 Hi good morning So we're gonna have a change of pace from yesterday there's gonna be Well, I'm not sure what it's gonna be there's gonna be less chemistry I hope that I don't I because I don't know any chemistry. So whatever I'd say would be gibberish I want to start by showing you this picture We heard about DNA linking yesterday and how does DNA do this in fact? There's some things this kinetoplast DNA Which is actually linked on purpose and actually the little loops there's some purpose and the big loop for the for the processing of the Cell so there are macro cycles and there are little cycles and the little cycles are in the right place So not only do you have these things that duplicate they duplicate and they stay linked together It's not just that you have how do they manage to duplicate while they're linked together and then come apart These actually go back together in a special way So I'm not going to talk about this kind of linking All right, but this is the kind of linking I have in mind I'm thinking of linking of things that are physical links Where you can't pass things through each other yesterday somebody said who said it are you here? Yesterday somebody said oh, well, I'm not going to talk about energy She's already she's already gone. Okay. She's all right But of course she was talking about energy if you try to like pull if you pull two strands of DNA hard enough They will break and go through each other. So topology and energy are always, you know, deeply interconnected It's just a question of scales I'm going to talk about work that started some time ago with Gareth who's up there and Brian Chen who's not up there and Ended recently with some work with Ricardo Mazna from Brazil So Here's a story. Here's your first introduction of the crystals. Who's ever seen a look at crystal before Come on, you all have one in front of you Are you serious? I've never seen one. Okay, but this is what looks oops. This is what oops. Oops It was so much easier when you had transparencies you could just drop them. All right, anyhow here Is a nematic look at crystal When you look at a nematic look at crystal Through cross polarizers and I'll explain everything in a moment you see places You see these black lines and you see these places where the black lines come together into four fold places And sometimes into two fold places. So let me tell you what you're looking at in case you don't know So this is how it works. There's a light. There's a polarizer There's another polarizer that's at 90 degrees, which is called the analyzer if you send light through cross polarizers you get You get Nothing exactly Okay, but the liquid crystal molecules in there the nematic liquid crystals are Elongated rods from the point of view of mathematics. They're just line fields But they're molecules that are long ellipsoidal and long and stretched and they're birefringent And as a result if they're in here They actually help rotate the light if you if you like light you would say well There's the ordinary wave the extraordinary wave and they go at different velocities and they get to the other end of They don't cancel or if you like polarizers You can think of them as a third polarizer inserted between the cross polarizers The upshot is is that when you see light coming through it means that the molecules Which are lying in the plane of the polarizer and the analyzer are Letting light through and it means that they're not along the polarizer direction of the analyzer direction if they were Along the polarizer direction you'd have two polarizers like this and one like this that would still give you nothing right the second polarizer would be redundant and so These black lines are the pre-image of the direction of the polarizer in the direction of the analyzer So there you go to apology in a nutshell right you take a picture with cross polarizers and you can actually see the root the pre-image of these two directions and Why do you see four? Well, I told you it's a line field So I'm drawing a line field in here notice that I don't have any heads on these lines if I have a Defect here a place where I can't tell you which direction the things go everywhere else I have to have a well-defined line field. So as I go around this point The line has to rotate by 2 pi one way or the other way And you get four brushes why for I get black whenever the molecules are along P or along a So you get four brushes because it goes through the polarizer once in the analyzer once in the polarizer again in the analyzer again four brushes four times It doesn't matter how as long as it goes around by 2 pi you get four brushes By the way fancy pants it means that these here where you get to Tell you that the molecule is symmetric under rotations by pi look at that I took a picture with something that was invented in the 1800s polarizers with light this is a picture taken with a microscope something that Galileo could have built and You can tell something about nanometer size molecules That they're the same whether they're like this or like this. It's pretty cool, right? They don't have to do that much Once you know mathematics, okay? This again, so that's the pneumatic phase if I were a mathematician I Would classify The defects how would I classify them? I'd have my sample here Here's the sample. It has a bunch of defects in it Here is the direction