 Okay, thank you very much. Well, I'd like to start by thanking the organizers for inviting me. It's my second time not at the ICTP and It's been really wonderful. It's everybody's good if not better than they the first time I was here Especially because this time I have a sea view last time. I was looking at the car park. So it's a big improvement So today I'd like to tell you about knots and liqueur crystals. This is all joint work with Tom Machin. He's up at the back there and it's Motivated in large part by Sobaland zoomers experiments at Ljubljana the stuff that he's been doing with his group and Igor Musovic's group that Created knots in liquid crystals for the first time about five years ago now. Maybe six for actually doing it and the question that I want to talk about is What on earth does it mean to have a knot in a pneumatic liqueur crystal? What is it like? What are its properties? so I'll start by saying a few words about liquid crystals and I'll repeat something that Randy said yesterday one of the ways of explaining to you all what liqueur crystals are Is to show something like this and say if you're old enough to remember this Transition then you have no excuse for not knowing what a liquid crystal is But maybe I'll say some things that are a little more scientific also So if you look at them under a microscope, this is one of my favorite ways of explaining to people what liquid crystals are Apparently this is what liquid crystals are And it's remarkable Randy talked about how you could extract Molecular size information from this picture. It's also the case that Back in 1910 George Friedel was able to understand with molecular resolution What's going on in this material just by looking at this picture? So I invite you all to think about how he might have done that apparently it was something that he learned in high school So I think that's another excellent way To convey what liquid crystals are but if you look in a textbook you see something different So here's a sort of standard textbook in the field And it will tell you that liquid crystals are composed of long rod-like molecules And then those molecules can do a bunch of things either they can be disordered That's an isotropic phase or they can all point in the same direction And that's the stuff that's in your displays or they can have these States with higher degrees of order in it like these smectic states that Randy talked about There's one other aspect that I like to emphasize Instead of talking about the way in which they're ordered. Maybe I'll go back a slide You can compare also this picture with the previous one and ask yourself How closely do the textbooks reproduce what you actually see in the real world? And one of the features of the sort of simple textbook descriptions is that everything is really perfect The layers are flat the molecules all point in exactly the same direction And so another characteristic feature of these materials and many others is to try to capture the essence of the defects in them Where the order breaks down in some way so pictures like this are extremely instructive for conveying what liquid crystals are You see locally you have an alignment of filaments and then there are these points There's a couple of them. I've highlighted there are a couple more that I haven't Where you can't assign any actual orientation to the molecules at those points and more than that You can characterize their character by thinking about how the orientation winds around them So these are called Disclanations and the main feature that this shows is that the rotation of the molecules around one of these points is not an Integer multiple of 2 pi but a half integer multiple of 2 pi and so it's telling you visually Directly without any other input that this is not a vector ordered material. This is a line field that you're looking at Okay, well, I'll say a few more two more slides with sort of general introduction to the theory of liquid crystals how to think about them and I wanted to say this because This way of presenting things has increasingly had a more and more profound influence on my own work It's remarkable. This is a paper of Charles Franks from 1958 Where he really set in stone the elasticity theory for liquid crystals It's one of the most famous contributions in the entire subject And it also happens to be the only paper that he published on liquid crystals So he did two completely seminal things in just one paper One of them was that he described the elasticity and the feature here the thing a contribution that he brought to it Was not to do it in some kind of algebraic fashion, but to think about it from geometric purposes So I want to describe the distortions of a perfectly aligned state in some geometric way I'll give you a little flavor of just one of these if I look at this bend distortion I can describe this as a curvature by saying well if I imagine the integral curves of The line field so that's some curve in space It will have some curvature and that curvature is exactly the bend distortion of the line field That's one way of thinking about the fundamental distortions that go on the other ones correspond to a mean curvature of layers that the line field is the normal to or a mean torsion the way in which things are twisting and the insight that he brought to this was that these ones were Somehow already known about from Osane, but by thinking about it in terms of curvatures He was able to realize that there's another curvature that he knows about which isn't on the list and which you can add to the elasticity theory, so of course there's also some analog of a Gaussian curvature for the line field and By thinking about it in the right way He could really get the right elasticity theory for liquid crystals So another feature of liquid crystals that I like a lot is that they're really very highly Geometrical and thinking about them in geometrical terms is really useful Yes, so if you if you take a curve whose tangent is everywhere the line field Then it will have some curvature and it's exactly this quantity actually as an aside Random aside just about vector calculus everybody in the field writes this band distortion as n cross curl n and I find that Obscures this geometrical interpretation whereas if I say that it's the gradient of the line field as I move in the direction Of the line