 This video will be a mathematics talk on the following topic. Is the Conway not a slice knot? This topic is a little bit unusual for me. In fact, it was requested by an audience member, and I should warn you in advance I'm not exactly an expert in this area. In fact, I was learning about it by hastily reading up Wikipedia earlier today, so you're kind of being warned that not everything I say is necessarily actually correct. So, this question, is the Conway not a slice knot, was an open question for quite a long time. It was answered very recently by Lisa Piccarillo in the following paper, who showed that the Conway knot is not slice. I should also warn you, I'm not quite sure how to pronounce her name correctly and I'm sorry if I've got it slightly wrong. So, what I'm going to talk about is, first of all, what is the Conway knot? And secondly, what is a slice knot? And then I'll say a little bit about how Lisa Piccarillo was able to prove that the Conway knot is not a slice knot. So, first of all, let's discuss what the Conway knot is. Well, first of all, we should just sort of review what a knot is. So, a typical knot might be something like a trefoil knot, which looks something like this. And if you've got two knots, you might want to tell whether they're different. So, is the trefoil knot different from an unknot? And it sort of looks intuitively obvious that they're different. And for simple knots like this, it's not very difficult to tell whether they're different. But in general, if you've got a really complicated knot and another really complicated knot, it can be really hard to tell whether they're the same or not. And one way to show two knots are different is to look at the Alexander polynomial. So, the Alexander polynomial is a certain polynomial. Actually, it's really a Laurent polynomial, usually denoted by delta of t, depending on the knot. And you can calculate the Alexander polynomial for knot. For instance, the Alexander polynomial of this sort of unknot is just one. And the Alexander polynomial of this trefoil knot is t minus one plus t to the minus one. And these are different polynomials, which shows that these are different knots. Of course, for this particular case, it's trivial, but in more complicated cases, it's much easier to tell whether or not two polynomials are the same. And for complicated historical reasons, there's a variation of the Alexander polynomial called the Conway polynomial, which is usually denoted by delta of t. And these are related because the Alexander polynomial of t squared is equal to the Conway polynomial of t minus t to the minus one. So, how do you figure out what the Alexander polynomial of a knot is? Well, you can work it out using the Skyn relation. And the Skyn relation is a little bit easier to do for the Conway polynomial, so I do it for that. So, the Conway polynomial of the trivial knot, sorry, the Conway polynomial of the trivial knot is just one. So, here this isn't a zero or an O. It's an actual knot consisting of just a circle. And if you've got a knot like the trefoil knot, then what you can do is you can close in on one of the crossings in some plane projection and you can twiddle this crossing in several ways. For example, you could change it like this, where you put one piece of string under the other piece of string and you could just change it so it's going over. And another thing you could do is you could just cut these two strings and rejoin them and you would then get something like this. So, there are three very close related knots. You know, this one isn't actually quite a knot, it's a link, but let's not worry about that. And what's happening is you get three different sorts of links here. Here you get a link, sorry, a crossing which can be denoted by L minus. And here we get a crossing that looks like this, which is usually denoted by L plus. And here we get a crossing that looks like this, which is denoted by L zero. So this is really a picture of what's going on inside this little circle here. And now the Conway polynomials are related by the Conway polynomial of L plus of T minus the Conway polynomial of L minus T is equal to T times the Conway polynomial of L zero of T, where these represent the three knots where you've done this crossing, this crossing and this thing inside the knot. So this would be L plus, this is L minus and L zero. And you can check this recursion relation actually defines the Conway polynomial of any knot. So this is the famous Skyn relation. It was actually first discovered by Alexander in the 1920s, but everybody forgot about it. He put it in a sort of little footnote right at the end of his paper that nobody paid any attention to. So it then got rediscovered and it was only later realized that this was really first found by Alexander. So if the Alexander or Conway polynomial of a knot is not one, the knot is non-trivial. And you can ask the converse if the Alexander polynomial is equal to one, is the knot trivial. And Conway found an example of a knot with trivial Alexander or Conway polynomial where the knot wasn't trivial. So here is Conway's knot. I haven't dared to draw it because I'll undoubtedly get it wrong. So Conway's knot is the left-hand one. And as you see it's got, I think it's got 11 crossings and is trivial Alexander polynomial. And the knot on the right is another knot that also is trivial Alexander