A combinatorial description of knot Floer homology

0:00 Thank you for your attention. People debate a lot. There was a lot of debate. People who stand on the Yao side and some people who stand on the Chinese side. There was a lot of debate. I was not sure. Yeah, yeah, they knew this. People at NSF were also aware of this. Just in one place, somebody is giving such and such. Yeah, Yao is quite generous. He doesn't get any pay. You've answered my question. I have no sense of direction at all. That is the way back to the... Yes, I'm sorry. You wondered what I was doing, just hanging on your words. Okay, right, that's it. Good, good. Yes, I don't know how that happened. This paper, you know, looks very stupid. Yeah, it looks bad. It's pretty bad, isn't it? Yeah, because they are saying, OK, you got everything. What do you want? What do you want? What do you want?

2:30 In fact, it is. Yeah, so it's... What about this one? This paper is quite... Interesting, not probably from a mathematical point of view, but from general. There are various people in mathematics. One type is Yao, another is Hamilton, and another one is Perrin. They are absolutely different personalities. Yeah, and they really show you the real human side of things. Who was the second person you mentioned? Hamilton. Hamilton? What, as in, what, not as in THE Hamilton? Sorry, sorry, which Hamilton? Richard Flore. Richard Flore. Oh, that's okay. Not a classical. No, I thought you were going back rather a long way. I didn't think you meant Hamilton, you know, of the Hamiltonian. Very seriously. But he would also have been a very different kind of person, I would say. He spent a lot of money on women, you know, he doesn't really probably have a lot. He has a girlfriend, he's keeping a girlfriend. I can't give you this article, but it's my... Yeah, but this is all for real. Okay. Yao went to China, gave some talk, and he used the opportunity to attack Hamilton, he said, in America, he said, even Hamilton couldn't get a grant in one year, and this shows that, you know, in America, the system is unfair, but Hamilton doesn't have... I don't know, I can't say much about it. What was the article you wanted to say? You were saying that Perelman told him some ideas and he didn't catch them. Yeah, yeah, he wasn't working hard. When Perelman was given part, he didn't ask any questions. He had this brilliant idea before, but he never really worked hard after that. It's strange, I was at the Newton Institute at a meeting until yesterday, and so I came out here on non-committal geometry, but there was a young Russian representation theorist there, Bajov, who apparently knows Perelman very well, I mean, you know, they were contemporaries. What's his name? Bajov, Yuri Bajov. He's a representation theorist. Very bright guy, obviously, because he's got a visiting fellowship at the Newton and he's also at the IHES. But he was saying a little bit about Perelman's personality having undergone quite a sharp deterioration in the direction of paranoia and complete withdrawal from any kind of social interaction, even in the last four years.

5:00 He's known him since they were in school together. They go to the Maths Olympiads together. He didn't say that it had been connected directly with that, it's just that he had become, but he was, I mean, you know, there are obviously so many parallels, look at Grotendieck, look at the example of Grotendieck, although in his case, the withdrawal and the isolation came, you know, at the end of his. Well, he had already done tremendous work, but it was kind of... Actually, Grechtig's a very interesting case because when he did his early great work in functional analysis, which now tends to be overlooked because it was in some sense rather eclipsed by what he did in algebraic geometry, which was obviously so... It's incredible, it just transformed the whole nature of 20th century mathematics, but he had already done work of extraordinary depth in theory of nuclear spaces and functional analysis before that, and at that time he had the reputation of being completely non-political, rather a... You know, rather a socialite by the standards of mathematicians, almost, I wouldn't quite say a playboy, but I was told by Cartier, who of course knew him very well at that time, they were both in Burbaki, that Andre Weil, for instance, had a very rather poor opinion of him as not being a very serious-minded person, and then, of course, he became the director of the IHES and he became very political. Very, very idiosyncratic in his, you know, his rejection of any kind of connection, particularly between mathematics and physics, but in his case, and then ended by withdrawing, leaving the kind of public practice of mathematics. Although he has continued, he has continued.

