Hello everyone, and welcome back to GLeaM! I announced in my last post that we are now moving into a monthly article schedule, so this is the April edition of GLeaM. I apologize for the wait (it was largely entirely out of my control), but I assure you it was worth it; I think you are all going to love this article! I have been extremely eager to write another article, and I'm very excited to bring you more content in the coming months. As you know, I am always available over email if you have any questions or would like to schedule an online meeting.
I received a lot of positive feedback on GLeaM's March edition about hyperbolic spaces (and crochet!), so today, I thought it would be exciting to walk you through another engaging, visual field of math: knot theory.
Knot theory is extremely expansive, so this article will, of course, be limited in the scope of the field it can cover, but I hope it serves as an effective introduction to the fascinating world of knots. Knot theory is also still very open—with SO many problems yet to be solved—so who knows, it's possible you might find a question in here that you'd like to tackle someday!
What is a Knot?
You might think you know what a knot is. When you tie your shoes, you've certainly created what you would expect to be a knot: a methodically tangled piece of string that keeps your shoes from sliding off your feet.
You've maybe even seen sailor's knots, intricate combinations of manipulated rope for nautical means, perhaps tying a loop to attach a boat to a dock or creating a structure to successfully hang the fenders (aka the "boating bumpers") over the side of a ship.
However, it turns out that none of these images are actually mathematical knots. All math knots must be a closed loop with the ends fully connected, and a shoelace or a piece of rope starts out as merely a straight line. Though your intention with knots in real life is usually to be able to untie them at some point, mathematical knots cannot be untied; unless the knot is equivalent to the simplest knot—a closed loop—it will still have tangles and crossings even if you twist it and bend it into its "most untied" form.
In this way, mathematical knots are objects that are actually physically knots at all times; they are not merely pieces of string that only "act like knots" when you need them to. We can also think about a mathematical knot as any object that can be formed by looping string around, interlacing it under and over itself, adjusting it in any possible way, and then gluing the ends together. Let's look at some visuals to further illustrate these ideas.
The notation that we will use for knots looks like this: we draw out the entirety of the knot's loop (as it looks from above if you were to lay it on a table), and we leave a break whenever one portion of the string goes under another in a crossing.
The above image is called a trefoil. It happens to be the simplest knot that is not just a closed loop. (We call the closed loop the "unknot" or the "trivial knot" because though it makes sense to include it in the set of knots, it does not actually act like a knot as it has no tangles.)
Here's a question: What type of knot is the following knot? Is it still a trefoil? Is it something else? (Following the approach of my 'solve-as-you-go' articles, think about the answer before you scroll past the image, and see what you can come up with.)
This knot is actually an unknot! Imagine picking up the large loop that lays on top of the knot and untwisting it; you can easily form a simple closed circle: the unknot.
Knots can be very deceptive. Our example very much looked like a trefoil, but yet, it was not. As you imagine, as our knots increase in complexity, it becomes even more challenging to figure out the type of knot we are dealing with.
This takes us to one of our first important concepts: knot equivalence. In knot theory, two knots are the same if one knot can be moved about smoothly in (usually 3D) space without intersecting itself until it coincides exactly with the other knot (and vice versa).
This is exactly the process we conducted with the knot above: we decided it was an unknot (or more formally, it is equivalent to an unknot) because we can move it around until we're able to create the unknot. On the other hand, the trefoil cannot be moved around until it looks like a simple closed loop, so it is thus an entirely different knot than the unknot.
To be more specific, there are three types of moves we're allowed to use when transforming one knot if we want to remain equivalent to the same knot. They are typically called R1, R2, and R3:
R1) We can untwist a loop or twist a loop.
R2) We can pass one string over another string to form two crossings or undo this same move.
R3) We can slide a piece of string over a crossing.
A knot is still the same knot, no matter how many R1, R2, and R3 moves you apply—even if it looks completely different.
How Do We Classify Knots? Using Knot Invariants
As straightforward as the discussion above may appear, we actually landed on a question that mathematicians have yet to fully answer: how do we tell when two knots are the same?
For the scenario we just examined, this process is relatively simple: we find a sequence of moves to transform one knot into one another without allowing it to intersect itself, proving that our trefoil-like image is the same knot as the unknot.
However, as our knots become more and more complex, they might look something like this:
This is, in fact, an unknot, but it is very difficult to realize that it is—even with the assistance of a computer.
The complexity of this problem made it increasingly important for mathematicians to find ways to classify knots, and thus, knot theorists conducted an extensive search to find characteristics of knots that would allow us to understand what the set of possible knots looks like—and if two knots are the same or different.
