Hebrew University topology and geometry seminar

Abstract: I will talk about two different results whose proofs are very similar in nature.
    Theorem 1. Any knot in a simply-connected 3-manifold is isotopic to a Fourier knot of type (1,1,2).
    Theorem 2. Any knot in $\mathbb{S}^3$ is isotopic to a knot that bounds a minimal disk $D$ in the unit ball $B^4 \subset \mathbb{R}^4$ that is branched at its center, immersed away from the center and with possible self-intersections which are double points.
In order to explain these results, we recall first the relation between branch points of minimal surfaces and knots. In a small neighborhood of an isolated singularity -or branch point- a minimal surface is homeomorphic to an embedded cone over a knot. But it is still unknown whether there are topological restrictions for these knots. In joint work with Marina Ville, we constructed a family of simple minimal knots that bounds embedded minimal cones. It enabled us to show that minimal knots are not necessarily fibered - contrary to branch points of complex curves. A second notable difference is that true branch points of minimal surfaces can be bounded by trivial knots as well as singular knots (curves with self-intersection) - which is impossible for complex branch points. However, the family of simple minimal knots cannot a priori represent all minimal knots since they have, by construction, many symmetries.
Theorem 2 is more a “middle range” result about minimal knots in the terminology of Rudoph, but still sheds new light on the local nature of minimal knots.
This situation is similar in many ways to the one encountered in the study of Lissajous knots. These knots are the simplest examples of Fourier knots - i.e., closed embedded curves whose coordinate functions are finite Fourier sums - where each coordinate function consists in only one term.
The family of Lissajous knots cannot represent all knots. But it was conjectured that any knot can be presented by a Lissajous figure with a height function consisting of a Fourier sum with a fixed number of terms, or even, as suggested experimentally by Boocher, Daigle, Hoste, and Zheng, with a height function consisting of only two terms. We prove the conjecture in theorem 1.