OK, well, thanks very much, Martin. Welcome everyone to Oxford. I'm going to be as Martin said, talking about knots and knots are pretty much exactly what you think they are. So you you take a piece of string or in this case, a skipping rope and you tie it up some way. And then you glue the two ends together, you glue the twins together to form a closed curve. And that is a knot, and you allow yourself to say, well, like wiggle the knot around.
That doesn't change the knot, but you're not allowed, obviously, to cut it or make it pass through itself. So here are some knots as the simplest possible not the UN, not which is just the round circle. Then after that, the first non-trivial knot is is the trefoil and then the Figure eight. And so surprisingly. Mathematics has something to say about knots, but you may ask yourself why, why should we study knots?
Well, they come up, it turns out all over the place, they come up in the physical and biological sciences. So let me just give you one example of that which is in. DNA. OK, so DNA is, as we all know, it's shaped famously like a double helix, so it has these two strands which curl around each other. And as you run along the the strands, there are various chemicals which encode our complete genetic makeup. They give the instructions for for life. And so.
Clearly. Not only do these two strings tangle around each other, but it turns out the DNA ends up by being extremely knotted. And this is the particular relevance in the replication of DNA. OK, so sitting inside each of our cells in our body, there is a complete copy of our genetic code and every now and again, the cells divide divide into two. In fact, this happens incredibly often. It happens a thousand so a million times a second in each of our bodies.
And what happens when the cell divides is that this genetic code or DNA has to and it has to be copied so that one bit of DNA ends up in one of our cells and one ends up in the other. So the way it actually works is that the two strands of this double helix get separated out from one strand ends up in one of the cells after division, and the other strand ends up in the other cell. And then the cells can rebuild the entire DNA from this process.
Because as you run along the double helix, there are these different chemicals which specify the code that called a C, G and T and G only binds to C and A only binds to T. So if I have just one half of this double helix, then I can use that to recreate a complete copy of the double helix. So. These two strands, they have to pull apart, but this is actually a quite a complicated and difficult procedure. And the reason is that DNA is incredibly convoluted within our cells.
So each cell nucleus is typically about five or six microns across. And if you were to take the DNA that was just sitting in that cell nucleus and stretch it out into one long line. It would be two metres long. So something two metres long has to end up like getting scrunched up into something just five microns across, so it's incredibly knotted. And when the cell replicates, these two bits have to somehow get from one one has to end up in one cell and the other one has to end up in the other.
So clearly the not the two strings have to pass through themselves, pass through each other. And this is done by various enzymes and importantly, the enzymes that work for bacterial DNA, a different from the enzymes that work for human DNA. So in fact, lots of the antibiotics that we use, they they specifically block this untangling procedure in bacterial DNA.
And so that means that the bacterial innate DNA cannot unlocked itself and so cannot replicate and so dies, whereas that has no effect on us because we use a different type of enzyme to do this. So clearly, understanding this unlocking procedure is important, but it's not completely well understood. And there's been a very productive collaboration between biologists and not theorists to be able to understand that in more detail.
So that's just one example of the occurrence of knots and the importance of knots in nature. I'm a pure mathematician, and one of the reasons that I find not so interesting is because they are somehow intrinsic to three dimensional space. If we lived in two dimensions, then there wouldn't be enough space to be able to form nuts, whereas if we lived in four dimensions, there would be enough space so that every knot could be undone.
And maybe if this time at the very end of this lecture, I'll show you how you do that, how you are not any, not in four dimensional space. So somehow, the existence of knots is one of the intrinsic features of the three dimensional space in which we live. So not theory is now part of topology, so part of the aim of this talk is to introduce you to topology and explain a little bit about what it's what it's found.
So topology is a bit like geometry. So in geometry, you look at spheres and cubes and triangles and squares. But to a janitor, a sphere is different from a cube. Whereas to its apologist, they look the same. So in topology, you're allowed to deform objects with as long as you don't tear them apart. So you can deform this sphere, you just squish it down into this cube. So to it's apologists, these two objects are the same. So famously so its apologists, his coffee cup is the same as his doughnut.
So there you can see the two deforming one into the other. So it kind of then makes sense that because when we're looking at knots, we allow them to deform but not pass through each other through them selves. That that is exactly the right sort of subject. Topology is exactly the right tool to be able to study knots. So the first person to realise that knots were actually part of mathematics was the great Carl Friedrich Gauss.
