 Hello everyone good morning. I know it's hard to get up. It's even harder for me because I have to get up earlier and you guys do. So thanks for the speaker of the walk-alizer for inviting me to over and you think you have got a very, the conference has been going very well right. Nice talks and everything you know but you have to wait until you have heard my talk. Then you probably have a different perspective and some people are smarter than you are though so they probably know that so they just decided not to show up. So what I'm gonna talk about something I really don't have a whole clue. Okay so if you look at the the list of the co-authors of mine who made a contribution to this to this work so there are four of us. Eric did most of the work and he's in the audience that's good. So if I there's something I don't know or most of since I don't know you can ask him okay. And Klaus and Wuta they did all the analysis the number crunching you know all the computer things and I basically I'm here presenting it so if I don't know what's going on you understand. Yeah all right so what this is about well the main idea is that we know the random knots occur right and so and we have a model to generate the random polygons as a model for the random knots but the random knots per se I mean the way we generate it's you know it's the equivalent of random polygons you know it has no confinement which is not confined in any given volume which is quite the contrary in reality right in reality what do we have this random things usually they are confined in a cell or whatever right so they behave quite differently and what we know is in the unconfined case that if for the random knots you have is highly knotted then that actually affects its geometry for instance if you think of the radius of generation now how big this thing can grow in the highly knotted case this thing tends to be much much smaller right it's more tighter right so and so that's the kind of thing we're thinking well so if we generate the random polygons within a confinement like in our case it's going to be within a sphere of a certain radius then we ask the same question for those kind of things whether the topology still plays a significant role or not or the the confinement itself is the major factor that changes the geometry so the geometry we're talking about we only look at those four things the total curvature total torsion and the mean ACN and the rise the mean rise so let me start with a simple definition of what a random polygon confined in a sphere is how we define it so this is the basically the definition of equalized random polygons so you have those vertices between any two consecutive vertices the distance is exactly one and without confinement when you take the first step you walk out the the direction you go is the uniform you have equal chance to go anywhere but you are conditioned on that eventually you have to return to where you start and then you add a condition that whatever you generate has to be within a sphere of radius given radius okay and the sphere in our algorithm the center of the sphere and the starting point of our polygon is the same although we actually later changed it somewhat to so they can be different but that complicates our algorithm a lot which slows down quite a bit which means that we were not able to collect a larger sample of data to to for the not to be statistically meaningful okay so that's what we've been using so when you define this there are more details in there exactly how you're going to sample it there are different ways to define how you're gonna sample it and actually the way you do it it affects the distribution so you may not get exactly the same distribution so it's just an example here they are on the right side you see the confining sphere in this case the radius of the confining sphere is three and within there you see a random polygon okay magnified on the on the on the left side which has the step length of 20 so length of this is 20 is just an example to show you you know an idea what it looks like so and this is a short description of how we actually do it so I don't want to go to the detail because it's quite a technical so how exactly you do it but the point is that we can do it and we can do it with a rather reasonable speed so that we can collect many many samples we can generally those things fairly quickly as long as your lens is a controlled actually I think we can go over that we can go probably to 200 but when you go to that length it's the next step that we cannot we are not able to do when you analyze what kind of notes you get then we get the we get the hang up there so we actually stop there in our study so the next thing is that the kind of who the geometric quantities that we we want to study right the first one is the total curvature it's the the angles that so defined so at any given point this is your x i this is the next point right so if you take the orientation that's a vector and then from there you have a next point okay that's your x i plus 2 and then if think of this two vectors right so if you move that over here then this two vector has an angle right and that angle is the theta i so so you make that rotation so that angle is between 0 and the pi and you add all those angles up that's the total coverage that's how it is defined and now for the for the torsion on the other hand for the torsion if you take the two steps there's a two things with the probability one you know these two things are not co-linear so it defines a plane and this plane these two guys defines a plane then it has a a normal vector right and then the next two steps if you think of the next two that also defines a plane also has a normal vector you you measure the angle between those two normal vectors that's how you define the angle you used to define the total torsion so that's the two things that's how they are defined okay so that angle is also between zero and pi then the ACN is you take a random polygon you make a projection in the projection you count how many crossover the crossing points you have you make a note of that but then you take another projection you have different crossing number there are then you just take the integration of all this over the sphere over all the possible projections so that's the ACN and rise on the other hand is when you take a projection you look at it in the projection the things is oriented right so you assign the positive one and negative one according to the sign convention and then you add all those plus and minus ones and that's the rise of that particular projection and again you have to take the integration of all those numbers over all possible projections that's the spatial rise of that curve so those are the four quantities that we are going to analyze so the what are we going to do is you generate those random polygon then you try to identify what kind of not it is and this is a description how we try to do that