 Thank you very much for inviting me to the stimulating environment where we have a lot of different fields which are coming together and on a kind of similar topic perhaps from the different aspects. And today I would like to talk about the knots in the more pneumatic system. So it's a simpler system compared to the semantics what we heard just recently. But there are also possible complex structures which I would like to show some of them. And it's a kind of overview of the work which was going on now for like five, six years in this field. And now most of the work on this simulation site was done from my younger colleagues like Simeon Chopper, Tina Porenta, Michal Raunik and David Setch. They all have been my former students. And now it's very important that a lot of this work is tightly connected with the experimental research efforts, particularly from the group of Mushevich in Ljubljana and partially also by a group of Ivan Smalyuk in Boulder. So without experimental support from one site, I think this modeling which I will present would stay in some kind of in the air. But now with experiments which show that some of these things really work, some of these things have been supported in some way. Now to say a little about the broadness of this complexity of structures in even pneumatic or cholesteric systems which are here just as a few examples. So these are kind of knots in the planar system of colloidal particles in the pneumatic. And that's a kind of Hopfian structure which was also a Garrett who is here strongly involved in this. And then there is also a knotted colloidal particle, then magnetic colloidal particles, then cholesteric shells. These are thin layers of cholesteric in a spherical shape. Then blue phases confined by the presence of polymeric network. And then topologically interesting structures on surfaces which induce defects and then here defects on the simplest spherical drops but penetrated by a fiber. So these are just examples. But in all together all of this area is a kind of topological soft matter which we try to use for the characterizing the systems where we have defects in the orientational order which are stable. Stabilized by external effects like external fields or confinements and on the other side by chirality of the system if it's present. And just pneumatics in these conditions or chiral pneumatics are very nice examples of this topological area in soft matter. So here are just some examples like liquid crystal in blue phases where you know a case and then confined liquid crystals in tiny drops and then colloidal liquid crystals and this the last one will be mostly the topic of my talk. So first I will go through a little of the modeling also we heard a little already from Randy so I can do some of this even faster. Then I will shift to this colloidal systems in particular to the discrimination entanglement there which is then leading towards systems where knots and links in these pneumatic braids are possible. And then also more complex colloidal particles which are itself knots will be interested while the last topic which is a kind of inverse geometry of cholesterol drops will be presented on Friday by Simon Chopper. So if I go to this our basic pneumatic system which is probably known to everybody here is a system of fluids where you have molecules of elongated shape that spontaneously in certain temperature range exhibit the pneumatic phase or if they are chiral exhibit the cholesteric phase. And so this is a kind of schematic presentation of such a molecule and also then schematic presentation of our pneumatic and of cholesteric. And for this pneumatic it was very important this orientation order is characterized by a director which is not a real vector because we have both there is no polarity present so that means that up-down is the same. So in some way beside the usual description by using just a vector and then the degree of order by the scale up order parameter is better complemented by a kind of tensorial approach where we have the tensor where this direction is no problem because we are talking only about the axis of this frame of the tensor. And then we have the degree of order and then also by axiality the order local is not symmetric around the principal director that's important probably in the defects or close to surfaces. So these things are a kind of base for the simple description which we will use later. And perhaps it's important to stress that confinement in the system is bringing a coupling with this interface which is usually an energetic question and also a question of direction of what the molecules do on the surfaces and this is important for the further. And then in the enigmatic what one observes also we have seen before some pictures that these deformations are something what occur very easy particularly long scale deformations where on the several microns scale or even larger scale they are very cheap and they are terminally excited so visible a lot while deformations where you have really variation more molecular scale where order parameter and axiality changes are more localized and these are in principle in the areas of defects. Now we have seen also before these descriptions of elasticity of the smectics here it's very important that we have deformations of the director field which are like splay twist and bend which are the basic type of deformations and then each enters in the energetic considerations and on the other side is a coupling with surface where it's important at what angle director is coming toward the surface and usually it's taken as a kind of contact interaction then one should add also external features and so on. But as we heard before there are defects in the systems and we call these topological defects and if there are point defects which for instance like this hedgehog here or hyperbolic hedgehog one can define topological charge and so on. But for us will be perhaps more important defects which are lines so line defects which can then form these more complex things as we are our aim are links or not and line defect here perhaps should be stressed is characterized by the winding number before in the cross-section it's like this 2D defect which Randy was introducing and this line for instance the basic one have this winding number one-half means that director turns 480 degrees going around and what's typical this is typical for the pneumatic