 Oh, okay, we're getting towards the end of this of this great week. Oh, forget it. This is still just myself So before I really start I want to say Thank you to a lot of people because this to mean I think a really great conference You know when you go to a conference, which is as widely spread out in terms of topics as this You can't expect to understand everything if you understand Half of what a quarter of the people said you're not doing so badly I think of these things and I think this is an exception because I reckon I understood about half of what three quarters of the people said And I think that's a wonderful event to have happened I learned a lot from this meeting and I'm sure where I'm sure we all did And I get compliments from my friends So I want to thank the organizers that's Pateo and Christian and do it and Enzo who did a fabulous job of putting all this together And I want to thank I want to thank ICTP for the for the facilities and And Erica for all her efforts in this I don't know how much we all had to interact with Erica over this but every time I had an interaction with her I felt she was Above and beyond the call of duty, and I think we need to you should say thank you to both ICTP and to Erica Okay, so Now I'll start talking about what I'm supposed to talk about So This is a bit of a switch and and it's a bit embarrassing that he hardly fits at all in this conference because I'm going to talk not about knotting in filaments and rings and such like and it has absolutely nothing to do with biology And very very little to do with physics but it's about the fact that you can also have knots in surfaces and It's a problem. I've been working on actually I've been working on it for a long time But I've been working on it quite quite a lot in the last few years. I want to try to persuade you that at least Entanglements in surfaces are interesting and sometimes you can prove things about the surface problem that you can't prove about the these rings not in three space So this is joint work with four of my friends My sheet at a poor and Chris Sotteros who are both at University of Saskatchewan She'd started working in this general area. I think when she was a Graduate student maybe she was a postdoc. I think a graduate student with Chris Buck's van Rensburg who's a York University in Toronto and with Summers my friend in the audience who is You know is at Florida State University They're the the cast of thousands Well, why should we care about surfaces? So I want to give you just a few examples of where surfaces turn up and where they might be interesting so the first of these is that Surfaces in clothes vesicles and red blood cells are a good example of a vesicle and In the case of red blood cells they the the surface is always a sphere. I mean topologically is a sphere You don't get red blood cells which are tori so We have this this example and vesicles We've also seen something about vesicles in Giovanni Dietler's talk. Do you remember he had These pictures of circular DNA on a surface and there were other expanded objects If it was low density and if the density went up then they became more compact because there was a kind of an external pressure So that's like a vesicle where you have an osmotic term If you change the ionic strength of the solution outside the vesicle then the the object can compress or or it can expand And red blood cells can burst if they if you change the osmotic conditions enough A polymer membranes is another thing on the same lines. You can have these form various kinds of vesicle systems in Magnets if you think of a an ising problem Series of up and down spins in the magnet you have little regions of up spins And if I'm in two dimensions, these are enclosed by by simple closed curves in the simplest case Or maybe more complicated things where the where the curves can cross or can touch I mean three dimensions that are enclosed by well three dimensional surfaces in four dimensions by by three manifolds So there are examples in in magnets where where surfaces might play a role But for me, I like them because they're just a natural exam a natural extension of simple closed curves. I Wanted to remind you of two things about Simple closed curves at least our lattice models of these if you if you take at the simple cubic lattice And you think of a simple closed curve in the simple cubic lattice So often called polygons in the lattice Then we know that as these polygons get very big the chance of them being knotted becomes very large But we don't know anything at all about the relative probability We don't know anything rigorously about the relative probability of a trefoil or a 4-1 knot or or an unknot for instance for these things so that that's an old problem been around 30 years and The rigorous level we don't know anything about it. It'll turn out with it with surfaces That's a problem. We can say something about under some circumstances So, um, here's some examples Oh, I should say that I'm going to talk about four or five problems If you get lost in any of the problems wait a minute because there'll be a new problem and each of them you can start again Because they're they're essentially independent of one another That's true to a very large extent so here's some examples of Surfaces and first of all, I want to think of surfaces in three space I'll pretty soon I'll be on the simple cubic lattice Z3 Z3 But just think of them for the moment in three space and these surfaces are going to have a single boundary curve So they're they're not closed surfaces So the simplest thing you could have is is a sphere something with a topology of a sphere And I just cut out a disc in this and so I'm left now with a punctured sphere and the punctured sphere is a disc So I've decomposed it into two discs So now I have a disc with a boundary curve and if somebody forgotten who reminded us this morning that