Abstract:The genus of a knot is the minimal genus of any of its Seifert surfaces. This is a fundamental measure of a knot's complexity. It generalises naturally to homologically trivial knots in an arbitrary 3-manifold. Agol, Hass and Thurston showed that the problem of determining the genus of a knot in a 3-manifold is hard. More specifically, the problem of showing that the genus is at most some integer g is NP-complete. Hence, the problem of showing that the genus is exactly some integer g is not in NP, assuming a standard conjecture in complexity theory. On the other hand, I proved that the problem of determining the genus of a knot in the 3-sphere is in NP. In my talk, I will discuss the problem of determining knot genus in a fixed 3-manifold. I will outline why this problem is also in NP, which is joint work with Mehdi Yazdi. The proof involves the computation of the Thurston norm ball for knot exteriors.