 OK. Če sem prišla za pomečnje, narednjamo vse, da sem prišla vse, da sem pomečnje, da bomo priječnje vsi, da sem priječnje vsi, da sem priječnje, in sem priječnje, da sem priječnje, da sem priječnje, so neko biologijne, je softej metar. I dovolj da bi se zazakvačil in nekaj neko tudi nekaj vzela. Zato bilo, kaj mi zelo, sve nekaj je tako zelo. in nekaj nekaj, ki smo so videli v izgledanju. Prvno, izgledanje izgledanje je izgledanje nematikovih, kar ostevih, v dropletih. Kaj je ostev? Ostev, ki so ostev, Vse različite, da so nekaj izložiti, da je vse izložiti. Vse vse zeložite, da je vse izložit. Zeložite, da je ta vse izložit. Hvala. Vse razložite, da je pa vstupnila, ki je na vse zašečnje. Sve vse začne, da je kot način, ki je način, ki je začn, ki je začn. struktur. So instead of just having a loop, you have a framed knot. So there is a framing that is supported by this director configuration described by the enomatic molecules. And this is why we have additional invariance there. OK. This is a general outline. Of course we have, in enematics we have deskination lines and deskination points. And the lines are those that can make knots. And the points are those that actually turn out to be quite as interesting as well. But are not knotted because they are points. But around points you can also talk about knotting if you observe the topological structure of the director around it. So if you trace the preimages of certain direction in space, you can also arrive at linked states. And in our systems mostly we have these minus one-half deskination lines. This just means that in a cross-section even line goes like this. And you have these molecules arranged in a certain direction. Molecules, of course, not polar. So this is why you can get this profile that actually turns by 180 degrees. And mostly we have this conformation. This conformation is important because it has a structure. It's got three-fold symmetry. So this is why you can actually define a third integer invariant when you come around. If you take this profile, this profile rotates when you come around the entire link. And this is what is conserved. So when you have transitions between different states, we have to observe that not only will a knot, if you revire it, transform another knot which is at most one reliant, one crossing away. So you have neighborhood in the space of all knots. But it also has to conserve these invariants. OK, so what's the self-linking number? I will just recap this. Every defect line has this cross-section. And when you close the loop, it can close at any multiple of 120 degrees. This is actually a topological invariant known as the self-linking number, which decomposes into ride and twist. And here it becomes interesting. Because ride is actually a shape invariant. It's just the returns on the shape of the discrimination line. And the twist is the internal structure. Twist is this twisting when you go along. This is not allowed by the free energy. Not much, right? So we can say that in pneumatically cut crystals, where this is discouraged by free energy, the shape of the knot is prescribed. The ride has to be a multiple of two thirds. And this is why when you do this on colloidal systems, which I will not talk about today, you can always see the same shapes. All the shapes are similar. And they have to go this three-dimensional shape. It has to follow a trajectory that conserves this ride into. It quantizes the ride. And it quantizes it in a similar way, which also the tight knots actually conserve this approximately, at least, right? And what's more is that the self-linking number satisfies this conservation law. So the self-linking, sum of self-linking numbers of all the knots involved, plus twice the linking numbers between the loops, of course, this normalizes to an integer, plus the number of components equals the topological charge, which is in this case just a number of these colloidal particles, but it's modular, too. So this is even not effect. This is related to the up and down symmetry of the pneumatic molecules. OK. Where do we see knots? In a uniform cell, you actually have loops like this and you have these rewiring sites, right? And these rewiring sites have this tetrahedral symmetry, which, of course, also geometrically restricts this rotation to two-thirds, or one-third, or something like that, because this component happens also when you go into different systems. So locally, all the rewiring will look the same, if you have a system which doesn't have these stabilizing spheres, you will also have, just before you get the rewiring, you will have this characteristic local shape that always has to be the same. And we did a lot of this in pneumatic cells, so we just put colloidal spheres between two plates and fill it in the crystalline and you observe what you get. And in this particular system, 90-degree twist, which basically has a wire mesh of these collidations on top and bottom in 90 degrees, and then you have rewiring from top to bottom, we can actually prove that you can get all links and knots provided the colloidal grid is large enough. We also played a little bit with three-dimensional systems, but today I will talk about kinematic droplets. So this is easy, right? You put a sphere in there, a sphere forces a loop to exist there, and then you rewire one loop with the other loop with laser tweezers, or you just cool it down from a high temperature. But here, in pneumatic droplets, you have a much stricter rule, right? Basically, we only care about droplets which