boundary curve is a knotted But it doesn't have to be a sphere. I could take a torus and Cut a hole in the surface of the torus and now that could be knotted because now I have a Genus one surface and so it's a possibility now that I have knotting and the third one I want to mention is the Mobius band and The distinction between the first two the punctured sphere and the punctured torus well spheres and Tori have an inside and an outside and so I Remember that information when I puncture it and so these are orientable surfaces the Mobius band is non-orientable and My main reason for mentioning this now is to say that everything else I'm going to talk about from now on is about orientable surfaces We do have results on the non orientable case, but I'm not planning to talk about any of that today So it's all orientable now from time to time I'll forget and I won't make this proviso that my surface is orientable and you will quite rightly say hey, what if it was a projected plane and I'm telling you now in case I forget later. Everything's orientable Okay, so I'm going to be talking about these sort of things for the moment surfaces with a single boundary So here's a punctured torus. So in this picture here, here's my torus and here I cut out a little piece from the from the punctured from the from the torus So I start with the torus and I cut this out now. I have a punctured torus And what I want to do is to peel back this the boundary of this of this hole pull it back this way and this way and this way and This way and what I get I get I get a disc with two bands on it So I've got a disc and I've got two bands sticking off this So that picture there is a better from me from my point of view That's a better picture of a punctured torus than that one. So this is what you should be should be having in mind so that picture you could You can check has a single boundary curve and it's an unknotted As an unknotted curve. It's living in three space But it doesn't have to be unknotted. So here's another picture. So this is also this Surface with the shading on it is a is a punctured torus There's a single boundary curve and the boundary curve now is the is the figure eight not is for one So we can have we can have unknotted boundaries I go to another one. There's another one that I'm going to compare. So the picture on the left is Is a punctured torus. So this is the surface that you see here This is the disc if you like and here are my two bands, which are which are making this this punctured torus There's a single boundary curve and in this case the boundary curve is unknotted But I could take I could disconnect this I could cut this band and I could install some twists in it And then glue it back on again. That's what I've done in this case. I Keep taking this thing apart inadvertently. I just can't if the batteries will fall out if I'm not careful so in this case I've Take it apart install twist in both of these in the same sense and I've got a new surface and it's not difficult to check I mean you just do it that the single boundary curve still but now the boundary curve is a trefoil and in fact It's a it's a plus trefoil If I'd install the twist the in the other direction But the same twist in both of them how to get the mind I don't get the minus trefoil If I install the plus twist in one of the bands and minus twist in the other I've got the figure eight knot so I can get a bunch of knots of course They have to be genus one knots because I can't have a genus two knot and in on a That's the boundary of a of a punctured torus. I'd have to have a higher genus object to do that So now the kind of question that We were interested in this is mainly work with Bucks van Rensburg a long time ago The kind of question we were interested in was asking how likely is this compared to this if I do this at random You know in a lattice I embed these things in the simple cubic lattice and I ask for the likelihood of this to happen Compared to the likelihood of this what what proportion of the surfaces will be of one type and what kind of will be of another type So the tools we have for doing this only in fact answer questions to exponential order All of these things grow the number of embeddings grows at an exponential rate as you vary the area of the of the surface And you can ask Do these guys with unknotted boundary grow at the same exponential rate as these guys with a with a trefoil boundary? That's a question which in the in the case of simple closed curves. We don't know the answer to it Okay, so you you get the you get the question which is involved of the five problems or four or five problems I'm going to talk about technically. This is my father most difficult I'm not really going to tell you anything about how you do the proof for this case I'll try to give some idea of the proofs for the other cases. This one is a is a complete mess To try to to try it. Well, it's an enormously long proof. So I don't want to try to say much about it So I embed this in the simple cubic lattice With area n Which means it has n unit squares n plakets I embed this guy the with a trefoil boundary in the simple cubic lattice Area n and I ask how how does the number of embeddings of these as the area changes? Compare the number of embeddings as these as the area changes and It's a theorem all these two. This is a statement and then I'll give a theorem so I want to consider these Orientable surfaces now with one boundary curve, but I don't I haven't fixed the genus yet and Now they have some genus G zero so Taurus it will be genus one Embedded in in Z3 the simple cubic lattice And I'm going to call Sn of Kg G naught the number of embeddings of these with area n Modular translation. I don't care if two things I won't count two things as different if they can be translated into one another Which have where the boundary has not tied Kg and the genus of the surfaces G