Knot genus in a fixed 3-manifold

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.

Mathematics

Audience: researchers in the topic

Regensburg low-dimensional geometry and topology seminar