The most useful characteristics are called knot invariants because they are the same for any equivalent knots: the "Gordian knot" above, our trefoil-like knot, and the circular unknot would all have the same value for any knot invariant because they're all the unknot. And, perhaps most importantly, if two knots have a different value for any knot invariant, they must be different knots! This allows us to definitively sort knots into categories.
One such knot invariant is called the crossing number. It is the minimum number of crossings a knot can have in any potential diagram of an equivalent knot. For example, the unknot has crossing number zero and the trefoil knot has crossing number three. There also is only one knot with crossing number four: we call it the figure-eight knot based on its shape in the diagram below.
You can move a knot around so it has more crossings than its crossing number, but it will never have fewer crossings than its crossing number. Hopefully, the diagram below helps you understand what the crossing number is.
Using the crossing number, mathematicians have tabulated the set of knots, working to discover exactly how many knots have each crossing number.
The amount of knots with each crossing number increases rather quickly. This diagram includes all 21 knots with 8 crossings and all 49 knots with 9 crossings:
Each knot is named with its crossing number and a subscript, where the subscript is an index value each knot is given relatively arbitrarily. It took several years of progress, and the discovery of several errors in the original knot tables from around 100 years ago, but we have reached an exciting point in knot theory.
For the first 16 crossing numbers, we have discovered the exact number of knots for each crossing number. For many of these numbers, we have proven that the tabulated knots are the only knots and that they are all different from each other: no knot listed here is equivalent to each other and thus, they are all separate objects. These discoveries are still far from fully understanding knots, but they are certainly a start.
Because it is still extremely challenging to know when you've reached the minimum number of crossings, the problem remains extremely open. Computer programmers have attempted to figure out how to write programs to tell which of the knots above a 3D image is equivalent to, but these programs are only successful up to a a crossing number of about 10 or 12. A more successful method requires knowing both how to deform your knot to get to the minimum number of crossings and then how to reshape it so it looks exactly like one of our labeled classified knots in the table—strategies that still remain out of our reach.
Even so, the crossing number gives us a very valuable handle on the problem. If we can prove one knot has a crossing number of, say, 4, and that the other knot cannot be manipulated into a knot with 4 or less crossings, we know that they must be different knots.
(One more side note about classification: We don't list the mirror images of knots. The trefoil is actually not the same knot as its mirror image as one cannot be manipulated into the other, but we only list one of the two mirror images for the sake of conciseness. They are called the left-handed and right-handed trefoil, and this property of handedness—or chirality—is important to their classification. Some knots, like the figure-eight knot, are indeed equivalent to their mirror image though.)
In addition to the crossing number, there are many other knot invariants that mathematicians have developed to understand the field. Values like the bridge index, linking number, stick number, and unknotting number, are also numbers that remain consistent for all equivalent knots, allowing us to tell if two knots are different. I won't be able to discuss them all in this article, but we'll talk about the unknotting number in a bit.
In fact, not all knot invariants are numbers. The Alexander polynomial and the Jones polynomial are some of the most powerful and fascinating knot characteristics used today. Just like the crossing number, we can conclude that two knots are different if their polynomials are different, which is an extremely valuable tool for our toolbox.
However, we still cannot easily figure out if two knots are the same; even if all of these characteristics match, we cannot conclude that two knots are the same knot! Knot equivalence is a much more elusive property than you might have previously thought, and the question of proving for certain that two knots are equivalent is still very much unsolved.
Prime Knots
I did leave one detail out of my explanation of the classification table above. The only knots we classified above are those that are called prime (or irreducible) knots.
Prime knots are knots that are indecomposable and cannot be described as a sum of two other knots.
Here is a knot sum (typically denoted by the symbol #). You take two knots and connect one section on each knot together like this, extending the string on both sides of a loop on one knot to connect to a loop on the other knot.
The opposite of knot addition is unintuitively termed knot division (not knot subtraction). You can think about it as taking a knot and squeezing a portion of it together until it "bottlenecks." After cutting the knot at that part and tying the ends back together, you have two knots.
Prime knots are knots that cannot be separated into two nontrivial knots through knot division. If you do try to perform knot division on a prime knot, you will always end up with the same knot itself and the trivial unknot, so you will not have succeeded in dividing it into two nontrivial knots.
Knot theorists focus on prime knots for much the same reason as number theorists focus on prime numbers: since the composite knots can be formed from prime knots, one only needs to understand prime knots to understand how the set of all possible knots behaves. The table above, then, only includes the prime knots.
Coloring Knots
Before I move on from the discussion of knot classification, I want to mention my favorite knot invariant: tricolorability.
A knot is tricolorable if you can color it with three colors.
When you color a knot, note that each segment must be entirely colored the same color. This means that each string that goes over an intersection remains the same color. In addition, each string that goes under an intersection must either be all the same color as the string on top or it needs to contain two entirely different colors as the string on top. In other words, every single crossing must contain three different colors coming together OR all of the same colors coming together.
