What is knot theory about, exactly?

Solution 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.

Solution 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?

Solution 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?