By Will Stavely, Monash University
You may have noticed that if you twist the end of an old-fashioned phone cord, it will start to bunch up into a clump somewhere in the middle of the cord. A mathematician might wonder whether they can explain why this happens. In fact, we can, using a branch of mathematics called Knot Theory.
To do this, we will use a couple of useful bits of information about our knot (in this case, the phone cord). There are formal definitions for these, which are a bit too complicated to put here, but I’ll try to give a rough idea of how they work. First we have what is known as the Twist number, which measures how many times the cord twists around. Secondly we have the writhe, a number that tells us how flat the cord is. If we have a low writhe, the cord lies mostly flat on the table, whereas a high writhe means it curls up and around a lot.
Now for these to be useful we would like to understand how they change as we do things to our cord. For this, we need a third quantity, the Linking number. The precise definition is not important. The key idea is that, whatever this number is, it will never change when we add twists to our cord. What’s more, it satisfies a very simple relationship:
Linking number = Twist number + writhe
What does this mean? Suppose we have a very twisted knot (that is, a knot with a high Twist Number). Naturally, the cord would like to untwist itself, thereby lowering its Twist Number. But for the above equation to still be true, if the Twist Number goes down, the writhe must go up. So as the cord untwists, it must start to curl and bunch up, just as we expect. So we have found a mathematical explanation for why this happens.
Now this might all seem a bit pointless. Why should we care about modelling how a phone cord twists? In fact, the same mathematics at play here can also be used to investigate how DNA behaves. The shape of DNA, a pair of twisted strands, is quite similar to the shape of a phone cord. Biologists believe that, while DNA is being used in the body, twists are added to it. This causes a phenomenon known as “supercoiling”, which is identical to the phone cord – the DNA gets tangled up into a small clump. This is but one example of how results from pure mathematics can play an important part in understanding the world around us.
Will Stavely was one of the recipients of a 2013/14 AMSI Vacation Research Scholarship.