An example should make this clearer. This knot is tricolorable:
All of the crossings either have red, green, and blue meeting at the same point or they are entirely green. Note that an entirely red crossing or an entirely blue crossing would be acceptable too. However, for a knot to be tricolorable, NO crossing can contain exactly two colors coming together.
Because it is a knot invariant, a tricolorable knot remains tricolorable no matter how you manipulate it, as long as you are using R1, R2, and R3 moves.
Here's another solve-as-you-go challenge for you: Once again, here are the three types of moves that you can use to transform a knot:
Can you prove that tricolorability is a knot invariant by showing that a tricolorable knot remains tricolorable under these moves and that a non-tricolorable knot remains non-tricolorable?
I won't explain the answer here, but you're welcome to leave a comment, post in the forum, or email me about it.
This gives us yet another tool to analyze knots: If one knot is tricolorable and another isn't, they are definitely not the same knot!
In fact, the unknot is not tricolorable; no matter what position of the unknot you use, you will never be able to color it with three colors. Therefore, if your knot is tricolorable, it is NOT the unknot and is thus tangled in some way.
How Do We Untangle Knots?
We've spent much of this article talking about knots as objects that cannot be untied. If one moves a nontrivial knot around in a valid way (meaning, in a way where the knot won't intersect itself), it will never become the unknot.
But, what if we break that rule? What if we allow ourselves to perform a move that represents the knot intersecting itself? This move is called a crossing switch, and it is important to note that it does inherently change the knot to a different knot. It requires cutting a knot's intersection and gluing it back together in another way, taking the string on top and moving it to the bottom. Thus, this transition represents a different way of thinking about knots, and it poses fascinating questions.
In fact, you've already seen this concept. When we took the trefoil and redrew it so the upper-right crossing had the strings crossing in the opposite direction, that was a crossing switch.
This is where the unknotting number of a particular knot comes in. It is the number of crossing switches it takes for a knot to become equivalent to the unknot.
For example, the trefoil has an unknotting number of 1 because it only takes the one crossing switch shown above to create the unknot.
Thus, the process of unknotting is a series of crossing switches until the knot becomes equivalent to the unknot and one can just rearrange it (with R1, R2, and R3) into a simple closed loop. That's all there is to unknotting, and it sets up a really amazing application of knot theory.
Unfortunately, just like every other knot invariant, we also don't know a lot about the unknotting number. It is very difficult to figure out how many crossing switches we need to unknot a given knot, but as usual, if we can prove that two knots have different unknotting numbers, we know that they are different knots.
An Application of Knot Theory: Bacterial DNA Reparation and Preventing Infections
If you've ever taken a biology course, you've probably heard that bacteria have circular DNA. What you might not know is that bacterial DNA can easily get tangled, especially when bacteria replicate. Because DNA is so critically important to a bacterium's survival (after all, DNA is the "executive" molecule in the cell that directs protein synthesis and everything else), tangled DNA would cause the bacterium to die.
Using an enzyme called Type II topoisomerase, bacteria are actually surprisingly adept at untangling their DNA, finding the optimal locations where DNA crosses itself to make crossing switches: cutting one piece of the DNA at the spot and attaching it together on the opposite side of the other piece. This means that bacteria—organisms with only one cell—are somehow better at untangling their own complicated DNA than even the best mathematicians are at untangling a simple knot. In fact, we have yet to even figure out how many crossing switches need to be made for most knots, let alone the best locations they should made at!
Admittedly, the scenario is slightly different: Bacteria are interested in returning their DNA to a relatively neat, easily accessible looped unknot even if that requires first transforming the knot into a more complicated knot first, while mathematicians want to use the fewest crossing switches possible even if they require dealing with a tangled mess until the end result is reached, but it's still a remarkably similar situation.
In fact, mathematicians and scientists are currently looking into applying this knowledge to prevent infections. Because bacterial topoisomerase is chemically different than human topoisomerase, it is possible that we could create a drug to target the bacterial topoisomerase and kill malignant bacteria, repressing a spreading infection. This would be a revolutionary new treatment for bacterial diseases, and it relies extensively on our knowledge of mathematical knots.
Knots and the Universe
Knot theory has been a cornerstone of the math world since the 1800s, forming the backbone to a famous scientific theory that has now been dismissed. Before the Periodic Table was created in 1869 by Dmitri Mendeleev, there was an enormous argument in the scientific world about whether all matter was made up of atoms or if all matter was made up of... drumroll please... knots. In the words of Matt Parker, "it was argued that the universe was filled with an undetectable ether and that the matter we experience is merely knots tied in that ether. Molecules were formed when these knots became entangled and linked."