7:30 He's been living in the Pyrenees for some years now, in complete isolation. So he was not... He was very close to Montpellier. Yeah, he lived in a windmill. He lived in a windmill. He lived in a converted windmill near Montpellier for many years. And then about ten years ago... He was showing up at university sometimes. He was given a position at Montpelier at one point, briefly. He did actually hold a position at Montpelier after he decided to come back into the practice of mathematics, but it was very brief. He only held it for about three years, and then he went completely into it. His last student, Jean Magloire, who is visiting him maybe once a month, and Jean showed me the list of signs that he has all the rights for anything from mathematics and so on. But on the other hand, if you look genetically, the parents of Grotendieck were very left and very politically engaged. Left or just Stalin? Far from Stalin, it's just the opposite. His father was a member of the Left SR. Left? The Left SR, the Left Social Revolutionary. They were one of the most important revolutionary groups before the October Revolution. They were crushed by Lenin. They were crushed by the Cheka immediately after. When the assassination attempt was made on Lenin in January 1918, the Left SRs were all rounded up and a lot of their leaders were executed. And Grotendieck's father, whose shame was given name was Shapiro, was one of the leaders, but not, as Cartier says in the memoir that he wrote about Grotendieck, I don't know if you've seen that thing, he published a mad day's work, it's published in the American Mathematical Society, it's also on the web, it's an interesting memoir, and he has a lot of interesting things to say in it.

10:00 But there are one or two bits of information he gives which are completely wrong. And I've actually asked him about this. He insists that Grodendieck himself gave him this information, but it's the case that Grodendieck told him things which he maybe wanted Cartier to believe. I don't know. It's a very interesting thing. But he told that in the memoir it says that Grodendieck's father... ...was the Shapiro who appears in accounts of the ten days of the Bolshevik seizure of power in St. Petersburg or Petrograd in 1917 as the Shapiro, Lev Shapiro, who was the leader of the Left SRs. Now this is simply wrong because the Shapiro who was the leader of the Left SRs in Petrograd... Grodendieck survived, went into exile, and died in the United States in, I think sometime in the late 60s, possibly 1968. Rather like Kerensky, they gave him a teaching position at some American university. Grodendieck's father, unquestionably. ...was gassed by the Nazis in Auschwitz in 1942. He was deported by the French, by the Vichy French, and died in Auschwitz, so there's no question at all that it's not the same Shapiro, although the father of Grotendieck, Shapiro who was the father of Grotendieck, may well have been, it had some independent evidence that he was an active member of the left-left class, but he certainly wasn't the leader. So this is again an interesting distortion. There's no reason whatever that Cartier would have made the story up. It seems very plausible that he would indeed have got it from Groeningdijk. So Groeningdijk was already fantasizing about his own parentage, you know, even when he was speaking to Cartier. So there's a very, very strange going on there. And of course he took his mother's name because in Germany in the 1930s it wasn't a very clever thing to have a Jewish name. Yeah, exactly. But that's the sort of background. The only reason I know that is because I... Is it written in this memoir why Grotendieck appeared in Montpellier? No, no, I mean when he was a schoolboy before, why he was studying at Montpellier. Yes, because when they were hiding from the Nazis, his mother and he... ...were given shelter in this village in the Franche-Comte, in the French Alps, the Huguenot village, where a lot of Jewish children were sheltered by the Huguenots. It's a very, very fine episode and should reflect very well on the Huguenot community in France. They were all given sort of false names, and they hid. And, uh...

12:30 He went afterwards because the schoolmaster in this little village had been an acrégé of the University of Montpellier, but then later, when he went to Paris, he went with a letter of introduction, and she went with a letter of introduction to Élie Cartard. Because the schoolmaster had been a student of Elie Kartan, and of course by that time Elie Kartan was very old, and so he handed the letter to Henri Kartan, and the rest is the history of centric mathematics. But that was why he didn't have a conventional school career because he was, that was the period when they were hiding. So Montpellier was in the occupied part of the zone? Well, it was in the unoccupied part to begin with. Yes, yes, but then the Germans came in and occupied all of that zone in November 1952 anyway. So by which time the deportation of the Jews had begun. And the Rishim regime was deporting the Jews. In fact, they were doing something more than see what they had to do. All of this was done by an artist himself, so it wasn't as if being in the unknown at that time was any great safeguard, if anything, it was almost, in some respects, because that part of the past was really important, more historically, than the directors. It wasn't a great advantage to be in the unoccupied zone if you were in some ways in the worst position, but after November 4th it made no difference anyway, it was a change. The only difference was that I obviously had a little bit more time to prepare for, you know, to go into hiding.

15:00 I see where Alex's handouts are. Look at those. I see. I'm afraid I got such an impression of him on that subject. After he looked at me, don't say that, don't tell anybody. But after he looked at me, I actually lost my mind. He, of course, Colin persuaded him to look at the videos of the talks on my side. You can imagine the ones he likes. These people haven't been listening to anything for about 60 years. I like Charlie Boesel, though. The Chinese and Bioneurologists, you know, really like it down there. Yes, I have to say, it's always difficult, isn't it? New book? Yeah, by somebody that André knows quite well in Switzerland. He's a kind of logician. And you're doing that? It's a formal coffee break. So do you know each other? Of course I do. I'm a G.W.U. I went to a very nice conference there. Long long time ago, almost 20 years ago. It was actually a cosmology conference. Cosmology and E.P.R. So who was organising it? Gosh, so now, as I say, it was 1988. It was... No, no, 18 years ago.

