  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 ${\sf
NP} \cap {\sf co\text{-}NP}$, assuming the generalized Riemann hypothesis. We
also show that satellite knot detection is in ${\sf NP}$ under the same
assumption, and that cabled knot detection and composite knot detection are
unconditionally in ${\sf NP}$. Our algorithms are based on recent work of
Kuperberg and of Lackenby on detecting knottedness.