So Gauss is widely viewed as one of the finest mathematicians who ever lived. He worked at the end of the 18th century and the beginning of the 19th century, and his mathematical notebooks contain lots and lots of pictures of nuts. He realised that they were mathematical objects. And what he did was he developed a way of encoding knots, using a sequence of letters and numbers called a Gauss code.
So although he realised that knots were part of mathematics, he didn't really make much headway with them. He did it a little bit, but topology did not exist when he was alive. It was created at the end of the 19th century, the beginning of the 20th century. And so the real headway, the real, real progress and not theory had to be made by people after Gauss. In fact, the first person to really make any progress with knots was was not a mathematician, but a physicist.
So here he is. Peter Guthrie Tate looking very stern and Victorian. He worked. He was a Scottish physicist working in the 19th century, and he got into knots via smoke rings. So I think you all know what a smoke ring is. Here is an example of a smoke ring that actually was belched out of Mount Etna. So coming out of the side of Mount Etna, there are various vents and every now and again, one of them. And look, if, if, if you're lucky, it'll form this.
This smoke ring so tight it created these experiments to to create smoke rings. And one of the things about smoke rings is that they are surprisingly long lived. They fizzle through the air and don't break up instantaneously, as you might expect. So his experiments on smoke rings caught the eye of this man, Lord Kelvin Lord Kelvin is an extremely famous physicist for which the Kelvin temperature scale is named.
And he was quite taken by Tate's experiments on smoke rings, and so he came up with a brilliant leap of imagination. He came up with a theory which was brilliant, but completely wrong. So his his his his theory was as follows was he thought that perhaps nots could be used to explain atoms? So at that point, atoms were known to exist. It was known that matter is made up of different atoms, but it had people had no idea what was inside an atom.
So we now know, obviously, that it's protons and neutrons forming a nucleus surrounded by a cloud of electrons. But at this stage when when Kelvin was working, they had no idea what the structure of atoms was. So he thought the perhaps atoms were knotted vortices in the Aether, and that the long lasting nature of the smoke ring would explain the long lasting nature of the atom and the different types of knots would explain the different types of atoms.
And maybe, for example, some of the structure of knots could explain some of the different properties of atoms, for example their absorption lines. Wonderful idea. Completely false. But what it did was it sparked Tate to investigate knots and he kind of got hooked on them and he created it, started creating tables of knots, which is a sort of very Victorian thing to do. So biologists would create tables of plants, and he was creating tables of knots ordered by then, not Guinness.
So here's an example of he got a bit carried away, to be honest. Here is he's just one of the paper pages from one of his papers. Let me just zoom in a little bit. You can see this is just the top left of that page, and you can see there is the the truffle oil and the Figure eight and more and more complicated knots, and you can see the title of the page. The first seven orders of nuttiness. So what is nuttiness?
So what we what what he called nuttiness is now called the crossing number of a nut. OK, so the crossing number of a knot, what you do is you look at the different possible projections of the knot and when you project the knot onto the plane, some bits of the knot have to cross over themselves, and the number of those is called the number of crossings. Different projections of the NOT will have a different number of crossings.
So what we do is we consider the minimum possible number of projections and that then is the the crossing number of the not. OK, so just so that we're all on the same page. Let's just. Ask what is. What's the crossing number of that not? To do I hate her. You have no zero, right, you. You mustn't consider just this diagram. You have to consider all possible diagrams of this not and look at the minimal number of crossings so I can take this not I can just undo this and undo this.
And I get. Just the UN not on this projection of the UN has zero number of crossings, and that's the minimum possible, obviously. So this has crossing No. Zero. OK, so that's the nuttiness of the knot or crossing? No. So it actually did a remarkable job. He has tables are amazingly accurate. So he didn't miss any knots out or at least within the type of knots that he was looking at. And I have very few duplications, so he or she did a remarkable job on it.
By looking at this data, he actually came up with some conjectures which actually weren't finally resolved until the 1990s. So he really is, you could say, the founder of Of Of Not Theory. OK, so what I want to do now is get things a little bit more mathematical and I want to introduce the notion of addition of two knots. And this is going to be we're going to do this to sort of develop an analogy and analogy between knots and and positive whole numbers.