Sounds ridiculous, doesn't it? In fact, this was a rather sensible option at the time because particles were still way too small to be detected, and there wasn't much evidence for atoms yet. Indeed, matter behaves in diverse ways and knots have inherently diverse characteristics that could explain the vast differences between different substances. Much of the drive to tabulate and classify knots, then, actually came from a desire to understand the universe at its most basic level.
All that is to say that knots once held a very prominent position on the international scientific stage. Though knots seem to only be remembered by mathematicians today, they are still extremely applicable in everything from DNA reparation to managing money at the quantum level to the surface of the sun.
Who knows? The secrets of the universe may still lie in knots in one way or another. I mean, why knot? :)
CONCLUSION
I hope you enjoyed learning a little bit about knot theory today! I am by no means an expert, and this is simply the tip of the iceberg of the subject, but I hope you gained something valuable and interesting from this article. I would absolutely love to have more conversations with any of you if you'd like to learn more about the subject; I have so much more I'd love to share that I wasn't able to share in this article. If you'd like to see a sequel, just let me know!
One of the challenges of writing about math is rigor. I tried to strike a balance between making this article as rigorous and accurate as possible without disclosing so many details for the sake of technicality that the main messages get lost. If there is an explanation in this article that you'd like to see with more or less rigor than it is currently presented, just let me know! I'm always willing to receive feedback on how the content is coming across and/or to provide you with more angles on a topic.
I'd also like to take a moment to point out one connection. At its heart, knot theory is a branch of topology, a subject that studies invariants that are preserved when objects are deformed by bending them, stretching them, and moving them without creating or closing holes. Topology looks at classes of objects, like all the objects without any holes, just like knot theory looks at classes of knots, like all the knots that are equivalent to the unknot. (If you've ever heard that a donut is equivalent to a coffee cup, that's topology!) If you want to learn even more about knot theory, I recommend starting with topology. Here's one video that could help (https://www.youtube.com/watch?v=p2ofJPh2yMw) and I'm happy to recommend more.
Here are some other resources if you'd like to learn more about knot theory, in no particular order:
This Numberphile playlist of videos about knots: https://www.youtube.com/playlist?list=PLt5AfwLFPxWLyfD4nhZCX_3UZdSSpkBTs
I especially recommend this video "What is a Knot?" which was one of my sources for this article: https://www.youtube.com/watch?v=aqyyhhnGraw
Andrews University's "Introduction to Knots & Invariants": https://www.youtube.com/watch?v=EBWP1POPc2A
Olivia Fugikawa's "Why Knots? Knot Theory in Three Minutes": https://www.youtube.com/watch?v=WXVItJnZiWc
Things to Make and Do in the Fourth Dimension by Matt Parker: I've recommended this book many times before, and the eighth chapter is all about knot theory! This was my most important resource in writing this article (especially the last two sections), and I highly recommend it.
The Math Book by Clifford A. Pickover: This book covers so many topics in math, and it includes well-written succinct overviews of milestones in knot theory.
This article about Lisa Piccirillo's discovery last year in knot theory: https://www.quantamagazine.org/graduate-student-solves-decades-old-conway-knot-problem-20200519/
I'd recommend checking out Quanta Magazine for all your current math news!
Thank you so much for reading! I really appreciate all of your support, and please share this article to allow GLeaM to further our mission of supporting more girls in STEM. Since you made it to the end of the article, I'll also post one more exciting question for you to try!
The Knotted Molecule (from Martin Gardner's Wheels, Life, and Other Mathematical Amusements)
"Enormously long chainlike molecules (long in relation to their breadth) have been discovered in living organisms. The question has arisen: Can such molecules have knotted forms? Max Delbrück of the California Institute of Technology, who received a Nobel prize in 1960, proposed the following idealized problem:
Assume that a chain of atoms, its ends joined to form a closed space curve, consists of rigid, straight-line segments each one unit long. At every node where two such 'links' meet, a 90-degree angle is formed. At each end of each link, therefore, the next link may have one of four different orientations. The entire closed chain could be traced along the edges of a
cubical lattice, with the proviso that at each node the joined links form a right angle [see Figure 10]. At no point is the chain allowed to touch or intersect itself; that is, two and only two links meet at every node.
What is the shortest chain of this type that is tied in a single overhand (trefoil) knot? In the answer, I shall reproduce the shortest chain Delbruck has found. It has not been proved minimal; perhaps a reader will discover a shorter one."
As Martin Gardner admits at the end of this problem, even he didn't know the answer fully at the time. Let me know what you come with if you try it, and see you on GLeaM's May edition! Make sure to become a member or a subscriber to get an email when the article is published.
Cover Image courtesy of HelenKlebesadelArt
Bonus Question: Did you notice that the cover image is not actually a mathematical knot? Why not?
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