The Basics of Knot Theory

What is knot theory? While some might say “the theory of knots”, the topic goes much deeper than that. In this article, I will explain knot theory, its concepts, and even some of its use cases.

First, knot theory is the study of knots. It categorizes them into ways that make them easy to understand. Constructing a knot mathematically is similar to constructing a knot in real life, but instead, once you have tied the knot, you glue the ends together so it becomes a closed loop. Notice that you can rearrange the string while still keeping the knot the same. One of the biggest questions in knot theory focuses on determining if two knots are the same or different.

The most important part of knot theory is how to rearrange knots. While you could probably rearrange a knot by just messing around with the string, it is useful to name a few moves that you can perform to switch between knots. These moves are called Reidemeister moves. There are three of these moves. The first one is that you can untwist or twist a string, like when you tie your shoes. The second states that you can uncross two strings if one is completely on top of the other. The third rule is that you can slide a string above or below a crossing if it lies above or below both strings in the crossing. These three simple moves can transform any knot.

There are a few notable types of knots. The first and simplest is the un-knot, which is just a circle with no crossing of string. This knot is useful and easy to deal with since it is so simple. Note that by using the three Reidemeister moves, you can make this knot look more complicated than it actually is. This creates fun problems where you have to figure out if a given knot can be simplified into the un-knot.

An interesting property of knots is tri-colorability. This means that you pick three colors and color each segment a different color such that no similar colors are touching. A segment is defined as a section of string that starts from a point when it goes under another string and ends at another undercrossing. Note that between these two undercrossings, it has to go over all other strings it passes. Since all three of the reidemeister moves conserve tri-colorability (try to think why that works), it is a good way to tell knots apart but not too good for seeing if two knots are the same. Also, this property of a knot is fun to use, since you get to think if you can color a knot.

Now, I will introduce adding knots. At first, this might be hard to comprehend because “how can you add knots!” Adding knots can be very simple. Take knots A and B. If we add A+B, we take the furthest right point on A and cut the rope. Repeat the same process with the furthest left point on knot B. Now, each knot should have an upper and lower end. Connect the top string of one knot to the top string of the other knot. Repeat this process with the bottom string. Now, you have one knot which was created from 2 knots! Note that B+A is different, meaning that they are not commutative . This is an interesting property since lots of topics in math are associative. Another interesting property is that you can always take an un-knot out of a regular knot since any knot plus the un-knot is just the knot you started with. This means that the un-knot is the additive identity for knots! It can almost be thought of as adding 0 to a number, since it doesn’t change the number and you can do it as many times as you would like.

Knot theory is a very interesting field that has lots of potential; knots are used in lots of different subjects such as examining how proteins fold or how to tie your shoe. It can help us classify knots with other operations such as division and multiplication and is a great subject to dig into on a weekend when you are bored. There are lots of other sources that can give you more information about this! Just look at the Wikipedia page for inspiration (although it might be a little confusing)! There are also great videos on Youtube that can teach you more about knot theory. Good luck on your knot theory journey!