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Pure Math

Knot Theory

If you’ve ever tied your shoes or made a bracelet, you probably have heard of knots. You start out with a line and then fold it over and under itself until it forms a shape for you to use. However, this is not quite correct in the mathematical world. In mathematics, a knot is an embedding of a circle in the 3D space. You may be confused about this definition, but the gist of it is that a knot is a connected strand that goes over and under itself. Here is the simplest kind of knot:

Figure 1: The unknot

This knot is called the trivial knot or the unknot. Here is the unknot, but in a slightly different form:

Figure 2: Also the knot

A little harder to see, right? We consider this to be the unknot because it can be twisted and turned in 3D space to get the unknot. These twists and turns form equivalence classes for knots. When twisting and turning, there are three moves called Reidemeister moves that we can make:

Figure 3: The Reidemeister moves

It turns out that all possible transformations of twisting and turning a knot can be expressed in these three moves. But how does this have to do with proving two knots are equivalent? Notice that if we had some characteristic of a knot that was constant under these three moves, then every knot in a knot’s equivalent class shares the same characteristic, called an invariant.

We can easily find some simple knot invariants. For a knot, we define an intersection to be a place where the knot goes over or under itself. One invariant would be the minimum number of intersections in all knots equivalent to it. Of course this characteristic would stay constant for all knots in its equivalence class; they all point to the same knot!

There are many complex and invariants in knot theory, for example tricolorability, or even knot polynomials such as the Jones and Alexander polynomials. Knot theory is a fascinating subject that bears a lot of similarity to braid theory, and thus has applications in both physics and cryptography.