zero Okay, of course and G the genus of the knot has to be less than or equal to the genus of the surface So now you might like to make a guess from what you know about the business of random knotting Whether How the exponential growth rate of these things depends on the knot type and how the exponential growth rate depends on the genus so you are Pause for a second to let you make a make a guess No, and I'll tell you the answer It's a kind of quiz that students like in lectures where you ask a question and that promptly tell them the answer But here's the theorem So if I look at the number of these guys the number of these embeddings with Kg and G nor fixed I take the logarithm. I divide by n and let n go to infinity. So this thing this limit is the exponential growth rate Then first of all the limit exists, which is not easy to show and it's independent to both Kg and G naught so this says that the exponential growth rate of Tori whose boundary curve is the unknot is the same as Tori whose boundary curve is the trefoil is the same Those whose boundary curve is the figure eight and it's the same as the exponential growth rate of punctured spheres So it's an interesting question because are an interesting I think an interesting answer because that's exactly what we expect That's analogous to what we expect in the in the case of simple closed curves in in Z3 But we haven't got a hope we have no way no mechanism for proving that in the in the case of the simple closed curve Okay, so that's the that's problem number one So now I'm going to go up a dimension. I Believe this is the first talk where we've talked about anything above three dimensions. So now I'm in four space Now in the same way that simple closed curves cannot in three space spheres cannot inform you simple closed curves are Circles that one sphere so two spheres cannot enforce space You just need two extra dimensions for this to be this to be a possibility So I want to look at knotted two spheres in four space so we know if we're on the simple cubic lattice that the The number of embeddings of simple closed curves in the simple cubic lattice which are knotted is enormously larger than number which are unknotted So do we get the same thing in in this high dimensional problem? So let me tell you what these curves are going to look like So I'm gonna I'm gonna do this by spinning. So let me do spinning one dimension down to get the idea So if I'm in two dimensions, I Take a line in two dimensions and I draw a curve which starts and ends in this line And I spin it about this line So I now go I just rotate it rotate the curve about that line Then what do I get? Well, I get a sphere in and in three space I've just gone up a dimension by doing this So now it's the same thing when I when I spin about a plane So I start in three space And I have this this plane here at two points in the plane And I have a curve which starts and ends in the plane and otherwise lives in the half space defined by the plane once I do the other of the half space and I spin About this plane in much the same way that I spun about this line here So I go up a dimension each point here on this curve becomes a circle in four space So what I get is a two-sphere in four space And now there's a there's a nice theorem which says that This is vague, but I'll say it vaguely that if this bit here is knotted It remains not of when I spin the fundamental group is invariant under spinning is what it's saying So I know how to construct knotted two spheres in four space here's an example of that's the most famous example the spun trefoil. It's been known for a long time and We can't do that. Oh go back. I'd like to be able to ask the question if I'm now putting these The surfers in the hype in the four-dimensional cubic lattice z4 What can I say about the probability of knotting among these among these two spheres in z4? So That's the question you'd like the answer to it's time to quit because I don't know But I can do a sort of a toy version of that problem and let me explain what I mean by that so instead of doing the whole of I'll do it in d-dimensions because I will need it both in three and four dimensions Instead of working in the whole of the of the lattice. I'm going to look at going to work in a tubular sub lattice So if I'm in zd, I'm going to take I've coordinates x1 up to xd for the their integer coordinates in zd one of these is going to be Half infinite, so it's going to be z plus and the rest of them x2 to xd are going to form a d minus 1 cube so what I've got is It's a high-dimensional cube cross an infinite line That's the that's the idea. I don't care how big this cube is you can make it any size you like but I do need this boundedness in these directions and I'll explain probably on the next slide why this is why this is handy and why it allows us to do the problem The wire tube is easier. It turns out. Well, they control the The space in all except one of the dimensions things can't get too big in all except one dimension and Secondly because we now have a sort of a quasi one-dimensional object turns out we can use linear algebra techniques to solve these problems And I don't know whether I'll have time at the end But if I have time I'll say some something more at a technical level about why those two things work But you should think of the fact that because we have Because it's bounded in in d minus one dimensions I I never have to do too much when I'm doing things in those dimensions And I have a sort of a preferred direction that I can run along and it's that preferred direction Which allows me to to prove some of these results so if I have These and two spheres in tubes in z4 and I put them so I have this tube I don't care how big the tube is l by l by l by l by l three dimensions And then infinite in the other direction Then well how you can read it yourself all except exponentially few Sufficiently