What is knot theory about, exactly?

Question

"In topology, knot theory is the study of mathematical knots."

This is how Wikipedia defines knot theory. I have no idea what this is supposed to mean, but it does seem interesting. The rest of the article is full of examples of knots, their notation and such, which I understand a little bit better, but I still fail to understand why and how they are studied. So, what exactly is knot theory about?

What characteristics of knots are studied, and how does it connect to the rest of mathematics? Full disclosure: I have experience with analysis, know at least the basics of abstract algebra, and I think I could understand elementary topology. So feel free to give me only a moderately technical description.

Answer 1

A knot, for our purposes, is a (well-behaved) "loop" in 3-dimensional space. Mathematically speaking, we could think of a knots as (injective, differentiable) functions from the unit circle to $\Bbb R^3$ (or equivalently, the image of this function in $\Bbb R^3$). Without losing any real structure, we'll suppose that these loops fit inside the sphere of radius $1$.

That's the easy part. Now, the tricky bit: what does it mean for two knots to really be the "same knot"? Intuitively, we'd like two knots to be the "same" if you can strech/squish/twist one to make the other without "tearing the rope" or passing the rope through itself. The way we encode this mathematically is to say that two knots are the same knot if they are ambient isotopic (or, for a weaker condition, ambient isomorphic). In particular, two knots $K_1,K_2 \subset B$ ($B$ is the closed unit ball) are ambient isomorphic if their complements $B \setminus K_1$ and $B \setminus K_2$ can be continuously deformed from one to the other (they are ambient isomorphic if these complements are homeomorphic).

Note: It is not enough to check whether two knots are homeomorphic, since all knots are homeomorphic to the unit circle. I believe that ambient isomorphism implies ambient isotopy in this case, but I'm not sure.

With that, the central knot theory questions are

• How can we tell if $K_1$ is the same knot as $K_2$
• How can we break complicated knots down into smaller knots that we understand

Another helpful way to think about knots is in terms of their knot diagrams. In particular: we take a knot, look at its projection onto a suitable plane, and keep track of all over/under crossings. It turns out that two knot diagrams correspond to the same knot if and only if one can get from one diagram to the other using Reidemeister moves.

So, how do we tell knots apart? Usually, we do so using knot invariants, properties that a knot retains no matter how exactly it's stretched, twisted, or smooshed. For example, we know that the trefoil is distinct from the "unknot" because the trefoil is tricolorable, but the unknot isn't.

Answer 2

Knots are embeddings of the circle $\mathbb S^1$ into $\mathbb R^3$. The basic question is, when can one knot be continuously deformed into another?

Answer 3

EDIT: This post is not really correct (see comments), but I'll let it stay because I still think there's some value!

When would you say, intuitively, that two knots are the same knot? Your intuitive idea probably coincides strongly with them being homeomorphic topological spaces. (i.e. stretching or bending a knot without cutting or gluing does not change its fundamental knot-soul.) It's important to note that by "knot" what is meant is a loop tangled in some way. Check this examples from this image I took from wikipedia.

All of these knots are $\textit{fundamentally}$ different in the sense that to transform one of them into another one of them you must cut and glue the string (which is the idea "topological homeomorphisms" try to capture.)

Some easy questions are already interesting. For example, are there infinitely many different knots?

Answer 4

As was mentioned numerous times in the other answers and comments, to knot theory is usually studied through the lens of knot invariants. So to give an idea of what invariants are about, I will give some examples of what they can do and can't do.

1. Being able to compute an invariant is useful for telling knots apart, but not always good for being sure you have the same knot. The easiest to explain is tricolorablity. This is a simple yes or no invariant. A knot is either tricolorable or not. So, if you have two knot diagrams, and one is tricolorable and the either is not, they definitely are not the same knot. But if they are both tricolorable, then we can't tell if they are the same knot or not. So we often are looking for specific classes of knots we can tell apart based on a certain invariant.
2. One problem with the first item is how easy it is to compute the invariant. Some are relatively easy, but they usually not as useful for telling knots apart. The ones that are are "easiest" (as in, there is an algorithm) to compute are the polynomials - Alexander, Jones, HOMFLY - and they are rather good at distinguishing knots. But the bigger the knots get, the harder the invariants are to compute. To be a little more precise, they all grow exponentially in the number of steps required based on number of crossings in a diagram. (Of course, for a big knot, you might have no chance at computing any invariant, so they still might be the best option.)
3. Invariants are interesting to see how they act under changes in the knots. For example, we can make new knots from old by a process called connect sum: We just combine two knots into one, by breaking them apart and connecting them into one continuous loop. The bridge number of a knot $b(K)$ is an invariant which gives a number: The bigger the number the more "complex" the knot is. But when we connect sum a knot $J\#K$, we get the nice relation $b(J\#K)=b(J)+b(K)-1$. We think of connect sum as addition of knots, so knot theorists find it pretty and exciting when we almost add in the invariants when we "add" the knots.
4. We often look at all the knots of a certain value for an invariant. Example: we can look at all knots which are bridge number 2, since I mentioned this invariant already. These knots often have exploitable characteristics which can let us say something about another invariant. Again, keeping with our 2-bridge knots, every two bridge knot has a knot group (another algebraic topology invariant) $\pi_1(S^3-K)$, has a presentation with two generators and one relation.

Obviously, this is just how I look at knot theory, and I am sure you will get different opinions about it from other knot theorists, but I hope this might give a flavor for what it is we do.

Answer 5

We can study knots algebraically also because Quandles are complete knot invariant and quandles are algebraic object. For more information see this http://aleph0.clarku.edu/~djoyce/quandles/aaatswatkt.pdf