OK. So just as we can add two whole numbers together and get another whole number, we're going to add knots together. So how might you add two knots together? So given two knots, K1 and K2, we're going to add them to form this new not called the connected sum, so K1 plus K2, and that's defined this way. So here's K1, for example, the Figure eight, his K2, the trefoil and the way we form the sum is you just cut the two knots and then splice them together.
And it turns out that's a well-defined operation, and this is the then the sum of those two knots. And so I want to develop this analogy further between knots and and whole numbers, because whole numbers, as we all know, have this decomposition into primes. So the special hole numbers, these prime numbers and any hole number, any positive whole number can be written as a product of prime numbers and this can be done.
This is unique, right? So if I say fifty seven and work out, it's prime de composition, well, that's three times 19, I think. And if I were to tell you to go away and do the same, you get the same answer, hopefully. So we want to come up with the same sort of result for lots. So what? First of all, what does it mean for not to be prime? So not as cool prime if it cannot be written as a non-trivial connected sum?
So just as a number is prime, if it can't be written as a product of two smaller hole numbers, so not his prime if it can't be written as a connected sum of two two smaller knots. And in fact, just as there's a prime decomposition theorem for numbers, there's a prime decomposition theorem for knots, so any knot can be written as a connected sum of prime knots in an essentially unique way, just like crown numbers.
So this is a theorem it was proved by Fields medallist John Milner in some of his early work. And what I want to do today is to explain an intermediate step in his theorem. So I'm going to explain this theorem here. I'm going to write it down so that because it's going to be something I'm going to refer to on and off throughout. So the theorem is. That if I take two knots together. And I add them. And suppose I get the simplest possible, not.
Then, in fact, K1 and K2 must have been the simplest possible not. Namely. They're both. They are not. OK, so something like this has to be true if there's going to be a prime decomposition theorem for knots. So suppose it were false. Suppose that there were two non-trivial knots that when you added them together, gave you the andnot. So suppose that you could take two non-trivial knots K one plus K two and get the andnot.
Well, now take any other prime, not say K three. And addicts take one plus K2. Well, then that gives me K3, some the are not, and if I add they are not to anything, I just get three. Well, maybe K1 and K2 weren't prime, but if this theory is going to be true, they'll have a decomposition into prime summons. And so that'll mean that I have this not here, which is prime written as in a different way as a sum of prime knots. And I would have them for a failure of uniqueness of prime decomposition.
OK, so if I'm going to have this uniqueness theorem for prime decomposition, then I'm certainly going to have this results here. And in fact, this is an intermediate step in the proof of of this theorem. So I call this this, this this theorem, the garden hose theorem, and the reason is as follows so suppose I'm out watering the garden with my hose and being somewhat absent minded.
I managed to get this hose knotted up, save with a Knot K1, and I'm busy thinking about theorems and not really focussing on what I'm doing. And I get over here and I notice over there there's some annoying, not one which has got all tangled up. And I think, OK, well, I want to get rid of that knot and now being extremely lazy, I can't be bothered to go back and untie it.
Instead, I think maybe if I just stand here and tie another knot K2 and then push that down there and hope that K1 will cancel out with K2. And this theorem says that's doomed. The connected sum of K1 and K2, which is what I would have just formed, would never be the andnot unless both K1 and K2 were the UN not to begin with. OK, so this is the garden hose theorem, and this is what I'm going to show you how to prove today.
OK, so if you're going to try and prove a theorem like this, the first stage is to think, well, OK, let's let's to think why it might be true. OK, so what we want to try and do is we just want to think, think of a strategy. So why might this be true? Well, it seems just like let's just go back a bit. But when I formed this connected sum that somehow. This new note I've created is somehow more complicated than each of the original ones.
So suppose. That were true, but somehow K1 and K2 was at least as complicated as each of the original knots. And that what I've created was the UN, not the simplest possible thing that must mean the K1 and K2 must therefore have been even simpler. No more complicated. Therefore, they are not. OK, so that's the idea. But how are we going to formalise that?
What we want to do is we want to have a notion of complexity that behaves well when you add two knots together and you think we have such a notion, the nuttiness, the crossing number of the not so you're naturally led to think about the crossing number of connected sums and you think, well, well, maybe the crossing number of a connected sum behaves like this, that the crossing number of the sum of two knots is the sum of the crossing numbers.