of the line field, so this is an angle phi And what I would do is I would look at some path around this defect and I'd make a map from this path To this space. All right, so this is a map from the circle to the circle So it's a map from the circle to the circle and so I can classify those defects by the fundamental group of This thing which I'll call the ground state manifold It's the place that tells me all the directions that are ground states after all the ground state of the liquid crystal the ground state of the line field is all parallel lines and Now I tell you that as I go around some defect the direction rotates and I can tell you how much the direction Rotates because it's an angle and there's maps from the fun from this path Which I went around to that path and I see a black line Any time I'm here or here or here or here for instance But I don't want to talk about pneumatics the next talk. We'll talk about pneumatics I want to talk about something which is a little more subtle which we don't understand And I hope that I can entice you to work on it with us or work on it yourself solve the problem and Mail us a copy. All right here is a smectic a smectic is A crystal and what I mean by that is that there has one dimensional order It's like a building right so there's a stack each layer of the building can do whatever it wants Each layer of the building can be a liquid no organization on each floor, but the floors are separated Like a department store. Okay, but by the way, this is a real picture of a smectic This is what a theorist writes down they draw always parallel layers But real smectics when you look at them and cross polarizers are very complicated things But Everywhere that you see the smectic Locally it looks like this Locally a smectic looks like this So how would you describe a smectic one of the reasons energy is not that interesting here is that the Ground states or the things that you see not the ground states the things that you see are quite complicated Let me just tell you what we do we write down energies We'd write down energies that say things like the layers want to be equally spaced and they don't like to bend But I'm not interested in those what I'm interested in is how do I describe the layers and I describe the layers by Writing down level sets of some phase field phi The density of stuff is proportional to cosine of phi Phi equals zero is the first layer phi equals a is the second layer phi equals 2a is the next layer and so on and Because phi cosine phi is the variable of interest you'd say this is exactly what I did before After all phi, which is telling me where the layers are is also lives on the circle Because phi and phi plus a to the same so you'd say oh, it's the same thing. I Would classify defects and objects like this precisely the same way I would find some point where I didn't have nice order. I would go around the point in the sample That would be a circle measurement here, and I would have a circle measurement here But that's not true. It turns out smectics are more subtle and The reason that smectics are more subtle is because this order is periodic in space. It's actually in space It's not in some internal manifold like telling me what the direction is So unlike the pneumatics So one of the things about pneumatics Which you could realize for looking at this is all I need to do if I actually am classifying things in a fundamental group Pi one of s1. Oh by the way, I should I shouldn't keep it a secret from you These are just the integers and so since they're just the integers the fundamental group I can add things together, and it's true if I take two defects where the rotation is 2 pi and bring them closer together Far away. I'll get eight brushes It'll look like I've gone around 4 pi and if I've gone around plus 2 pi and minus 2 pi as I bring those four brushes together They're all cancel with each other. They'll all link up, and I won't have any brushes at infinity. So in fact This is not nice a billion group. I can add the defects together because they're classified by this object But smectics are different. There's a real theorem by Ponaru. It says look suppose that you have a system Wait hold on Measured foliation Measured foliation means locally You can always zoom in on whatever the thing is and it looks like this Looks like a smectic except for a few points, which are singularities That's what the all these words mean measured foliation is the important one It means that you can locally make things look like layers and Pornaro proved that you can have a Defect that looks like that see here is a texture if I asked myself. What is the normal direction to the phase? Right the normal it goes around by 2 pi. I Could talk about this here the normal goes around by pi That works because the normal and minus the normal are the same here. Nothing happens here It goes around by minus pi minus 2 pi minus 3 pot minus 3 pi and Ponaro prove that you can go all the way this way You can have any negative charge you like But you can't have any positive charge in fact plus ones as high as you can go Now You can read this it took us a long time to understand it here's the problem a group is a group and We say this thing in the United States what state what happens in a group stays in a group and so here's this group I Should be able to combine two objects