Quantum groups are useful for making knot/link invariants: for example, $U_q(\mathfrak{sl}_2$) you get the Jones polynomial. This boils down to the fact that $\mathcal C = \operatorname{rep }U_q(\mathfrak{sl}_2)$ is a braided monoidal category, which is **not** symmetric, hence gives us interesting knot invariants.

A deformation theoretic path to the quantum group $U_q(\mathfrak g)$ from a Lie algebra $\mathfrak g$ is as follows. Fixing a Casimir element $\Omega\in\mathfrak g\otimes \mathfrak g$ endows the category $\mathcal C=\operatorname{rep }\mathfrak g$ with the structure of an infinitesimally braided symmetric monoidal category. We can infinitesimally deform $\mathcal C$ to an honest braided monoidal category $\mathcal C[\hbar]/(\hbar^2)$ with braiding given by $\sigma\circ(1+\hbar\Omega/2)$. The next step is usually to use a Drinfeld associator to integrate this infinitesimal deformation to a formal one $\mathcal C[[\hbar]]$ with braiding given by $\sigma\circ\exp(\hbar\Omega/2)$. After some twisting/scrootching/variable replacement we get $\mathcal C[[\hbar]]=\operatorname{ rep}U_q(\mathfrak g)$.

QuestionFrom the vantage point of knot invariants, is there any benefit to integrating the infinitesimal deformation $\mathcal C[\hbar]/(\hbar^2)$ to the formal one? The infinitesimal deformation is already a non-symmetric braided monoidal category.