(Cover Artist: Sophie Cheng)
If you’ve ever had to deal with a huge, super-tight knot in your shoelaces or headphone cables, you’re probably familiar with the feeling that there’s no possible way to solve it. Indeed, the only apparent solution is to follow the path of each of the individual strands. If you wind up in a confusing spot, you can try to just pull the strands apart at this knot, often revealing another knot. Now what if after all that work, you found that the ends were connected to each other tip-to-tip, like a circle or the closed loop? That would be the simplest mathematical knot: an unknot. Why are these unusual knots useful, especially in the field of math known as knot theory?
The fundamental question knot theory seeks to address is whether you can convert one knot to another using any transformation except for a cut — one common case is going from the trefoil knot to the left to the unknot above. Logically, that should be feasible as the trefoil’s ends meet each other, but how is that possible – what transformations are required?
First of all, what are the simple transformations? In 1926, mathematician Kurt Reidemeister figured out a way to boil down the intuitive ways to change the knot to just three possible transformations, called the Reidemeister moves. If you have a rubber band, see if you can try them out. You can turn a simple loop into a straight line or vice versa, you can cross/uncross two strands, or you can move a straight part of the rubber band across an intersection (see below).
Again, all of this is fairly intuitive, but where can it be used in STEM? In the 1860s, Lord Kelvin, famous today for the scientific temperature system named after him, proposed a physics theory popular at the time. He postulated that different elements consisted of different knots in space, meaning they could not be changed to different elements (this was later disproved because of the discovery of radioactive elements). Going further, Lord Kelvin suggested linking knots could be used to make molecules. Today, we believe in the model of electrons revolving around a nucleus, but this compelling theory helped spark interest in the field and the endless possibilities of knots. Today, we can use computers to plot complicated knots on the X-Y-Z coordinate planes, allowing us to model them in 3D. This can help us learn more about real-world knots – the most famous of which is the DNA molecule.
This doesn’t really look like the double helix shape you may be familiar with, but if you look closer, the whole molecule is made up of those double helixes, tightly bound and following one another. Unlike a regular DNA molecule, which isn’t a mathematical knot, since it has a beginning and an end, this is an example of a bacterial DNA molecule, which is circular, and so can be condensed to the unknot mentioned before. It’s common for DNA to get tangled, which is an issue when the bacteria wants to replicate its DNA to reproduce. Only after separating the strands can each individual segment, or nucleotide, be replicated. Bacteria, and even our own DNA molecules, possess special helper enzymes called topoisomerases. Their function is to help untangle these twisted strands by making cuts and then reattaching nucleotides to change the double helix shape to one more similar to a ladder. Because topoisomerase uses cutting and rejoining, it transcends the Reidemeister moves, but the general idea is still present. Bacterial topoisomerase’s efficiency is so advanced that figuring out a way to stop or decrease its function could be a potent anti-bacterial.
Another example of knot theory is with the corona of the Sun, a sort of crown around it consisting of extremely hot plasma. It appears looped and often random, but is very important as its knotted shape results in its magnetic abilities and tremendous energy emission. Not only would an accurate model of its constantly-changing knotted shape help us better predict phenomena like the Northern Lights, it could also better help us understand what portion of climate change on Earth can be chalked up to aberrations with the Sun’s corona and not human activity.
Of course, while there is more complicated calculus and probability involved in knot theory, in its most basic form, it can show how something as seemingly simple as the study of mathematical knots can affect things from bacterial life to Earth as a whole.