# Parameterized complexity of quantum invariants

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^3/2cwpoly(n)) time algorithm to compute any Reshetikhin-Turaev invariant—derived from a simple Lie algebra g—of a link presented by a planar diagram with n crossings and carving-width cw, and whose components are coloured with g-modules of dimension at most N. For example, this includes the N^th coloured Jones polynomials and the N^th coloured HOMFLYPT polynomials.

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