order which as I said is not vectorial and it's different from what we are used from electrical systems or magnetic systems and so this one-half lines and also energetically the lowest so they appear the most but then the other side for instance the higher lines where the director turns more like for 360 degrees are observed that they are not stable in fact they are too costly energetically too costly so it happens that in the central part the director turns out in the third dimension like here or here and then it becomes non-singular. For us it's important that when we will have some results to treat or to demonstrate then what will be how to present the situations and therefore we cannot use always director field like here the cross-section particularly if you want to show something in three dimensions it's not very easy to use just director fields so therefore for instance if we have a defect line we use many times simply a kind of tube which is characterized by an isosurface where other parameters have some prescribed value usually this thing is going around the core of our defect so that's for instance one of the presentations but the other thing is that one should be considering here is this inclination line here is an example of this minus one half line does have some detailed structure in the three fold symmetry so we have here this bend deformation and display deformation in different areas and for instance using colors one can show display and bend deformations which exhibit a three fold symmetry so the disclinational line is like a general ribbon which is important later on when we want to make links or knots out of this and the measure for this is for instance some derivative here second derivatives used to measure this so there are also ways how people observe them and the simplest are this which we heard before polarization optical microscopy and we have here these collections which Randy have shown as well but now there are many more other more fancy techniques which can reconstruct the director field in a more in the space in three dimensions so one thing which perhaps should be mentioned are these bontriagin term surfaces which are simply a kind of extension of these brushes here to the systems where for instance director has more complex structure and this surface is usually constructed in this way that you choose a certain direction usually symmetry direction of your structure and then you go perpendicular to this axis and observe where director is laying in this kind of surface and by colors for instance here you then tell in which direction in the plane is the director for a simple droplet for instance which have this so called this kind of structure that director is here perpendicular to surface but here this clinical ring runs around so from the top looks like here it's here mostly up and here is planar and then if you construct as a surface then you get something like that and colors are simply illustrated where director is and this illustrations particularly from Randy's group have been introduced more in the field and it's really coupled very well with new techniques and can be of use often so perhaps the base of our simulations which I will show later are in fact that we use phenomenological approach where we for the base have a Landau type of expansion of the free energy in order parameter so in such way that you construct the lowest invariance out of the product of order parameter and then also gradients so gradients again are put in such a way that you have invariance and then in the simplest description one have just one constant here we say one elastic constant and then this term here but if system is chiral you need to have this breaking symmetry term for instance we were in both pitch of our system is hidden here so that's the usual thing also as a kind of lift sheet stern norm from other fields and then coupling on surface is a quadratic type so just coupling the order parameter with some preferred value by the surface and then one can introduce fields we usually it's a dielectric coupling what's relevant and this one that's the whole system which is probably enough for most of the simple descriptions and one looks for the minimum of this free energy of such a system by a numerical procedure which here I will just skip these details and then we go to examples so the colloidal system where we will go and look for our knotted braids, pneumatic braids first one should understand when that one simple colloidal particle can do in a pneumatic environment so a particle which have a homotropic boundary conditions so somehow preferring perpendicular orientation on the surface produces for instance in a bulk something what's a kind of radial director filled around but when it's confined to a thinner layer which is usually in experiments this thinner layer means in general homogeneous far field of the director then you get two known structures like the Saturn ring defector here or point defect here accompanying and all this is satisfying the topological condition that in general environment charge topological charge should be zero so that means particle together with the defects either this or this are really yielding topological charge zero perhaps it should be mentioned that there are also planar anchoring where defects also appear like here you have two surface defects but this one is coming as a more less interesting case because defects this doesn't stimulate creation of this inclination lines they are mostly point defects accompanying this and then the variety of structures is weaker smaller so for instance here are this now with the simulation showing these two structures which I mentioned and in the micron size for typical thermotropic liquid crystals both have very similar energy so they appear both in experimental situations now depending to some extent also how thin is the layer of an emetic where they can find this but this structures or colloidal structures of the particular particles interact together effectively by being attractive or repulsive depending how they come together and the source of this attraction is the minimization of the free energy of the distortion so if you put this so-called dipoles looks far-field looks similar to somehow dipole they come together in this way and at certain distance they have optimum energy and the similar thing is here where by symmetry of the field is globally