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

Leertaste = Abspielen/Pausieren
m = Stumm
f = Vollbild
Pfeil-links = Zurückspulen
Pfeil-rechts = Vorspulen
Lukas Lewark

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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