 Okay, so welcome to GSS, and this week we have Antoine, who is going to talk about knots. So, my first time, for example. And I have to start with, I guess, an apology or disclaimer, perhaps. I was introduced to knots working with physicists, so we were looking at this as a magical tool box, which allows us to say a lot of really great things. But I have to admit, the goal, when I started looking at this stuff, wasn't to try and prove anything, obviously. So, I'll have a few baby proofs in there, but mostly I will state a bunch of really, in my opinion, interesting facts about knots and what they allow you to do and bring all these tools that they allow you to do. So, I guess, first of all, I should tell you what a knot is. And really, the key idea here is that we're going to define them in such a way that we can only worry about simple diagrams we're going to draw, right? Because they're going to be curves in arc three, which can get messy, so we're going to restrict it so that all we have to care about is knots we can draw. So, to maybe not just motivate the definition, let's look at a couple of things that can go wrong, if we're not careful, I guess. So, basically, because you want to have the right notion of knot equivalence. You want to have a sensible idea of when knots are the same, because, for example, if I look at something like this, so on something like this guy, and knot to which I, I mean, I just take a loop and I start knotting it at the top. But then I could say, okay, clearly this knot has to be the same, the same thing but with a smaller loop up there. I mean, just as this is sensible so far. I mean, the loop doesn't say the same, but the knotted part gets smaller. But surely these two things should be the same. The problem is, if our notion of equivalence is too weak, then by taking the size of this little knot there to zero, we would get that all these things are equal. So we have to make sure we rule this out, because obviously they're not the same. And the other thing we have to be careful about, because they can get nasty, are those what are called wild knots, because they get pretty wild, pretty messy. And it's essentially the same problem again. We have this pattern here, but let's, and then I add some sort of big loop and then I'm knotting things. But let's say that I keep adding tiny little knots like this. I keep going on, what do I get? I promise you they will do lots of pictures, so. And I keep adding little things, I keep adding them smaller and smaller. And again, I don't want this to be a knot, because it's going to be a horrible mess. So that seems like you've got no reason for me. Anyway, so this is basically two things we have to be careful about. So now we can define knots in such a way and knot equivalents in such a way to rule out these things. So we will say that a knot, a continuous injection of S1, and you can play the same game if you go up in dimension with both of these guys, but obviously it gets harder to draw, so I'm not going to do that. And then we want to know when two knots are equivalent. So two knots are equivalent. And the idea is that it's not enough to say the equivalent when they're curved, when you can deform their curves into one another. It's a wing more than a wing to be able to deform the whole space into one another. And this is called an ambience isopic. You have to modify the whole space with it. And that's a equivalent when there is an ambience isopic between them. Your one doesn't seem to rule out this knots. Yeah, there's going to be a creep for that. We're going to say this is a wild knot, and then we're only going to care about the other guys, which are whole tables. So this is still a knot, but it's wild. And we're very tame, because we're mathematicians at the very least. No, it's still a knot. And for the rest of the talk, when I say knot, I really mean tame knot. We're in an isopropy, a manual isopropy, so I e some, let's say, k1, it takes the knot k1 to k1 at n0, and to k2 at n1. Well, essentially it was the problem here. We have infinitely many knotted things. So it's going to be tame when it's equivalent to a polygonal curve. It's just a bunch of three lines. It's tame when it is equivalent to a simple closed polygonal curve, so not self-intersecting. And I link them with three lines. It's not self-intersecting, and that's closed. Okay, so I know what a knot is, but that's not... Do you require any continuity in the bubble T? Yes, we do. I should have said that. The whole F is continuous. Yeah, but that's only when I fix T as a map from R3. Yeah, otherwise it just has a bunch of math that don't need to enter there. This is like really a composite problem. So k1 and k2 are the knots, right? k1 and k2 are the sub-function. The sub-function is R3. You can say F of k1, 0. Are you ready for composing? Yeah, okay. This guy is going to be the image of k1. Oh, you just required the image. Yeah, just the image. Oh, okay. I guess the right method is something. Yeah, sorry. So, again, in the definition, the knot is... I mean, it's the same thing when you deal with curves in general. You just end up talking about the subset, that is the curve, not the T. But that's a good point. So I guess when I say knot, I mean tame knot, and I'm just looking at the subset, unless I read the embedding itself. Okay, but basically this is... Okay, we know what we're working with, but we don't know how to work with it. And how to work with it is