17:30 No, I was... Oh, no, no, sorry, it was before. It was definitely before. It was 18 years ago. Yes, I think that... I think that's why I want to put it on the last... So then maybe he already was not in UW. Later our university somehow went to teaching university and he left for Colorado, I believe. But he was standing there for 20 years. As I say, this was in 1980. This was almost 20 years ago. Oh, so you're from GW as well. Hi, nice to meet you. I was just saying to Joseph that I happened to go to a conference. It was actually a cosmology conference at your university about 20, almost 20 years ago now, when Gamow was still... Uh-huh, yes, yeah, yeah. They had John Bell there, the usual suspect. Nothing like this. Rosen. Rosen was there. No, the Rosen of Einstein and Podolsky. David Boehm. Yes, it was a very interesting lecture. I think Gamow died soon after. Yes, it must have been the last conference or the last thing he organized, I would think. I don't know about physics, I think. Anyway, it went down. I think it was not only physics. I think the president decided that he should be teaching at the university. And Gamow left and went. And when Gamow left, of course, his home collapsed. Well, it was the only time I'd been there, but it's a nice campus. It's just in the middle of town. Yeah, and it was right in the middle of town. Exactly, and it was in the heart of Washington, which was very interesting. It was my first time in Washington. Halfway between White House and Washington. Yeah. I know, actually, but I think that the conference was something on symbolism or something. I remember we had a reception in the National Gallery of the National Gallery of the National Gallery of the National Gallery of the National Gallery of the National Gallery of the National Gallery of the National

20:00 Yeah, yeah, brilliant. But when there's a rating, when there's a rating, then they have to go for some backup plan. They don't let them have it in the Oval Office, it's a bit small. Yeah, no, not there. I hate to say it, I don't think Bush would probably approve of it. I'm just saying that when they have this conference, this is what... They had a reception at the National Gallery, you know, the National Gallery, which was a great place, but they didn't have a reception. Yeah, we don't get that. So that was Gamow. I think that was where, gosh, he was in. No, I don't think it was anything, I can't think it was. It was in the White House that I've forgotten. I think it was Fush's farm, it wasn't it, or it might be the school of the brain, but I can't remember now. Well, I guess we're ready to go.

27:30 The identification of the Alexander Poisson in the sense that sum minus one to the m is exactly the Alexander Poisson.

30:00 Okay, so here are some properties. The property is that it detects the genus of the knot. It's maximal such that k in that directs sum over all of m, but in Vs it's non-zero. It detects the genus, it means that it detects the unknown. The property is that it also detects whether a knot is fiber in the top degree, i.e. the genus of the knot. One, the equivalent of K is fiber, which is the theorem of P and E. And this implies, and in fact it was proved earlier by Vigini, that H of K had a particular H knot. Some knots have the same H of K as one of these, and it didn't. Another property, so not only detects the genus, but it says something about the slice genus, so for this you need, so there is a sequence, similar to this, like this sequence. The topology in this case, A hat, and it comes, so it starts with E2, so it determines that it's A hat, and the E infinity term is just one copy of hat.

32:30 It has the property that it provides bounds for the slice tree. So these are properties like the application of Knob Theory. Originally it was developed as a building block for Heegaard Globology. So it connects knobs to three manifolds. It has enough information to help... You can use them to recover the Heegaard Floer homology. Heegaard Floer homology is an invariant of three manifolds. It was invented by O'Connor himself when it's related to gauge theory. And H has enough information to give what it is for the surgeries from that time. It's defined for links as well. It's the simplest way to define it. Now, well, this observation is only true for knots, as of now. So, for lengths it's a bit more complicated. It's not clear that you can recover. You should be able to, but it hasn't been done.

35:00 Okay, so one term of this theory was that it was, so the original definition, well, it's called cohomology, and it's called cohomology. Meaning that these definitions involve symplectic geometry, so you can define it, but it's not clear how to compute it. So maybe I should say a few words about how you define topology. So this part was a Tegel diagram for the math inside the 3-sphere. In fact, it's more the complement of the math. The Tegel diagram consists of a Riemann surface, sigma, And some curves, alpha 1, alpha g, beta 1, beta g, and two points, w and g, so sigma is the surface of genus g, alpha g, homologically, I mean, are homologically linearly independent curves on sigma, and then a subset is quite a handle body. Basically, for alpha, you get a candle body H given by adding a cell to the group, and beta1, beta, and B are other curves, and they give another candle body H prime, and you want this, so without the W and the C, you want this to be a pegadiagonal for S3, so S3 could be the union of...