that are classified by This winding how come I can't join them together the group operation would tell me I could join them I must get something with winding number two and a winding two or an index two zero Why don't I get one and put our approved you can't do it and it's this thing that locally I always have to have my layers look like this And we didn't understand the theorem I mean we could read the all the words and We could read even sentences, but we realized to us what was going on was much simpler in our minds is I Told you we have this field Phi level sets of Phi tell me where the layers are So I'm going to talk about a two-dimensional Smectic because I can't draw the three-dimensional one this green thing is just the graph of Phi It's not a physical object It's a mathematical object and level sets are easy You intersect this green surface with the surface by equals zero And that's this level set and the next level set and you get all the level sets and you realize that Asmetic is nothing more than a topographical map And it's important this thing about it being measured means that at every place there's a slope that isn't zero And so Here you go, right? You get to basically use the idea as a Morse theory to talk about the Smech-tik and it's different when you talk about a Smech-tik instead of talking about Layers as the objects or the direction as the object the phase you think oh it's really something that lives on a mountain range and The mountain range level sets of the mountain range tell you where the layers are and it's great. The layers never cross The layers never end You can tell that you're at the top How can you tell that you're at the top of a mountain good? In Europe and you're you know you yeah in Europe. It's easy because you can buy a coke Right. There's always a coke. Yeah. Yeah, it's not like that. There's a coke. There's a chop You can buy lint chocolate everyone Switzerland the top of every mountain you can buy lint chocolate, right? It's true right but in the u.s. It's what you said you get to the top and every way else is down Okay, all right, and you know all right and so here's the and if it's every way Oops, oops. Sorry if every other way is down it means that as you go around you have these closed loops Which are the plus one index zeroes? And the thing is you say how come I can't bring two of them together And you know the answer because there are mountain passes in between and there's this thing the mountain pass theorem That tells you if you bring try to bring two mountains together You always in train a pass and a pass is precisely a minus one index and So as you try to bring the two mountains together you lose the pass in between them and the charges all cancel So it's not so simple. However, it's not as simple as you can bring the two plus ones together They always in train this extra minus one in there How can you have any negative charge you can have a situation where you have you know 35 mountains? And you have one pass That goes down to the 35 valleys that's not allowed in Morse theory that would be a degenerate Critical point, but it's allowed in my picture Okay, so it's not quite Morse theory. I should probably have some other symbol here that isn't equals So the idea is you can have any negative charge, but you can't have any positive charge It means that classifying these defects is not as simple as having a group If we're a group I'd be able to take two plus ones and add them together and Has something to do with the fact that you have this spacing and that you have this measure everywhere now Smectics get to enjoy something else now Nobody stopped me You guys all nodded when I said oh, it'd be just like this problem The original problem thematic and here's a situation where it is the same problem as the pneumatic because now I Have a phase field and the phase field rotates by 2 pi or the phase field changes by pi or 4 pi And what do I mean by that? These are called dislocations. These are different than those index defects That was something about a defect in the normal to these surfaces But the normal to the level sets now. I'm talking about the level sets themselves and In the US, I'm sure it's true everywhere. Oh, I know where it's true. It's true in Bristol Who's from Bristol? Excellent. Okay. It's definitely true in Bristol Sometimes you have these buildings that are joined together One was built like before the war and the other was built after the war I'm talking about World War two and and the war they come together and There's more floors in the new building Somehow over time they improve the technology you need less space between the floors and people got shorter or Whatever they didn't use much space, right? And so what happens is you have nine stories here and you only have eight stories here The buildings are the same height and you have to like mush them together somehow you do it like this You relabel them one two You always call this the mezzanine Right one of them is the mezzanine with three and so forth and The weird thing would be is that if you weren't paying attention to what you were doing You could be wandering around the building. You could start here. You could go up one floor Here you are here's the ground floor, but you start on one See I'm adapting for you guys. Okay. You start on one you go up to two You come over here. Now