 Thank you, Mariel, and thank you very much to the organizers to give me the possibility to talk in this beautiful meeting. First of all, a little warning about these vortex knots. Vortex knots, the talk, the main results, can be, let's say, seen, useful for a number of physical systems and not necessarily vortex knots. So don't focus on vortex knots and think of whatever physical system you like, so polymers would do probably well as well. Anyway, to keep things focused and to give you ideas about my work, I prepared the first few slides as a recap of some 20 years of work just to consider a particular physical system I will deal on vortex knots. And the objectives are to determine relationships between structural complexity of physical knots and energy and to quantify energy or illicit transfer in dynamical systems. Both topics are fundamental in at least trying to tackle this big problem of turbulence that is still an open problem. So vortex tangles might be the skeleton of far more complex systems, but they will allow us to do some analysis in terms of geometry and topology and possibly relationship with energy. So for this, in recent years, I moved to consider to apply the results of knot theory, in particular knot polynomials. And on this occasion, I will present you some results about this kind of approach, knot polynomial as new invariance of physical invariance to quantify topological complexity. And let me stress here to quantify. So we know knot polynomials are able to to a certain degree of accuracy to qualify topology, but here I want to stress the idea of quantifying, adding, attaching numbers and possibly estimates for physical quantities. And then something that they will pursue in the future years is to extend and apply new topological techniques to study complex systems. In particular, I have in mind homological theories, but it won't be the topic of this of this talk anyway. Okay, so my reference tangle is a vortex tangle, and I will keep this in mind for the rest of my talk. I want just to give you a brief idea that these approach using topological methods to study complex systems. In fluid mechanics has a long history and in recent years, you know, this dates back at least 20 plus years. Actually, I'm quite happy to see in the audience the people who who taught me non-theory do it some others and can be led back to Santa Barbara's time. We're going back 25 years or more. And it is in the last more or less 25 years that I've been pursuing this. Of course, in recent decades, we had a very high resolution of direct numerical simulations of Navier Stokes equations to the the most real fluid dynamical description you may have. And this allows us to pick up anything that we like like a pressure, density, vorticity or whatever magnetic fields, if we have magnetic fields, and to a high degree of resolution to control that, to understand and follow the dynamics and control the processes to a certain degrees understand the processes. Even easier, in a sense, would be to consider the skeleton of the vorticities. But this is hard to determine in a precise way in classical fluid mechanics. However, it is easier in a super fluid helium where the vorticity is so highly localized, there is no diffusion at all. And so the vorticity is highly localized on basically defect lines in the in the ambient space. And this also has benefited a great deal in doing numerical simulations to a very good degree of accuracy that are reliable and consistent with experiments. So what I'll do for simplicity, I will stick to this last model. So superfluidity, superfluid vortices as defect lines in a fluid that is otherwise irrotational and kind of perfect. So in this very simple approach, I remind you the approach is extremely simplified. In this context, I will take homogeneous incompressible and inviscid fluid in R3. There is a velocity field that somehow drives, governs the motion of the particles, and I identify the particles with position vector x and time t, and I'll use this divergence less condition for you and you at infinity. Vortex line is, as I said, this kind of defect. So for me, vortex line is basically a space curve, and I identify the tangent unit vector with vorticity as simple as that. And the cross-section of this vortex is so small, we're talking amstrons with a length of centimeters, that I take it as a constant. Circulation in quantum fluid is quantized. So even better, we can take the circulation that is constant in ideal fluid mechanics as setting to one, as one. And just disregard that in a sense. I will keep it in formally present here and there just to keep track of it for a number of reasons. So my vortex tango is just the union of these say filaments. These filaments may form knots, knots and links, and because of vorticity is the curl of view. There is also a kind of unassumed consideration about the fact that this filament should close on themselves, but not necessarily actually because of a number of reasons, physical reasons. The kinetic energy by looking at lambs of old formula can be nicely written in terms of double integral over the the filament length, and of course another important quantity is total length. Well, if I said the first few slides are a kind of recap of the first or the last, I should say, decades in fluid mechanics where topology starts to play a role, and this is basically linked to the discovery