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0:02
So, welcome to the 22nd lecture in this series on algebraic topology. I'm Norman Wildberger at the University of New South Wales. Today, we're going to talk about knots and surfaces. So it's going to be the first of two lectures on this topic, and we're going to just introduce this very big and interesting theory called knot theory, which is properly a branch of algebraic topology. So, what is a knot? Well, a knot is something like this, made up of a piece
0:41
of string in three-dimensional space which is tied together at the end to form a closed loop, right? But it's not necessarily like a circle in the sense that it's not in an obvious way or maybe it's knotted. And what we want to do is want to study these kinds of objects, but we want to study it from the algebraic topology point of view, which means that all the manipulations that I'm doing now, namely continuous motions, keep it the same. It's always the same knot.
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So this theory entered mathematics in the 19th century, and it was a physicist perhaps who introduced it first. So, in the 1860s, William Thompson, who otherwise later became known as Lord Kelvin, a famous physicist, introduced the idea that atoms might be knots. So, it's assumed that there were small, indivisible kind of objects that created molecules, but they didn't know what they actually looked like. And he thought maybe they were really knots
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of something rather, and that the various physical or chemical properties of atoms might be reflected in the way these things were knotted together. This was prior to the nucleus, the Bohr theory of a nucleus with electrons floating around. A little while later, his friend Peter Tait began a classification of relatively simple knots — ones that can be represented in the plane with 10 or fewer crossings. And he was helped
2:37
by this by C. N. Little and used work of the Reverend Thomas Kirkland on curves. So let's have a look at some of the knots on Tait's list, which was around the beginning of the 20th century. Beginning of the 20th century, it was more or less completed. And so there are knots like this appearing. So, the simplest knot is the UH knot, which is simply a circle. And then the next simplest knot is a trefoil. And the trefoil comes
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in two different kinds. One is the mirror image of the other. So this one here is a trefoil knot.
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And this one here is also a trefoil knot. It's the kind of simplest knot that you can teach a child to make. But there are mirror images of each other. And if you start fiddling around with this one trying to make it look exactly like this, you find that it's hard to do, or in fact, it's impossible to do. So you suspect that these two knots are actually different in some fundamental way, but maybe it's not so easy to prove that. So there are simple knots:
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the figure-eight knot, pentafoil knot, gingerbread man knot, a square knot. So, sort of like two trefoils, one on top of the other in opposite directions. And they're roughly ordered in terms of the complexity. The complexity could be measured by the number of crossings. That's got three crossings. This one here, it's got one, two, three, four crossings. This one here has one, two, three, four, five, six crossings, and so on.
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So let's ask, well, how do we mathematically model what a knot actually is? It's not just that a knot is a topological space that matters because topologically, all these knots are circles. They're all actually homeomorphic to a circle. What's interesting about a knot is how that circle embeds in the 3-dimensional space. So how are we going to answer this question of what do we mean by a knot? What is a knot?
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Well, one way is to try to discretize the problem and to think about a knot as a bunch of points in space connected by line segments. So we could make the kind of following picture.
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Okay, so this is supposed to represent a certain number of points in space with the line segments joining them so that all together we have a circuit. As we go around, we come back to where we started, and we want them to capture the idea. Well, how can we move this knot and keep it unchanged? So, there's a basic idea of movement called an isotope. And that can be explained in terms of combinations of a simple kind of isotope called an elementary isotopy. And
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that's what we get when we take one segment, let's say this one here between these two edges A and B, and we are going to replace this one segment with two other segments and a third vertex, C. Okay, so then we're going to get... oh, that way over. All right, all right. So far, it's much the same as it was before. Here are the vertices A and B, and now is that having one vertex, one line segment joining NB, two line segments with a new vertex. Now, the requirement,
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of course, is that this new triangle that we've added on, that we've got to be able to move from here to here without crossing the knot. All right, so this triangle here had better be free of any of the rest of the diagram. And if we allow that... so here's where the formal thing was. If we allow that kind of elementary isotopy as a basic way of moving from one knot to an equal knot or not that we'd like to consider the same, then we can compose that and create a
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whole sequence of elementary isotopes isotope is from one knot to a related knot we call equal.
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So, a finite sequence of such things, find a sequence of elementary isotope e's, is just an isotropy of knots. This is our notion of equality. Now, in this story, the polygons that we start with have to have only a finite number of segments. Okay, so we are only considering a finite number of line segments. So, no so-called wild knots. Sometimes in the theory, people like to get to their knots which are wild which have an infinite number of little
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pieces that do some repetitive thing, but we're not going to consider such things. We consider those things that kind of thing is perverse, and we're not going to consider those things. Okay.
