Why All Knots are Actually Useless
(...if you have a fourth dimension)
Sometimes, thinking in four dimensions leads to some surprising results. One cool fact about four dimensional space is that in four dimensions, you can undo every knot. It doesn't matter how complicated you tie a knot - just by moving pieces around you can always turn it back into a smooth piece of string, without cutting or breaking the string.
How? Let's find out.
But before we start thinking about knots in four dimensions, I want to tell you how mathematicians think about knots. Knot theory is an entire branch of mathematics which studies, well, knots.
Take a tangled piece of string. Can you undo all the tangles, or is there a knot that makes untangling it impossible?
How do we describe knots?
How many different knots are there?
If you have two knots, is one a tangled version of the other, or are they different?
If I asked you to think of a knot, you'd probably think of this, called the "overhand knot". Can you untangle this knot?
Turns out, undoing this knot is easy: just move the loop over the edge of the string and it comes apart easily.
If the string's ends are just random points in space, then you can untie any knot this way with enough effort.
Take two knots. Earlier, we asked the question "are they the same knot?" Now we have one answer: if you untangle both knots, they always become the same thing: a straight piece of string.
...but that's not helpful, or interesting. So mathematicians study different, more interesting, types of knots.
Let's define knots like this: a knot is a closed loop of string, with both ends attached. If we join the ends of our overhand knot, we get a new type of knot, called a "trefoil knot".
Defining knots as loops is better, because before, every "knot" could be turned into every other "knot" with enough effort. Now that's not true; you can't turn this knot into a circle without cutting it. (If you don't believe me, grab some string and try it yourself!)
OK, now we're ready to talk about undoing knots in four dimensions. Let's start by focusing on this crossing, right here. Can we undo this crossing? It sure would be nice if we could pull one string through the other, like a ghost walking through walls. That would make undoing the rest of the knot much easier.
Unfortunately, our strings aren't ghost strings. Two strings can't be in the same place at once, so if we try to pull the string downwards, it bonks into itself and can't go through.
How can we describe "bonk" mathematically? One way to do it is to choose a coordinate system, and give each point on the string some coordinates.
Now we can see the problem from a new angle: when we pull the top string downwards, we get two points with the same coordinates. That's not allowed, because two things can't be in the same place at once.
But what if we had a fourth dimension?
Before we try to undo this crossing, we can move one part of the string into the fourth dimension.
And now, when we try to slide the strings past each other, the two points have different coordinates, and hence are in different places. Therefore, there's no collision, so there's no problem sliding the strings like this. It may look like they intersect in 3D space, but that's just because of the way we're visualizing the fourth dimension.
Finally, we can move that second strand backwards in the fourth dimension. And now we've won - the two strands are on opposite sides of one another from when we started. Now we know how to move two strings past each other, whenever we want.
To undo the rest of the knot, we just repeat that technique as many times as we want. Once we undo enough crossings, we're done! Without cutting the string, we've unknotted a knot. In fact, this technique works on any knot you can tie in a piece of string. In four dimensions, knots aren't knotted.