In a nutshell, my question is:

Q0:is there a classification of invariant of (framed) tangles arising from the Reshetikhin–Turaev construction?

I will now make it more precise. One could define a *functorial invariant of (framed) tangles* to be a ribbon monoidal functor
$$
F\colon\mathbf{FrTang}\to \mathcal{V},
$$
where $\mathbf{FrTang}$ is the ribbon monoidal category (or tortile category) of oriented framed tangles and $\mathcal{V}$ is a ribbon monoidal category. By a result of Shum, $\mathbf{FrTang}$ is the free ribbon monoidal category on one object, so that such functor is determined by a choice of object in $\mathcal{V}$. Hence, classification reduces to classification of ribbon monoidal categories.

On the other hand, the Reshetikhin–Turaev construction states that given a ribbon Hopf algebra $A$, its category of finite-dimensional representations $\mathrm{mod}(A)$ is a ribbon monoidal category.

This leads me to the following two questions:

Q1:Does the Reshetikhin–Turaev construction giveallribbon monoidal categories, and does $\mathrm{mod}(A)$ determines $A$, taking the ribbon structure and fiber functor $\mathrm{mod}(A)\to\mathrm{vec}$ into account?

In other words, is there a Tannakian duality for ribbon monoidal categories, akin to the one for rigid (or autonomous) braided monoidal categories and quasi-triangular Hopf algebras?

Assuming a positive (or at least optimistic) answer to Q1:

Q2: is there a classification of ribbon Hopf algebras? Or at least of those of the form $U_q(\mathfrak{g})$ for some Lie algebra $\mathfrak{g}$?

Any partial answer would be greatly appreciated.