A study on knots recently revealed a surprising feature of the mathematical system describing the electrical activity that plays a role in some heart attacks. This work could help us better understand the physical context of these heart attacks, and also demonstrates a new approach to one of the fundamental goals of knot theory.
Most of us only think about knots when they frustrate us. A shoe comes untied on an escalator. The ribbon on a gift is annoyingly tight. Earbuds are a tangled mess. Despite our lack of attention, knots have a deep impact on our world. They provide insight into how DNA replicates, how light and heat reach us from the sun, and how our bodies react to certain types of medications–all phenomena essential for our survival.
By modeling knots mathematically and correlating them with physical situations, we can better understand physical processes and use them to our advantage.
Mathematical knot theory dates back to the 1880s. At the time, scientists thought that empty space was filled with ether, a material through which light and gravity traveled. William Thomson (Lord Kelvin) suggested that atoms were knotted vortices, or swirling regions tied in knots, in the ether. He proposed that each atom was a different kind of knot. Although this idea fell apart when experiments showed that ether does not exist, it had already inspired an elegant field of knot theory. This remains a vibrant field of study that has provided many important insights across the physical sciences.
The mathematician’s knot is different from a shoelace knot in three main ways:
|An overhand knot in a string (top), and the
same knot as treated in mathematical knot
Image Credit: Talifero. (CC BY-SA 3.0)
- The ends of a knot are always connected—as if your shoelace was knotted and then the ends were glued together.
- Knots are endlessly stretchy and flexible.
- There is no cheating—you can’t cut a mathematical knot.
Although Lord Kelvin’s concept of atoms as knotted vortices of ether is dead, knotted vortices are still an important area of study. Vortices form in fluids or gases when a region swirls around an axis, as seen in the air around airplane wings, smoke and bubble rings, tornadoes, and hurricanes, among others. A knotted vortex is simply a vortex that has the geometrical shape of a knot. Knotted vortices have implications for studies of the sun’s atmosphere, superconducting materials, sub-atomic particles, and other areas as well.
|A vortex created by the passage of an aircraft wing, revealed by colored smoke,
during the Wake Vortex Study at Wallops Island.
Image Credit: NASA Langley Research Center. Public domain
As part of a project to better analyze, exploit, and apply knot structures to a wide variety of physical situations, Paul Sutcliffe and Fabian Maucher from Durham University recently used a new mathematical model to determine whether a knotted vortex always remains knotted.
In most cases, says Sutcliffe, knotted vortices can unknot themselves into a circle through a process called reconnection. A vortex is different from regular string in that a vortex string can pass through itself and reconnect on the other side. Mathematical models showed that knotted vortices all seemed to eventually untie themselves through a series of these reconnection events. Sutcliffe and Maucher set out to see if there were any exceptions—vortices that didn’t reconnect.
Last week they published results
in Physical Review Letters
highlighting three examples of knotted vortices that stay knotted like regular string. Sutcliffe and Maucher then showed that although the examples in this knotted vortex system don’t reconnect, remarkably they can still untangle themselves.
To a mathematician, tangling means pretty much the same thing as it does to the rest of us—wrapping something around itself so that it looks messy and complicated without knotting it. Knowing whether something is knotted or simply tangled is one of the fundamental goals of knot theory.
Because most knotted vortices reconnect to a circle, it is impossible to tell if any given vortex is knotted or tangled. However, by identifying vortex strings that don’t reconnect but do untangle, the researchers found a new way to test whether specific geometric shapes are actual knots or just tangles—by making them out of these vortex strings.
This isn’t the only surprise.
The system turns out to be a model for certain types of electrical activity in nerve fibers and heart tissue. The vortices in this case are swirling patterns of electrical activity that can cause the heart to beat too fast and erratically, which can lead to sudden heart attacks.
According to Sutcliffe, whether such a vortex is knotted probably doesn’t matter very much from a medical perspective, but understanding the physics that governs a vortex’s behavior is certainly important for designing potential treatments.
|The vortex string (red) of an initially tangled knot, together with a cross-sectional slice to show
the spiral waves of electrical activity that emanate from the vortex string.
Image Credit: Paul Sutcliffe