We tie our shoelaces to ensure that our shoes stay on tight, and we do these by tying a knot. There are different ways to tie your shoelaces, you may have learnt the “around the tree” technique, but somehow, they still always come undone, why? This all has to do with knot theory.

If you have ever gone camping or been on a boat, then you will know that there are many types of knots, and each are used for different things. Knot theory began in 1867, where scientists were keen to understand what kinds of matter make up the world. Inspired by the strange behaviour of smoke rings, Lord Kelvin began to believe that atoms behaved as smoke rings, like vortices in the air, knotted in various different shapes. These vortices were supposedly an invisible medium that allowed light through, so that people could see. Of course, we now know this is not true, but this did trigger an investigation into knots. Lord Kelvin’s ideas were not far off some of the behaviour of atoms, as through this investigation, scientists discovered that knots, with all diversity and intricacy, model the behaviour of many chemical elements. In fact, as of today, there are 352,152,252 knots, each with different properties. Knots are also more useful than just for sailing – they are used everywhere. For example, knots are used to understand DNA structure, or are used to make strong materials, or can just be a pain when you take out your headphones to see they are in a tangled bunch – thank God for wireless headphones!

Before we can delve into knot theory, we need to define what a knot actually is. For mathematicians, a knot is slightly different, namely that a mathematical knot is in a closed loop. What this means is that a mathematical knot is formed by taking any knot, and sticking the two ends together. This way, we cannot undo it, but we can put it tighter and looser to examine its shape. Thus, the simplest knot we have is a circle, this is called an “Unknot”. From this, we can make more and more complicated knots. Then, to add a new knot to our list, it must be of unique shape, so no matter how many twists and turns we try, we cannot get it into one of our previous knots, or untangle it to the “unknot”. So, as you can imagine, there are lots and lots of knots (no rhyme intended!). We can also combine different types of knots to create a new knot that has a unique shape, so the possibilities are endless.

Now, how do we really know if a knot is unique? Yes, of course we can try endlessly to untangle it into the unknot, or into a form that we have seen before. But, just as some of our friends are brilliant at untangling necklace chains, and others terrible, this is not an efficient way to find out. If I gave you a knotted necklace, would you be able to quickly tell me if you recognise this knot or not? I don’t think so. So, mathematicians created a new way to decide this. This idea is called an “invariant”, which is a property, quantity or algebraic entity in the knot that we can easily compute. Then, we can label each knot with a number of properties, and use this to compare them. Just as we can differentiate between people by hair and eye colour, we can differentiate between knots. Of course, knots may share the same invariants, but just as two blonde people can have different colour eyes, we can still differentiate using more of these invariants.

So, now we have a bit more understanding of the complexity of knots. This is a little daunting, so now let’s discuss apply some of this knowledge to a simple case: how should you tie your shoelaces? We start always with my criss-crossing the knots and pulling them tight, then holding a loop in one hand, and using the remaining lace to go around the loop and tuck it into itself. If this knot is done correctly, it should remain tight and in place even as we walk, but if we tie it incorrectly, we have a knot that gradually unravels as we walk, leaving us with retying our shoelaces every five minutes. The way to tell if you tie your shoelaces the “correct” way, other than judging if your shoelaces are frequently undone, is by looking how your knot sits on your shoe. If tied correctly, the knot lies front and square on top of the shoe, but if your knot is wonky or at the side of your shoe, then you may be tying your laces wrong.

The key to tying your shoelaces is understanding how the knot tightens and loosens. In a good knot, if you pull on the ends, they become undone, but if you pull on the loops, they become tighter. This is the same for a “bad” knot, but the key difference is this: in a good knot, if you pull only on one of the ends, the knot becomes tighter, in a bad knot, pulling on one of the ends results in undoing the knots. So, how can we ensure that we always tie a good knot?

Let’s dissect the process of shoelace-tying. I think all of us begin with a criss-cross tuck, and pulling this tight to ensure we have approximately equal length laces to create the rest of our knot. There are two ways we can do this, we can either cross the left lace under the right lace, or vice versa. In practice, many right-handed people cross left under right, and left-handed cross right under left. Neither of these options are wrong, but it does influence your next steps.

The next step is to pick a side to create a loop, and then use the other lace to go around this loop and tuck. Which side is better? Again, in most cases, people pick their less dominant hand to create this loop and hold it, so that our more dominant hand does the more complicated step of going around the loop and tucking. Again, neither of these options are wrong, but this combination of first and second step will change your third steps.

Now that we have our loop in one hand, we need to go around the loop and tuck. Again, there are two ways to do this: clockwise, and anti-clockwise. Unlike our previous steps, there is no real evidence that dominance of hand has an effect on the direction of this loop. This step is, however, the step that differentiates a good knot from a bad knot. Let’s discuss the correct combinations:

Criss cross left/right under right/left | Hold loop with left/right | Go around loop clockwise/anticlockwise |

Right | Left | Anti-clockwise |

Right | Right | Anti-clockwise |

Left | Left | Clockwise |

Left | Right | Clockwise |

All of these combinations will give a correct knot. Any other combination will result in a knot that will unravel. Now, writing this article made me realise I have been tying my laces wrong all my life, but now I am retraining myself to tie a correct knot. Have you been tying your laces wrong? Hopefully now you can see the correct way to tie your knot, and understand how knot theory is extremely applicable in our daily life, and is definitely worth the headache of trying to figure them out.