# Quantum Groups for Generic q and 3d-TQFT. What breaks?

I've just started looking through Quantum Invariants of Knots and 3-Manifolds by V.G Turaev and want to understand what exactly is breaking in the construction of a 3d-TQFT when one considers the quantum group $$U_{q}\mathfrak{sl}_{2}$$ when $$q$$ is not a root of unity.

In particular if convergence of expressions for the invariants of all manifolds isn't guaranteed this wouldn't be so much of an issue for my interests. For example something like $$S^{1}\times\Sigma$$ having infinite invariant isn't so bad for me (i.e having infinite dimensional vector spaces on boundaries).

• Hyperbolic knots and their volumes are what I'm trying to understand. A TQFT can't have infinite dimensional vector spaces with the usual definition of a TQFT. Is this the only thing (and its consequences) that goes wrong in their construction of a TQFT? Seemingly they can define invariants for some links in $\mathbb{R}^{3}$ at arbitrary q is it just that occasionally when they glue they will get infinite invariants? I'm not sure if the anomalies they fix will be a bigger problem for arbitrary q for example. – Campbell Dec 14 '18 at 20:48