  For a banded link $L$ in a surface times a circle, the
Witten-Reshetikhin-Turaev invariants are topological invariants depending on a
sequence of complex $2p$-th roots of unity $(A_p)_{p\in 2\mathbb{N}}$. We show
that there exists a polynomial $P_L$ such that these normalized invariants
converge to $P_L(u)$ when $A_p$ converges to $u$, for all but a finite number
of $u$'s in $S^1$. This is related to the AMU conjecture which predicts that
non-simple curves have infinite order under quantum representations (for big
enough levels). Estimating the degree of $P_L$, we exhibit particular types of
curves which satisfy this conjecture. Along the way we prove the Witten
asymptotic conjecture for links in a surface times a circle.