more quadopolar type so often this is called quadopolar defect dipolar defect and interesting in that you can form chains out of this in a 2D so and lattices for instance a few years like now 10 years ago there was a first experimental proof how one can construct these lattices of such or such particles so both can make this and what's more interesting that this can be done also in 3D also it's much more demanding because this interaction in 3D is not helping too much to self-assemble while this one self-assemble to large extent so that means that this is the situation with both kinds of defects you get this thing but what is then further more interesting to how one can create more complex break that's again a story which is known for quite some years so if you bring two such quadopolar particles together then the disclinations are a little deformed and nothing happens but if you put if you change the temperature either in simulation what was first done to go to isotropic phase and then you wait what will happen with your solution you will see that beside this solution appear other which are entangling two particles together by a single line more symmetric figure of 8 or less symmetric or by two lines where one they are orthogonal and the experiment are really directly showing that this really happens and it happens also with certain probability when they do experiment because the energies of this system are different the idea is now how this discrimination line when you go together with the two will somehow couple and that's something what this question was also addressed on the DNA case yesterday and here I would like to show what is here relevant how these things couple together so if you look at the loop so you have a minus one half discrimination line making simply a loop and if you connect it back it should fit together so because it has a cross section of this refold ribbon and so that means the rib this ribbon can be twisted as we heard here before the previous talk so but twist is not arbitrary this twist if you come around could be 120 or more depend left or right and this thing can fit together that you can make a loop and that's a question is energy there how much you must have but because we have this complex situation it's not only the twist present things are more than twist because we have a 3d mean mention a situation so this situation is like that that there are areas where two discrimination lines are close here here and this here here here these are all these examples which I showed before but more schematically and here is this kind of the 3d space where the discrimination lines comes with certain angle together and if you look in details here when two lines goes close also direct or filled in between is such that it really fits well so it's directly orthogonal to this direction to the both lines in this and what can one see that reconnecting lines just in these small points you can reach all these different different structures by simply visualizing this twisting of this either in different way what they do in experiment or could do in experiment is simply use a laser light which will manipulate these points but in principle this was not done but one can for instance see if shining light with the right polarization direction because this enforces the director can switch here such a reconnection what happens in fact you must first go to isotropic phase or nearly isotropic because otherwise it's an energy barrier in between and then you have a present field in the right direction in the right connection will happen but if you do just by heating and waiting you will get different reconnections depending on the energetic difference now topological question of this linking together or different discrimination lines is important that one as we heard before I saw some of this is that something should be conserved as I said before we need a twist which would be right that you can make a loop but beside the twist we have some kind of three-dimensional right which happen which is present in the systems because it's not in plane it's everything is in 3D so both things are changing so and what is important here that we have this we stress here self-linking number which is simply some of our right and twist it's a well-known thing but now if one could connect these things more in a general form for instance for minus one-half network why just network of minus one because there we know how much you should twist these twists are 120 degrees which are needed at least to reconnect or 0 or 240 and one can put together that if you have more loops which could perhaps link together or be not it and if we have a total topological charge of this system and we have then also besides this self-linking number also the linking because two loops can link because you can have such an object for instance and then you put everything together and what happens that the sum of this self-linking linking numbers plus number of loops should be equal to the topological charge but unfortunately model is too so it's no precise difference so it could be 2 0 2 minus 2 but that's the limitation which is then here possible form topological consideration so now if we go and see this case which I showed before so in this case one first observe that in most cases when you even numerically analyze twist is practically 0 but what is relevant is right so if you have in this case here and in this case right is also 0 because they are mostly planar while if you go to this case where you have a figure of omega right is non-zero so it's plus minus 2 3 depending on which direction it goes and if you put these things together in our formula then it works now within this that it could be you get minus 2 and it fits in this so that's topological simple analysis of what is going on on the other side is also energy if you looked both of together what is the energy of this thing so this energy is here in my coincide objects are high so several thousands KT so termally will not happen too much so but helping with the laser tweezers so if we go here further yeah so if we just simply extend this and you can extend and at some moment this breaks or this one the same so it's a miracle explain but it's also experimental evidence so by laser tweezers they do the same thing so they extend and then release and measure the force effective force and also energy in this way and you see that