basically because I've drawn these things now, like these already examples. I mean, each of those sectors is an example of a knot diagram. So I just want to know what we're allowed to do when we're drawing these things. And basically then say that we only have to care about this. You know, it's going to be... Since we're only considering very simple knots, it's going to be... It's pretty clear that... The problem is going to be what? The problem is going to be... I want to... So this thing is not this in R3, and I want to make sure that there is a projection that is good enough where... If I'm looking at this guy, this circle and projecting along, looking down this, then I'm just seeing a line. Obviously I don't want that. But you can always move your projection a little bit and have something that's good. And by good, that's an definition that's seen what projections are allowed to use. And the idea here is going to be the same spirit as requiring knots to be tamed, which is essentially having finitely many things here we want. We only want to worry about finitely many points where the projection is degenerate where it crosses. So this is a quick sign of why it's not interesting to talk about knots in R4, because basically you can always unknot it, right? So this is not going to feel knot. This is going to be a link, right? I have two circles, so they're sitting somewhere in R3, but now if I put them in R4, I can cheat and say that I can... Because I can just say they're in R3 cross zero, say, and then I can move this part like R3 cross one, and just move it up somewhere, I move the other part down, and now they're in completely different spaces, so I cross it and it's nothing interesting. So in the definition of knot, we're going from S1 to R3, and if you go up in dimension, like the same amount in both, you get more interesting knots, but if the difference is not two, things are not interesting. Okay, a projection from R3 to R2 is regular. Potentially only finitely many bad things can happen. So finitely many... So like every time a point is not two, there's only finitely many points that are not two more than one, more than once. So regular is like pertaining to some things to knot? No, it's a projection. Yes. Otherwise... Yeah. Oh, and actually, okay, this is just vocabulary, but I'm going to keep talking about knots. Most of the same things can be said about links, which are just taking copies of S1. This, for example, is a link, because I have two copies of this to just use the word knot throughout. So finitely many multiple points, and each multiple point is only a double point, so why am I ruining out, like, every time? I don't want... I don't want two strands to look something like this. This is bad, but of course I can just change the orientation of it and I'll leave it. And the second one, what I don't want to go wrong is, I don't want to have something like this, like this, this is bad, change the orientation of it, I always get only a double point, so every multiple point is only a double point. What else do I have to worry about? Yeah, I have to worry about this, that I don't have this strand going like this and then another one doing this, like, every time they intersect, I want the intersection to be transverse. All intersections I'm doing is building up to saying, all I have to worry about are knot diagrams. So, let me tell you what a knot diagram is. It's actually a knot diagram where we can make sure I'm not, yeah. So, it's going to be a regular projection, but I have to tell you more, because that's a perfectly acceptable projection of a knot, but I don't know how to reconstruct a knot. So I have to make sure that I know how to label the crossings and for some purposes, the orientation will be important as well. So, a knot diagram will be a regular projection of a knot where I label the over and under crossings and the orientation. A regular projection with enough information to reconstruct a knot, such that under and over crossings and the orientation is over. And now, here's why all of this works. Essentially, two knots are going to be equivalent if the knot diagrams are equivalent in a very nice way, because essentially, there's only, I mean, right, so a knot, well, it's still there. Knots are equivalent when there's some taking one to another. But actually, it's going to be, but I'm going to have a lot of things to check, I have to check all of those. So instead, we're just going to look at, it's enough to look at any knot diagram of each knot, and then we have a sort of finite number of all these three of them, three moves that we can do on the diagram to just get from one to the other. So there's not much that we have to check, but that's fine. The proofs are not terribly, there's nothing necessarily deep. It's just going to take, it would take like half an hour or 45 minutes, and we wouldn't get to this fun stuff, so they just think about that. To knots are equivalent, of course I should tell you what you can do with knot diagram. So those are called Ryder, Meister, and the surprising thing about those is not that they're allowed, like that's pretty obvious. The really important thing is that there's only three of them and they're enough. That's all you have to do. So the first two are pretty obvious, just telling me that if I have this or that, it's the same. It's pretty straightforward. The second one is the same thing. If I have this, it's the same as just two