you want to leave the building So you go down two stories because you were on two you're just you think you're on the ground floor, but you're not You're on the mezzanine. You've messed up. You can't get out. You're trapped. You have to get a coffee and so That's a dislocation. You're not where you thought you were because there's this extra layer That is the phase field changing. What am I doing? I'm looking at grad the gradient of that phase field Phi and I'm adding up the changes And I went like this and I added up one and then I went over here and I went down to So when I did some integral of grad Phi It wasn't zero These are classified by maps from the circle to the circle So, how do I mix these two things together? I just told you that a smectic so they get to enjoy this kind of defect I told you that a smectic is a Topographical map I told you topographical maps can't have ends to their lines So how do I manage this that I'll be allowed on a topographical map. I Can ask you a question How is it possible even draw a surface that does this What kind of earth or what kind of planet or what kind of map or actually what kind of mountain range would give you this Here's a hint remember the graph is up. It's away from the surface and you're so here's the surf Here's the XY plane and I'm telling you that as you walk around the circle Phi changes by Or to pie or whatever units you like That means you've gone up though. That's a staircase. It's a spiral. It's a helicoid You say, how can I work a helicoid in here's a helicoid? What's wrong with this helicoid if you get the answer wrong I get extra fun. It's a negotiation The problem is far away the helicoids flat So it's not measured. It's not a smectic. I have to have a slope Another way of saying is I want the ground state to be equally spaced lines So I don't want to have something which goes to flat. I want something a mountain range. It looks like this infinity So I can I combine the two things together? Of course I can Right. Here's the helicoid and I can stretch it and there it is It's a stretched helicoid and I stretched it up. You know why I stretched it up I stretched it up so that on the circle Phi and Phi equals zero and Phi equals one and Phi equals two are Exactly the same thing. I'm making this space the mountain range live in a periodic space Instead of living in R. It's living on the circle. Haha the circle right And so then I can fit it in and it works and so here. Oh, I don't know what's going on Oh, but you get the other version of it. I don't know why okay Here's a picture of it. Now watch me take level sets of it. You ready? Okay, so there it cuts through you get these two black lines you cut through you cut through you cut through Do you see it? Do you see there's two extra layers coming in? There's the dislocation watch there it is see It's a dislocation two extra layers And I can go and I can even get the other geometry If I want the one where I get one extra layer is a little more singular Because I have to have only half of a helicoid, but you can do it, too So here you have the situation where you have dislocation. So if you make a mountain range That lives on the circle instead of the real wine Then you're good then you can actually completely describe the defects and smectics And you can say why am I telling you about this? What's this workshop on anyhow? It's on links and knots, but I'm going to try telling you that the story of links and knots of This object is very complicated All right, I'm not talking about vortices and superfluids Or defects in nematica crystals I'm talking about defects in a very complicated system because it has all these extra problems The defects don't add even the blame simple story the defects don't add nicely. There's no group I can't use homotopy theory so easily So we've been exploring this system. Our goal is to really understand it Our method is to just find problems that we can solve that we can understand So let me go from two to three dimensions. Oh Do you have a question? You have question ask me you you it's okay. Who says that? Well, of course, I have a theorem for you everything shown at this conference is too deep Okay, but I'm not but here is a picture. See there's the extra dimension y Okay, this is the xyz plane. This is a section in the zx plane I have complete symmetry in the y direction. So this is an edge dislocation Look at the light blue line only the light blue line. You have one come in and you have two going out to infinity Here's the dislocation these different colors Phi equals zero is the blue color Phi equals point one is this darker blue Phi equals point two is this even darker blue Phi equals point three is this pinkish color and so forth I'm drawing all the values of Phi all the level sets of Phi not just Phi equals zero and Phi equals one all of them They're all there the phase to wrap around has to be defined everywhere and This point here is the place where I've added an extra layer There's an extra two pi of Phi in here. I've squeezed in an extra whole two pi I can write down a phase field if you like, but I noticed something Suppose I draw the normals to the surfaces The normals to the surfaces around this point wrap around So this is an index one Defect in the normals, but I have this funny boundary condition. It's that measured foliation boundary condition