field. It's much more obvious The other contribution that he made was a classification of the defects in the material He was thinking about a sort of simple situation where I have a two-dimensional material and there are isolated singular points Where the orientation is not defined and this is figure 2 from his paper where he drew all these examples And illustrated all the different things that can happen It's a line field So it comes back to an equivalent state in rotations that are half integer multiples of 2 pi rather than integer multiples of 2 pi But other than that it can rotate by any of those half integers whatsoever And those all correspond to different types of point defects in a two-dimensional or quasi two-dimensional pneumatic material And there's a slightly sort of fancier way of wrapping this up in terms of a lots of the groups that Randy talked a little bit about yesterday and one of the things that they're good for is That they tell you what the analogous classification the analogous to what Frank did is in three dimensions If I take line defects, and I try to classify them with the same homo to be 2 then The homo to be classification is just Z mod 2. There are exactly two of them There's a line that is no defect at all and there's a line that is a defect and Then it's interesting to look at this picture of the cross-sectional profiles And for instance convince yourself that there is a continuous deformation of something that looks locally like this picture Into one that looks locally like the minus one-half winding And if you haven't done that for yourselves, then that's another nice geometrical exercise that I encourage So this is a still from a video that I got from Miha Raunig at Ljubljana Of some of the experiments that were done there associated with work that he did This is a quench experiment. You take two colloidal particles. You put them in a liquid crystal You locally melt the material around it with a laser beam Then you turn the laser beam off it forms a collection of defects and they settled on into some nice Configuration around the defect and I pulled out a still before it settles to the final state to emphasize That these really are nice elongated lines and they have interesting shapes And what I'll try to do today is to give some kind of generic classification of the properties of such lines in pneumatic materials from a sort of a global Perspective of trying to characterize what they're like and especially what they're like when they're not simple loops But have knots in them this is just a sort of Summary slide to try to convey to you very briefly that Since the experiment came out of the Ljubljana group about five years ago There's been a number of people around the world trying to make similar sorts of things in liquid crystals So there's quite some variety of sort of knotted structures that you can study these days and there's quite a significant amount of Experimental work that is available where you can really genuinely make these things In the lab in some kind of controllable or reliable way by a number of different techniques So we want to contribute some kind of theoretical Understanding to the sorts of things that they're doing in these experiments and that's going to be one of the goals for this talk Now I want to give some kind of contrast one of the things It's wonderful about coming to seminars or workshops like this is the great diversity that there is in the talks You get to hear all sorts of things that are not in your own subject area and that's something that I like a lot So everything of course is about knots But there are many different types of knots. So here just a small selection of things. I thought It hasn't really come out. I made it too Faded in the background, but in principle the background is this knot table from Tate's paper That a couple of other people have shown and I thought I would supplement that with another sort of artistic rendition of knots So here are some beautiful ones from the book of Kells, which is one of the ancient books in Ireland Trinity College Dublin And then there are a couple of flavors so that we've heard about knots in the first day about knots in molecules in molecular structures molecular ordering there are knots in DNA and then there are also these knots in Continuous fields like these vortex knots that Renzo was talking about and knots in optical systems That you might think of as being slightly different because they're not just knots in a little strand They're not in a continuous material and you have to worry about the entire field It's a random Specifying some quantum wave function or something like this. How are you supposed to do that? What does that mean? And so it's interesting even though you might always be producing the trefoil knot for instance Every time you see it it has a slightly different character and a slightly different flavor to it and you learn something new about it I thought I would show a little microcosm of this Again is a sort of general motivation for trying to understand what knots are like in different physical systems This is a recent one that I heard about when I visited Durham at the start of the year and talked with Paul Sutcliffe So these are two systems that loosely speaking are vortex knots They're slightly different one of them is the ghost pitaevsky equation that Renzo was talking about so that's like a Bose condensate or a superfluid vortex knot. The other one is a vortex knot in a reaction diffusion system It's the Fitzhugh-Nagumo equation So it's slightly different systems, but as a loose sense They're both vortex knots and the question is do they behave to see him or do they behave differently? So William Irvine's paper was exactly about The sorts of things that Renzo was talking about that under evolution you have these reconnection events There are pathways the knot type changes. He starts with a complicated 917 knot it goes through some sequence and at the end of the day you end up with an unknot or some collection of unknot So that's one way in which knots can behave in continuous fields And I think Paul Sutcliffe finds something very interesting. Here's his simulation with the Fitzhugh-Nagumo equation So it simplifies and simplifies and simplifies and you end up with the unknot So you start with something that looks complicated in both cases and in both