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Unfortunately, the physicists soon realized that as a model of the atom, this was inadequate. And so the physicists lost interest in the mathematical theory of knots, but mathematicians started to develop the theory independent of physical interest. It's interesting that the circle has kind of closed. Now, in the last decade or two, knots have become very important in modern physics again due to the connection with quantum field theories and with string theory. So,
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there's been a real turning around. But for a long while, there was only mathematicians pursuing the theory for its own intrinsic beauty. And one of the earliest important contributors was a German mathematician, Kurt Reidemeister. He said, well, how are we going to study knots? We're going to study knots by drawing planar pictures of them. But not a diagram. We're to assist a projection or not onto the plate just the same as the ones I've just rubbed off here.
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And those were projections of knots. They weren't actually the three-dimensional knots themselves; they were just pictures of knots. And a natural question that immediately strikes one when thinks about the projection of a knot is: How do you tell whether two projections are the same? All right, if I take some make it take more complicated knot. Okay, so there's some kind of knot. It's actually one of the ones on my list. Maybe it's a figure-eight knot. Okay, so I could project
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it onto the plane in a whole number of different ways but with over-and-under crossings. So the question is naturally: How do we recognize it's the same knot that we're talking about? That two different projections are really the projections of the same knot? And Riedemeister was able to answer that question, a very important question then. So his theorem is that any two projections,
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all the same or what we're calling isotopic, same or isotopic knot, are related by a combination of three relatively simple transformations of the planar diagram which have since been called
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Riedemeister moves. And here there's the first: If you have a twist somewhere in your diagram, and I'll indicate that by putting a circle around here, so that means that what's outside here is it may be complicated, but we're only focusing on this part of the knot, then we can untwist this, obviously, and it just becomes something like that.
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That's kind of an obvious one. Let's call that Omega one, first Riedemeister move. And the second one is when we have two parts of the knot that cross like that. And again, this is just a local picture that just we're just looking at a very small part of a planar representation of a knot. Then we're allowed to move those two strands apart to separate them. So what you just get two lines like that. We should make them a little bit like this. Well, that's the second Riedemeister
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move. And the third one is the slightly more complicated one. It's one where we have three
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parts of our knot crossing like that, crossing of two lines and another strand lying on top of the crossing. Now, what we're allowed to do is we're allowed to move that top strand, which is right now above the crossing, across the crossing below it. So the crossing stays where it is, but now this thing here is now looking like this, and the crossing is as it was before.
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So there's the crossing thing, and this strand here has moved from on top of the crossing to below the crossing. That's the third Riedemeister move, Omega three. And so we prove that if you take any knot and you project it one-to-one fashion, and then you do all kinds of distortion to it and project in some completely different way, you could get from one projection to the other projection by just slowly performing these three operations step at a time.
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So, this is quite a nice result, and it's tempting to think that now the obvious problem in the subject might be accessible. What is the obvious problem? The obvious problem is how to classify knots. How do you make a list of all the knots so that's certainly one main problem? It's the main problem number one, is how to classify knots. And a related problem, let's call it main problem number two, is how to tell if two knots are the same. And by the same,
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we mean isotopic. So, Riedemeister's theorem tempts us to think, well, what we can do is we can take a plane of representation of knot, and we can simplify it. Why don't we just try to simplify it so it has the minimal number of crossings? So, whenever we can do this, we do it. Whenever we can do this, we do it. And what we can't do one of those, we'll try to do one of these. Yeah, this is not increasing the number of crossings, and if we only
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go in this direction or in this direction, then the number of crossings will keep on decreasing, and eventually we will have the simplest possible form, and we'll have a surefire way of getting a kind of canonical form for a knot. Unfortunately, this doesn't really work, and here's a problem that illustrates that. Some problem 22 knot theory gives us lots of chances for interesting problems here. So, show that the following example is an unknot by performing
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Riedemeister moves, but note that you need to increase the number of crossings at some point.
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Okay, what's the knot? It's this one here. Okay, I think that's right. So, I hope you can see where the overs and unders are, and the claim is that's an unknot, meaning that if you shake it and do enough to it, you can turn it into a circle. Yeah, I don't want you to prove that because that might be hard. I just want you, in whatever method you use to go from this to a knot, not I want you to simply observe that at some point you had to increase the number of crossings. Sorry,
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I just want you to sort of observe or note that you have to increase the numbers, all right? So, these are very difficult problems, and they're not solved by knots. There even though it's very sophisticated subject now and highly developed, but mathematicians have made a lot of progress towards these questions, and most of that progress has involved the idea of invariants of knots.
