KNOTDYNAPP · Knots in dynamical systems with applications to electromagnetism and quantum systems

Knots in dynamical systems with applications to electromagnetism and quantum systems

A knot is a closed loop in 3-dimensional space that does not intersect itself. Knot Theory is a branch of mathematics that studies such objects up to continuous deformations. In this sense two knots are considered equivalent if one can be deformed into the other without any cutting. Knots and their mathematical description play an increasingly important role in biology, chemistry and physics. Funded by the Marie Sklodowska-Curie Actions programme, the KNOTDYNAPP project studies the time evolution of knots in different dynamical systems. Of particular interest are solutions of differential equations that originate in electromagnetism or quantum mechanics. Such solutions correspond to mathematical functions that can contain knots in one way or another. More concretely, an electromagnetic field is a mathematical function that satisfies Maxwell's equations, a set of differential equations. A closed field line forms a knot. Such an electromagnetic knot is not made up of any physical material and usually disappears almost instantaneously. However, for a small family of knots, the so-called torus knots, solutions to Maxwell's equations have been found that have a closed field line in the shape of a torus knot not only at one moment in time, but for all time. The knotted structure is "stable". As the field changes with time, the knot may move around in space, but it will always maintain its knot type, its topological shape. One objective of the project is to prove that such solutions exist not only for the family of torus knots, but for all knots, and to study the dynamics of the knots in these fields. Similarly, a quantum system can be described by a "wavefunction", a time-dependent function that satisfies Schrödinger's equation. The zeros of this function can form knots, so-called quantum vortex knots. As the wavefunction changes with time, the knot may move around. Parts of the knot may also pass through each other, thereby changing the knot type. The time evolution of a knot is thus described by a surface in 4-dimensional space. In fact, it is known that for any surface in 4-dimensional space there exists a quantum wavefunction whose zeros are knots that undergo the time evolution described by the given surface. However, finding such solutions to Schrödinger's equation for a given surface is very challenging. The project aims to develop a construction that produces for any given surface a corresponding quantum wavefunction. Given a physical system that contains a knot in one way or another we would hope that topological properties of the knot are somehow reflected in physical properties of the corresponding equations. Another objective of the project is to establish such a relation in the case where the physical system is the magnetic field that is induced by an electric current through a knotted wire. A knot is called fibered if the space around the knot can be filled with topologically equivalent surfaces that are attached to the knot. Not every knot is fibered. We will prove that a knot is fibered if and only if it can be placed in 3-dimensional space such that the magnetic field that is induced by an electric current through the knot vanishes nowhere. This would produce a new characterisation of the family of fibered knots in terms of physics. Knots (and more generally topology) play an important role in modern science due to their (topological) stability. Many possible applications of the study of knots in physical systems have been suggested, such as topological quantum computing as well as new communication and information storage possibilities. Understanding which knot types can appear as stable solutions in certain physical system and describing the topological changes that can occur therefore is expected to lead to more discoveries in the research area, resulting in progress on applications of these theoretical results.

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