so within the scope of our just random polygon we generate we are able to basically identify almost all of them there are very relative few of those we have trouble but even those you don't count it because they are statistically they are not significant so so that's how we handle those things so once you have those not identified of course then you can actually put them into separate beams right so the way we do it is at each at a certain given length and at a given confinement radius we would generate a bunch of those random polygons so you actually have to do a lot because you have two parameters you have the polygon length you also have the confinement radius so the the our entire data set has more than four point eight million of them sounds a lot but they are not for just one radius or one length they are spread out right the length we studied is from 10 to 90 and the confinement radius is from one to four point five and with a certain increment and it's not uniform because they are certain cases where we want to collect more data to have a closer look so to have so for certain of those we actually have a larger data set all right so the the special ones are those for those two cases we generate a lot more data for those okay so that's our data structure so that's how we got our so but if we each data set we have for we have for is it true that we have at least a 10,000 Eric you listen okay for each data for each given one is it a 10,000 the the the data set I think it's at least a 10,000 yeah for those two there are a lot more right yeah thanks all right so now I'm going to present lots lots of pictures which I cannot explain and now what I want you to do is when you think of it as a total curvature and total tossing whatever and they are I mean we're gonna look at them by they are not types you know so for the specific like you know for the trifle for the four crossing lines for the five crossing lines how they're going to behave compared to each other so whether that matches with your expectation or not and I'm sure after a few of those you soon you get tired so by the time you get tired as I give up or you raise your hands I will shut up okay then we'll come today so in the unconfined case when you look at the total curvature and the total torsion the mean of those okay so this is the total curvature the mean that's over all length n lines right so if you take over this then this is a theoretical result due to a growth back so it grows like n times pi over 2 so I mean which is understandable so in the free case each take each step you take the angle is between 0 and pi and pi over 2 is right in the middle so on the average is right in the middle right so it's over two for each step right and so that's why you have that term for the torsion it's kind of yeah similar but the correcting term is a little bit different so that's the that's the unconfined case so the question is in the confine the case what do you expect what do you think well if it's confined in a sphere so the total to our curvature before I show you the picture okay think what it looks like okay well you think it's a extreme case if the confinement is the sphere is really really big and then it's like the confinement is not there right then of course this behavior should still be there but on the other hand if this confinement is really really tight imagine you are in a very tight space you take a step the next there's no space to go you have to go right back then the extreme case would be like a pie right every time you take it will be close to pie right so the confinement would increase that the force you to turn over more right so you therefore you would expect to see a linear pattern like this but with a larger coefficient right alright so so here are some examples this is a larger confinement radius so this is down here see that's the lens right and you can see the two curves so the yellow is the torsion and blue is the total curvature and see it's quite a linear and so so it's a little bit more than pi over 2 now and the behavior is quite similar to the unconfined case right yeah this guy is a little bit more than this just like it's the unconfined case so you're no surprise here and tighter okay to R now equals 2 so difference seems to be a little bit bigger huh yeah so eventually looks like over here they're getting together no idea and then it becomes pi over 2 yeah so actually it does not go linearly actually because it doesn't have to go to infinity I mean once it reaches a half of the lens it's already like a confinement right so it flattens out and R equals one and our in our algorithm the R the smallest R is one because we started from the center you have to take one step out so we cannot generate anything with our less than one so that's the that's our so so it looks like that so I mean it does give you that feeling right so it it behaves linearly as you expect and it has a larger coefficient so no surprise here which one comes yeah isn't it yeah but don't ask me why that's that's yeah yeah yeah how do you prove it I don't know the same all right okay now so that's basically this picture that shows you the effect of confinement it's not the effect of who not complexity right it's not about nothing yet so the next ones would be about nothing right so oh this just tells you how how we actually proposed this okay so yeah your receipt picture so and those are how the fitting parameters and bees are fitted okay now how do you measure that first so you know the things grow linearly if I'm gonna draw the pictures I mean because n goes up right so the slumber is actually quite a bit so we decided to measure in in in this way so in the per per age way so you you you look at the particular not class like you know all the north ways the seven crossing the four example here right and you look at the average the mean total coverage of those guys and then you look this is the overall mean curvature you take the difference and those are all of length ends okay within the same length then you divide by n which gives you the idea for if this is a bigger then this tells you how much increase you have per age right that way give you a pretty pictures more scaled and same thing for the torsion so that's how we actually measured the effect and let's look at the pictures so this is measuring so for the fixed length 30 that's the one we actually did the most studies right so the data down here for the small confinement is seems to be noisy here and then for the larger ones is that what we expect for the total coverage you see for the so this one they are all more than the the overall right because the difference is positive right so this is a down here is below below means that here they are actually less than the average they are they are more than the average so what does that mean the lens is a fixed if for a fixed lens if the radius is one it's a tight confinement then for that lens those things actually tend to be what