37:30 And the math, you can see it by, so W and Z are the two points on the surface, which join from the alphas and the betas. So you can join W to Z in H interval by some interval I, which is the size of everybody, and you can also do it in H prime. And you get an interval I in prime. And you want K to be the union of these. And this gives you a knot in S-thick. And for every knot, you're going to find such a figure of the angle. This way in the original. I mean that then, the knot was hidden in one of them. It was? That's the definition I know. Okay, so, well, I don't know, maybe I should just give a simple example. I don't know, let's say we just have the simple functions. For example, we have the course of genus one. We have one alpha curve and one theta curve. This is a diagram for S3, and let's say we put that what you want to see here, or these are close to each other, then you can blend them by intervals on both sides of things, but you just get one.

40:00 If you change beta so that it intersects alpha in three points rather than one, it still has three, but if you put W and C here, you actually get the square root. Okay, so you do this, and then you consider the following for us. You take the product of all the alphas, alpha 1 times alpha g, and the product of all the betas. These are all G-dimensional products, and I look at them as half-dimensional sub-manifolds of the G-symmetric power of sigma minus these two base points. I take the complement of sigma, and I take the G-symmetric product, and then basically these are all Lagrangian, they can be made Lagrangian, Floor cohomology construction, and you can apply the standard floor cohomology construction as inflected geometry. Basically, we define a chain complex, so this is the floor complex CF of T alpha and T beta. This is really generated by, these are half-dimensionals, so they intersect in a number of points. The random generators by this formula were n, x, y, so given two intersection points, I will draw it here, just schematically, we have T alpha and T beta, and we have two intersection points x and y, n, x, y is the kind of pseudo-photomorphic this.

42:30 I don't want to get into details, but basically... It counts disks like this, satisfying a certain PDE, like an analog of the Cauchy-Riemann equations, and this is where the symplectic geometry comes in, I mean it's not, it's not a combinatorial problem to count such things, right, and then the flow, the natural homology of K is the homology of this complex, this differential. So this is one way to define it. There is a slight variation on this definition, which is due to a number of people, I think, more or less than several were the first to know this. Basically, you don't have to just, I mean, you can do it not just representing the knot as a union of two intervals, but you can do it using the knot as a union of several intervals. The variation is that you want a diagonal, so let's say the Riemann surface of G knows G, but then you want G plus K alpha curves and G plus K beta curves, and this should span, this should still have only a G-dimensional span, and this also should have a G-dimensional span.

45:00 What does that mean? If they do that, okay, so maybe you can add another curve here if you want, but it has to be in the same... Basically you want this to still give a 10 to the 5th, and this to still give a 10 to the 5th A prime. And you still want S3 to be A union A. But you have more curves, and not only that, you also have more phase points. So we're going to consider K plus 1 W's and K plus 1 C's. And then you can take the thermology of combined theta alpha with the product of the alphas, the data with the product of the nations, and you want to take the thermology of these things just as before, but now you work in the G plus K. Symmetric product of sigma, minus the union of, so the complement of all w's and z's. And this one turns out to be almost, not the homology, so it gives the same answer apart, but you still, you have, with one difference, you have the answer with v to the k, v is just add to 2, so it's a two-dimensional vector space.

47:30 And if you want the gradients, these should be generated by some generator in M0, 0, 0, so Maslow and Alexander's gradients 0, and this in Maslow and Alexander's gradients minus 1. I didn't say, but you can define these two gradients on chain complex. For the M-naught, you have alpha and beta adjusting the first magic vector in the surface, and they have just one generator, so the vertical model of the M-naught is just one generator by its form. Here's how you can do it this way. For example, you can work in the flame. You can say it's, you know, zero and k equals one. So I think of s2 as a plane, and this is an alpha curve, a beta curve, and you have the w's and the z's like this. And then basically this is the unknot, what you have to do is join the w and the z in the complement of the beta, and then separate them in the complement of the alpha, and see what knot you get. And this is just the unknot. But now you have two intersection points.

50:00 And basically we've got the plot model in the U of K is 2 to the minus 1, or B times 4, F. So every time you add something to K, you cancel with B. Okay, so now we want a combinatorial description. So this is based on a number of observations. So the first observation is that we want to... So we want to come to the holomorphic disks, and it's easy to split them apart by homology classes. Let A become a relative homology class of the topological data, and you're looking at some homology class. Okay, then basically we have, so we have sin g sigma here. This is covered by sin g minus 1 sigma times sigma. If you have a g minus 1 point to sigma and then another point, you get g point to sigma. And this has a projection to sigma on the second factor. Let's go this way too. So you have some homology class in here.

