We show that the problem of recognizing that a knot diagram represents a specific torus knot, or any torus knot at all, is in the complexity class $ \textsf {NP} \cap \textsf {co\text {-}NP}$, assuming the generalized Riemann hypothesis. We also show that satellite knot detection is in $ \textsf {NP}$ under the same assumption and that cabled knot detection and composite knot detection are unconditionally in $ \textsf {NP}$. Our algorithms are based on recent work of Kuperberg and of Lackenby on detecting knottedness.