have perpendicular anchoring, because this is restrictive enough to actually force a boundary condition, because if you make the boundary conditions free or just to vary in the plane, then it can just relax. So you have to force it, and then we use chiral pneumatic. It means that the pneumatic wants to introduce a helical structure in it. Why is this interesting? Because helical structure requires a non-vanishing helicity, which is completely incompatible with the surface condition, which of course surface connection requires there is no twist there. So this is a deep geometric restriction, right? You have a twisted thing, and you explicitly forbid all twists on the surface. So what does the liquid crystal do there? Then we just run a simulation, right? This is what you get if you have free boundary conditions. So here you can see that it's not complicated enough. You have a lot of optical interesting stuff, but not defects, because you have a free boundary condition. If you go for chromatropic, you get something like this. And every time we run a simulation from a random initial condition, you get something different, because of course this is a frost-stated state, and it just kinetically freezes the diskination lines, which cannot relax, because between the diskinations there are these helixes, and it's just like a spring. So what we get is something like this. OK. So we get a zoo of nuts, right? This is not so recent paper, this is two years old. This is from simulation, right? And then we have a lot of questions here, right? What are the transitions between these systems? Because of course you can always, now you have a field, it's not just a knot. It's a machine line, which has a field around it. So there are free energy transitions between those. So if you have two states, you can always ask yourself what's the easiest way to transform this into that. And this is actually still an open question, because we don't have enough students to do this work. But of course this can also lead us to another question, right? Are the knots that are similar in terms of closeness in general topological rules without the field, are they also close here? So you have something that can mathematically be on the paper transitions from here to there with one crossing, can you do it here? Or is it forbidden because of the field around it? So we have to do some complicated transition which involves like four crossings or something. So we want to do this in the future. What's also interesting here, we have a parameter. It's the cruelty parameter that tells us how much this wants to twist. And we have a known boring ground state for nomadic. There is just a single point effect in the middle which can be actually represented by a small loop, depends on the exact free energy and temperature and everything. And then how do you get from this to this, right? You can imagine increase in clarity, you can do this in experiment, you can put in there an azodi, it means that you have a molecule that changes conformation under the influence of UV light. And you can do this transformation from a carol to carol state continuously and the question is how do you get there? And mostly the answer is you don't. These are metastable states. You can always start here and make it less and less carol and you will always get there. But the energy barriers are such that when you start with this you always get one of those structures that just have a discrimination loop stuck to the surface. It just expels this loop away and then it starts winding on the surface. So these are metastable but hidden states, right? You can get there by quenching by creating a random initial condition but not otherwise. So the plan is to start with a complicated condition because then you can visit all the intermediate states and see how they converge. This is work in progress but there is another important issue. This is all simulation. So did we observe these states in experiment? And the answer is not yet. At least not conclusively proven. So let's check that. We first had to develop a experimental method that can actually observe the states in three dimensions which is a challenging thing because this is like 20, 30, 40 micron droplet and you need three dimensions and you need direction, not just density. So this is why we used confonkone microscopy. For the experiment we made droplets but we had to use a certain mixture that allows this macroscopy to function because you want to avoid too much light diffraction because otherwise you don't see anything and we use this fluorescent confonkone polarizant macroscopy which is something that just excites the molecules of that direction depending on the direction you get a different signal of the emitted light. And this is how it looks like. You see there's a layer structure and this is under different polarizations of the incoming light. So it clearly works. You can reconstruct the angle from here for each slice. You see if the polarization is like this it's signal is strong and it's weaker if it's blue so there's a formula that can instruct the spherical angles from here. The two-dimensional angle, the flat angle it's clear. But the problem is the formula for the polar angle is different. It has a fourth power which kills the angle. So we don't know this from this. We only know the absolute value of the projection so what to do. But of course this is just a sign. And we do have the simulation code that we have the free energy which we know describes very well this system. So instead of simulating