So this statement in red implies the golden host theorem. Right? Because suppose I take two notes and I add them together and I get the other note. That means the crossing number of the left hand side is zero. But if this were true, then that would mean that the crossing number of K one plus the crossing number of K2 would have to be zero. They're both not negative numbers and so therefore they must each have been zero and hence K1 and K2 must therefore have been both.
They are not. OK. So this statement here implies the garden host there, unfortunately. This is a famous unsolved problem. OK, so this is this is I think this is bizarre and amazing that such a simple thing crossing number of the summer to Nazis, to some of the crossing numbers that should be unknown. But it shows you that mathematics is full of these simply stated conjectures, which seem. Beyond our reach. So you start to think about this and think, well, look, how could this not be true, right?
I mean, it just looks like it has to be true. And you realise that actually your intuition is playing tricks with you, that actually your intuition actually only gives this inequality. So let me explain how you get this inequality. So you take K1 and take a minimal crossing. No diagram for K1 and K2 in a minimal crossing, no diagram for K2 and splice them together.
That gives you a diagram for the sum, and the number of crossings in that diagram is the right hand side, and therefore the crossing number is of this connected sum. Is that most that. But maybe there is some other diagram with a few a number of crossings. Seems unlikely no one's ever found an example of such a thing, but no one has been able to rule it out. But all is not completely lost, so I have a theorem which is relevant to this, which is a slightly bit of a comedy theorem, really.
What does it say I approved about 10 years ago? It says the following that the crossing number of the connected sum, it's a most assumes the crossing numbers, and it's at least the sums are crossing numbers divided by a hundred and fifty two. I know it's yeah. How can a theorem be funny? But it is unfortunately so. So, but this actually is powerful enough to prove the garden hose theorem, right? That so if K1 plus K2 is the andnot and this thing in the middle is zero.
And therefore, this thing on the left hand side is less than or equal to zero, but therefore multiplying three by one hundred and fifty two. The crossing number of K1 plus across the number of K2 is therefore less than equal to zero. And therefore, each of the individual things is zero and therefore K1 and K2 are both the not. OK, great. It looks like the theorems proved, but that is a cheat. And the reason that's a cheat is because I use this theorem in the proof of that.
So I'm not allowed to use that in the proof of that. That was. That is not allowed. OK, so we need a different strategy. And the different strategy is using topology. OK, so topology is. So now the proof this term is going to be an extended tour through topology. So let me explain what topology is all about. So topology is, as I say, the study of spatial type objects like cube or the sphere rarely seen them or the Taurus.
But there's a particular type of object which plays a big role in the subject, namely an end dimensional manifold. So in end, dimensional manifold is a topological space. Near each point just looks like ordinary Euclidean space. But globally need not. So your basic example of a of an end dimensional manifold in this case and equals two is is the surface of the Earth.
So if I'm standing on the surface of the Earth and I just look around me say like a mile radius, then it just looks like a flat Euclidean plane. Maybe there's some hills and valleys, but politics don't care about that kind of thing. But that's only true to a certain scale. It's not a flat Euclidean playing the surface of the Earth curves round on itself in a non-trivial way. So globally, it's not a plane, but near each point it looks like a plane.
So that's the defining property of a manifold, and manifolds are fascinating objects. And one reason there are many reasons why mathematicians are interested in them. But one of the main reasons is that we live in a manifold. OK, so according to Einstein's theory of general relativity, the universe is a manifold. So you look around you and it looks just like ordinary flat. You like Euclidean space. But that is only true locally.
So what do we mean by locally? Well, in the relativistic world locally means maybe, I don't know, within 10 billion light years. So but if you allow yourself even larger scales, then maybe the universe curves around on itself in a non-trivial way? We don't know, actually. We don't know what the large scale topology of the universe is. We might never know. People have been trying to find out by looking at.
Patterns in the microwave background radiation, but so far, no definitive answer as to what the topology of the universe is. So manifolds are interesting things. We're going to be focussing on things that we can visualise nice and easily, namely two dimensional manifolds such as the surface of the Earth, and they're called surfaces now surfaces actually a bad name.
