How do we tell when two knots are the same and when they are different? That is, if we start with one of them and fiddle around with it for a while, might we eventually make it look like the second knot?
In order to start answering this question, it’s important to be clear on what we mean by a knot. If we just mean a string with free ends, then the problem becomes rather stupid: take the first knot, untie it, then tie it up like the second one.
So, that definition is clearly inconsistent with our intuition about knots. We expect knots to be tricky to untie. (Isn’t that the point?) We remedy this problem by gluing the ends together. Hence, rather than considering objects like this we consider ones like this, the trefoil knot: Let’s also point out one special knot: This one is not tied at all; we call it the unknot.
I’ve actually already done something a bit sneaky: I drew a picture of a knot, which lives in three dimensions, in two dimensions. The picture I drew represents the view we’d have if we were looking at the knot from above. Hence, when we see something like this: it means that the strand starting at the top left and going to the bottom right goes above the one starting at the bottom left and going to the top right. (Note that when I draw this sort of picture, with a bunch of ends hanging loose, I mean this to be part of a larger knot. I don’t need to draw the rest of it because we’re just focusing on one small section of the full knot.)
We’ll always draw knots in this way, but we need to say a few more things about what is allowable. We don’t want to have one strand directly above another one. We also don’t want three or more strands to come together at one point in our diagram. We can always arrange this with a bit of wiggling.
It turns out that, not only are these diagrams helpful for visualizing a knot, but they are also useful for doing computations and analyses. We’ll study knots by analyzing their crossing patterns. Of course, the crossing patterns can change when we wiggle the knot, so we need a way to cancel out that effect.
In order to understand what we’re up against, we have to determine the sorts of changes that can take place in the knot diagram as we wiggle the knot. The simplest thing that might happen is that we vary the shape of the diagram without influencing the crossings, like this: Since we’re only going to be studying the crossings, we will not concern ourselves with this possibility.
Kurt Reidemeister showed that if we have two configurations of the same knot, then we can move from one to the other by a sequence of Reidemeister moves. Here’s an example of this process, showing that a figure-eight knot is the same knot as its mirror image. (This is not true for all knots; it’s false for the trefoil, for example.) This picture is from Colin Adams’s The Knot Book.
We’d like to tell when two knots are the same based on their knot diagrams. This really means that we have two tasks in front of us. The first one is, if we’re given two knot diagrams representing the same knot, we’d like to know that they are the same knot. (Even better, in fact, would be a way of wiggling one of them around to get to the other one! That way, we could actually deform one knot into the other, rather than just knowing that it’s possible, somehow.) The second one is, if we’re given two knot diagrams representing different knots, we’d like some way of proving that they’re different. We’ll focus on the second task, mostly.
One thing we might try is to assign numbers to various knot diagrams. One particularly natural way to do this is to count the number of crossings. Unfortunately, this can easily change, for example by applying an R1. A slightly more refined version of this would be to look at all possible knot diagrams corresponding to a given knot and track the minimal number of crossings in any of these. We call this the crossing number of the knot.
The problem is that it’s very difficult to determine the crossing number. We might have some configuration that looks minimal, but how can we tell that by excessive fiddling, we can’t make it smaller? It would require a lot of work to prove such a thing, as there’s no obvious way of doing it.
While the crossing number of a knot is indeed an invariant — that is, it doesn’t depend on which knot diagram used to depict the knot — it isn’t necessarily a very useful one for distinguishing different knots, since we have no clear way of computing it.
So, let’s look for something else. One pretty natural thing to do with the crossings would be to uncross them. Of course, we can’t generally do that without changing the knot, but it’s something we could do with a knot diagram. Perhaps we could relate a knot invariant associated to a knot with invariants associated to the knots we get by modifying the crossings by cutting the strands at the crossing and gluing them in a different way. In particular, we’d like to relate the following crossing (and noncrossing) patterns:
Well, actually I said something a bit imprecise in the last paragraph. When we uncross a crossing, we may not end up with a single piece of string; we might get two. Technically, if this happens, we should refer to it as a link rather than as a knot. Hence, when this possibility comes up in the future, we’ll refer to objects as links rather than knots.
A natural way to relate the invariant of some link to the invariants of the links we get by changing some crossing would be as follows: We need to put in , , and in order to make them actually be invariants; random values generally won’t work. We’ll determine appropriate values for , , and in just a moment. If is a link, then will denote the invariant associated to it. We don’t know what it is yet, but we hope that something will work. Also, recall that if we have a small piece of a link, we can put that into brackets and we mean the full link. The rest of the link is unchanged by our changes in crossings. We’ll refer to as the bracket polynomial of , for reasons that will become clear shortly.
