# Turning the Knots in Resonators

Image courtesy of Charlotte Leakey.

What do piano strings, air particles, and colliding comets have in common? Each of them is an oscillator, broadly defining anything that can vibrate. Oscillators are ubiquitous—they can be electrical, mechanical, optical, and astronomical, varying from smaller than an atom to larger than a planet.

Every oscillator has a discrete set of frequencies at which it naturally vibrates. These are known as its resonance frequencies or eigenfrequencies, collectively known as the object’s spectrum. You may have encountered these in physics class in the form of standing waves, which are the various waves that naturally “fit” into a given object. For a very simple object like a guitar string, these are just sine waves, which “fit” whenever a half-integer number of their wavelength fits into the length of the string. Each wave will vibrate with its own frequency or eigenfrequency, which can be changed by adjusting anything that affects the system. In the case of a guitar, factors such as the tension of guitar strings, the type of wood, and the temperature of the environment can be used to tune its spectrum.

Physicists know that any oscillator’s resonance frequencies are always given by the roots of a polynomial equation. When the oscillator is free from friction, all of the roots of this polynomial are real numbers (which is quite reasonable given that frequencies are usually thought of as real numbers). However, when friction is considered in the model, these roots become complex. This means that the root contains an imaginary number i, equivalent to -1. When the model for an oscillator’s resonance frequencies includes friction, it is known as a non-Hermitian system. In contrast, a Hermitian system does not include friction in its model. The complex root takes the form (a + bi), where a is the real part of the root representing the resonance frequency, and b is the decay rate, or how quickly the oscillator stops oscillating. One example is a guitar’s strings coming to rest after being plucked.

To visualize the relationship between a system’s parameters and its spectrum of resonance frequencies, it is helpful to use two graphs: one showing the parameters that are being changed and the other showing the system’s resonance frequencies (with each frequency being a point in the complex plane). This pair of two graphs can be seen in the figure below with pairs of plots “c” and “f,” “d” and “g,” or “e” and “h,” where plots “c,” “d,” and “e” are the parameter graphs, and plots “f,” “g,” and “h” are the resonance frequencies graphs. Continuing the example of a guitar, the parameter graph would have the tension in its strings, the type of wood, and the temperature of the environment, while the spectrum graph would have time as its vertical axis, representing how far along the control loop we are, and the complex eigenfrequencies in the horizontal plane. The spectrum graph can be thought of as plotting the complex eigenfrequencies in the horizontal plane and then stacking these planes on top of each other as time passes. When the parameters are gradually changed and then returned to their original values, a loop is formed in the first graph,  known as a control loop. This control loop can be seen below in the figure as the green, red, or blue loop in plots “c,” “d,” and “e,” respectively. Such a loop may or may not enclose points corresponding to a choice of parameters that would produce a spectrum in which two or more eigenvalues are equal, known as a degeneracy. These points can be seen below in the figure as the points along the yellow “trefoil knot”-shaped structure in plots “c,” “d,” and “e.” When the parameters are varied around a control loop, a “braid” topological structure is created in the spectrum graph. Much like one can braid hair into different styles, these spectral braids also can twist and turn in a variety of ways. The specific braid that is created depends on the manner in which the control loop encloses the degeneracy points. These braids can be seen below in the figure as the triplet of green, red, or blue lines in plots “f,” “g,” and “h,” respectively.

This relationship between polynomials and their roots was previously well-understood for a system with two oscillators (N=2). The braid would twist once if the control loop encloses a “degeneracy”–a point in the parameter graph at which two or more resonance frequencies of the system are equal. If the control loop does not enclose a degeneracy, the braid, in turn, does not twist.

The relationship becomes more complex for a system with three oscillators (N=3). Mathematicians have known for a long time that the degenerate solutions of polynomial equations result in a curve/structure with non-trivial ‘topology’–the degeneracy curve forms a trefoil knot in the parameter graph for N=3. Topology is the branch of mathematics that deals with the shapes of geometric objects. For example, a donut has a different topology than a sphere, as the former has a hole while the latter does not. This topology had historically been almost exclusively explored in mathematics, not physics. Physicists knew that the braids twisted in various ways, but they did not know why, though they knew it had something to do with degeneracies.