things in there tends to be more complicated in general so the overall thing is complicated they are for those guys compared to the overall picture they are actually relatively simpler that's why they are total curvature actually are less but as the confinement radius increases well search length 30 within this is the confinement radius at the end is no longer that much confidence I in fact it's almost no confinement therefore well most of things they are turned out to be much simpler notes and those guys in comparison actually become more complicated so now it takes over so it implies that well not in is still making difference but in order for this to be actually a path they are you at the not actually more complicated and still not quite the case right for the 7 up to 10 crossings you see down here they are still below yeah so the picture that's right yeah the knots I think the knots here actually are simpler than the average well for for age 30 for 30 ages actually you got a quite a complicated ones done over here so that's one thing about it the London polygons within confinement when the confinement is a tide it's much easier to generate the things that are much more complicated which is not the case in the unconfined case so this is in in terms for the torching part the torching goes the other direction so it goes down instead of up and but here I think that they actually followed the more or less the order of the nodes this one yeah the next one is actually on a smaller part of it because the radius did not go to 4.5 you see this one on your store we stopped at 2.4 to give you a closer look at the tail so this is a negative point 02 but down here that's see does this is actually up there I think it's the it's the data size I think it's the noise I think no we did not if we generate a whole lot more we probably would be able to yeah so but may not take so much time and Eric's time okay don't mind so that's the torching and for the other so for the higher crossings similar behavior so don't know whether this is what you expect now I if I fix the radius out of three and I look at how it would change here according to the lens and this is how it is look you mean to me I don't think it is that much difference actually because those not for in terms of total courage I think actually they behave very similarly actually it means that the the change in the not complicity in this small not it's really not that large so it's hard to see really significant difference that's how I take it I mean I think those it means total courage is not that sensitive to small changes like this so you see they all behave similarly which is probably a good thing right means our calculation is correct it means this they should so this is just a lot of portion so you can see that they go up and down since probably because the noise is but on the other is also because they are closely charged that's why you it's easy to cross over all right now this one it's about torching other pictures are about it so the torching and the curvature and torching see they always behave exactly opposite this one goes down and that one goes up okay and but again I mean it's just it's still similar they are quite closely each other and higher crossings same thing okay look at the tail it's more or less same however you know the order of the nots are you know in this case more or less preserved and one thing we notice that once we divided the north up into the two classes the alternating ones and non-alternating ones we were looking of course those guys in terms of how to realize them lengthwise there's a difference right so we were wondering whether you can see us you know a clear difference there and so when we did that actually it did show the difference you see here you can see clearly the alternating ones has higher total curvature than the non-alternating ones throughout this is a very consistent right so it means that the alternating ones are more complicated right harder to tie on the average and this is ten crossing it's a similar so and the torsion then this turns the other way around correct torsion is going up and the non-alternating one is on the top and yeah we don't we don't quite understand why it's a complicated more complicated not actually tended to have why is it this if the yellow one is on the bottom and they are the yellow ones on the top so the non-alternating ones has higher total torsion than the alternating ones but not the curvature so these two things actually is it the artifact of that that correction term that is causing all this I have no idea what it causes such a big but this is the throughout the whole thing I mean it's not just it doesn't look like it's going to disappear all things okay so that's about the total curvature total torsion so I really did not see any really big surprises I mean I think they are more or less within our expectations all right now about the mean SCN so again in the unconfined case this is a known result so we know it behaves it grows like this the length of the polygons is n so that's the growth rate and in the confine the case it's known that the growth rate would be n square and with a fitting function awful of that form so that if this guy if with the confine radius really large like go to infinity this term actually basically no this is not right something is not quite right here this whole thing should go to zero when R is large but I guess the feeding function only for small for small R so in our case we use this fitting function with our data and it did fit very well so we we actually throughout our data if I just look at if I don't look at the not types I just look at the overall sense we generated and then with the certain choice of the parameters so the fitting is very good we if you are thinking about a similar questions about the topological crossing number of those guys then yes it's topological now it's not a geometry right it's not geometrical and this guy is actually unknown because if we generate a random polygon without a calculating I just ask you like what is the probability you know what is the topological crossing number what can you say about it then there's nothing you can say right now it's an open question so so if we generally just confined the random polygons we cannot even prove that the knotting probability will go to 1 as the length go to infinity unlike it I'm confined the case in which we could prove that so yeah so that's one of the things that are possibly right you think that this is so obvious that it's going to be noted right but we just don't have the proper tool to prove it okay so but in our case that you using our data we were able to actually fit it with with a format like this so it's also of n square growth so for that's for the topological crossing number and the way we measure that the we measured using a similar things by per age thing right so this is like similar to what we did for the total curvature and torsion so