cases It simplifies and you end up with the unknot. What's different about it is that this started out as the unknot There were no reconnections. There were no crossing events. It was nothing like this and I showed a movie for the unknot But this was true for actual knots for trefoil knots and higher type knots that they studied in this system So I think it's very interesting that although in principle they're both types of vortex knots they behave very differently Because of the physical system and I think it's a problem Well, I asked him if he had any understanding of this and at the time he didn't really and I think it's a wonderful question So I'm going to talk about knots in pneumatics with in mind a feeling for what does the pneumatic tell me about the knot Why is it what is special about it being in a liquid crystal as opposed to being a superfluid knot and The two questions that I'm going to focus on are a classification question I'd like to know what are all of the knots in liquid crystals I would like to come up with an analog of Tate's table some tabulation of knots I want to know how many different knots there are how many different pneumatic textures there are for each given knot type So what is the homotopy classification and the second question? Which I'll start with is a simple one. I've shown you already that you can make these Experimentally experimentally you can make knots in pneumatic materials Well, I wonder can I do that as a theorist? I don't have a lab, but I can think very hard. Can I do it for myself? So it turns out that they're doing it for yourself is not so difficult There's a small trick, but it's only a small one What you need to do is get yourself a copy of this beautiful book by John Milner and Read the first three pages So it's not all that hard and he explains in those first three pages the following construction for torus knots Arbitrary torus knots you take a complex polynomial in two complex variables with integers p and q that are telling me the type of Torus not or link that I'm encoding and Then if you look at the following thing if you look at where this polynomial is zero so where it vanishes On the surface of a three sphere So I have two complex numbers. That's four real numbers. I take the sum of the squares to be one That's a three sphere so on a three sphere I ask where is this number identically zero and then it's a wonderful Exercise to think about that and figure out that it's exactly a pq torus knot Now we want more than this. We don't just want the curve I want to specify the orientation of a liquid crystal molecule at every single point in space not just on this curve But Milner also tells me how to do that. He tells me hey look away from this curve I have this complex number that isn't zero and if it's a complex number that isn't zero it has a fees and the fees is an angle and you can point your Liquor crystal molecule at whatever that angle happens to be so there's a prescription for filling up all of space except for this Torus not curve with some angle which is the direction that you orient your liquid crystal molecules at So that's nice. It's a fairly straightforward thing. We use this to good effect for some of our early work It's also a little restricted He's able to construct torus knots and they're useful for many things and they come up frequently in many applications But it there's certainly not all possible knots So you might like to do other things and you can extend the technique a little bit and do some other knots And it you can go quite some way in constructing things that are geometric breeds But there again, you're a little constrained that the shape of the thing is a geometric breed and not an arbitrary curve And so I'll remind you of these images from the simulations that William Irvine and Paul Sutcliffe were doing These are numerical simulations, but they still have to start them in the right state Otherwise, you have no hope of creating this spontaneously So they managed to solve this problem of constructing completely arbitrary knots You just take a random curve and generate some field from it And I'd like to tell you about that because I think it's nice story And also it's one that at least in principle. We should all be familiar with So the idea is to use the Biosavar law Basically turn it into either a magnetostatics problem or I'll talk about it in the language of fluid dynamics and vortex lines Because well, I find that slightly more convenient for my purposes So what you do is you pick an arbitrary curve You imagine it to be a vortex line and exactly the way that Renzo described in the previous Talk and then you can construct from that vortex line what the fluid flow field is Absolutely everywhere in the material by some kind of Biosavar law So if it was a current loop, this really would be exactly the magnetic field that you would measure So that tells you about some vector field that's circulating around this line And then you ask yourself if you can turn that vector field into an angle That is winding around the line and that angle of course is the velocity potential or the magnetostatic potential associated with those fields So there's a fairly generic prescription that says start with any curve that you care to dream of And end up with some angle that tells you how to orient your field, your liquid crystal Or your super fluid or whatever it is that you happen to be working with And I'll show this just again to emphasize that in these continuous systems You really have to specify what's going on throughout all of space what the field is like absolutely everywhere And not just on some curve and of course, this is exactly what this construction does Okay, these things are kind of awkward to think about and visualize so sometimes it's nice to see that sort of hands-on description Okay, and we can of course you can Take this and just plug it in immediately to construct some director field that winds Around the curves and turns them into defect lines in the liquid crystal and you can construct a whole bunch of textures in this way There's only one little drawback from this construction in terms of its generality when it comes to applications in liquid crystals as opposed to in Superfluids for instance and that is that the director field that I write down The way that I write it down it has components in the x