Even if we figure out what values of , , and work, though, we still won’t be able to figure out what the invariants actually are, unless we make some arbitrary choice somewhere to get the process off the ground. We’ll assign the invariant 1 to the unknot (i.e. the bracket polynomial of the unknot is 1); that’s about the nicest thing it could possibly be.
Finally, we need to check R1. It’s not invariant under R1 unless . So, we could force to be a number equal to 1, and that would be a pretty good solution. would be one solution, but there are other, more interesting, complex numbers with that property, and we’d get a better invariant if we were to choose one of them.
However, there’s a trick that can make the invariant even better. We can agree, right now, that the what we have, with the left standing, is not quite an invariant, but is off by a little. That’s because the rules we set up at the beginning were a little bit too simple. We can content ourselves by calling the bracket polynomial an almost-invariant and trying to find another almost-invariant with a similar defect. Then, maybe they’ll cancel each other out and give us an actual invariant.
When we draw a knot diagram, we don’t just end up with a knot; we also chose a direction to move on (the way our pen moved to draw it). While it shouldn’t matter which direction we traveled, it can be helpful to consider the direction as part of the data of the knot, and that can help us refine our bracket polynomial. So, we draw arrows on the knot diagram to indicate the direction we travel, like this:
Now, at every crossing, we see two arrows, one for each strand. The crossings come in two types: ones that look like this: and ones that look like this (We might have to rotate it in order to get it to look like one of these pictures.) We count the first type as +1 and the second type as -1. Then, we add up these +1’s and -1’s for all the crossings in the knot diagram, and we call this sum the writhe of the knot diagram. We’ll denote the writhe of a link by .
Let’s see how the writhe fares under the Reidemeister moves. Under R1, it changes by 1. Under R2, it doesn’t change. Under R3, it also doesn’t change. Hence, the writhe, like the bracket polynomial, only changes under R1. This suggests we can put them together to make an actual invariant.
The simplest way to do that is to set . Now, this is an actual invariant, and it’s equivalent to the Jones polynomial. For the sake of historical consistency, we’ll substitute for in to get , which is exactly the Jones polynomial.
It isn’t actually a polynomial in the usual sense of the word. Generally, we expect all the exponents in a polynomial to be nonnegative integers, but here there might be some negative exponents, and there might be some half-integers as well. Still, it’s close, so due to tradition, we refer to it as a polynomial.
So far, we don’t know that the Jones polynomial is at all useful. We know that the Jones polynomial of an unknot is 1, but perhaps the Jones polynomial of every knot is 1. If so, we’ve done a lot of work for nothing. Fortunately, this turns out not to be the case. Let’s compute the Jones polynomial of the trefoil knot
Here’s the computation of the bracket polynomial: We begin by applying our rules on the upper left crossing, so we end up with the following two diagrams: If we call the original trefoil and the modified ones and , we have . Continuing to break down and will eventually lead to . Here’s a picture that helps us to determine the writhe. Thus the writhe is +3. Hence we have , and . So, the Jones polynomial of a trefoil knot is not 1. Hence, we have proven that we cannot wiggle a trefoil around and get an unknot. We’re also very close to proving something else interesting, that I mentioned earlier: the trefoil is not the same as its mirror image, for the mirror image has Jones polynomial .
If you recall what we said at the beginning, we’d also like to be able to tell when two knots are the same. Does the Jones polynomial help with this? Somewhat, but not completely. It turns out that it is possible to find two different knots with the same Jones polynomial, such as these two, the so-called and knots.:
However, this behavior seems to be relatively rare; when two knots have the same Jones polynomial, we at least suspect that they are the same, although this is not guaranteed. There are other invariants that can be used that might distinguish knots that the Jones polynomial cannot, so using several of these in conjunction can be a good idea.
Vaughan Jones won a Fields Medal in 1990 for his discovery of the Jones polynomial. As you can see, it isn’t so hard. To the best of my knowledge, this is the most accessible work that has been awarded the Fields Medal.
The Jones polynomial is relevant in other branches of mathematics as well. If you are interested in learning more, you would do well to look at this paper by Jones.
There are still open questions related to the Jones polynomial. Perhaps the simplest is the following: is there a knot K, which is not the unknot, that has Jones polynomial equal to that of the unknot, namely 1? In other words, can the Jones polynomial distinguish the unknot from any other knot?
A knot invariant that can distinguish the unknot from any other knot is said to detect the unknot. There are invariants that are known to do so, including a wonderful generalization of the Jones polynomial called Khovanov homology. A good place to learn about Khovanov homology is from Dror Bar-Natan’s paper “Khovanov’s homology for tangles and cobordisms.” Of course, it requires far more background than is used here.