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What is an invariant? An invariant is the idea is to assign to a knot diagram a number or maybe a polynomial or maybe in fact something else which is invariant under each Riedemeister move. So we have to invent some way of associating to a diagram a number or maybe a polynomial or maybe something else so that if you perform either of the three Riedemeister moves and then it doesn't change, now if it doesn't change under those moves, it means that any planar
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representation of the same knot is going to have the same invariant, and that's going to be then a number that's intrinsically associated to the knot and with the aid of such an invariant then you can say that one knot is different from another knot because the invariant for this knot might be 5 and the invariant for this knot might be 7, and so those knots have to be different if the invariants
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are different. So, people have created a lot of invariants but they are some of them are less, some of them are very attractive but a little bit hard to compute question.
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Is that saying they say that's right? Yes, so it's the same thing that we're trying to associate a way of assigning numbers to knots so that isotopic knots have the same invariant. So what are some examples? Well, the first invariant that people naturally thought about, it's probably the first one you think about too, is the invariant that you get by drawing pictures of the knots and noting the number of crossings. The number of crossings in a
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picture is some kind of invariant, well you have to say that in the right way, it's the minimal number, the minimal number of crossings in a diagram.
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Almost by definition, that's an invariant, but the problem is that it's hard to compute. How do you know that your particular picture is the most efficient way of representing that knot? Fortunately, in the 1980s, three prominent knot theorists, Kaufmann, Murasugi, and Thistlethwaite, proved a lovely theorem that says that a reduced alternating diagram exhibits the minimal number
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of crossings. So I have to explain what those words mean. So the notion of an alternating representation or an alternating diagram is very important one. If we go back to one of our pictures, let's say figure-eight knot, if we traverse this knot in some direction, we walk along it and we record at each stage whether we're going under or over the other strand. Okay, so here we're going under the strand and then here we're going over that
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next strand and then here we're going under, and then here we're going over, and then under, and then over, and then under, back to where we started. So, alternating means that the crossings alternate, over, under, over, under, etc.
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An alternating representation of knots are particularly pleasant. Once this representation that we had over in this diagram was not an alternating representation because here we went under and then we went under twice in a row. So that's all stating. What does reduced mean? Well, reduce means that you don't have a picture like this. Okay, the knot cannot be separated into two pieces where there's two strands that do this. Have you look at your picture and it
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looks like something complicated, here's something complicated here with one twist, there obviously you could just take this part and flip it over not changing the number of crossings here but then this crossing would undo itself and you get a smaller number of crossings. So obviously this kind of thing is not going to be minimal. So, we say that a knot diagram is reduced if it doesn't have any of this kind. So reduced if this doesn't occur.
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The trefoil, this time, they have the same invariant, your eyes. So for the trefoil, well, let's say this figure-eight knot. It's not too hard to see that this is not, this is reduced and it's alternating. So the theorem says that the number of crossings here, one, two, three, four, that's actually an invariant of the knot that any diagram for that knot will have at least four crossings. And that, meaning, particularly, it means that this
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is actually not a trefoil because the trefoil has three crossings in its standard representation.
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Or this theory doesn't take two notes. No, definitely not, right? All of these invariants that I'm going to tell you now are only partial. They only give you partial information. No one has yet discovered an invariant that will distinguish between isotopic knots, whether they are or are not isotopic. If the invariants are different, then you can conclude that the knots are not isotopic, but if the invariants are the same, you can't necessarily
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conclude that the knots are the same. That's a good test, right? It does. Your invariant distinguishes between the trefoil and its mirror image, the crossing number. This first invariant doesn't. It's not sensitive enough to tell the difference between those two knots, but it is sensitive enough to tell the difference between the trefoil and the figure-eight. In what sense?
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Alright, there'll be lots of knots. There'll be lots of knots that have happened to have except of 23 crossings. Yeah, there's probably thousands and thousands of knots that have 23 crossings as their minimal crossing number. Well, there's only one mirror image, and it turns out the Pentafoil and its mirror image are the same. So some knots, the mirror image just looks the same as the knot does. Alright, so the second
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invariant is actually the oldest invariant in the subject. In fact, it actually goes back to Gauss.