far away The normal points up The normal isn't zero if I were a superfluid. I Would say far away grad Phi is zero But here grad Phi isn't zero grad Phi has to point in some direction there has to be a slope And so now I can use the mountain pass theorem again if I want to call it that that's I think overkill I have a situation here where the line points down Near this defect and I have a lot and I know out here the line points up Somewhere the line must vanish There's another zero Grad Phi has another zero grad Phi has a singularity here and it has a zero right here And that means that here I have a place where the I have a plus one index Dislocation or plus one index zero of the director field or the normal But here I have a minus one index one there are the two points the plus one and the minus one and they're separated And they make the dislocation Did you see that you have these? Disclanations the things you saw in the pneumatic and put together they make this extra layer in the Psymectic, but I have to have them they always have to be in pairs in other words if I want to add an extra layer I am forced to have a disclanation pair a pair of Singularities in the orientation It's unavoidable Yes No That's right, but I'm just talking about the topology you're saying it could be that the energy is so high You just get a giant spot in the middle that blocks all of them. I agree Well, then I have to worry about what I do on that boundary And typically as you know, it never really goes to zero. It just decay it goes it gets it gets smaller So what happens if I look at this system? So here's this edge dislocation going through the sample I've added an extra layer suppose now that I want to tilt it. So instead of it going along this way now It's going to be more 3d Okay, now the dislocation goes like this This is still where the layers are this is the layer normal So the whole thing gets tilted a little bit But it's still true when I look at a section perpendicular to the defect line. I will still get these Layers there'll be a different spacing. They'll be stretched out But I'll still have this hyperbolic point and this Dislocation point or if you like the mountain and the saddle until eventually I have a helicoid Remember that helicoid. This is the boring helicoid, right? This is a defect. Look at this. Here's Phi That's the level the phase field. I look at a level set Phi equals zero So Z is arctangent Y over X. It's a nice helicoid. This is a defect in the smectic Which is allowed a 3d smectic and This doesn't have to have this hyperbolic point because these guys just go around. There's no condition on grad Phi in the XY point Right here. There was a condition in grad Phi in this funny tilted XZ plane But here there's no such condition and So you lose this property a screw dislocation, which is what we call this because it was like a screw Doesn't have to have a hyperbolic point, but this one does So I become interested in how do you link together defects in these materials? Are they linked? Are they linked the way that DNA is linked or? Are they linked in land-out theory where you have two superfluid vortices and you say well They're linked because there's some energy costs to have them pass through each other But that's a very small energy Compared to breaking apart two pieces of DNA and what I want to know is are they topologically linked when I mean That is can you actually pull them apart or just apology force you to leave a thread behind? as you pull them apart So here is my screw dislocation You see blue might be the level set Phi equals zero and red is Phi equals, you know Two-thirds and green is Phi equals one-third and blue is again Phi equals zero and Here's the screw dislocation and out at infinity it fits on the flat layers, and that's okay Now suppose I have an edge dislocation going around it Watch carefully So what do I mean by that an edge dislocation is a place where two red layers become one red layer? The mezzanine in the second floor just joined to be the second four So here's the edge dislocation. It's a closed loop in the XY plane What do I have to do? I have to pinch together the two red layers over here But as I go around because the screw dislocation is changing I have to pinch together the two green layers here and in back. I'll have to pinch together two blue layers So what I see is that on this edge dislocation There's actually a winding The winding is what color do I pinch together as I go around? You could say to me why don't you just keep pinching the red together? Well, I can't I would have to stretch everything in some terrible way it would effectively I would have to tear the smectic to do that. I'll make this more precise. Here's a picture I don't know which picture helps more. I think the picture on the left Doesn't help me at all. It's like some drill that was badly made, right? So here's a screw dislocation in infinity with one less layer than the screw dislocation that's going up the center Another way of saying here here are the level sets. What I'm doing here is I'm rotating around the central axis So here's where the screw defect was that helicoid. I'm going around it Here are the two here the here is the edge dislocation. Do you see as you go around the color changes? In other words, what's happening is there's a winding on The edge dislocation that's being induced