Non-collapsing knots could reveal secrets of the Universe

A computational trick that stops imaginary knots collapsing could help us understand how to unravel a loop of DNA or reveal the true nature of elementary particles, research published today suggests.

In the New Journal of Physics, published jointly by the Institute of Physics and the German Physical Society, Phoebe Hoidn and Andrzej Stasiak of the University of Lausanne, Switzerland, and Robert Kusner of the University of Massachusetts, USA, explore the mathematical complexities of particular kinds of knots. They highlight characteristics that could help physicists develop new theories that explain some of the things observed with elementary particles like electrons.

Mathematicians have for many years been intrigued by knots. Thinking about them, the way they are formed and the way they can be mathematically described have often been the subject of international conferences. One way in which mathematicians think about knots is to imagine a long, practically weightless, silk fibre that has its ends spliced together so that it forms a ring. This silk fibre is charged electrostatically (for example, by rubbing it) and then released so that it can relax – ideally in a gravity-free environment such as aboard Space Station Alpha. As the ring relaxes, it forms a perfect circle. It remains in this perfectly balanced, minimum energy configuration.

Next, the mathematicians imagine what would happen if they release an electrostatically charged trefoil loop forming the simplest knot known as trefoil knot (cloverleaf shaped). The “common sense” answer – that the silk fibre would be most stable keeping its three loops as large as possible – turns out to be wrong. Instead, the knot tightens into a very small region on a perfect circle. The same is true for any other knot.

So is there some way, mathematicians ask, that the knotted fibre could be mathematically stabilised so that it doesn’t collapse? One way would be to think of the knotted fibre as being immersed in a cloud of very short-lived charged particles that pop in and out of existence so quickly that they don’t have time to approach the silk fibre and neutralise its charge. This cloud – which is an idea borrowed from quantum physics – would hide the electrostatic attraction and repulsion over longer distances and hence prevent the knots tightening up. This is a very interesting idea for particle physicists, whose latest theories include thinking of electrons and other elementary particles as little loops of charge, maybe even knotted loops.
Describing this cloud idea mathematically is, however, extremely difficult. Instead, it is possible to use a relatively simple computational method developed by Robert Kusner and other mathematicians to keep knots stable. Normally, the repulsive force between two particles with the same charge is described mathematically as decreasing proportionally to the square of the distance between them (the Coulomb law). If the repulsive force is instead described as decreasing proportionally to the third (cubic) or higher power of the distance between them, the knots get stabilised. This result was met with huge interest by mathematicians and physicists working in the expanding field of knot theory.

Using computers, Hoidn, Kusner and Stasiak have used this numerical method to study configurations of knots which can be internally stabilised in this way. They have gone beyond just seeing how these knots look and have systematically investigated different families of knots (torus knots and twist knots with even and with odd numbers of crossings). Within different families, they have found intriguing characteristics such as the quantization of energy – precise energy differences between knots were observed – and the quantization of writhe (a measure of the chirality or of the extent of left or right-handedness of a knot).

“This is potentially very interesting“, says Stasiak. “In polymer physics, we’re often interested in the shape a polymer reaches in equilibrium, when it might be knotted and charged. If we have a charged polymer molecule in solution, like single-strand DNA, it could be enormously helpful to understand how to unknot it and manipulate it to isolate different forms and investigate their properties. It’s also fascinating that we are finding energy quantization. That’s something we’ve seen before, in atomic physics. Is there perhaps some direct link between elementary particles and self-repulsing knotted strings?”.