  We give a general fixed parameter tractable algorithm to compute quantum
invariants of links presented by diagrams, whose complexity is singly
exponential in the carving-width (or the tree-width) of the diagram. In
particular, we get a $O(N^{\frac{3}{2} \mathrm{cw}} \mathrm{poly}(n))$ time
algorithm to compute any Reshetikhin-Turaev invariant---derived from a simple
Lie algebra $\mathfrak{g}$---of a link presented by a planar diagram with $n$
crossings and carving-width $\mathrm{cw}$, and whose components are coloured
with $\mathfrak{g}$-modules of dimension at most $N$. For example, this
includes the $N^{th}$ coloured Jones polynomials and the $N^{th}$ coloured
HOMFLYPT polynomials.