direction and in the z direction But not in the y direction So these are all completely planar textures the liquid crystal molecules live in this plane and they don't point towards you or away from you at all and Okay, you can do that But the molecules are allowed to do whatever they want so we should also be able to describe states where they can Rotate more freely if you like we need a second angle That will tell you the angle that they are towards you or away from you in addition to this angle in the xz plane so I'd like to think for a moment or two about how to do that and In doing it I find it convenient to translate the originals or BIOS of art construction into one in terms of Differential forms but the key part is that when you're trying to construct this angle that's winding any angle That's winding around some zero It winds by 2 pi and you have to decide Whether you want it to be between 0 and 2 pi and then there's a cut at 2 pi And you start again or whether you want it to be minus pi to plus pi and there's a cut between minus pi Plus pi or whatever it happens to be you have to decide to cut things on some cut surface In order to define a an angle in some kind of uniquely to single-valued fashion So the key trick in this is to introduce a surface whose boundary is the knot and Of course, there's tremendous freedom in the choice of exactly what that surface is and if you make different choices You might get different knots An example of that here's the simplest example of that. Here's the hop flink I show you the hop flink with two surfaces They look fairly similar But if you stare at them for long enough you can see that they're different and they really genuinely are different from each other One is linking number plus one the other one is linking number minus one and that's going to be a Topological invariant that will be preserved so long as you always have a hop flink So it really matters whether you pick this surface or that surface you get something different The other thing is that the surface helps us out in trying to choose or construct a second angle that will bring The orientation of the molecules out of this XZ plane to pointing towards you in some way And that's because it provides us with some additional structure to the space the space that is not this surface Is an interesting space it has all these holes in it that you can go through and come back up some other hole and so on So here It turns out there are there are two cycles the homology of the complement has dimension two so I show choice of bases for them here and What you can do is you can pick another curve So a second curve again completely arbitrary curve It could be a knot if you like and it should correspond whatever it is It will live in the complement of this surface and it will correspond to some homology class and again The homology class will be a topological property of the thing that you're constructing So it matters which class you choose other than that Doesn't matter so much Having done that you can then construct an angle that winds around it by exactly the same trick as I described previously So that now gives us two angles and if I have two angles that covers all directions on the sphere I ought to be able to get the molecule to point in any direction I want So that's the idea you use those two angles you write down a slightly more general Expression for what the direction of the liquid crystal is pointing in and then I Encode this second angle that I introduced by some kind of color winding Here's the schematic that tells you this color winding that's going from in this plane to pointing out towards you And you can construct a whole bunch of different things in this way But still there are questions about these things. There are at least two serious questions about this We can construct lots of things But you should still ask yourself. Can we construct everything is this really completely general and Are these all different from each other or? Some of these secretly equivalent under some appropriate Homotopy of the texture So it's nice to have a Robust I mean in principle this construction would give you a robust answer to that But in practice our knowledge of Cobordisms of surfaces was not quite strong enough to carry it out in this way So we had a slightly different approach They're coming up with a knot table if you like for textures in nematically crystals The approach is essentially Traditional calculation in algebraic topology using the methods of obstruction theory that I won't tell you about I'll just tell you what the answer is So the answer is that the number of different Neumatic textures that you construct with some particular fixed knot as the defect set is in one-to-one Correspondence with the elements of some group. This group is the first homology of the branch double cover of the link complement And then there's some equivalence relation that's associated with the fact that I have a line field rather than a vector field So up is the seam is done So that really the point about this is that having identified it this this thing This is a knot invariant and you can look that up in all sorts of tables and tabulations And it's not so difficult to calculate it for yourself by a variety of different little techniques And so when you've done this you can write down that if I have a 4-4 tourist link There are this infinite number of different things that I can do in a pneumatic liquid crystal There's not just one of them if I have this collection of three things a trefoil knot an unknot and a hop flink And they're all together in the same material Then there's this number of different pneumatic textures that I can make and if you make the baron man rings Then there's only 16 of them suddenly. There's a finite number And they come with some structure associated to that 16 of them To give you a flavor of the sort of variety that's out there and some of the things that we Find out about the properties of knots when they're in liquid crystals as opposed to other materials Here's a sample of the torus knots. So these are the PQ torus knots for P and Q up to 12 And I write the What this group what this magic number is that? Encodes the number of distinct pneumatic textures that there are And there are a few things that come out So one of the things that comes out is that some of them are