  We compute the braiding for the `principal gradation' of $U_q(\hat{{\it
sl}_2})$ for $|q|=1$ from first principles, starting from the idea of a rigid
braided tensor category. It is not necessary to assume either the crossing or
the unitarity condition from S-matrix theory. We demonstrate the uniqueness of
the normalisation of the braiding under certain analyticity assumptions, and
show that its convergence is critically dependent on the number-theoretic
properties of the number $\tau$ in the deformation parameter $q=e^{2\pi
i\tau}$. We also examine the convergence using probability, assuming a uniform
distribution for $q$ on the unit circle.