The surface of the Earth is clearly the surface of something, namely the the the the solid ball that forms the Earth, but which some of the things that we're going to look at won't actually be the boundaries of anything, but they're still called surfaces. OK, so here are some surfaces. The Taurus is a surface. This is a surface here. It's called pretzel or Jenness to surface. Here's another surface this is called the infinite Loch Ness Monster.
So if you take a flat plain and you drill out little holes from it, there are those holes and then you attach onto each hole a little tube. Then the resulting thing is a surface. These guys go on forever and some infinite long. So we're going to focus on surfaces of the first two types rather than the third. The first two are what's called compact, which means they're nice and bouncy.
They live inside. Some roughly means they live inside some finite portion of of of space, whereas this thing here goes on forever as non compact, and we're not going to attempt to try to understand this. So the reason why we focus on on compact ones is because there's a classification theorem. So one of the the early high points of of topology is the classification of compact surfaces. So it says that any compact surface is one of two infinite lists.
So the first infinite list starts with a sphere and the tourists and the pretzel, et cetera. And each stage we just go from the previous one by adding on another handle. Or there's another infinite list, which I won't attempt to draw, but includes things like the Klein bottle, which I'm not going to focus on too much. This is what's called non oriented. All those are going to be bad, guys. I'll focus a little bit more on on orange surfaces in a little bit. OK. So those are surfaces.
Now I want to generalise that to surfaces with boundary. So the surface with boundary is, by definition, a point on it. Either has a little portion around it that just looks like a desk, just like ordinary surfaces do or has a point on it where the disk just stops like a half disk. OK. So for example, this is a surface with boundary and the boundary curve is is is just this this round curve here? So this is this is a surface with boundary. It's just just a disk.
So let me give you another surface with boundaries, so no talk on policy would be complete if I didn't include this example. So this I'm going to build you a Mobius Band. So Mobius Band, what you do is you take a long strip like this and you glue the two ends together. OK, so instead of doing them like this, this would form what's called an annulus. We're not going to do that to form a Mobius strip, you glue the two ends together.
But with a half twist. OK, so that's a Mobius Band, and it has all sorts of interesting and paradoxical properties. So, for example, it has a single boundary curve. All right. So if I start here and want around, I come back to the other side and then keep on going. So that's just as a single boundary curve. So the boundary curve of this is actually a not. Any ideas, what not that might be?
It's the unknown, it turns out I could have I could have made this a bit more complicated, so I could have instead of gluing it with one half twist, I could have made it with three half twists that would still be a Mobius Band because I would have glued the two parts together in exactly the same way. It just would be sitting inside three dimensional space in a different way.
OK, and now the boundary of that is is the track. While it turns out OK, so Mobius Band is a black eye because it's non oriented. So what Oriental means is that so this is oriental because it has two sides which you can colour with different colours, whereas this it starts off before we do the gluing with two different colours red on one side and black on the other. But when I glue it up. You get a discontinuity like that.
And that means that there's no basically this is a one sided surface if I start when the Reds come back and end up on the other side. And so this is a will surface. So it's a bad guy, right? So we're going to focus on the oriental ones because, well, the non-renewable ones are classified as well. But the oriental ones, they are very simply to state simply class simple classification theorem that any compact Oriental will surface with. Boundary is exactly one of these guys here.
So you take a long strip and then you attach onto it to G. Alternating bands for some negative integer G. There may be an isolated bands like So. So actually, in our case, we're going to focus on the case where any zero because we're going to want this to be just a if. So, each time you put on an isolated band, you end up with a new boundary curve. And I want this to have connected boundary. In other words, I want this to have a boundary.
I'm not. So at the no, when you have to try alternating pens, G is it's called the genesis of the surface. So. This surface could be embedded in three dimensional space if I wanted to in some complicated way, so I could have those bands and I could tie little knots in them and that kind of thing. Still be the same surface. OK, so we have this classification theorem for surfaces with boundary compact Oriental surfaces with boundary.
So we want to use that classification theorem in making progress with knots. So surfaces with Boundary. They are related to knots via the notion of a cyphers surface. Sifford surface for a not OK. It's a compact oriental surface sitting inside three dimensional space whose boundary is equal to K. So for example, this is a flat surface for its boundary curve, which is the, um, not.