powerless times. That's pretty obvious as well. The third one is much trickier, but what happens when we have something like this. Essentially, all I want to say is that this guy is underneath the white one and above the blue one, so I can shift it to still be underneath the white one and above the blue one, so I can shift it there. That's all I'm saying. So I have... And two knots are equivalent when you can go from one to the other by finitely many of those moves. It's not surprising that these things are true, not equivalent when they're related by this. What is surprising is that this is all you need. You don't need anything else. It's the same after finitely many or I even match the moves. Okay, so now we can get to work and we can actually draw some knots and say things about them. So this is really nice. All I have to do is I just have to draw pictures and then I'm good to go. The problem is it would be really nice to say that starting with any diagram or the knot, I have a nice way of just simplifying the diagram at every step to go towards. Like if I have something that is just a really complicated way of the unknot, I want to know that I can just simplify my diagram. The problem is that there are some examples of knots where you start with a certain number of crossings in your diagram and you have to go up before we start going down. And I'll give you such an example now. So this is nice, but then there's something that's likely better than this. So given the number of crossings that your diagram originally adds, you know how many moves you'll need to go to the simplest form. But that bound is really just a big upper bound. So you know it's far into many, but it's not very useful. Statement that they don't know how many moves you would need. They don't have a good upper bound for how many moves you need to simplify knots. It's a pretty elementary question, but it's not going to be very hard. Because you have very simple examples, so let me see if I can draw this one. So that's enough, right? And the problem here, I mean the reason why this is an interesting knot is that it's quite clear that this is a really easy way to simplify the whole thing. I have this one here that's underneath the whole thing. I just want to push it away, like push it all the way up there. Like that's easy to see. But it's an easy example of why you have to make the knot more complicated, the knot diagram more complicated before you make it simple because I can't simply push this across. What I have to do if I do one of those at a time is I start by saying okay I want to take this trend here and put it there and then I want to take this trend and push it across here and so on and so forth. I do that step by step which makes it trickier. I mean it means that I don't have a simple monotonic sequence in terms of number of crossings and the number of crossings has to go before. Right, so I know I have knots. I know what I can do with them. I don't actually told you that I have anything that turns out not to be the one knot. It could be that they're all equivalent to the other knot. And there's a really... It's not hard to show that there's a non-trivial knot but there's a really cute way of discussing that that is due to William Thurston that was a huge geometer and he's got to really easily explore it. So I'm not going to look at it. Okay, so first of all I want to use this discussion to show you that there's a knot that's not the other knot. So let's talk about the... Let's talk about the other knot first. Find interesting properties of this and show that the other knot has different ones. And we're just going to consider this loop to be some sort of portal. Okay, let's say it takes us to Narnia. Okay, so if I go through the portal I'm now in Narnia. I'm walking around in Narnia seeing lots of weird things. And then when I walk through the portal again, it doesn't matter which direction I'm going. If I walk through the portal again I'm back in the real world, right? So the knot doesn't do a lot of interesting things. It's just Narnia and the real world. And that's it, right? So now let's see if we can do some more interesting stuff with a more interesting one. All right, so now let's see what happens here. So let's say that I walk through it again I'm back in the real world. Then let's say I go through this one. And then I meet lots of adventures. And then you come back. You come through this portal and you come back. And now I want to see where these other two portals lead. Do they lead anywhere interesting and do you relate nicely to the others? And, okay, so let's look at this. Let's say that I go through let's say that I go through the portal A first. So I'm going through A and then this. But if I just move this trend through here this is the same thing as if I go through the middle one immediately from this color, right? I'm just taking this guy and moving it here. I'm doing this because this trend is above this one and below this one. So if I move it here I'm still going to be above this one. So this portal here is the same thing as doing A, B, right? And now let's see if I can play a similar game here and it's not a surprise. I can. Let's go through one here I get that I can get this. So if I go through I'm going to come back through B and underneath here. And this is the same thing as just going through this guy, right? Same thing as... So this guy is also equal to what? It's also equal to doing B minus one