Reshetikhin-Turaev and Turaev-Viro invariants of 3-manifolds are in FP exactly when the category (or its center) is pointed; otherwise #P-hard.
Rowell and Zhenghan Wang,Mathematics of topological quantum computing, Bulletin of the American Mathematical Society55(2018), no
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A Complexity Dichotomy for Quantum Invariants of 3-Manifolds
Reshetikhin-Turaev and Turaev-Viro invariants of 3-manifolds are in FP exactly when the category (or its center) is pointed; otherwise #P-hard.