of one classical now, invariance of fluid mechanics, that was proven to be a conserved quantity under ideal evolution only in 1958, and this is helicity. Helicity is just the integral of a field at times the curl of the field, whatever is the field. In our case is the velocity and the vorticity integrated over the domain of the vorticity. Now this in case of very thin filaments or actually almost space curves with some some thickness anyway, can be reduced to a line integral, obviously, by working here, where you, mind you, you is just the induced velocity asymptotically close to the vortex by the vorticity on the line, on the chi line of the vortex in space. Well, there is a fundamental result due to Keith Moffat 69 that relates helicity to linking numbers, in particular Gauss linking number in his work, and then was extended to include self-linking number, and if we consider a structure present in these filaments, and we do indeed, the linking number is the Gauss linking number, and the SL is the self-linking number, which I remind you, we saw this already many times, is the caliguran white invariant, when you consider for modeling this physical system a ribbon, and you identify your filament with the ribbon, with the axis, and the other edge of the ribbon, and then you define the Gauss linking number of these two edges of the ribbon, then you take the limiting form for the width of the ribbon going to zero, then you find this self-linking quantity, and caliguran prove that this quantity is a topological quantity. So this is a topological quantity, and as we saw can be decomposed in writhe and twist, and these are also playing an important role in fluid mechanics. I remind you, as we saw many times, that the writhe is taking care of the say the coiling and some torsional distortion, the coiling of the filament in space, and it depends on the axis of the filament, so it's a geometric quantity that depends only on the axis of the filament, and the total twist, the total twist depends on the ribbon, or if you like the structure of this filament, not only on the axis, and it combines a contribution from total torsion plus intrinsic twist. Well, just relying on this information, during the years I proposed a number of experiments to detect complexity, so I show you just in one or a couple of slides, a kind of a brief recap of what has been achieved during the years, of course, it's pretty pretentious because I will focus on work that I've been, I was involved in, and so of course is very very biased, but this is the idea. We start to consider superfluid tangles back in the 90s where the software was pretty, pretty good, and then said, okay, let's, with time, provoke this entanglement, so we put energy in the system. I will refer to this test case that for me is still a very, very educational example, so we put energy, mind you, we put energy in it, and we provoke this entanglement to a point where we start to say, okay, this is a mature tangle, and when it's mature, for us we are ready to take measures. And what kind of measure we could take? Well, we took the measure we had at the time, so of course all these linking numbers and rive and average crossing numbers, maybe the correct average crossing number, the exact integral definition, or the estimated average crossing number, similar for the rive, estimated means that we evaluated according to three mutually orthogonal projections, something like that, there is an error bar associated with that, but all these measures, you see, they scale almost the same, the jumps in the linkings due to the fact that, of course, there are reconnections because the system reconnects and devolves is a software that simulates or emulates the real evolution of a superfluid tangle, and of course there are jumps in the linkings, but all these evolutions of whatever measure you take are roughly going twice as much faster than the evolution of energy or length, which is usually taken as a measure of the evolution of the tangle. So remember, you put in energy to provoke the tangle, so these increases, and we came up with this relationship that was tested in many, in many cases afterwards, and of course this is just a nice possibility, if confirmed analytically and if confirmed in many other situations, to say something about energy by considering algebraic complexity of the system. So this has been done and confirmed for system that decay, in a turbulence that decay naturally, so energy decay is there, but still you find something like this. So of course this is an interesting thing, but there are a number of things that are not so satisfactory, and one is, first of all, that you have electricity functionally depending on linking numbers, and okay, I put the circulation that just reminds us the structure of this functional dependence, but the functional dependence on linking numbers we know historically from our theory that that is not so satisfactory. People were looking for other invariance, and so we did, and other invariance, okay, but then also there is another big limitation. Limitations that there are systems like well known the Boromian rings and other systems, even Maxwell noticed that 1867 in the private letter to Tate pointed out that this is have a gas linking number zero, and so gas linking number has a limitation clearly, but even more so for us, for, say, fluid dynamics or in general classical