Whereas on the other hand, this thing here where the Mobius Band with the three half twists in its boundary is the trefoil lot. But it's not a cipher surface because we want to focus on the Orient, but once. And so it turns out that any not nevertheless has some surface surface, so every every, every mathematical talk should have a proof. And so let me show you how to build a cipher surface for any not. So there we are. Any not has a siphoned surface, I'm going to build one for the travel.
So here's the track oil and the way you build a surface is you start with any diagram for the not so like so. And then you give it an orientation, so I've put arrows on that, not. And then that orientation allows you gives you a way of getting rid of each of the crossings. So you resolve each of the crossings using that orientation, so so when if you like that, you just go like that.
So this is a from a not theory point of view. This is a terrible thing to do because I've completely changed the knot type. In fact, I've now just got a collection of simple close curve sitting inside the plane. So I view that plane now as as lying inside three dimensional space, as just the horizontal x y plane, and there are my three well in general, my oriented curves. And now what I do is I now attach disks above the plane onto those curves.
OK, so now I've got some disks sitting inside three dimensional space and the boundary is equal to those curves. And now that's not quite what I wanted, I wanted to have a surface whose boundary was the not so to what I end up by doing is I re-instate. I enlarged my surface as follows. I add in half twisted fans at each crossing.
And then when I do that, so I put one this way where the original crossing was there, one there, one there, so three half twisted bands in this case and I've created a surface with boundary and the it's exactly equal to the knot that I started with. And because I took with the orientations, the resulting surface really is oriented. You can trace that through and therefore really is a cipher surface with a knot.
OK, so this is good because what we've got now is we starting to introduce some of the tools from topology, particular understanding of surfaces to provide some information about knots. And so that leads naturally to the notion of the genesis of a knot.
So the Genesis G.K. of ANOC is the minimal possible genesis of a cipher surface for K. OK, so just as you know, so before when we're looking at crossing number, we looked at all possible diagrams of the knot and minimised the number of of crossings here. Instead, we look at all possible surfaces, cipher surfaces that the knot bounds. We look at each of their genesis and we take the minimum. And the resulting thing is the genesis of the note, and this has some nice properties.
So in particular, like crossing number or nuttiness, it's the K is the unknown if and only if the Genesis zero. That's just because if case the are not a LB disk and that has Genesis zero. Conversely, if the knot has Genesis zero, will the only surface with Genesis zero and one boundary component is the disk. And if not bounce a disk, it's it's the output. So we have that property there.
And now, because we're dealing with something more mathematical, something topological, this really does behave well with respect to connected. So it's possible to prove that the genesis of the sum is the sum of the genesis genesis of each of the knots. And that is enough to prove that God knows there because if K1 plus K2 is the andnot. Has Jenna Zero, and therefore each of those numbers must have been zero and therefore K1 and K2 must have been me, I'm not.
And so we've proved that my strategy in the back garden of trying to undo mine not hit home is doomed to failure. OK, so that's an example of the use of topology in and not theory. And so you can say, well. What where does the subject go from here and some actually taped when he was writing down his tables had really quite an interesting perspective on on the subject before he even knew that the subject exists. So you can ask you, can you classify knots? If so, how many are there?
And you immediately come to the problem, not how do we decide whether to not of the same or not? So, for example, we are at the very beginning. I gave you the the trefoil and the Figure eight. And I said, those are different knots. But but how do we know for a fact that the foil and the Figure eight are different?
And Tate realised this, so he says, though I have grouped together many wildly different but equivalent forms, I cannot be certain that all those groups are essentially different from one another. So this is slightly convoluted Victorian way of saying that we don't really know whether the Figure eight and the Truffle really are different. For example. Now, it turns out that we can now proof that they're different.
But you could ask yourself, like, is there a general procedure if I give you two different knots? Is there a general procedure for being able to decide whether they really are different? And interestingly, this particular question was taken up by Alan Turing. So Alan Turing is famous for his work at Bletchley Park, and he's also famous for his work in the foundations of computer science.
And one of his key, his key contribution is that he established that there are fundamental limits to what computers can do. There are some questions that mathematical questions that computers will never be able to answer. So, for example, here's a system of polynomial equations. Does it have a solution in integers? Computer will never be able to answer that question. But Alan Turing didn't realise that at the time, so he he he knew that there was some fundamental limits to what computers can do.