completely unique There's just one single knot So all of this stuff that I told you of it whether it was planer or it had these funny windings And so on and so forth those were all equivalent to each other you could do that But as far as the topology is concerned those were all equivalent In other cases There is something funny that happens that I'll try to say a couple of words about in a few slides time Which is that the group that appears is just a bunch of different a bunch of copies of Z mod 2 And it turns out that the twos are important For some other property that pneumatics have so I'll come back to this in a moment But these these things where everything is just a copy of two So there are eight copies of Z mod 2 for a 99 torus link turns out to be an interesting observation from this table And the last thing is that some of times there are an infinite number of distinct textures that you can construct And I guess in a sense we were kind of surprised about I guess we would have been surprised either way if if You know, they had all been infinite and only a few finite we would have been surprised the other way around We would also have been surprised so we scratched our heads for a long time Wondering what was going on that give us these infinite numbers And then it was wonderful. I don't always think about liquid crystals. Sometimes I do experiments with soap films I persuade my students to take time off from serious stuff and do homemade experiments with soap films so I'm going to show you an experiment that Convays or explains why there are an infinite number of states in these systems so this is an experiment was done by Tom and Davade and Another one of my students. It's not here Dario Vasili couple of years ago with a rig that they 3d printed with the 3d printer that Matthew provided. Thank you, Matthew And set up this little experiment with a 4 4 torus link in soap and Then you try you create a surface on it So you create a spanning film then you have to pop some holes until it's a proper surface So that's what we do first Okay, so now we have a surface Whose boundary is the link and then you move the rings around and the surface changes and every now and then it's topology changes This is a real-time movie, so it's quite fast in real time and then suddenly this happens So what's happened here? It's maybe a little difficult to see it on this screen But if I tell you you might be able to pick it out What's happened here is that I know I have not one surface, but two There is a surface Here that connects these two components and that forms a hot flink With a nice little hot flink surface and then there's a completely separate surface here It connects these two components, but it's otherwise identical And so this is the magic fact that if you can construct a surface whose boundary is the link And it has more than one component Then it's Alexander polynomial is trivial and you can compute this group and the order is infinite So that's where the infinity comes from and it was really nice to see it in a home-made experiment So here's Some clearer pictures of the different surfaces the transitions that you saw in that video We started with this surface then it went to this one About halfway through and the end state was this one on the right This This one is with surface of over Okay, well, I gave a classification of what all the different types of knots are So it will be nice to apply it where possible to some experimental data The experimentalist can make these things it would be nice to say which one they made Of course identifying knots is difficult and identifying them in pneumatics is no less so But if it's simple enough then we know some things and we can Make some progress so one of the things it's simple enough this is a beautiful experiment of Liquor crystals that are embedded into toroidal droplets is done by even schmalyuk's group at Boulder And you have normal anchoring conditions on the surface of the droplets and they form defect lines in the interior because of those anchoring boundary conditions and they can make a variety of essentially tourist knots With this construction you have them in a tourist Geometry with the defect lines winding around and then it's a question of how many times they wrap over themselves as To which tourists not or link you end up with so they can construct trefoil knots and sometimes hot flinks And I'll tell you about the analysis of the hot flink So of course first of all I should tell you a little bit about how to think about hot flinks and Identify how many of them there are and how we'll try to identify them So it turns out that there are just two of them and there are two ways in which you can think about them and think About the differences one is in terms of these planar Configurations that I started telling you about things that just depend on one angle and have no Component that's pointing towards you And I told you there I showed you an example that there were two surfaces that you could choose one gives linking number plus one The other one gives linking number minus one and it turns out that that's a way of describing the two different states That exist in the system and usually I think about it in those terms so I'll map everything onto one of these planar Configurations and refer to the two different textures as being either linking number plus one or linking number minus one That's one way of assigning Another way of doing it would be to use this second angle that I introduced and have some winding around Zero line that links with this hot flink Okay, and I represent that angle winding that angle variation by some color That's going around a color wheel on the surface itself. Okay, so that's telling you what this angle is Just map it on to color on a color wheel And so you should ask yourself How can I relate one of these colored descriptions to one of the planar ones and there's a process? of Deformation that says well you can make the rotation very rapid you can collapse it Into a small region of space and have it more or less constant everywhere else. That's a continuous deformation And then I can think of this rapid little thing as just a little line a tether That connects one link component to another and then it's an exercise in visualization That I have never managed to do successfully in full three dimensions But you can do