this energy is here several thousand KT but it's what's interesting here energies are practically linearly depending on distance so it's a kind of force which is string like so it's not the usual force if you would have these two objects separated then the force very much depends on distance while if they are connected entangled forces as long as they are connecting practically constant not within and working well in the modeling and also in the experiment now let's go further to this systems which can really link together into a linked system or not a system here if we started with two particles you need more of them to somehow support a kind of nothing so this way perhaps a simple example is if you take four particles and then if you bring them together for instance this is experimental picture when you analyze what this inclination line does you observe that there is a link behind so for instance another case here is this experimental evidence of three objects three already interlinked colloidal particles here and here and then there is one free and then this one has a set on the ring but this is less clear what it is but it's a single line connecting all of them and now if we do by laser tweezers a little helping so this was done in Mushovich lab few years ago and if you do this with some patience and perhaps some feeling for the system you create something what's more complex and now this object really re-analysing then mathematically what does the single declination line you observe that out of this you get a three-fold knot so it's a complex structure supported by the particles it's in fact the knot but this knot would disappear if you would remove particles particles are here stabilizing we heard before for some yesterday about metallic particles metallic ions which were helping to stabilize some of the knotting systems so in some way that's here this into similar okay so that's in but the question is where to realize think in a larger scale so one can realize this in a simple structure like a twisted cell why twisted cell because in a twisted cell declination lines are running in appropriate way that you easy do reconnections on a broader system so here is a 2D arrangement of particles interconnected by single lines running like that here or in this direction or in this direction but I here mark these areas where reconnections are possible because here these two are close or here this is close to the one which runs down or this one is close to this and if you reconnect these areas you can and the question is how usually what they do they just use laser tweezers and here are examples for instance of this a little bit bigger system mark here and here you see different local connections and if you analyze the lines here is a more complex this composite knot or 3-4 knot or prime link all realized in the same 4 by 3 particles but here just simply doing reconnection in these points or schematically here down is this shown how it looks so these areas are reconnected so this goes to this so this and that works also the numerical example or obviously experimental one so now if you play more with this more things could be done so in this way it was also connected this description of this interconnected interlinked or knotted system by description of graphs but in all cases you go from this which looks complex to this mathematical description of our simple knots but as I said without colloidal particles knots are not stable in this system and now if we go even more further introducing our formalism of linking and self-linking which I was showing before so one can make a kind of graph which showing if system is 2 by 4 and then adding something or how you go to this 4 by 3 system and here you have if you look for solutions you see that this small system 4 by 2 can have simple loops and have a trip or not and then we have here a link and here two other links and here another link and that's all what could be realized on such a small system and then one can also calculate this self-linking number and then the number of loops and so on and looking all satisfies our rules which I showed before so if system is bigger here we see much more possibilities so for instance we have here all kind of different knots from 3 4 to higher some also higher and a lot of different links which all should be possible on this 4 by 3 system which I showed before and then how you go from one to another behind always this reconnection on some local position in our array and here is a 3 by 3 system which is smaller and have left only one 3 4 knot possible and knot more so if we go and we see here that here then this links also possible so this system is somehow a 2D system where this interconnect, reconnecting of the inclination lines induced by laser tweezers in experiment or numerically by certain local simulation is fitting everything together quite well. The question is what to do if this can go into 3 dimensions so 3 dimensional system so that you have an array of particles which are 3 dimensional so there was not very few experiments in this direction and perhaps one which is already 15 years back was a form-clugged group where they had opal structure of this face center the distribution of particles which have chromatropic boundary collision on surface and then infiltrated by pneumatic and here we tried to make out of this also kind of understanding how one can describe the networks in such a system and again we were looking for minus one half this clinician lines and here first one can observe that there are areas in between this spherical particles which are C-trader and some which are cubic like and now the whole problem of finding the possible structures here is simply analyze what can happen in this cubic object or in this C-trader we know from before so if you put this together and then analyze what kind of lines of minus one half type can go and you observe that there are seven different states possible here with this kind of this clinician lines and in one case also point defect is together and that is then that together with this you can have more possibility because you can take rotations out of this but it's in principle the same object while this our C-trader system from before we know it has three different three different possibilities so that means that here in this case putting everything together you can have an answer like for numerical solving I