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Gauss actually thought a little bit about knots. He realized that a certain integral sort of could be interpreted to what is essentially an invariant of links. So this we'll call this the linking number of a link of two knots. So it's a little bit different now. We have two different knots. When you have more than one knot, it's usually called a link. Okay, a link in general is perhaps two intertwined combination of knots.
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So it's a more general object. And so now we suppose we have two knots forming a link, and what we do is let's call one of them is red and the other one say is blue. We give the knots an orientation. So suppose we have so each of the knots has an orientation. Now there's two pieces of rope, there's a red rope and a blue rope, and each of them is a knot, but the two knots are linked in between each other. So this is part of the blue knot,
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and this is part of the red knot, and the red knot is going in this direction, and the blue knot is going under in this direction, and there's a crossing right here. Let's call this crossing P, and we're going to give a number to this particular configuration. We're going to call this sort of a positive crossing and assign it a plus one. And if they do the opposite, so if the red one is in this direction, well, maybe I'll put the red one here, if the red
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one goes over the blue one like this, then we'll say that this crossing is like negative, has a negative one value. And so this, what we just done, is we define the sign of the cross.
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Pardon? Yeah, so in order to make this work, we've given each of the knots a direction. We've given each of the knots a direction, and then the link between our two knots, let's call the knot. It's just, it's just this. It's just an arbitrary convention. We just agree that whenever we see the red going over to the right of the blue, we'll say that's Plus, and whenever the red is going over, or whenever one is going over the other,
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it doesn't matter which one's on top, into there, to the right, to the left, we'll call that negative. And then we define the link between the two knots to be the sum over all the crossings. Let's call the crossings P, all just the sum of all the individual signs of the individual crossings. So for example, suppose that our two knots are like this, they're just circles.
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And we'll give that one that.
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Okay, so there are two crossings here. Okay, so remember if we go over to the right, that's given a plus one. If we go over to the left, that's a minus one. So at this crossing here, we're crossing over to the right, that's a plus one at that crossing. And at this point, well, this one's going this way, and this one's going this way, we're going over to the right again. So if we're looking at the bottom one, the bottom one is this one. So if we're
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going along the bottom one and the top one is crossing over us over in the right direction, it's like going underneath the freeway. The freeway is, you're going this way or this way, it's either in that direction or that direction, and it's going to the right. So that's also plus one. So there's two crossings there, and the link between these two then is 1 plus 1, which is 2.
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Alright, so here's problem number 23. Prove that no matter how you orient the knots, the link of these two is zero, and the two knots are both of them are unknots individually. So we have two unknots here individually. They're not interesting, but the way they're linked together is sort of interesting. You can't pull them apart. For example,
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alright, so our next invariant, very number three, is the Alexander-Conway polynomial. So, Alexander, who worked in the 1920s, discovered a polynomial, and then in the 1970s, John Conway, the same John Conway that we've talked about already, discovered sort of another way of thinking about this Alexander polynomial. Okay, particularly, a pleasant way which perhaps Alexander already sort
36:20
of knew about. So, the idea is we start off with a knot, okay? And what we want to do is we want to obtain an invariant, Delta K, which is actually a polynomial. It's a polynomial in, say, variable X.
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And the way we define this is through a certain relation, and I'll diagrammatically explain this here. So, we're going to look at just a part of the knot.
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Alright, so, what this, so, our knot, first of all, we give it a direction, so we decide on a direction to orient it, and then we look at a particular crossing. So, this is just part of a bigger knot, this stuff outside here, and we're looking at sort of three possible things that could be going on at this crossing. So, when somebody rubbed off the crossing and you weren't sure what was going on there, you want to reconstruct, you could, we maybe we constructed
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like this, or you could reconstruct it like this, or you could reconstruct it like this. And this invariant is designed to have the following two properties. Now, first of all, when you apply it to the unknot, which I'll represent by a circle like that, you get one. So, the Alexander-Conway polynomial evaluated at the unknot is by definition one. And the second property relates to these three things here. Let's give these a name, let's call these K plus, K minus, K zero.
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And so, the second relation, well, right, it's symbolically like this, that Delta at K plus minus the Delta at K minus equals X times the Delta at K zero. And the way we interpret this, we think that this is a relation that tells us a relationship between the three different values of this polynomial on three different knots. One, the first knot is where this crossing looks like this, the other, the second knot is the
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one where the crossing looks like this, and the third one is where the crossing looks like this. And we insist that the polynomial has to have the property that its value at this and on this knot minus its value on this knot equals X times its value on this knot. And it's a magical kind of fact that this makes sense and that it turns out to be an invariant.
