A Beginners Guide to Tying Knots (With Math!)
Originally published on my Quora blog
For a bit of context, I’m currently diving into an area of quantum computing known as topological quantum computing, which involves the braiding of world-lines of non-Abelian anyons in order to perform computation. In order to understand this concept of braiding, one must understand some basic knot theory, so that is what I am going to discuss in this short blog post! It is important to note that I am just beginning to explore knot theory, so my knowledge is practically non-existent at this point, but nonetheless, I thought it would be fun to share some interesting concepts that I have come across so far! (P.S. Brace yourself for terribly sloppy diagrams of knots that I drew on my computer).
Quantum Noise
Tolerance to noise is one of the greatest challenges within quantum computing. Noise is essentially external environmental factors that can affect the coherence of the quantum states in which qubits must exist in order to perform quantum computation. Interaction with the classical world leads to a phenomenon called decoherence which basically means that the qubits is no longer useful.
Now, there is a method of quantum computation called topological quantum computing that hopes to eliminate this factor of noise greatly. Rather than encoding quantum information in the qubits themselves (which is very precarious and delicate), topological quantum computing braids the world-lines of anyons to encode information in the local topology of the system. You can imagine this is encoding information by tangling together rope. You can still make lengths of rope really messy and twisted, and still braid them in the same way, which is exactly the reason why topological quantum computing could be so useful for reducing noise.
I’m going to talk more about topological quantum computing at the end of the article, but for now, in order to understand how topological quantum computing works, we need to understand the theory behind braiding world-lines and tying knots, which is an area of mathematics called knot theory.
Knots, Links, Braids, and Reidmeister Moves
Knot theory is the study of mathematical objects called knots, which are essentially closed loops embedded in three dimensional space. There are many examples of knots, with the most simple being the unknot, which looks somethings like this:
Despite seeming like a very archaic topic within pure mathematics, there are actually uses of knot theory within fields such as quantum computation (which I will discuss later), and quantum field theory.
Aside from knots, a link is defined as two or more knots (closed loops) that are joined together. A very simply example of a link is simply the one where we join two unknots together, which is called a Hopf Link:
Finally, a braid can be defined as adjacent parallel strands that can be crossed over and twisted around each other. A collection of braids would look something like this:
When studying knots, we are very interested in determining which knots are equivalent to one another, and which are different. In order to find out which knots are in fact equivalent to each other, we first must define criteria for the transformations of knots that preserves ambient isotopy, which is basically criteria that dictates whether a transformation on a knot will leave it mathematically unchanged, or will alter it in some way. Specifically, the transformation that preserve this isotopy are called the Reidmeister moves, and they are defined as follows (the best way to understand them is through pictorial representations):
For the first picture, we basically see that f we have a string in a loop, it is equivalent to a straight string, without the loop. This makes sense conceptually. If we have a piece of string, and we make a loop in it, we can simply pull both ends to get rid of the loop and straighten the string. For the second picture, if we have two strings with one passing over the other at two intersections, we can simply pull the strings apart into two separate strands. Again, this makes conceptual sense, as we can easily see this happen in the real world (take two string, place one on top of the other as shown in the diagram, and then pull them apart). Finally, the third diagram says that we are allowed to “pull” a string down to any other point along the lengths of the two strings that both lie above or below it.
The Kauffman Bracket Polynomial
Ok, now that we have established exactly how we are allowed to transform knots and maintain their equivalence, we can now define quantities called knot invariants. Knot invariants are quantities based off of properties that remain invariant under transformation of knots, therefore allowing us to have the ability to equate two knots as “the same” when their knot invariants are equivalent.
One of these “knot invariants” is called the Kauffman Bracket Polynomial. Now, the Kauffman bracket polynomial is not actually a real invariant, as unfortunately, its invariance does not hold for the first type of Reidmeister move (the loop on a linear string). Luckily, there is a version of this polynomial called the Jones polynomial that can be used to maintain invariance under this condition. I will not be talking about the Jones polynomial in this blog post (as I haven’t looked into it in depth yet). Instead, I will be discussing the Bracket Polynomial, as it is simpler, and is necessary knowledge in order to eventually understand the Jones Polynomial.
Essentially, the Kauffman Bracket Polynomial It is calculated in a recursive fashion, by breaking a knot down into structures called 0-resolutions and 1-resolutions. These are defined as follows:
This basically means that we are taking a crossing of two lines, and are joining horizontally components of the ends of the strands:
Let’s now look at an example, which will also illuminate how the Kauffman polynomial stops working at the first criteria of the Reidmeister moves. Consider the knot that looks like this:
It seems simple enough. Now, let us break this knot down into its 0-resolutions and 1-resolutions:
Great, now all we have to do is represent this mathematically. For each of the newly-separated resolutions, all we have to do create an expression of some variables and inverse variables, and for any two unknots in the same resolution (like we can see in the 1-resolution), we replace one of the knots with another special expression, with combined variables and inverse variables. We eventually get that the Kauffman bracket polynomial is equal to -A³.
And so we have calculated the Kauffman bracket polynomial for a simple knot. Notice however, that by the first Reidmeister move, that a knot of this form is already equivalent to the unknot, which is just equal to 1. Therefore, by calculating this Kauffman polynomial, we have shown that it fails to be invariant under the first Reidmeister move, as promised.
Applications of Knot Theory: Topological Quantum Computing
Surprisingly, this area of mathematics has huge applications in the field of topological quantum computing. Topological quantum computing is essentially a protocol for quantum computing where qubits are realized as collections of non-Abelian anyons, which are particles that obey very general statistics and occupy a 2+1-dimensional space. Calculations can be performed to entangling the world-lines of these anyons, therefore adding certain phases to anyons within the system, allowing us to perform quantum computations.
As you may have already guessed, the way that the entangling of the world-lines of anyons actually works is through the literal crossing-over of sad world-lines into braids, much like is demonstrated in the image near the top of the post. There is a construction in knot theory called Alexander’s Theorem that states that any knot or link can be represented by a certain closure of a braid (matching up the end points of braids to create closed loops). Because of this, we can use ideas in knot theory to help understand these braids, and ultimately, how we can cross-over world-lines to perform quantum computation.
This section of the post is vague on purpose, as I have just begun looking into the field of topological quantum computing, and I have so much more to learn before I can talk about it in a blog post. If you want to learn more about topological quantum computing, I’m currently reading an excellent paper on topological quantum computation with non-Abelian anyons, which can be found here.
Thank you for taking the time to read this post! As this is the first time I have ever written about knot theory, please do not hesitate to let me know if you noticed that I made any mistakes, and I will correct them as soon as possible! Also, if you enjoyed this post, stay tuned for more content, as I will be posting more about math, physics, and quantum computation in the coming weeks!