#### Marc Lackenby: Knot genus in a fixed 3-manifold (RLGTS, 21 July 2020)

*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.

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