The computational complexity of knot genus in a fixed 3-manifold

We show that the problem of deciding whether a knot in a fixed closed orientable 3-dimensional manifold bounds a surface of genus at most $g$ is in co-NP. This answers a question of Agol, Hass, and Thurston in 2002. Previously, this was known for rational homology 3-spheres, by the work of the first author.