field theory, because in the end stages of this turbulent evolution where the tango interacts, etc., etc., reconnects and structures are produced, there are lots of small rings produced, and the small rings are traveling out like bullets in a fluid interacting with other structures, and the end result is the scale down of vortexes smaller, smaller scales, till dissipation kills it off, and this idea of moving in the turbulence energy landscape towards a kind of, I'm now thinking of dynamical system approach in a kind of basin of attraction of rings, of vortex rings of smaller, smaller scales has been with us in the last 30 years. Now if we have so many small rings that are shot off from the bulk of the turbulent region, and we use this illiciting in terms of linking numbers, we are not doing great deal in detecting these things, because linking is zero there, and unless we can say something about internal linking, then if we rely only on the external linkage, illiciting would be zero, no matter how many rings we have, so this is very unsatisfactory, and so we moved to consider non-polynomials. Of course, we benefited from the progress of the results in quantum field theory, rely on the Chern-Simon's applications interpretation in terms of polynomials, and so the idea is why on earth just stick to linking numbers, let's go to non-polynomials, and I will focus here, work started to consider Jones polynomial, but I'm now happy to tell you something about the Humphly PT formula, and Humphly PT allows us to go back to Jones, so we understand also the difference, at least the physical implication maybe of this difference, so the idea is to use Humphly PT polynomial on a tango, so what you do, you are on a computer, on a computer they have everything, all the data are available, so you take from your tango, you freeze the tango, you extract the component, from this component you project the component, you analyze the diagram of it, you put the usual crossing signs at crossing sites, and then you can implement this Humphly PT computation. The computation or the polynomial is constructed by using recursively these two skin relations, and now in front of experts I'm there to give you a crash course, half a slide on how to compute this polynomial, and how to interpret this relation, this important, so I'll start with the first one, and the very simple configuration is the un-not-unlinked circle, and U1, as you say immediately, well the render is nothing to compute there, because the polynomial of this quantity is 1, indeed is 1, so if I deform it continues into this figure of 8, with a positive crossing here, you would say that probably this is 1, because it's just topologically equivalent to one another, so indeed this polynomial is 1, okay, so I'll take one of this configuration, let's say gamma plus, and I will do the same, I'll try now to apply this second relationship to this quantity, and again you may say, well this P is here, P gamma plus is 1, so there's nothing to compute here, correct, and then I go from the positive crossing to the negative crossing, so I switch this crossing, and I go to this side, and this configuration is just the gamma minus, which has a polynomial that is also 1, so I plug one here and one there, then I have to smooth out these crossings into this, and when I smooth out these crossings, what I get is two unlinked circles, okay, so I can use this relationship, because the P of the smoothing is the P is the polynomial of two unlinked circles, and I ended up with a different polynomial than 1 by this information, you know, A minus 1 divided by Z, so I have this P of U2, so I can compute polynomials of all north, and we have these tabulated in north tables on internet, so you go there, you take your preferred north, and you have your polynomials, I mean polynomials, so Alexander Kaufman, Kaufman Brackett, our polynomial, and of course, hopefully PT Jones, and hopefully PT, and others, okay, so the result, the main result is this, if we deal with a north that is a fluid north, say, but as I said, it's not so important for this meeting, we take this variable E2 self-linking, E2 self-linking, I appropriately rescale because I have to take care of the circulation there, satisfied with a plausible statistical hypothesis, the skin relation of the hopefully PT polynomial. I will very briefly say something about the proof, but this is the main statement, okay, so what we have from the main statement as an immediate result is that we can consider this skin relation, and if we consider the skin relation and we're doing all the algebra there, we can relate these two relations with these two variables, A and Z, to A to the twist, and Z through K to the writhe contribution. Twist and writhe of what? Well, I put there a angular bracket, and I refer here, you have to be patient, at the end you will understand very well what I mean, but for the moment I will refer to, let's say, we would say a gauge field, a reference field, a reference field can be something like this, you do your simulation or your experiment, and you average out the amount of writhe or twist present in the whole system, and you refer to those values, okay, so let's stick to this proposal for the moment, and then to be extra careful now I can relax easily the little algebra there because I can see the naivety, but anyway at the time to be extra careful I can put