  We show that the problem of showing that a cusped 3-manifold M is not
hyperbolic is in NP, assuming $S^3$-RECOGNITION is in coNP. To this end, we
show that IRREDUCIBLE TOROIDAL RECOGNITION lies in NP. Along the way we
unconditionally recover SATELLITE KNOT RECOGNITION lying in NP. This was
previously known only assuming the Generalized Riemann Hypothesis. Our key
contribution is to certify closed essential normal surfaces as essential in
polynomial time in compact orientable irreducible $\partial$-irreducible
triangulations. Our work is made possible by recent work of Lackenby showing
several basic decision problems in 3-manifold topology are in NP or coNP.