large embeddings of two spheres in sufficiently large tubes in z4 are not it so that's that's exactly analogous to the Lower-dimensional theorem about simple closed curves in in z3 I Say whether or not this is true without a tube constraint is open that's something Do it and I have been working on for 30 years. I think it's a hard problem. It's it's difficult to do well either. It's difficult to do or we Just didn't quite catch on to what the clever trick is that somebody else will spot it before the lectures Oh, I once heard Paul Urdosh give a talk you know he used to ask these famous questions in his lectures and Somebody said to him and he would offer a monetary reward if you solved his problem Somebody said to him what was your biggest mistake in this and he said I gave a problem for a hundred bucks and a guy in the back Have back row solved it before I finished lecturing so So you should get to work on this This part So how do you prove this so I'm not going to say very much about this but Tell you two sort of facts which are which are key ideas So if I think of of two spheres and I connected I'd take the connect some of those two spheres I get a sphere So if I take a sphere and I connect some with a sphere, which is knotted I get a knotted two sphere if you like because Alexander polynomial multiplies under connect some that's one that's one argument which would do that So if I could prove that all two spheres Contained in their connect some a knotted two sphere. I'd be home and dry Now it turns out that the solution that is a pattern theorem I'll explain in a second what I mean by a pattern theorem, but here's another way to do it If I'm in four space, and I take slices Perpendicular to this long direction. What will I see? I'll see bits of three-dimensional space. I might see a knot a single Curve in that not not a link and not something more complicated, but a single simple closed curve It might or might not be knotted now There are lots of knots which can't occur as cross sections of the two sphere in four space The knot has to be sliced so three one can't occur but lots of knots can occur Some of those knots when they occur could occur in an unknotted two sphere So the square knot can occur as a cross section of an unknotted two sphere But there are some knots proved by by do it in 1960s sometime which if they occur in a cross section of a two sphere ensure that the two sphere is knotted so six one will do the trick and nine do it What nine forty six will do the trick? And so what I another way to prove this is to take these little slices and show that with high probability At least one of these slices is six one Or is nine forty six and if you can prove that you know that it's not it So if you can prove that that happens in all except exponentially few cases, then you know that it's not it in all except exponentially few cases So what you need then is to prove that this thing happens and that's That's a pattern theorem. So what is a pattern theorem? think of a zero one law in probability imagine you have a sequence of independent events and Something could happen every time If it could happen every time in the events are independent It will happen with the probability which approaches one that you have more and more events So you might even flicking a coin and saying I'd like a hundred heads heads in in succession Well, if I split my sequence of coins into groups of a hundred a hundred heads is not zero Probability that's probability one half to the hundred. That's not zero And so if I now instead of doing it a hundred times I do it an enormously large number of times the probability that never happened is one minus the small number to the power the number of Groups of a hundred that's going to go to zero and so the probability that it happens is going to go to one pattern theorems don't have the advantage of independence and So the trick in proving a pattern theorem is to control the dependence in the problem And that's where all of the difficulty in the all the technical difficult difficulty comes in Anyway, it turns out that For these two spheres in tubes you can prove a pattern theorem and you're home and drive by one of those two arguments Well, why two spheres? Why don't we look at two manifolds without boundary? So now I have oriental two manifolds without boundary for this talk. So now I have spheres Tori and so on any anything as a two manifold with no boundary just that I mean don't worry about it In a manifold we probably don't even care about it being a manifold ourselves We could probably do it if it was a manifold with singularities and things, but it's just sort of a simple nice smoothish surface or piecewise linear surface And so here's the theorem They're all except exponentially few large embeddings of sufficiently large embeddings of two manifolds without a boundary in Sufficiently large tubes in z4 and knotted. So that's the theorem But you might very well say what what does it mean? What I mean by a knotted two manifold Well, this is this question So any two manifold can be seen as sort of the connect sum of a bunch of things and If if I take a two manifold and I connect some with a knotted two sphere, then I get something which is knotted I don't destroy the knotting by doing this connect some so if I can show that a Two sphere appears somewhere in the connect some then I know that the object is knotted And so it's the same argument you use these pattern theorem tricks You have to prove the pattern theorem for this case use these pattern theorem tricks to show that a spun trefoil say will occur Somewhere in this as I've run up the tube Is there a corresponding theorem without the tube constraint? I Well, there isn't we we don't know it and we would dearly like to be able to prove the corresponding thing in the absence of this of this tube constraint Okay, so let's come down a dimension oops We are so