an uncertainty factor in front of these average values of writhe and twist of the reference field, okay, so if I do this I can relate, as I said, one variable to twist and the other variable to writhe, and these are interesting physical implications because first of all I can interpret the skin relations as a generic relationship between twist and writhe, but when I couple these two variables to go to Jones for instance, as we do here, we get to Jones, so we get to the one variable polynomial of Jones, and then we interpret Jones physically as a polynomial for framed systems, for physical systems that have the writhe and twist absolutely locked together, all right, so this is a nice way to interpret maybe one polynomial to the other, and maybe to prefer one polynomial to the other, regardless the topology we want to tackle, I just sketch, brief sketch of the proof, I'm afraid this proof I will show you two proofs of two results, but this proof is long, so I cannot go through that in details, I just give you an idea, the idea is that first we derive the Kaufman bracket for unoriented knots, and then we have, you remember in the statement I said for plausible statistical hypothesis, the statistical hypothesis turns out to be also quite interesting, we didn't discuss that at all, but with discussions with Luke Kaufman and others came out to be quite an interesting problem, so we consider crossings as a virtual crossing in a mathematical sense, there is no physics there, so the crossings is like this, and we decompose the crossing in a left right and the up down situation, and now we didn't give any preferential way to these decompositions, we assume that both the compositions are acceptable and they weigh the same, but in statistical mechanics there are certain systems where you can skew your statistics towards one or the other of these, let's say re-labeling of the crossing, and it would maybe give different interesting results for appropriate physical models, but for the time being this, we give this kind of a gothic assumption that is statistically the same, we are in the knot, and then we derive one, the skin relation for Z in terms of the writhe by going from the Kaufman bracket to the R bracket, the R polynomial, and here you have the direction of the writhe that plays a role and enters in this example, in this equation, then we noted that at a certain point in the derivation, we have this relationship present, and this is just relating what you see through a pre-factor here, and at the beginning we didn't pay attention in Jones case of this pre-factor, but then we gave a little thought about this, and we could interpret or we offer as an interpretation this pre-fact in terms of then surgery, so if you plug in through then surgery the twist, then you can reinterpret the rather master type one move as twist action on the strand, and so this becomes crucial in giving A the relationship with the twist. So you get to this, let's say to this transformation, E to self-linking, okay, so if you do this, what you get? Well first of all is a polynomial and adapted on flea polynomial, then now takes into account the physics through the circulation, and of course takes account of the topology fully and more complete way than the linking numbers. So let's do an example, okay, because we didn't perform any computation at the moment, so it's just, I'm sorry, it's just a kind of mass talk with the hope to stir up some interest and plug in this approach to some numerical data. Anyway, homogeneous super fluid tango, okay, we take gamma one, and then is a thought experiment, all right? So thought experiment, okay, let's be simple, mathematicians are very simple people. So we take an average write, an average twist of a half. You will please bear with me because it is not so essential, but okay, let's take this as a half and then be extra careful and add this other half for the uncertainty of these values, and then you come up with these numbers. So these are numbers, right? Okay, so we are numbers, because we have numbers, we start to use these numbers to compute the values of these polynomials. So we have the n collection of unlinked unnoted loops that scales like this, then we have the op, positive op, and negative op, etc., etc., etc., etc., all these numbers, sorry, all these numbers. First of all, when I saw these numbers, I wasn't that happy for a number of reasons. First of all, this is less than one to the exponent that is going to be much larger, and so I said, it doesn't click with a physical intuition. You would like something that you think is getting more kind of, you know, is interacting, rings are interacting, and this is the more rings you have, and this is getting down and down and down. So I wasn't that happy. Plus these numbers, you know, there's no relationship. Some are positive, some negative, you don't see anything there. All right, but then I was very lucky to have Duit, one of my talks, and he pointed out the results of Marielle and others, and at the time we had also the results done by the group of William Irvine on vortex filament interactions, as Marielle showed, so I go back to vortex experiments. So we have a vortex trifoil cascade process in water. So these are real, as Marielle said, these are real vortices in water. They are very thin. This is not superfluid, okay? So it diffuses and dissipates quickly. However, you