first and then A, B, right? So now I found out what did I find out that so this now turns out to be different than the other one, right? It's creating a lot more portals but they're related a nice way. I guess Matt I start here. If I go back if I go through it again I could just get back here. There's nothing interesting. Or I can do this and then up in A, B I can do this and then up in B, A and then the whole point is there's only one... Then what can you do? You can see this guy as a shortcut basically. It's taking me straight from this to there but it's kind of a boring shortcut because I can just go back and forth. And this guy is actually much more interesting because if I go along the middle I now have to go through the portal three times if I want to go back home. It's a different not a different situation. This generalizes to nice way of describing that because essentially all we're computing here is the fundamental group of R3 minus the other. If you're in Narnia and you want to go back home does it matter which side of the portal you go through? No it doesn't. It's just the portal like Narnia at home doesn't matter which way it is. Seems reasonable. If you accept that Narnia exists it's not a group at all. Can you briefly mention this is the fundamental group of R3 minus the other. So does that mean that the fundamental group is generated by A and B subjects of the A and B? Yes, that's exactly it. There's nothing else to it. Alright, so now this is just like some basic generic not stuff. I want to talk a bit more about essentially more of what drew which is that really nice knot you can actually say that you can knot the whole space using the knot. The idea is that you can decompose the whole space in a way that is consistent with the knot. I'll make that precise in a second. First, as an intermediate step between this idea and what we've done so far let's talk about another way to distinguish knots which is to talk about the whole structure you can do which is given any knot you can find a surface whose boundary is that knot. That surface is going to be orientable as well. You're not going to know what the commode is for. The construction is simple. This is interesting because basically what we want to do now is what we want to do. What we're wondering is if it's possible to have a knot sitting in R3 and then to have a whole bunch of surfaces whose boundary is the knot that fill up the space without self-intersecting which doesn't seem obvious that you can do that. I mean the knot might just twist in a weird way and so on. But first of all we can check the case where we just want one surface and that is something turns out you can do for every knot these are called C for surfaces. Essentially I just want to find a surface such that the boundary of the surface is the knot or link. And the construction is very simple so let's talk about the same knot again. This is my knot. And I have to get an orientation so let's take this. Now I want to construct a surface whose boundary is that knot and essentially the only thing I have to worry about is what I do around these crossings. And there's a simple rule which you follow which is that you're going to glue the strands following the orientation so I'm going to I'm basically going to delete the crossings following the orientation of the knot so I want now this thing instead. So I end up with what? I end up with two disks and think of this thing that this is an R3 so think of this as I have one disk here and then the other one is kind of like sitting somewhere on top of it. And so and now I have to essentially glue the two disks in the right way such that the surface I get its boundary is the knot I started with. And again the algorithm like the recipe is always the same whatever knot you have so I'm going to have essentially little little strips going from one surface to the other but twisting because I want to make the the overall surface orientable at the end of the day so let's see so I I mean again I just hold the orientation right? If I'm along here I'm here along the original knot so I just want to go to this guy and I do the same thing everywhere I had to cross it so now I follow this one which gives me this okay and you get there and now you can see that at the edge I always have the same rotation so I can just orient this and essentially here I'm always looking at the same side of the surface for the other side I don't see this picture and of course I mean it seems like I didn't really do anything but I'm going from here to there because these crossings they're the same that's the whole point I want the surface whose boundary is the same knot so of course when I just look at the curve there it's going to be the same thing otherwise I'm going to stop this one I want to do this in different ways so you can see the twisting a bit more because here you always see the same surface but the twisting becomes a bit clearer with a second another example and the twisting is just to ensure that the final thing you get is orientable is that twisting in the top right to the right around let's see yeah because I start from the outside and go inside so you're right it was the wrong way the others yeah because it was going the other way the other way the thing probably wasn't even something interesting well it wasn't so much I know it no no no you're right I was going from in to out which is not what so no you're right and now so I'm right here then I