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We show that the problem of determining whether a knot in the 3-sphere isnon-trivial lies in NP. This is a consequence of the following more generalresult. The problem of determining whether the Thurston norm of a secondhomology class in a compact orientable 3-manifold is equal to a given integeris in NP. As a corollary, the problem of determining the genus of a knot in the3-sphere is in NP. We also show that the problem of determining whether acompact orientable 3-manifold has incompressible boundary is in NP.

The efficient certification of knottedness and Thurston norm