O Time, thou must untangle this, not I;
It is too hard a knot for me t’ untie.
-Viola in Twelfth Night by William Shakespeare
The knot Viola speaks of in Twelfth Night is a complex love triangle. Knots are often used to symbolize complicated situations, in addition to anxiety and lasting commitments. Like Viola, when most of us think about knots our focus is on how tightly they are tied. For the scientists who study them however, knots are much more—they represent a unique approach to understanding the universe.
Last month in the American Physical Society’s journal Physical Review A, a team of scientists from Leiden University in The Netherlands demonstrated just this. They used the mathematical theory of knots to explore electromagnetic fields. In doing so, they realized that some solutions to the fundamental equations of electromagnetism can be described as knotted structures.
|Visualization of a mathematical cable knot.
Image Credit: Albertus de Klerk.
The mathematician’s knot isn’t quite the same as a shoelace knot. Instead of being made from a single piece of string, you can visualize a mathematical knot as being made from an elastic string with the ends glued together. The origin of knot theory dates back more than 125 years, to when Lord Kelvin suggested that atoms were swirling regions of the ether tied in knots (more on that here). The idea of ether was abandoned not long after, but it kicked off a field of study that has produced important results in many areas of physical science.
The connections between knot theory and physics are remarkable and deep. “Knot theory, which started out by the studying the knots we know from everyday life, turns out to be useful in constructing non-trivial solutions to physical theories,” according to Albertus de Klerk, lead author of the Physical Review A article. “The widespread occurrence of these knotted structures in wildly different theories motivated us to better understand how these structures come about mathematically.”
About twenty years before knot theory was developed, James Clerk Maxwell developed a theory that describes how light, electricity, and magnetism behave and interact. He is the one that suggested light can be described as electromagnetic waves traveling through space. Maxwell’s groundbreaking work can be boiled down to a set of mathematically sophisticated equations—often called “Maxwell’s equations”—that are the foundation of classical electromagnetism, optics, and circuits.
Recently, physicists realized that studying electromagnetic fields from a knot point-of-view leads to intriguing results. This new work focused on points in space called optical vortices, where the strength of an electromagnetic field is zero. Imagine a ray of light twisted like a corkscrew that is traveling toward the wall in front of you. The twisting causes the light to interfere in such a way that when it hits the wall, you see a ring of light surrounding a dark spot. The dark spot, where there is no light, is called an optical vortex. Like the calm in the eye of a hurricane, within that circle there is emptiness; the electromagnetic field is zero.
In two dimensions, like on the wall, optical vortices are points. In three dimensions, they are lines. This new work shows that optical vortices in real electromagnetic fields can be described as knotted electromagnetic field lines. Put another way, the researchers show that knotted structures can be exact solutions to Maxwell’s equations. In addition to this finding, the team mathematically describes how these structures emerge.
“We [the research team] believe that it is always a good thing to approach a subject from an unconventional viewpoint as it makes one more likely to find something new,” says de Klerk. Instead of trying to solve Maxwell’s equations directly, he continues, the team looked for general constructs giving rise to electromagnetic fields and then used knot theory to give structure to these solutions. For those of us not used to thinking about knots as anything other than a way to keep our shoes on, this is certainly a new approach.