polynomial. And it's something called a mutation of Conway's knot that we'll come to later. So just explain what a mutation is. You see inside the red circle the two, the knot's the same except that the stuff inside the red circle has been sort of twisted round a bit. So that's roughly what a mutation of a knot is. You take a sub-piece with four bits of string sticking out and you twiddle it round a bit. So next I'm going to talk about what is a slice knot. So the first attempt at defining a slice knot. So we can think of a knot as being a circle inside, circle S1 inside three-dimensional space. Well, for various reasons, knot theorists don't use three-dimensional Euclidean space. They tend to use a three-dimensional sphere, but it's just a minor technical difference. But I want to think of S1 inside R3 for the moment. And what you can do is you can think of R3 as being embedded in R to the four. And what you can do is you can take a copy of a sphere, S2 inside R to the four, and you can intersect this sphere with R3. So what you're doing is you're really taking a slice of a sphere by the standard hyperplane R3. And most of the time this intersection will be a copy of S1 inside R3. And if that happens, you might try calling it a slice knot. Well, it turns out that's a completely stupid definition because every knot can be obtained in this way. And you can see this quite easily as follows. So suppose I take a copy of R3 and I'm going to draw R3 as a plane because I'm going to need a bit of extra room to move in. So you have to pretend that this plane is three-dimensional. And now suppose we take a knot in R3, and I can't really draw a knot very well because this is a plane and there's not room to draw a knot. So pretend this is a knot in R3. And what I do now is I take a point in R4 and I just sort of join it up to all the points on the knot. And then I can do the same thing on the other side. And if you think about it a bit, you will see that if I do this, I get a copy of a two-dimensional sphere S2 whose intersection with R3 is just the knot I started with. So we can't define slice knots quite like that. We would like to define them as slices of a sphere, but the obvious definition just breaks down. Well, if you notice there's something rather funny about this construction. This point here is rather weird. In particular, it doesn't look locally like the standard R2 contained in R to the 4. So you can think of R2 as being a vector space inside R to the 4. And you'd really like the sphere to look locally like that. So what we say is that it's locally flat. And this construction isn't locally flat. So we can try saying that something is a slice knot. So we can say something is a slice knot if it is a slice of a locally flat copy sphere in R to the 4, or possibly S to the 4, depending on what you want to embed things in. So this gives a slightly better definition, but there's still a slight problem with it. So there's a theorem of Friedman from 1984 which says that if the Alexander polynomial equals 1, this implies that the knot is a locally flat slice in R to the 4. So it means you can take it by intersecting a sphere with R3. Well, if we use this as the definition of a sliced knot, then this would show the Conway knot is a sliced knot because it's Alexander polynomials 1. I mean, that's the interesting thing about the Conway polynomial. It has Alexander polynomial 1. Well, it turns out that there are two slightly different concepts being locally flat. So we can either demand that it's topologically locally flat, or we can demand that it's smoothly locally flat. And a lot of the time, anything you can do with topological functions, you can do with smooth functions, essentially because any continuous function can usually be approximated as close to you like by a smooth function, but sometimes you can't. And this is one of the cases where it really matters whether you're doing things topologically or smoothly. So Friedman's result says that a knot with Alexander polynomial 1 is topologically locally flat. And this is sometimes called topological flatness. On the other hand, what Lisa Piccarillo proved is that it's not smoothly locally flat, and this is what's more difficult to prove. So let's just give an example of something that is smoothly locally flat. What I'm going to do is I'm going to take the following knot. See if I can draw it correctly. Real knot theorists are much better at drawing knots than I am. So what I'm going to do is I'm going to take a knot here. Notice that what I've done for this knot is I've taken a knot and kind of joined it together with its mirror image. So you can think of this as being a mirror, and then this is just the reflection of the knot in a mirror. And what I can do now is I can make this knot a slice of a sphere in four-dimensional space sort of adding semicircles in four-dimensional space from each point to its mirror image like this. So you should think of these orange semicircles as kind of sticking out in four-dimensions. And that gives me half of the sphere, and then I can get the other half of the sphere by joining up with semicircles going in the other direction in four-dimensional space. So anyway, if you do this construction, you find that any knot that's obtained by taking a knot and joining it with its reflection in this way is actually a smooth slice knot. Well, now