Benjamin Bode

Constructing knotted fields - Joint seminar with Theoretical Physics

The term 'knotted fields' can refer to many different physical systems, an electro-magnetic field with knotted field lines, knotted vortex lines in a scalar optical field or a fluid, knotted disclination lines in a liquid crystal and many more.

Despite this diversity the problem of finding a mathematical description of such a system which contains a given knot $K$ almost always reduces to finding a polynomial $f:\mathbb{R}^3\to\mathbb{C}$ that vanishes exactly on $K$.

In this talk I am going to present an algorithm that for any given knot $K$ constructs such a polynomial, whose degree can be bounded in terms of topological data associated with $K$, allowing us to theoretically construct knotted fields with arbitrarily complex knots.