start twisting and I'm still right here the color the color scheme didn't make sense before but no it's consistent okay and okay so I want to look at I exercise to the reader show that they seem not no there's actually there's a very easy way to see it in the same if I if I just okay I'm going to look at these two points here essentially what I'm doing by saying this is like I can imagine this is sitting inside a cylinder and then gluing the ends right I get the same thing and I can see that's actually exactly what I get here I can think about like my tourist going around here and have so let's say that my two blue points are right here and there what are these turns doing they're just coming over each other every time this and it's exactly what these guys do like this one comes over and then the other one comes over and then the other one comes over so they're they're the same they're the same not but we'll get a different representation of the surface which makes the twisting so again I get the same recipe like this and now two surfaces and now like this but now I have to add the twisted ends and again it's going to be the same guy just remember this guy with it somewhere and so my twisted ends look something like and now this one shows you a bit more going on with the colors because now so here I have some orientations called this positive orientation just rather rotation and now so see the this orientation just flips behind here as I go along the band I twist behind and so this is the opposite orientation and then it's just that it doesn't seem to be much going on here because it's coincidence we only see one of the surfaces and then that allows you to again distinguish not because you can define the genus of a knot to be the minimum genus of all the surfaces you can build on top of that knot so like these surfaces are going to have a certain number of holes like on those holes I take the minimum that I can do that have another boundary and then that's the genus number of holes that the knot has wait so just to be sure these two surfaces are the same up to uhhhh or quote unquote by is it the kind of how they're going to do a standard one well these two might no I think those two might I think yeah so let's see because I want to like take the bottom like the blue one flip it over and move these things around I think they're probably the same just flipping and twisting no right because like these three things remember these three twists exactly these three twists here so like these three I'm sort of confused by that picture actually because what's the middle section doing there like I could imagine a surface where you just see the surface part in the three connected components are on the outside but it seems like in the picture you drew at every crossing three of the four bids have a surface next to them I don't deny you three of the four bids over the four bids so when you have to one screen crossing over another that sort of divides the blackboard into four regions and you're drawing a piece of the surface of three of those regions and I'm not sure I see how that's physically possible so I have I have like this pancake here and then I have like three other things that can take I have pictures coming up at the end of this thing in 3D okay I don't remember if it's this one or that one I think it's no it's this one so we can have a look it's probably going to be easier otherwise we can stare at this for a while okay three pictures they're not of the same latitude right you need to have that as boundary that what is that at the middle of this yeah this is like okay yeah this is not all flat like this middle middle like sticking out this is like this would be like a good idea for a water park you have this middle thing and then you have like slides going down like these are slides it's like here it's like there and it's like here all right this thing that is sticking out and then if I it's like not you have a strip and twist it twice and then glue the end together to form the same like oh you want I know it's here but you won't get the same knot on the boundary well I guess you cannot twist this thing you cannot no no no you have a strip like with wrist and then you twist with the dough and then twist again and then twist again you have three of those and then you glue the ends together oh and you want to say you get the same thing can you form the same thing with just the strip yeah I think that's what you get like just here right yeah I feel like without the middle part you can visualize the like this oh the middle part makes it harder to do you can just take off the middle part then I can sometimes visualize the strip and now it's coming now you don't think this is the surface I want to know if it's the surface oh so you're saying you just take oh so that's for your boundary you don't have this going down is that the same thing that's also good so my question is first of all is this thing even though yeah you can have a problem because you have two different circles right I think it's a you start with a strip that has two boundaries ah yeah I guess if you do three of those I'm free to do so what I'm trying to do is some of us so you have one boundary it's a moment it's a moment it's a moment it's a moment it's a moment it's a moment it's a moment it's a moment it's a moment it's a moment so this either either its boundary is not the same as the component but two components or else it's not a