see, they managed to produce remarkably in the lab a trifoil knot. So if you look at the movie, you will see, as if you think of some smoke in the air, that is very evanescent, okay? Get destroyed very easily. But anyway, this is visible for some time, slow motion, and then you pay attention to the frames in this evolution. You identify this as the two-three, and then you go on and you see that, for instance, it forms a structure that you may call it a torus link, choo-choo, okay? So it's topologically torus link. And then you let it go. And here at a certain point, if you follow very carefully this, it goes back here. So it's actually a kind of loop, unknotted, unlinked loop to deform the in space. So I will call it the choo-one, torus choo-one. And then it keeps going at a certain point. We are at the later stages of this evolution where the structure dissipates and basically dis-blogate in the fluid. So we are at the stage where you may still see two unlinked and knotted loops. Well, I'm afraid that I've been very often not so close to lots of people. I know they do numerics. So I couldn't intrigue them to do numerics, high sophisticated numerics. They're all taken by their grants, et cetera, et cetera. But I had a friend that close by Milan who is a great expert in Grosspitei-ESK equation. And his name is Simone Zuccher. And so I asked Simone to use his code to simulate something like this. So, you know, we started from a stage, this stage, not here. We aim to go up in complexity. But for the moment, this is the result. So we started with a simulation of the Grosspitei-ESK, of, sorry, yes, this is Bose-Einstein condensate. And these Bose-Einstein condensates evolve according to the Grosspitei-ESK equation. This is a kind of good approximation for superfluid vortices as well. And so I asked him if you could spend some time to do this work on following up the evolution of an uplink under the Grosspitei-ESK equation. And remarkably, he found almost the same situation. So, again, you have a stage where the loop is formed. And then, again, a stage where there is a reconnection and a secondary smaller loop formed. And because it could follow to a high degree, remember, this system is also a real system. In this simulation, dissipation is very, very low. Very low. It's like superfluid vortices can stay there for millions of years, for 250 millions a year. And the only way to let them decay is to increase temperature, you know? From milli-Kelvin, you go to temperature, normal temperature, then they die out immediately because viscosity becomes so important. But under Grosspitei-ESK equation, these things can evolve still. And so you still have farther reconnections and farther contributions from smaller rings. So you enter in a new family of structures. Can we do anything useful with this Humphly PT? So, well, we try to set up a kind of mathematical problem. And so you have to simplify things. First of all, we consider cascade process where only one reconnection at a time occurs. Secondly, I confine myself to the topological class you picked, i.e., the torus knots and links. So I stay within that class. And so I assume that going from a torus knot to a link to a knot to a link, et cetera, only one reconnection takes place. Moreover, I don't even care of course I do, but I don't even care in this approach if what kind of reconnection is taking place. I'm just assuming there is a reconnection. That's it. I don't know if it is anti-parallel, parallel, et cetera. So this is the class of torus knots and links I consider and the assumptions. So all torus knots and links are standardly embedded in mathematical torus that makes our analysis simpler in close very form. Then I consider all torus knots and links to form an order set of elements according to the decreasing value of topological complexity. So crossing numbers just identifying the topological complexity. And as I said, any topological transition between two contiguous elements is just dictated by a single orientation preserving reconnection event. I pause for a second. I don't have any reason other than advertising that we did some work. We do it on the conservation of rife and the anti-parallel reconnection. And because I've seen this rife evaluated across during reconnection in the torus, for example, Professor Zoomer and the other, I think it might, this result might play an interesting role in their analysis, particular crystals. But anyway, I go back to my slides. So these are the assumptions. I'm not saying that they are anti-parallel or parallel reconnecting. So these are the assumptions. So what we do? So we have this result. The adopted on-fleet PT computation of this family generates for decreasing n a monotonically decreasing, sorry, a monotonically decreasing sequence of numerical values given by a formula that is a bit awkward to read where these quantities are totally determined, are known function of, if you can read here, twist and writhe, basically, of given twist and writhe with initial conditions prescribed by these two configurations. So the sketch of the proof is this. Because you embed your torus family on the standard torus, you can always do a skin relation on this braid pattern. You localize your sight, your crossing side. You apply your second skin relation and you just relax the braid in one form or the other. So you come up with this. You do it recursively. And