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So, let's see how it goes in practice.
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So, example time. What suppose we have this knot here, which is just an unknot. We're looking at this crossing right here. And so, that's like a plus, so we'll call that K plus. And then, we'll replace it with its negative, or its K minus. And if we replace the crossing with the zero version, then we would just get two circles. Okay, there.
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Now, the Conway-Examiner-Conway relation is that Delta K plus minus Delta of K minus equals X times Delta of K zero. And which of these do we know? Well, both of these are just circles. Both these are circles, and by the first property of the polynomial, they both have to be one. So, we have one minus one equals X times Delta of K zero, and from that, we must conclude that if we have two disjoint circles,
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then the Alexander-Conway polynomial is zero, because that's the right-hand side.
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For our second example, let's look at what's called a hop link. So, let me write it like this. So, our first two...
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That might be, we're just talking about two circles which are linked, and we're looking at, we can choose any one of the two crossings and apply this reasoning. So, let's just choose this top crossing right here. So, if we replace that with going the other way around, we get, well, there'll be one like this, and then the other one will be now behind like this, still looking at this part of it right there.
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And according to our rule, that should be equal to X times what we get when okay, what happens when we replace this with two sort of parallel things? Well, then this part coming up goes up like that, and this part going up goes up like that. It looks like this.
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So, which of these do we know about? Well, this right-hand side, that's just a circle, that's an unknot, so it's, sorry, this should be a polynomial here, X times the Alexander-Conway polynomial, that object. So, we're talking about X times one, which is X. On the other hand, what about this one here? This is two circles, disjoint circles, not linked, which we've just calculated is zero. So, the value here is zero. And so,
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that we conclude that therefore, our hop link, the polynomial's value at that is equal to X.
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And I'll leave it to you as an exercise, not a problem, just to check that if you reverse the orientation of one of these circles, so if you have one circle still going like this, but now the other one going this way, that you get negative what you got in the other case.
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And now we can have a look at example three, which is the trefoil knot.
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So, we're going to look at our trefoil.
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And let's look at one particular crossing, in particular, this one right here. You give this thing an orientation, so let's make the thing go this way, and then this won't be in this way here. So, this is actually the negative kind of crossing that we were talking about before, so I should put a minus sign in front of here. And then I'll add the one where we change the orientation here, and that will be...
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I'm still just looking at that part there.
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And that's X times if I go over here, that's X times well we get when we replace it with the crossing with two little parallel lines.
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Just that at this point here, it's just that, just that particular crossing we're replacing it with the strands going the opposite over whether it was under before, and now we're going to just replace just here we're going to replace it with two parallel strands, so the rest is going to look just the same.
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Okay, now it's parallel so that's going to go up like that and that's going to go up like that.
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That's the point that we were looking at.
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Okay, so which of these three do we know something about? The middle one, why do we know something about the middle one? Because that's none not. This loop on top here can just be moved over and flipped over, and that's a nun not, so this is actually equal to one. And what about this one? Well, this one, this is two circles linked, so it's either going to be X or minus X depending on the relative directions of the two circles. And this one is going in this direction and this
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one's going inside, it's sort of left-hand kind of rule. So, the Conway-Alexander polynomial is minus X on it. So, this is minus X on the left-hand side times our, minus X.
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So, if we solve for the trefoil, we get that the Conway-Alexander polynomial on the trefoil, well, that particular trefoil is, so that's the minus goes wrong to the other side, we get one plus X squared.
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So, problem 24, so, show that if you take the trefoil mirror image you also get one plus X squared. And be a harder problem for, for onerous type students, compute the exotic hallway polynomial on the figure-eight knot.
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Of course, if you like this, you can go off and try to compute the values for those other knots too. Okay, there's an extension of the Conway-Alexander polynomial, called the Homfly polynomial, which I'm not going to talk about, but you should just know that there is a somewhat, a slightly fancier version called the Homfly polynomial, whose only difference is that it's determined by this relation.
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It's a polynomial in X and Y associated to a knot, and if you're looking at a crossing, then the three values are related by X times this one minus Y times this one equals the polynomial on that one. And that turns out to be a somewhat more sensitive, more powerful invariant. You might like to compute some values for it too. So, next time, we are going to see how knots and surfaces are intimately connected. So, I'll see you then.
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