He could come up with problems that computers would never be able to solve. He did this by inventing the idea of what's now known as a a cheering machine, which is this super powerful computer powerful enough to emulate any other computer that we have, including quantum computers. And he showed that that's super powerful computer was not powerful enough to be able to answer certain questions. Unfortunately, the questions that it couldn't answer was somewhat slightly artificial at that stage.
And so he was searching for natural problems that computers might not be able to solve. And so he came to nuts. So in his final paper that was published in the year of his death, nineteen fifty four called solvable and unsolvable problems. He examined notes and wrote No systematic method is yet known by which one can tell whether two knots are the same. So this is basically the same statement that Tate said.
There's no how do we know whether to not say the same or not? So he clearly was envisaging, and this is a picture from an excerpt from his paper that you what you might do is you might encode your not in some mathematical way. So, for example, as a union of straight arc sitting inside all three and you'd feed these so you take you to different knots and you'd feed this mathematical representation of them into a computer and the computer would say yes or no,
or maybe. And it was pretty clear that he was thinking that maybe there would be no such way of doing this, that maybe this was an unsolvable problem. So it turns out. That it is a solvable problem, and this was proved by this guy here, Wolfgang Harken. So Harken in the early 60s showed that there actually is a systematic method for reliably determining whether two knots are the same. So he's he's a very famous mathematician, not just for his work on knots, but also for what he did afterwards.
So after he proved this theorem, he then turned his attention to two plain graphs and in conjunction with Kenneth Apple, they proved the famous full-color theorem for plainer graphs. So but before that, he looked at knots and he showed that there was this. There is a systematic method of deciding whether to not of the same. And he did this using the theory of three dimensional manifolds, so spaces that locally looked like three dimensional space.
But globally, like curved round on themselves in unexpected ways. So that's actually my area of research. I work in three dimensional manifolds theory. And so I find this this theorem very impressive. It's still highly relevant today. I should say that.
This this method, it's very much a theoretical method, but if you if you if you apply it to two knots, then yes, there is a computer programme that will decide whether or not they're the same or not, but it'll take the lifetime of the universe to run it. I mean, even the simplest ones. So it's very, very much a mathematical mathematicians theorem rather than an actual practical result, although. The search for a practical algorithm to do this is still very much an open problem.
OK, so let me just close by talking about one final question that we don't know the answer to, which is a really simple question, which is how many knots are the. OK, so we know, by the way, the infinitely many knots before I go any further, we've actually proved that implicitly today. So if I take the trefoil, it has Janice won because the non trivial knot in it bounds a cipher surface of Janice one. Take the sum of two trifles by the theorem about A.R.T. that has Genesis two.
So this is different not and then add on another trefoil as Genesis three, etc. So there's an infinite collection of different notes, provably so. But how many are there as a function of their crossing them? They're not in us. So let can be the number of prime knots with crossing. No end. OK, so we know Cayenne for some small values of. Here it is. So for example, like an equal zero crossing no0, there's just one, namely we are not. Then there's none with crossing number one.
There's none with crossing number two, we've already saw that crossing number two really isn't crossing number two at zero. First, non-trivial, not there is the trefoil, this one, and then by the time you reach 14, there's over a million of them. A lot of work went into that number, by the way. So how does this how does this sequence go? Well, pretty clearly, I mean, not provably so, but pretty clearly you're unlikely to get it like a closed formula for it.
But what about how does it grow asymptotically? So how does Cayenne grow asymptotically? So we don't know, is the answer. The best we can do is due to Thistlethwaite Samberg and Dominic Welch. Dominic is an emeritus professor here at Oxford, and between the three of them, they proved the following very strange theorem that says the Cayenne lies between, well, the top end.
It's twenty seven over two to the power and. The bottom end, it's this given by this very complicated formula, don't worry too much about this. This is really the dominant term. And this is growing exponentially with a function of it. So these numbers, they definitely blow up exponentially. But the precise exponential growth rate is not known, and to be honest, I think we have no prospects of knowing it in the near future.
So that is just one example of some of the things that we don't know in the subject, which I think is remarkably interesting and intricate and fascinating. I'll leave it there. Thank you very much. So thank you all for coming.
It's our tradition that we don't take public questions in these lectures, but the speaker case mark is more than happy to hang around and if you want to come forward and ask him any questions privately, but I'm sure you'll agree that really was a beautiful lecture, so let's thank Mark again.