from the boundary conditions at infinity The fact that in infinity I have layers is forcing me to connect those layers to the edge dislocation on the inside You don't get that usually usually when you talk about Superfluids or superconductors or even pneumatics at infinity everything's constant And you don't have to worry about this, but now you do Right. It means that in fact I have a problem. I Can't compactify my my sample Usually I could say that every point on the boundary of the sample is the same I can compactify the plane to the sphere But I can't do it here because at infinity. I don't have constants So I either have to do it in a finite region or I have to have some kind of really strange I have to have a defect at infinity Right, and so here's the structure It turns out it's in here here. You have a structure where you have a dislocation going around a screw dislocation. You see as You go around these lines over here. It looks like a dislocation over here It looks like a dislocation But as you go from the left to the right the lines of constant color have a shift in them They have a jump and there's no way to get rid of that So that if I tried to take these two lines and bring them back together I would have to have a singularity in the phase To me this means they're really linked There's no way to take if you had a screw this location here and you start running at it You can do this with dogs. You can do this experiment. I have my college roommate My college roommate his mother was a veterinarian. They went to the beach and They they they just they had two dogs and they decided to keep the dogs from running away. They would just tie them together Forgetting that dogs are pack animals. They don't run apart. They run together So they went around the beach taking everybody out with this rope in between them But that's exactly what this is here. The things are running. Okay. There's no way to take that rope Cut it and rejoin it that would mean cutting all these phase lines and rejoining them There'd be a terrible singularity in the phase lines So these are linked. These are linked like ropes tying dogs together They're not linked like superfluid vortices or vortices in a liquid which are energetically for you know suppressed from crossing These are topologically suppressed from crossing knots I've people have already seen this table, right? So I Feel better though. I can't imagine that any journal would ever accept a title like this anymore the first seven orders of naughtiness Q imagine that I mean Maybe the New York Times It turns out you don't know this in the US, you know, people aspire We want to publish in nature and science, you know all those things people really want to Publish in the Times literary supplement, right that that would be the pinnacle of your publication, right? So here are the knots. I'm going to explain to you how this invariant I'm talking about is closely related to knots and actually is much simpler from the point of view of knots so suppose instead of having a Tangled set of defects. I had a knot in my smectic Okay, Mark Dennis. He sort of introduced this idea to me this idea that you have a knot But the field around the knot It goes to the whole knot The knot just happens to be a simple part of the whole field, but the field has the knot in it So this is also Mark Dennis and and John Hane's beautiful picture. They say let's think about how link and twist and writhe work I don't know why this is Edinburgh, right? I'm bringing that this is Trieste. Okay Trieste. Sorry, so They said look everybody knows link equals twist plus writhe Here is a situation where I only have writhe And here is a situation where I only have twist So what does it mean? To have only writhe or to have only twist So here I have my belt Okay, so and You can see the two stripes. So now what I'm going to do is I'm going to rotate it give it a two-pile twist Like so Now you see these things. So this Configuration is twist. How do I know that? Because sometimes you can't see the red and the green stripes There are times when the red and the green stripes wrap around each other That's because there are places where there's always no matter what projection I give you There's always a place where you can't see the belt But this is writhe because you always see the red and the green Rithe is when the red and the green cross at the same time So that you always see them non-local crossings and twist is when they have local crossings And you can convert one to the other see how simple that is Okay, this beautiful way of interpreting things. I Lost my advanced sir It's okay. I'm wearing I'm wearing underwear. It's alright. Okay Boxers, okay, so anyhow Here we have link is twist plus writhe. Let me think about something Hello Maybe I've gotten too far away. I'll just touch the computer. So here's a knot. Here's a trefoil knot. Here's its cypherd surface All right, what I'm going to show you is That there's a natural framing of this knot There's all this work beautiful work about how how is link equals twist plus writhe What's that word? generalized to whole fields and to whole fluids and the Helicity of the fluid, but here I have a much simpler problem. I have a system where here is my knot Here is its cypherd surface the cypherd surface defines one framing of the knot. We see it The remote is dead. Oh, there