random moment yeah that's a good point no no this is a this is a yeah exactly I think there's another one happening over there as well okay so yeah I guess it's a few problems I guess but let's move to the more the picture that's even like this one I'm not even going to try and draw which is when instead of having one of the surfaces I have now a whole family of those that fill up all of our free in a nice way because the boundary is going to be the knot and you know and they're not going to intersect each other that's the thing that's the knots that are going to decompose our free in a nice way I'll have to do that in a second we can always think of I mean actually that's how they usually define them you always usually think of think of now as I would say for surface because this is not oriented or not oriented sorry I'm here to make get people in focus so that's good yeah so the problem with the construction surface the one you get is the one yeah so we need it for the tax technical reason but you can always everything we talked about is fine because our treatment channel just out of the point you get everything we talked about is just more general planning and that's how they define us in general like okay so this is what we want we want you to be able to construct knots that we'll say that next to our S1 because we're sure in a second it will be here family of seaford surface so seaford surface is just the way of saying their boundary is enough okay I'm going to draw the big example I mean S3 now so this is a legal knot because it closes up at infinity so essentially like a simple vibration in this case it's just going to look something like a book like these pages coming out actually these things are called open book decompositions so this is what you have so okay for this thing which closes up at infinity is is the there's nothing interesting there as soon as I close it up and as soon as I knot it up like this it's not clear at all that I can find a whole family of these things that are going to cover my space okay I wouldn't be talking about this if you couldn't but so it turns out you can and it turns out it's really the representation is really simple and this is a a theorem due to the sky the sky which is the sky sorry we're curious about what's happening there but don't I don't know because this building has a ninth floor so this isn't just coming through the roof I don't think the ninth floor exists above here I think it's central there's gravity inside so even if there was a ninth floor it would just go through I know full well because this happened in my apartment alright so what this theorem tells you is it's essentially it's not going to be true for every knot that I can do such a thing but given essentially you can represent those oh I didn't write it but a knot such that there is a knot for which there is a vibration is going to be called fiber and it turns out that you can get a whole bunch of fiber knots by just looking at complex polynomials in two complex verbs because if you have a complex polynomial in two complex verbs it lives in like S3 isn't going to be inside of this so then if I look at where that polynomial is 0 I get some surface living in let's say like R4 pretty much I have S3 inside I intersect my complex surface with S3 and that's what's going to give me my knot essentially and provided my polynomial is nice enough in the right way then because what is the polynomial going to give me where it's 0 it's going to give me the knot and then around 0 I'm going to have all my I've got the angle of my polynomial so that the angle of the polynomial is going to give me how I index this family of fibers so Miller vibration theorem tells you that then if I took oh and isolated 0 singular point then if I look close enough I want to 0 as a map from this small S3 minus where the polynomial is 0 to S1 that gives me a vibration because so this this S3 I can just identify I can I can project down from S3 to R3 using say stereographic projection and then I have saying F takes complex like signs of complex language everywhere in R3 or it's 0 it's going to be my knot and then everywhere else I'm going to get the the complex like any on which of those pages I have basically an example which we've actually already seen that this guy is that this the truffle or not that's it that's that's all the information I need to give you for you to be able to tell me to write down at every point in R3 on which of those pages I'm going to be so the reason why physicists think about this as the knot knotting all of space is that now you can talk about this document interested in which is knotted singularities of vector fields basically because what do I like if I have a vector field in the plane like my singularities are going to look like what you're going to look like something where orientation isn't defined so in the plane it's kind of boring but now a thing if I take this in R3 that my singularity could be along the line now if it's along the line I could knot that line and I could knot it in non-trivial ways and so now that I have like a vector field knotted around this can I still assign a direction everywhere else and still get something the only singularity I get is my knot well this tells me that yes I can for some knots some nice knots for fiber knots I can actually not necessarily every it's a vibration so I mean if I take the preimage of here this is a really cool theorem for them