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All Tangled Up: An Introduction to Knot Theory

Updated: Jul 26

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.

Photo courtesy of Source Kids.

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.

Photo courtesy of mallorcaphotoblog.com.

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 trefoil

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.)

The unknot

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.)

A trefoil?

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.

Photo courtesy of theGIST

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:

Photo courtesy of Math Overflow

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.

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