname{SL}_2(\mathbb C)$. Another object that does this is the $A$-polynomial $A_K(M,L)$, which cuts out a (possibly reducible) algebraic curve $\mathfrak A_K$ in $\mathbb C^2$ representing the eigenvalues of the meridian and longitude. I've seen a couple people mention that the "interesting" components of $\mathfrak{X}_K$ (in particular, the geometric component) are birationally equivalent to components of $\mathfrak A_K$. However, I don't have a specific reference with a precise statement or proof. Does anyone know of one?

Answer 1

Via email, Ian Agol pointed me to a paper [1] of Dunfield. For $K$ a hyperbolic knot, Theorem 3.1 of that paper proves that restriction is a birational isomorphism between the geometric component (the one containing the discrete faithful representation) of $\mathfrak X_K$ and the geometric component of $\mathfrak A_K$. The author also remarks that restriction need not be an isomorphism for the other components.

[1] Dunfield, N. Cyclic surgery, degrees of maps of character curves, and volume rigidity for hyperbolic manifolds. Invent. math. 136, 623–657 (1999). https://doi.org/10.1007/s002220050321