We show that the problem of determining whether a knot in the 3-sphere is
non-trivial lies in NP. This is a consequence of the following more general
result. The problem of determining whether the Thurston norm of a second
homology class in a compact orientable 3-manifold is equal to a given integer
is in NP. As a corollary, the problem of determining the genus of a knot in the
3-sphere is in NP. We also show that the problem of determining whether a
compact orientable 3-manifold has incompressible boundary is in NP.Comment: 101 pages, 24 figures; v2 contains some improvements suggested by the
referee, which have strengthened the main theorem