here is an overview of the data tells us so as this is a combination of all the data we have with all sorts of confining radius combined with a weighted mean and you can see they are it's very uniform so it this I mean it's nice to have a nice picture like this so they really go by the not a compressor as you expect exactly that order no surprise so it kind of tells you a saying is really good right in terms of determine where they are right and that's for fixed for fixed lens okay so we if you fix the lens you change the confine radius you look at the individual knots here and that's how they are being ACN changes that one is the I'm not okay it's the I'm not minors the the overall thing okay so it's again it's a measure to her age or no it's not purchase is this accurate self actually yeah this is not divided by and this is just itself all right so that's for you that's for crossing but then those guys actually got mixed it's a hard to tell but then if you divided them using the crossing numbers in the groups then you see a picture like this and they are the order and that's a five and that's a six yeah very nice right still kept that order this is for fixed radius so now we fix the radius of the three and we change the lens and see how it goes with the lens so each one of them the ACN the difference with the overall one okay it goes down so it means it means for smaller lengths the way we read it is for smaller lengths the North actually are more complicated right because the con with that confine no the North here are actually less complicated actually right because in that small confinement you know you would expect the North to be more complicated we just sorry take it back this is the lens change okay this is the lens 10 in the same radius so here they are simpler so the and this is measuring the difference with the overall ones now the overall ones are simple they are for those guys compared to the overall they are more complicated therefore they are higher but by the time you get to this lens well they are crowded right because the radius is still the same you have a much larger much longer pulling pulling guns so over here the North you will get a more company North on the average those guys compared to those would be simpler so now they actually have less mean ACN so that's also reflected here so those are the by the class same idea but divided by classes okay so once you divide by class of course our lot our data set is larger right because you have more loss in each class and the data seems to be smoother and again the order is very well reserved all right so that's about ACN and what about the rise the rise we do not have a theoretical formula in the unconfined case so I don't have something to compare with so instead we just did the data fitting so you can see that the squared rise the mean squared rise okay so here we call the fountain changes means the overall thing so so if you look at the overall regardless of what not type then they behave like this so or fix the lens right so the lens is fixed so as you increase your radius the North gets less and less complicated right because the lens is fixed you have less and less confinement so and that causes the pointer here okay excuse me yeah that's yeah that's over all of them yeah yeah so that gives you the overall behavior a sense so now if you fix the the confinement radius and you change the lens then it then it does go up like a in a square in a squirrel so it behaves like quadratically okay so this is the rise for the lens with the fixed the radius we just want to see when you look at the absolute the rise of those guys this after rice we did the rise actually this is not rice square I believe this is the rise so that that's market wrong I think so you look at the the number here the order here maybe that's right the graph is right okay so you know in the in the rice case those guys actually kind of give you a sense of the ordering right how complicated they are but you see in this case it's no longer like the ACN case because the sign actually this has to do with the chirality now because they are sign we look at the projections the signs would come to each other so in this case you see the lowest one is the figure of eight not and that guy you all know it's a it's a car so you look at a projection in many projections the rise would come to each other so so this is actually not that surprising it's down there what is a little more surprising is the trade the tribute one is actually slightly higher than it and then you got the purple one which is the six the crossing six one very close to with this this group is together so the four zero six those guys are together and this one is the travel and this one is a the travel and a they are like in one class and the rest they are in a class and this is actually quite interesting because if you look at it the rise from the ideal not rise this ideal not rise if you look at the ideal picture you look at the the mean rice from this ideal picture and then if you look at it the rise from the rough sense not a table you just take the rice because like for six crossing they are they are however how many so for the five right so you have the rice is five well how do you get a five how do you get it the average here well they are two not in there are you just take it a tool which is in this is just you know mathematically just added to two and a divided by two so this is a very rudimental right so for the ten crossings however how many not you have for each one of those you got those integers you added up you divided by the total number of not in there then that's how you get those numbers but if you if you look at this numbers that's how those guys are ordered in terms of they are their rise here and they are actually marching with the ideal not rising exactly for this same group and so you can see this three in the first group they are higher than the rest right well ten in this case you could go either way so in our picture ten actually was with this upper group and three in the eight in the next group and the six of four zero are indeed in the in the next group so they actually are reflected in here that order yeah so even in the confined case that that is to say that in those overall things are indeed reflected somehow yeah so the order because the since change you can see sometimes it calls over and we you know when they are close again you know there might be a data just error you know so noise right but you for the when they are far away you're pretty sure that actually is turning you the trend oh that is what I give to my faculty members whenever I have a department meeting so and that tells you how many people we have from different regions okay and if you see your name you don't see your language in there that means we need you because we want to be we want to be diversified you know apply so we are hiring five people in the next year so spread the word and yeah so thank you