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.
Categories: math.GT cs.CG math.QA