52:30 What you can do, you can put it back by half, say you have a two-chain, you take the pre-image and then you take the image in sigma. So you define D of A, this is the domain of A, to be pi 2 O. And this actually determines A. And how does the domain look like? Well it's a two-chain in sigma, and the boundaries are on alphas and betas. So basically it's alphas and betas split sigma into d1 and blah blah blah, connect the components of the complement of the alphas and the betas, then d of some homology class is some sum of m i d i with m i d n. So they have some multiplicities, but it's a two-train of that type. Well, it was known for a long time, ever since the definition of pegatomorphology, that for this domain can look like what? They can have like handles in them, the pieces of a surface, but in simple cases it's known how to counter pseudo-polymorphic things. Observation 2, if D of A is just a rectangle, so it's something of a general zero, and it has four... It has four acute angles. This is an alpha, this is a theta, the piece of the surface, then A has exactly one pseudo-holomorphic representative, and you have the same mod 2.

55:00 For over C, you have to do a sine count, but that does work in mod 2. And this basically follows from the Riemann mapping theorem. So this is the only analysis that in fact is needed for this problem. So here you're trying to count this to the holomorphic distance. You know that there's just one when u of a looks like this. That's right. Well, that's how you prove it, but then you know that it doesn't depend on the normal complex structure. Like, you do a deformation algorithm for any good normal complex structure, the count is 1. But it's part of that. So then came an observation of Dr. Richard Park, who noticed that if P of A So d of a is the sum of m i d i. In order for this to have a homomorphic representation, if you want all the m i's to be non-negative, so let's just take the sum to be made of m i's equal to zero to forget about the zero terms, if it is of this type and all d i's are squares And it has Maslow index one.

57:30 So I should mention that in this column, you want, so x and y are graded, and you want the difference in the homological difference to be one when you define the differential. And in fact, you can define that just based on the domain. It's an invariant of the domain. So if you put the condition that everything that is made of is squared, and it could look like this, So this, for example, is something like that, but it doesn't have mass of index 1. If it has mass of index 1, then in fact, d of a is a square. It could be broken up into a bunch of squares like a grid. It is something of this type. Of rectangles? Of rectangles, yes. Are there the same number of sub-rectangles along one edge as the other, or not? No, no, it could be. And, oh, I should say, this also slices here. You can have it like this. It doesn't matter what you do. I'm not assuming it's a grid. I'm not assuming it's a grid. The claim is that it's a grid. The claim is that it's a grid. If D of A is made of squares, like this, if the alphas and the betas break it only into rectangles, I use rectangle and square interchangeably. And it has muscle of index 1, then it is a grid. It doesn't make sense on an arbitrary surface. Yes, yes, yes. I mean, yeah, so I'm going to have to say, if I were, I could think something like this, like this on the surface, but this with four angles.

1:00:00 Okay, so now the point is to find the Pega-diagram for the knot such that you're always in this situation. And these are provided by grid diagrams, so the diagram for the page is an n by n rows and columns with sums such that each row has exactly one black dot, black dot, and one white dot. And the same is true for columns. Each column has exactly one of each. Each row has one of each. Right, this would be a diagram for K, meaning that if you join the dots below each column, then you get a diagram for K, an actual diagram for the knot, with the convention that, whenever you have an intersection of two segments, you want the vertical one to be an overpass.

1:02:30 So let's do it. The vertical and then the horizontals are on the passes. Okay, and you get the knot, which in this case is the triplet. Okay, so observation is that every knot has a grid diagram, right? Well, you can start with any diagram, say you have a triplet or... So this is a diagram for the triple. You make it so that all segments are either horizontal or vertical, and then you replace, whenever you see this kind of crossing, you don't want it, right? So you want to replace it by this kind of crossing. So you introduce additional things, but in the end you get some diagram where all the crossings look like this, and then you just split the plane according to each interval, and you get the grid diagram. Grid diagrams are the same as our presentations of knots. Okay, so basically the grid diagram is a tega diagram for the knot.

1:05:00 You have to make it into a torus. So you identify this side and this side and this side. So now it's a torus, and this is your circle sigma. Alpha i's, say, are the horizontal lines. The line of coordinates y equals i, and beta i is the vertical line. These are the white lines in your picture, right? Yes, these are the white lines. And, oh yes, and w's are the white dots, and z's are the black dots. So, right, this is the diagram for S3, and, well, you can check that... It's a diagram for the knot, because when you draw it like this, you basically see them.