think that you start with this kind of distribution in this white and then you calculate then further on by numerical relaxation something what is then given by numerical procedure so that was done for instance in some cases and it is immediately seen that even if you have very few whites you have a lot of possible states these calculators for some case of 4 cubic whites and 80 trade you have 10 to 14 different realizations so all different in energy some are higher some are lower but definitely there are many states the question is if one could use this for any kind of memory the question is addressing I don't know so that's now I am still okay so if we go further to the situation now which we had simple spherical colloidal particles and then they were in all cases practically chromatropic on the surface as I said that's the clue for going toward more complex networks and then the question is if one goes to more complex particles so there opens a lot of possible realizations of structures so there were many trials like elongated particles or just fiber-like particles and then faceted particles and again depending you do in 2D one can get some kind of quasi-crystalline here structures and well known I think genus particles where genus means that you have at least 2 different areas with different boundary conditions like chromatropic or planar and then again such a particle would create this particular defect forms and also then the interaction between them very particular but here I just put as an example something what was few years ago starting on some our meeting or in fact workshop on knots it was this meeting in Santa Barbara like few years ago and there was the idea that perhaps why not to have a knotted type of particle so particle in a form of one knot here and what would happen in a pneumatic if you have perpendicular boundary conditions on such a particle that you would get this inclination lines which would simply run around all the particle so in some way both are knotted so knot is physical and knot is in the inclination line in this case so this is the kind of suggestion and it's true that soon after that it was also realized in the lab of Ivan Smalyuk in Boulder they created such knots of different kinds simple more complex and simply by 3D printing and you see the objects were on the scale of 10 microns so small and uniform so that means now technology can do anything but now then you cover this with certain agent which promotes a certain anchoring type and then out of this if you put this into an pneumatic phase you will be getting structures so in the case of perpendicular anchoring so homotropic as I said inclination lines running around this loop and it's everything in some homogeneous field and then how this was tested then this complex fluorescence microscopes were used to really detect the director field and certainly showing that it really works in the way what was known less interesting is perhaps the situation when planar anchoring is because then only point defects appear on the surfaces on different spots and but everything is according to what happens an interesting question is here now if the knot is thin so that means not knot but our this cylindrical object which is formed in the knot is thin compared to the scale then this are very well separated these declination lines but if they come closer together then there are such crossings like mark here and this really restructure and that's something what is expected that should happen now here is an example of such numerical what happens here that simply if you fix the volume of this object but simply changing the aspect ratio between diameter and land and then at some moment you see that there was a transition in the structure and then it was somehow cut and then if you go further and knot it completely tight then the final object for instance here is more a sphere like topologically with features on the surface but it's no more has no more topology of this knot and then these areas which are marked here there were these restructuring perhaps in another presentation here where also it was used this reagentome surfaces to show the area where things are things still okay yeah so here I would like to mention that beside these colloidal systems which I showed two examples so one based more in details on spherical particles and the other on complex particles themselves already induced the knots knots but then they are coupled with these knots of the inclination lines one can ask if there is a possibility that the knots will stay without support of colloidal systems and here idea was that probably that should happen perhaps in a cholesterol system and some time ago we analyzed the cholesteric in a drop with planar boundary conditions and then some nice features appear well known so this is so-called radial defect of power 2 and concentric cholesteric layers so nothing very particular and the reason why it's nothing particular what can be catched in such a sphere spherical confinement is that when this clinician line comes toward the surface then if structure is complex can relax easy in some way transform to a simpler less energetic structure because of a boundary condition so this clinician line can touch the surface if you go to homotropic boundary conditions on a spherical again for simplicity this clinician line can come close to surface but doesn't touch so that means it cannot relax if you have a complex structure in a simple way except going over high energetic barrier so this was idea that perhaps here one can not in a cluster and on this topic there will be much more in the Simon Chopper stock on Friday which was there was a change in the order because he's coming from Friday but that's already in there so if I say what was done here so we tried to use phenological restriction for analyzing and presenting and understanding of complex pneumatic braids mostly in colloidal systems and starting with two particles going to more particles seeing that knots and links can be supported and that a role of particle is crucial here and particularly boundary conditions on them and then we have seen that with more complex particles certainly you can create even more complex structures so then the story of knots should go further and see if one can see a support of knots not only by particle but for instance colloidal cholesteric structure itself so here I would like to stop