so there is a little bit of algebra here. And if you follow the algebra at a certain point, you come up with these groups of factors. And in these groups, you can now plug in your information about the on-fleet PT variables. You plug e to something like writhe and you plug e to something like twist. So the formula is this and you know everything here. All the information is available supposing, you remember, to assume certain writhe and twist as an average field or reference field. I come to that in a moment. But okay, so you have the formula. You can compute things. Then we start thinking, okay, we considered here, if you paid attention, it was a positive sequence of torus knots and links. What about the others? And so very interestingly, you pay attention to these algebraic structures and you see that if you switch to mirror knots, basically you can generalize this formula to positive or negative torus knots and links. You use this formula to compute numbers. And what you see is this. You compute the number for the trifle. You get 1.5. And then you go to 1.11. And then you go to 1. And then 0.48. So it's decreasing. Now the other thing is let's be wild for a moment. I'm not saying that it's justifiable mathematically. But let's be wild and keep going to the different family. And even the different families capture is still decreasing. That's very interesting. Okay, so next step is kind of obvious. You have a sequence that is monotonically decreasing in values. And it captures this monotonically decreasing complexity. And as all we guess is probably the same for any kind of topological detector, point is that here we can associate the number. Now we dare to refer this to energy. And the referee, one of the referees was pretty tough and said, no, no, no, you can talk about, you can speculate about energy in the text, but you have to remove it in the abstract. We didn't say much. We just said that possibly is related to energy. Anyway, I cannot comment on energy other than, of course, it's very interesting. Other than saying that, for instance, for instance, if you take the curvature energy, what mathematicians like about knots and links, i.e. the integral of curvature square, which is in physics, means the bending energy basically. The bending energy for this, for a number of torus knots and links, sorry, for a family of torus knots, for a number of them, keeps decreasing with decreasing complexity. I can add, I have the diagram is published in 2010, also kinetic energy for vortices decreases consistently with these decrease. I am not quite sure to say something about magnetic energy, but magnetic energy, let me publicly say something loudly, maybe not correct, totally correct, but magnetic energy basically scales with geometry because Lorentz forces in magnetic fields is a great deal of that force is due to curvature, to curvature information. Like elastic systems have a bending energy that go down, I would expect the bending and magnetic energy would go down for the same reason. We didn't establish a mathematical relationship between these two behaviors, but it's clearly there. Now, what about the cascade that we saw in other context, and of course I mentioned the way it's interesting in this, because he triggered my interest in the work of Mario's work and Mario's group, and so this is the kind of cartoon we saw from her presentation. Then I was pleased to see from Mark Miller a kind of similar cascade or the other way around, the increasing complexity that resembles this kind of family. So I came up just with the last minute with this idea, of course, is very much in our minds now. We don't have computational data to work on, and so I'd be the last one to discover anything useful here, but at least let me end up with this just conjecture. So what would be the optimal path to cascade? Well, optimal with respect to what? Of course, Mario gave us a clear definition of cascade, so it would be topological in terms of topological complexity. Here I would be inclined to think of some energy path, some preferential path for energy, but of course you are not strictly limited to this particular family of knots. Fluids can do whatever they want, and we may think that the 2.5, instead of going to the 2.4, it might go to this connected sum, why not? So we just started to play with numbers, the little we could do, mind you, and so this is the process, this is the path where you have, if you follow the HONFLY PT analysis as I presented to you, but you may follow an alternative path, why not, where you go to the connected sum of the triphons, and then you go through this configuration, and then you end up with the same tripho. Now, if you do this path, you see here that according to our adapted HONFLY PT, we don't have any indication that this would be preferred to this, because even in this case, the numbers are keep decreasing. So I have to stop here other than saying that, of course, I'm terribly intrigued by the possibility to associate in a kind of approach that Mario proposed, a probability to each of these paths, and if we have a way to give to this path, then we could presumably just purely on a, how to say, on a statistical mechanics abstract approach, determine which path is preferential or not. Clearly there is energy to play a role as well, and probably would trigger one path from the other. Well, I stop here. Thank you very much.