let's look at two manifolds without boundary so close to many falls things like Spears and Torah and things but now in three space So can these be knotted well a sphere can't be knotted in in three space, but can we not? more complicated two manifolds So I'll draw some pictures So here's a Taurus and this is a Taurus whose centerline is knotted So I can't smoothly deform this into sort of the standard picture of a Taurus And so I can define this as as being knotted and I can ask about the relative chance of finding this if I look at look at say Torai in In three space But I can have more complicated situations I love this picture so this is Where the the other Generator of the homotopic group is is is knotted. So this is a Taurus. All right. It's got this This is a surface. This is the hole running through it And if you like the hole is a is knotted as a as a figure eight knot So who was the guy who wrote the paper about some cubes with knotted holes? Bing Bing. Yeah, so this is this is an example of an old example of many years ago by Bing Okay, so we liked I'd like to ask the question if I look at all Torai in Z3 in the simple cubic lattice with area n I let the area go to infinity What are the chances that the that the Taurus will be knotted or we'll have some particular not I maybe will be Will be am I stopped it to this guy or to the previous one the one in blue like there are two two Separate questions one can ask about this so I could either simply consider all Surfaces without boundary all too many folds without boundary in Z3 Well, I don't control the genus and so I'm going to I'm going to have spheres and Torai and higher genus objects there and Ask this question. What are the chances of it being knotted or I could fix the genus So I could just like for instance at Torai and the the techniques for handling these two are completely different So again, we we can't do it unless we have this tube constraint But if we have this tube constraint, then it turns out that we can say something about both those questions How's my timing I have 15 minutes or 10 anyway, yeah, okay Okay, so let's let's talk about surfaces where the genus isn't fixed so remember that these are these objects where they're going to be confined in this tube enormously wide to in in Z3 and If I could show that these surfaces with high probability Contained a knotted Taurus Something like this as part of they connect some then I'd be done So I can't unknot that farther up or farther down the the tube. I can't sort of get past it in the tube I have a little it'll be in a little region of the tube and that'll be knotted So that's the way to handle that problem and again, you do this by a pattern theorem Which says that any event will happen with high probability if the object the the two manifold gets big enough You And you're using the tube constraint there to use linear algebra techniques to prove the pattern theorem and in the second case Okay, so the theorem is that all except exponentially few surfaces in tubes in Z3 where the Where the tube is big enough and where the size of the Surface the two manifold is big enough and knotted with a probability which appears which approaches one Actually approaches one exponentially fast If you look at surfaces with fixed genus And again, I invite you to to make a guess about this so now I've got Let's fix the genus at one to get the idea. So now I have a Taurus in in this tube and now Remember the Taurus is is is this surface, but it's a closed surface. There's no boundary And now I'll tie this this center line in a particular knot It's tied as the Trefoil of the figure eight or something or it's unknotted so now Guess do you think that they are going to the not a ones are going to dominate or the unknotted ones will dominate or they're all about the same But the answer is that all of these things if I fix the knot if I fix the Genus say the Taurus and I fix the knot type of the Taurus not the boundary curve These don't have boundary curves anymore. I fixed the knot type of the center line of the Taurus Then all of these surfaces grow at the same exponential rate So again, that's analogous to this problem of simple closed curves in Z3. Well, we don't know that That simple closed curves with fixed with fixed knot type or grow up this exponential rate same exponential rate Okay, so this sorry. I didn't mean to do that. So in this case This is true and if I change the genus same thing Something with genus one tied in a knot or something with genus two with no knots all the same exponential rate fix the genus fix the knot knot type of the of the Of the center line the exponential rate is independent of those of those quantities so here's the Here's a certain kind of take-home message that if you have large flexible surfaces In lattices, they're typically embedded in complicated ways and that's surely is no surprise I mean everybody in this community must guess that if the thing is big and flexible It's going to be embedded badly in in in some sense so the The broad general picture is no surprise But the fact that you can prove some things For these surface cases which you can't you can't prove the corresponding thing in the simple closed curve case is It's nice So I just want to spend five more minutes Saying a couple of technical things So pattern theorems, so I've already said something about what a pattern theorem is. Yeah, you take some you take some Jim for a minute you take some geometrical objects some object and you take some little thing Which can occur in this object and you ask If it can occur will it occur somewhere where the object is big enough? That's the that's the idea There are a lot of ways to prove pattern theorems the these first these first appeared in the In the combinatorics literature in in 1963 in a wonderful paper by Harry Keston Which was called on the number of self-avoiding walks and he wanted