because now I can have a vector field in R3 there's going to be I'm going to have a knotted singularity and I can write down the expression is easy to write down because I know like look here for the truffle knot if my singularity is a truffle this is all my polynomial is and then I just have to project that from S2 to R3 but ok I just use thermographic projection that's not too bad so it's a really simple way to write down some vector field which has a bus promised this is not the worst case yes because one of us starting out at the level of the city and one is starting out higher beam and dropping like that as you can imagine there's so we have to use umbrellas in this case so it's well in fact you can like see the mystery love how there's also like plates missing so as a result you can have this thing where a drop starts falling another drop beats it to the ground no this is I need my block software mathematics mathematics you can have I'm assuming on everything at the bottom where it's at where you can do that alright so that's the truffle knot with ok so there you can see is the truffle knot it does look like a water park sorry it does look like a water park yeah see it does have the slides this is the this is the surface we were looking at before I have this is my inside disk and this is the now very not this like outside disk and these are my water slides right so that's exactly what we had before sorry we're convincing yes yeah I don't know is it like a fan yes does this inspect itself no no no no no it doesn't well you can look inside let's look inside that's what happens see so it's it's all bent and I guess it's really neat and this thing how do you get this thing well you take this guy and then you know you do like you look well no I go back I start from 3 I look where the point on S3 is like you pretty much under that direction and then that point I know what my the value of my polynomial is so this is the surface which where the polynomial its angle is like 0 and then I can cycle through all of those and I cycle through the pages of my book so now you can see what's it's behind the it's not very clear it's so you have to think about this as so my not as like kind of almost planar and then because I had was here and then this surface is going to start like bubbling up in a way and then it's going to go up to infinity and then it will start closing up the upload so that's that's when it starts bubbling up on one side I'll show you more details for another not in a second that's where it's that's where it's at infinity now so the water part before was sitting exactly here and now I have basically what's on the outside of that so is this like the complement of the surface we've done before in a sense I mean it's not complement because it's not great so ok what you could say is that here they're going to be like continuous well even more than that like smoothly like if you take two of them together you still get a smooth surface like they'll continue nicely across the knot that is proved so in that sense and also I mean technically you require see-through surface to be like compact this is compact because I mean it has to be so like the other guy which now you know it was bubbling up and now it's going the other way so I mean this is this is basically so this is when the fiber is looking at say pi over two and it's going to be like the mirror image of the ones I have when I have three pi over two ok but so now I want to show you so now the one this one is the hopsling which is just the very simple thing which is just this guy right and you can do I mean this drawing the surfaces I did it for a knot you can do the same thing for a link so essentially this is a fiber link so in this one I have a few more so we have a better idea of what it the idea of like it's cycling through the whole link right so it's closing up then it's going to come through down the middle and then back the other way are these two is this a link of two things like together or knot the it's not a knot it's a link it's this guy two circles but they're linked together I couldn't put them apart they're not just these own circles and so ok this is as you cycle through the pages and I mean you also have to when you look at this remember I'm using stereographic projection so like it kind of everything that's inside the knot it's kind of squashed down inside like if you look at like S2 like the whole S2 or to the stereographic project all the upper hemisphere is going to get squashed and it's going to blow up so it seems like it's spent forever sheer but that's just that's the deformation from that I mean now that you have this in hand physicists can go and enter this in this simulation and see what happens when they put knots in the quick crystals in the water this is basically the surface I get if I were to play the same game with just doing it by hand the last one I mean it's the same thing but these this is another ok so these these links are so this is just so essentially I have three links that are pairwise disjoint but when I add the third one now they're going to be links so what do I want to do I want to add one that's going to be so now together they're linked but pairwise they're disjoint and ok you can see in the game I guess ok and I mean again a lot of a lot of how it's wrapped is also due to stereographic projection but it's not ok and you have and it gets really really intricate but that's it that's the fault