you go So I'll remind you this is from Mark Dennis here is a trefoil knot Here is its whole vibration in cypherd surfaces, which is quite pretty see each of those as a smectic layer You see a fiber knot is naturally a smectic so let me make my vibration here is the unknot and I can make a cypherd surface for it and then I can the cypherd surface is orientable so it has a normal everywhere and What I do that I call this an epsilon ball around the line. Is that what you call it an epsilon tube, right? what a tubular neighborhood tubular neighborhood around this line and I intersect the tubular neighborhood with the cypherd surface to get another curve this one and then as Tom Michonne would tell me since the normal is everywhere to find I can take that new curve and just push it off the surface And that means that these two loops are not linked Okay, the link is zero, but fortunately I have another framing Right, so I can talk about if I talk about this guy. I'll call this the surface where phi equals zero It's one of the level sets, right? And so this is the line I get I'll call it eight zero. It's the line I get from that cypherd surface defined by the symmetric But fortunately I have another frame. I have this point here You see I don't know the value of phi at this point where the dislocation is because it has all values of phi But the place where there's a saddle where grad phi vanishes is perfectly well-defined and this guy always has one with it So I can use this curve as the other curve So I have a framing from H and I have a framing from H zero if you like and Because the framing is a such that I'm gonna put the H's and these the D is the green curve Here's the not the projection of the knot the framing is always in such a way That the H curve you can always see it. It doesn't twist around the green curve Right. It's because it's always just it has to do with the gradient of the phase field And so now I can talk about this and we know from my belt and from Mark Dennis that The red and the green curve it's pure Rive No link no twist all Rive So Here we go. I'm gonna take Link equals just plus Rive for two different curves. Here is D and H That was H is the induced one from the symmetric. Here is D and D zero. I guess I called it eight zero That's the one that I get from the ciphered surface This one has no link the Rives cancel. I can subtract And I discovered that the winding Which is my topological invariant that's telling you how the color goes around as you go around this location There's nothing more than the linking number of D with H So the smectic gives you a natural framing of the curve the defect curve Until something terrible happens Davide do you know You're taking up my time You know yeah Do you know the answer if I take my knot after all the knot is a three-dimensional object Why couldn't I rotate it? Why is this projection have to be in the XY plane? Why can't I rotate it so that the defect starts pointing straight up? When the defect points straight up I lose H Remember I told you that if you I have a helicoid the helicoid doesn't have a natural hyperbolic point And that's a problem because as I do it now the defect tilts up and it's gone I don't have my framing anymore. What do I do and I say to you here's the math question This is about the boundary conditions So I can't take my sample and Compactify it. I have to take a finite sample. So I take a cylinder All right cylinder fine. It might be an infinite cylinder this way and it doesn't have to be infinite It can be you never have to be infinite. It's just a cylinder And I have boundary conditions on the top and bottom and on the sides I have these graduations right the phase going through phi equals zero phi equals one and so forth As I take my edge and try to tilt it up That other curve. This is supposed to be green I picked red white and green. I was trying to be patriotic. So here are the two things this guy moves further and further away as I take this guy This line and tilt it out of the XY plane and try to point it up the z direction This guy moves further and further away Until it goes off to infinity That's why it's gone It doesn't disappear because it merges with this point. It disappears because it gets pushed away Now I can't there is no infinity in my system. I have a cylinder It means that at some point this passes through the cylinder and violates whatever boundary conditions I had on the cylinder So if I do the topology right and keep everything inside I Can never let this guy pass through the wall of the cylinder Another way of saying is I can't take my knot if I try to take my knot and rotate it arbitrarily Arbitrary rotations will violate the boundary condition on this cylinder Which has graduations right these are these are phi equals? one zero minus one two Three and so forth. So as I move up I have my finite cylinder if I try to rotate the knot arbitrarily. It'll mess up these boundary conditions So this doesn't solve the problem It just points out the same problem because I have this measured foliation Infinity is problematic their boundary conditions at infinity or on the cylinder actually come back in and create Linking on the inside that I can't get rid of and because I can't get rid of it. I can quantify it It's two pi times the linking number That's the end. Oh, I don't need that. Okay. Thank you