to prove I Won't tell you what I want to prove because I'd have to tell you too much about self-avoiding walks But he wanted to prove a result about the asymptotic number behavior of the numbers of self-avoiding walks and Somewhat sort of incidentally in the middle of this paper. He proves a pattern theorem and Then goes on to use it and this was more or less ignored by the community for let me think for 20 years and then Suddenly a lot of people all together realized that this pattern theorem was a very nice result I mean tucked away in the middle of a complicated paper as a very nice result Which allowed you to prove all kinds of other things That was the first way to prove it and since then there must be half a dozen different techniques for proving pattern theorems But I'm going to tell you how the idea of how to prove it in when we have these tubular constraints So use transfer matrices and transfer matrices means you use linear algebra. So here's the idea So this is my tube running up here. This is just some object in the tube I mean I mean two dimensions, but don't worry about that. Yeah, I'm everything is going to be so vague no matter what the dimension is So I can regard this object as a sequence of little sections across it I can pick the height of those sections to be any finite number I want it could be you know They could be a single lattice spacing or a thousand lattice spacings doesn't matter and I can build up whatever object I have by by stitching together these sections so that they fit So if I have some section Here it has to fit with the section below that means that this these edges have to have to line up and so on Typically you break a long half integer values so you don't have to worry about edges in this direction You just have to worry about these things lining up So since I can build it up in this way I can represent I can I can sort of represent this process by a matrix in which the rows and columns are indexed by the Members of those sections the things which make up the sections and I get If I can't have section I following section J in this building up process the element of the matrix the ij element is 0 If I can then the ij element isn't 0 It's some positive number and the positive number is designed to capture whatever information you want to capture in this theorem So it's not just one, but it's some positive number Now so now I have a a non-negative matrix So now I have the glories of perimphrobeneus theory that I can use Now what perimphrobeneus tells you is that if I have this matrix and I turn some of the positive elements into 0 The spectral radius of the matrix drops It goes down and it's that that I mean I can't tell you any more than that because it's all detail That's what allows you to prove a pattern theorem and that's why these tubular constraints are so nice because they they allow you to use these perimphrobeneus methods So when you sat in a linear algebra class in your first or second year in university and learned about perimphrobeneus and thought What the hell is all this about? This is why perimphrobeneus did it Okay, so I want to and say that's the pattern theorem But I won't even read it out the other technique another technique that I want to mention is about concatenation and connect some so concatenation is a way of stitching together two surfaces So you take these two surfaces you delete a bit in this one a bit in this one You run a tube between them and that that connects the two together and when you do that when you concatenate in that way That's that's doing a connect some operation in in topology so if I take a sphere and I am concatenate it with a torus I go from this Genus zero object genus one object to a genus one object now I can make my sphere enormously large and the torus little and so now I go from having a big sphere to having a big torus When I when I do this operation that allows me to prove bounds in one direction Between the number of embeddings of a torus and the number embeddings of a sphere That's that's adding the the genus increase in the genus of the other problem, but I need a way of going back Actually, you can do that in any in any dimension You don't need tube constraints. There's a very well-known technique, but going in the other direction. It's not so easy So in the other direction what we want to do is take something like a torus in this tube and find convenient places to cut it So that now I have sort of holes in the torus and which I can cap off to form spheres so that would be a way if I play my game correctly of Converting a torus into a collection of two spheres In whatever damage in three dimensions. I am for the moment The nice thing about tubes is that tubes have a height function and There is a bit like a Morse function and so the Morse function allows you to know where to do these cuts Where to change the homology and and that's why the tube constraint works so nicely for this It works beautifully in a lot of cases, but there is a snag and I want to mention this snag as the last thing I say the snag in four dimensions Because if I have a torus in four dimensions and I take a slice through it that's a three-dimensional slice and I'll see two simple closed curves in three dimensions. They could be knots in this case they're both trefoils one of those is a plus trefoil and one's a minus trefoil and You can't cap off That to form a sphere because a trefoil isn't a slice knot And so if you have this problem of a torus whose cross section is a non slice not then our technique for capping off doesn't work So in fact this technique we have works in three dimensions and every dimension five and higher But it fails in four dimensions and we're currently trying to find a way around this this problem because At least some of our results simply fail in four dimensions where we can do it in every other dimension Okay, thank you for your attention