Consider the ribbon category of finite-dimensional representations of $\mathcal{U}_q(\mathfrak{sl}(2))$, with twist $\theta$. If $V$ is the vector representation, then $\theta_V$ is multiplication by $q^{-3}$. As described e.g. in Snyder-Tingley http://arxiv.org/abs/0810.0084, one could use an alternate twist $\theta'$ with $\theta'_V$ given by multiplication by $-q^{-3}$, but these are the only two options compatible with the braiding.

As a result, the Reshetikhin-Turaev invariant of framed links obtained by labeling all link components with $V$ depends on the framing: different framings change the invariant by $\pm q^{-3}$. This invariant is the Kauffman bracket; the Jones polynomial, which is independent of framing, is obtained from the Kauffman bracket by rescaling by powers of $q^3$ to remove the framing dependence.

I'm wondering whether the Jones polynomial itself, rather than the Kauffman bracket, can be obtained from a ribbon category using the Reshetikhin-Turaev connstruction. A similar question may have been asked at Invariants of unframed, oriented links from Reshetikhin Turaev construction, but the answer isn't quite what I'm looking for: I know the framing-dependent invariants can be rescaled at the end to produce a framing-independent invariant. What I want to know is whether this rescaling can be built in earlier in the Reshetikhin-Turaev construction.

It seems like this would involve rescaling not just the ribbon element $\theta$, but the braiding / $R$-matrix for $\mathcal{U}_q(\mathfrak{sl}(2))$ as well. But I don't see a way to do this that still satisfies the axioms of a braided monoidal category.

Ideally, I'd like a ribbon structure on the monoidal category of finite-dimensional representations of $\mathcal{U}_q(\mathfrak{sl}(2))$, such that the twist is trivial on the vector representation, and such that the Reshetikhin-Turaev invariant associated to the vector representation is the Jones polynomial. Alternatively, I'd take an explanation of why no such thing exists, or some weaker version that satisfies different axioms but is morally what I'm looking for.

Here's some additional motivation for this question: in http://arxiv.org/abs/1308.2047, Sartori presents a ribbon structure on the category of finite-dimensional representations of $\mathcal{U}_q(\mathfrak{gl}(1|1))$ and shows how to obtain the Alexander polynomial from (a slight modification of) the Reshetikhin-Turaev construction. In this setting, the twist acts as the identity on the vector representation and its dual (Lemma 4.2 of Sartori). Thus, the RT invariants for the vector representation are naturally framing-independent. I'm wondering if there's a way to set up the RT construction for $\mathcal{U}_q(\mathfrak{sl}(2))$ such that the same thing happens.