in a sort of a cheating fashion if you collapse things down to a 2d version of this 3d picture And see what's going on to convince yourself that you can move this tether around appropriately To change the linking number plus one surface into a linking number minus one surface By a process that only does singular things in the actual defect lines itself Which we allow for So something that has linking number plus one With on the surface, but the surface has color on it There is a deformation that will take it to one of these completely planar things with no color, but with linking number minus one That's the identification you should think of So if we go back to the experiment It's nice because the experimentalists used exactly this technique that I've been trying to describe to you of Measuring or plotting what this second angle that's rotating is and plotting it exactly as some kind of color function On their experimental data So you can read off fairly directly from the images that they provide you with what the surface is what the color winding on that surface is And then it's just a process of deforming it to see after the deformation when you make it a planar Configuration is it linking number plus one or linking number minus one and it turns out in this particular experiment They made the one with linking number plus one So that's at least Success in one case of course Applying it as a general thing to identify Completely arbitrary states for completely arbitrary knots is still something that's a long way off But in some cases we can really work these things out. So a second feature or property of these knotted continuous fields that I'll show for you is that They really depend on the entire knotted structure around them and not just on the Lines that you see not just on the link itself It really matters that it's in a field and this is an illustration of this So I show you something or if I only show you the lines themselves these look identical as identical hop flinks But if I let this system evolve Under relaxation of the energy for the liquid crystal This costs a lot of energy because it has high distortion and so it will try to reduce the amount of distortion in it and there are reconnection events that happen and You can watch what they look like for the liquid crystal and they're pure Relaxational dynamics. So here's a little movie that's showing you what it's like in these two cases And of course the point is that I'm not showing you the same movie twice Something different happens in the two cases. So the dynamics is different in the two cases And it's different for a reason Although they looked to see him at the outset when you just looked at the lines You have to remember this is in some continuous field and the structure of the field It's around these lines is important and it was different in a topological way in the two cases So you can go back and look at this and the movie on the left was a planar Configuration with this surface Associated to it and you can check that has linking number plus one and the one on the right was again a planar Configuration with this surface associated to it. So it was the other topological type with linking number minus one And their dynamics is sense. It really cares about this difference So I'll maybe try to say very quickly before I completely run out of time About one extra little thing that we find that's an interesting topological aspect of the behavior of of knots and links in liquid crystalline materials and that concerns when it's Possible to squeeze everything down so that the liquid crystal lies in this x y plane and Has no component that's pointing towards you For this hot flink example, I could always do this Any texture whatsoever it was possible to push it around Continuously to form it and turn it into something that was lying planar everywhere The question is whether or not you can do that in general and it's not such a Crazy thing to think about in a liquid crystal context This is something that at least in principle you could promote in an experiment by applying an Magnetic or an electric field that likes to align the molecules in particular if it likes to align them perpendicular to the field Direction and if I apply the field pointing straight towards you they will try as best as possible to lie everywhere in this plate with a few little Blips if you like so there are four different types of blips and now I have these little These little localized regions where it's pointing towards you and the question is can I move those around and Recombine them with each other and get rid of them. Yes or no And so there's an answer to this there's an answer to this again in terms of the topology in the topological Classification it turns out that you can do this if and only if That texture corresponds to an element in the order to subgroup of this homology group So there's some again obstruction theoretic calculation It tells you that this is true and what it says for instance is if I look back at these examples Then there are an infinite number of states on the associated with the 4 4 torus link But there are precisely two of them that you can realize in these planar configurations and only two and if I look at this Collection of a trefoil knot and on knot and on length and again They're in an infinite number of states, but there are only two of them that you can realize in this planar configuration And if I look at these Borromean rings There are 16 of them and there are four of them that you can realize in some kind of planar configuration So I think I'll stop here I think I'll go to my conclusion slide perhaps the last thing that I'll say is that this is still a mystery to me We have an understanding of what these planar configurations are for the hot flink That was a linking number story that I told you about it's fairly self-evident as Renzo said it was known even to Maxwell of course Maxwell was a genius of course it was known to Maxwell That that there are no linking numbers pairwise linking numbers for these Borromean rings So what's the analogous way of thinking about the four distinct? Planner configurations that are associated with pneumatics with Borromean ring defects, and I don't know that yet Okay, so with that I'll stop. I'll bring up my thanks slide. I was gonna tell you about cholesterics. Okay. Thank you very much