from scratch we just took this and simulated just the angles. We used stochastic method so we just it's simulated annealing actually. Because it's sort of like a spin system. You have a up and down in each point and we annealed this numerically to fix it. First we had to of course clip the angles because you have to know what's horizontal, what's vertical. And then you observe here of course it's wrong. Here it's not normal to the surface of the droplet. Here it's correct, here it's correct and here it's wrong. So after this numerical process we actually get a reasonably correct state. And the important is that if you want to observe topology this is enough. We don't care if it's a little bit scattered here if it's a bit flat here. This is clipped too much. These are all parallel. They would actually have to be fanned out a little bit but topologically it doesn't matter. We just want to see this structure on the side. And once we had this setup this is also a lot of optical challenges to make this work. You observed something different. It's not exactly loops. We observed point defects for instance a point defect with this distorted bubble around it. And this is also a topologically interesting object because we observed that it's a building block. All the structures that we observed are made of this. If you take a cross-section here you see that there's a vortex that actually turns here it's like vertical and then it goes 180 degrees down. So each of these vortices is actually a skirmjum in the cross-section. But this is of course just the point defect moved to the edge and there is some escape some complicated three-dimensional texture but no defects there. And then you go a little bit further with corality you get two point defects and there is another point defect in the middle. So as we go we get to the topological restriction in a sphere because animatic as here it can be represented as a vector field because you can just add an arrow at each point because there are no loops if there are no diskination lines it's okay to do that. And some of topological charges the wrapping numbers have to be one because of the Euler characteristic. So here it's a negative it's a saddle point you could say it's a point defect you can actually see in experiment under the microscope. And you have again two bubbles and it gets interesting because you go to higher corality you get all these constellations like chains of defects which fit nicely together it's just alternating one minus one, one minus one, one minus one up to five bubbles actually nine point defects which is amazing. This is just in a single droplet and you can also see that there is a symmetry here. These are always in an equilateral triangle and this is always a tetrahedron the surface points the every other one. So this is actually maybe useful to get droplets which have some valence to build superstructures from them or maybe as a optical resonator for liquid crystal lasers but of course topologically we care about these defects now and this is we can ask ourselves why points, why not lines and it turns out we do see lines. So we also see this this is a different set of structures which are loops we have a diskination loop which can wrap around some complicated structures which are actually just like this bubble but double bubble you can also see here that it's the same repeated texture again but no knots and there is a reason for that. We wanted to see knots we didn't see any the reason is that in simulation we used smaller effective droplets because of simulation is easier for a smaller system and we used a model for a liquid crystal at a high temperature just below the transition so the defects are cheap and this is the challenge that we have to get over now because it's difficult to perform confocal microscopy at a high temperature because of thermal motion so this is why we have this at below below room temperature and it appears it's not enough it's not enough to observe the simulated structure we know that they should exist in some regime but not in this regime but nevertheless we get at least some loops so we can anticipate that if you go with parameters further we can actually get those also through the loop through the bulk for a knot because a loop confined to a sphere is never a knot it would have to be at least a tors, to get tors knots or some higher genus surface for higher genus knots, of course so this is not enough so we need bulk discrimination loops but we did see something that wasn't seen before and this is higher order point effects so topologically they are of course allowed but there is a paper that says that there is a note paper that says that the harmonic map cannot have a higher than plus or minus one topological charge in there and pneumatic director is a harmonic function it satisfies a nabla squared equals to zero equation but this is a colosteric this doesn't follow the same rules so here we can of course have it's not forbidden to have point effects of higher charge but this is the first time we've seen them in experiment and this is how it looks like we have observed this point here and it's a perfect three fold symmetry actually in the confocal image and if you reconstruct this you can just see these three bubbles I'm not using this green shading anymore but they're here and it's a stable point defect and it's a perfectly symmetric structure and you can see that it's just this structure with two of those merged together so it's the charge is conserved the wrapping number is conserved