the problem with telling whether or not with showing that the Conway knot is not a slice knot is the following. First of all, Conway knot is a mutant of a slice knot. And let's just recall what this means. Well, if you go back to this figure here, the Conway knot is the one on the left, and it's a mutant of the one on the right, meaning you can get from one to the other by sort of chopping out something with four bits sticking out and rotating a bit. The one on the right is actually a smoothly slice knot. Well, the other problem is that the Conway knot is a topological slice knot. And this follows from Friedman's result because its Alexander polynomial is trivial. And the problem is that before Lisa Piccarillo's result, we had some methods for showing that mutants of slice knots were sometimes not sliced. However, they all worked by showing that the knot was not topologically sliced. So they completely fail for knots that are topologically sliced knots. There were no ways to show that a mutant of a slice knot that is topologically sliced knot is not a smoothly sliced knot. So this was the sort of hard problem that she had to solve. And the way... I'm now going to sketch very roughly how the proof goes, and for reasons that become obvious in a moment, I'm not going to go through the details precisely. So the idea is as follows. First we've got a knot k. So I'm going to write k for a knot. And a knot is contained in s3. And s3 you can think of as being the boundary of a foreball. And what we can do is we can form something called the knot trace, which is denoted by x of k. And what you do is you think of the knot in s3 and we sort of add a fattened disk to b4 joined along k, which is in s3. So a disk has boundary circle s1, so we can join the boundary of the disk to k, and then we can fatten it out a bit and we get a foremanifold with boundary, which is called the knot trace. And then you can check that k is smoothly sliced, sorry, smoothly sliced, is equivalent to saying that the knot trace xk smoothly embeds in s4. So now these picarello strategies are now the following. So step one, pick a special knot, which is denoted by k prime. So I'm going to take k to be the conway knot. So first we're going to select k prime and k prime has to have the following property. We want to show that x of k is equal to x of k prime. And thirdly, we want to show that k prime is not smoothly sliced knot. And we show that k prime is not smoothly sliced by showing that s of k prime is equal to 2, where this is a certain invariant called Rasmussen invariant, which has the property that s of l is not if l is smoothly sliced. So if we can show all this, then this will show that the conway knot is not a smoothly sliced knot because our special knot k is not smoothly sliced because this invariant vanishes and these two have the same knot traces. So if k prime is not a smoothly sliced knot, then k is not a smoothly sliced knot. So there's no reason why this invariant s should be the same for k and k prime, by the way. So the clever thing is you first have to change the knot to a different knot and then calculate an invariant of this different knot. Well, that's the overview of the proof. The details are kind of really rather complicated. So first of all, we need to know what the knot k is and the knot k looks like this. Let me just check I've got it right. So this is the knot k prime and you can see it's already looking a little bit complicated. Now we have to do step two, which is to show that the knot trace of k and k prime are the same. And the proof of this looks like this. Okay, well, you start with these rather complicated looking diagrams which are indicating various constructions you do with knots. This is a sort of calculus of four-dimensional manifolds that knot theorists use. And it's worse than this because this calculation goes on a bit like this. So we've got two pages of these rather hairy-looking geometric manipulations. And I sort of look at these. You know there's a sort of goal which is to formalize all mathematics so it can be computer checked. And imagine trying to take these diagrams and formalize them in some formal language that a computer can check. I really don't envy whoever has to do that in the future. So that sort of does step two, which is showing that the knot traces of these two knots are the same. Then we have to do step three, which is to calculate the invariant of this strange knot k prime. And the calculation of that looks a bit like this. Okay, well, as you can see, this is a highly non-trivial calculation. You need to calculate this large table of numbers in order to work out Rasmussen's invariant. Okay, so the proof sort of is one of these where the step... If you just look at an overview of the proof, it perhaps doesn't look too difficult, but actually figuring out the details of these three steps is really rather hairy calculation. I must admit I have no idea how she managed to pick this particular knot k prime in. Once you've picked this knot k prime, then checking step two and step three is in some sense a reasonably routine calculation. The difficult part of this proof is thinking of this process in the first place and picking the right knot k prime. Okay, as I said, I haven't put in the details. If you want to check the details for yourself, I'll put a link to the paper in the description of the video.