but this is more symmetric and then you can build molecules from that you can take one of those take it away replace it with a chain with the same topological charge so you can build this stuff together it doesn't end there so you could also combine this with that so you could have a loop and a higher order point defect and any of these combinations so it's not exactly what you want to see but it's a new result so this is why I wanted to present it here because it's also it's still not published this last result it's a work in progress we submitted to nature communications so I hope this goes through and now of course this is minus 2 can we go higher and it turns out we can so this is a more interesting one because it's three-dimensional it's a tetrahedral structure there are four four point defects on the surface four bubbles very clearly seen this one this one and these two are out of plane and a minus three hatchcock in the middle and you can also combine this with another different structure you can remove this and put something more complicated there and this is geometrically interesting because we know what the building blocks are so you have these bubbles and you have to put higher all the defects around and this appears to use all the space now can you actually have a higher charge than that well yes if it's in the middle you could fit biper pyramid here or something like that but if you go to higher quality if you increase the quality here nothing can reach the inside anymore because you have this restriction you can only go one bubble away because then you have to come back here point and then there is something there and the next one has to be plus one next one has to be at the surface so these are the structures that are only possible for a range of realities which allows this distance from the farthest defect to the surface to be in some proportion close to the one helical pitch that this cholesteric requires here you can see this one is closer to the surface but you cannot go much further than that because then you have this all bulk to fill in and this bulk is filled in with regular layers because there is nothing else to do there so this one requires very precise calibration of the pitch the pitch to the diameter has to be exactly right and here we also then get okay this is a specific regime so are the explanations in the noted structures which is simulated also restricted and it turns out that yes if you have a drop that's too big if the clarity is too high you always get this structure expel to the surface there is always just a snake on the surface and nothing inside so these are all specifically reserved for this transition regime when the pitch is just comparable to the frustration radius if we go back here for comparison actually for higher clarity you only get this so this is sort of disappears when you go higher but how? this is also a question for the landscape simulations because does it just become unstable does it just collapse into one of those is a transition first order or second order and of course here all the invariants have to apply that I was talking about earlier so at each of these transitions here if you give a crossover the right has to change by two thirds there is no other way and also here the thing that I mentioned before this is usually twist equal zero not anymore because now you have a colostaric which prefers one particular direction of this twist and from work we did on the 90 degree twisted colloidal this array lattice of particles we know that chiral knots prefer chiral curves the ride of a chiral knot wants to be of a particular sign so we got all the right handed trefoil knots for positive self-linging number and all the left handed trefoil knots for negative self-linging number we have statistics here it's not in this talk so now that we broke the symmetry by introducing chirality we anticipate that a certain chirality of knots will be preferred to their mirror images so yes, this is still work to be done to go back to the end so now that we have all these building blocks we can of course try to also get diskination loops and observe the knots in experiment because then, if you do that you can also try to polymerize the core of the defect and then you can observe the knot or you can actually use this as a template for a knot you can polymerize monomers that want to go to the defect core and you can observe the structure and after we we try to get the knots we try also that ok, for conclusions I hope I'm not too quick I left 10 minutes for discussion but maybe 15 ok, so we experimentally didn't get knots yet we got a different also interesting system which has this topological defects of higher order that were never seen before and we now have a method that can observe the structures in droplets in three dimensions so now we just have to tune the parameters to observe this and then maybe report back and present the results then but these higher order defects are also interesting as a template for highly symmetric droplets for imposing three-dimensional crystals of these droplets and also as a proof that this theoretical result that holds for memetics of course doesn't apply here but now we can also actually see how the higher order defect looks like ok, thanks for the attention and here is just a set of my collaborators of course my supervisor Slobodan Zhumar the supervisor of the experimental group Igor Mushevich Gregor Posnjak who did all the experimental work and David Sejč who performed the simulations ok, thank you