 about Kavala, and I'll be, and I can work with you. Great, thanks very much for inviting me. It's great to be here, and it's honored to be here at this conference for Tom. Of course, watching Tom, like Tom's work has been very influential, and talking to Tom has been great for the last 20 years. And Tom's talks have been very influential, so I've gotten to see a number of them over the years. I think I remember one of the first ones that I went to, I thought it was in Louisiana, but Lenny tells me it was at UGA, at the Georgia Topology Conference, so I'm sure he's right. And Tom had just proved something great. I actually don't remember which great thing, maybe it was Property P, or some other, I mean, you know. Perfect, it's hard to remember which great theorem. We have too many great theorems, but so I was really excited to go to the talk, and lots of people there, and so on, and Tom got up to speak, and he started showing his slides, and he chosen a somewhat unusual color theme. I thought I would do that in honor of his birthday, so, ah, it's not working, hang on, there we go. So the whole talk looked sort of like this, which, well, you could understand what he was saying, but it was a little bit difficult to read the text. So, let me talk about this slide for a little bit. Okay, so here's the plan for the talk. Well, I'm gonna give you the statement of the theorem, then I'll just talk about the proof of the theorem, and then I'll talk about some applications of the theorem. The sum of the transition pictures, if you guys don't know, there's a website unsplash that has free stock photos. So if you ever need stock photos for teaching or something, it's really nice. And I don't know whether at the end I'll be able to convince you that the theorem itself is interesting, but I hope that I'll at least convince you it's related to lots of interesting stuff. So the talk's a little bit more colloquium-style than some of the others. Okay, so statement of the theorem. So I should start by telling you what a strongly invertible knot is. So a strongly invertible knot is a knot which is preserved by rotation around an axis that intersects it in two points. So let's see if the movie will work over this. So here's a knot in the middle, and okay, well, sort of works. If you rotate it by pi, it gets back to itself. So that's done two of the pi rotations. And if you have a strongly invertible knot, there are two natural diagrams to look at there. So you can look at a projection to a plane so that the axis projects to a line. Okay, so the axis basically lies in the projection plane. Or you can look at ones the axis projects to a point. And the, okay, but you still see the symmetry. So the non-symmetric around rotation, around the point. So if you look at the axis, the one where the axis projects to a line did not rotate transversely to the plane. So the terminologies by Keegan-Boerl, but he called this a transvergent projection, and the one where it rotates in the plane is introvergent. And both will come up a little bit in the talk. Okay, so just there are those diagrams again. So on the left, we have the transvergent diagram. The rotation is transversed to the plane. And on the right, we have the introvergent diagram, introvergent and transvergent. Okay, so given the knot, so it's strongly a vertical knot, we'll keep playing with this one. There's an associated, well, okay, there's choice of half axis. So here I've chosen the sort of middle part of the half axis. There's what they called in a butterfly links. So it's introduced in a payroll. Issa, and I'll say at the end of the, later in the talk, what they were doing with this construction. But you can take the place that the knot intersects the axis and connect them up by a band. Okay, so just connect a band there. With some number of twists so that you get a link. So if you wanna pin down exactly which link, and they did, though it's not important for the theorem I'm gonna give you, you can just require that the linking number of the two components is zero. So here there's a blue component, a yellow component, I've rigged up the number of twists on the axis and the linking number is zero. I should have said feel free to interrupt with questions. It's supposed to be sort of understandable. Of course it's hard to interrupt during a slide talk, but still. Okay, if you take the quotient then you get a knot. So this is the quotient of that. We just kept say the yellow part and then it gets connected up on the left side, outside the fundamental domain. And the quotient lives in an annulus because there was still this axis, okay intersect they're not there, but now it's disjoint from this link and so it descends to an axis disjoint from the quotient knot. So the quotient naturally lives in an annulus. Okay, you can do the same thing in the introvergent diagram. So there what this taking the butterfly link corresponds to is you take a resolution of the middle crossing, the crossing that intersected the axis of symmetry. And really I should have put in, I think I acknowledged that there, really I should have put in some number of twists in the middle to get the blue and yellow to have linking number zero, they don't, but the picture was already so complicated that I didn't want to make it worse. And then again it has a quotient. So there's a quotient drawn in the introvergent diagram. And the number of twists you put in the middle to get it to have link number zero will affect how many times it winds around the axis there. Of course in this picture you might as well do the other resolution also. The other resolution gives you a knot. That's why there's only one color. And the quotient is the same knot. So this quotient with the other resolution is the same knot, but the axis lives in a different place. So you'll notice that this part of the quotient moved across the axis there. So here's a theorem. In the next couple of slides I'm gonna tell you what some of the notation of the theorem means. But the theorem is that given a strongly invertible knot there's a spectral sequence starting for the Kevon ophthalmology of the knot, okay, blown up by a factor of Laurent polynomials converging to the kernel plus co-kernel of a map F plus from the annular Kevon ophthalmology of K1 to the annular Kevon ophthalmology of K0. So the map on annular Kevon ophthalmology there. Okay, I'll tell you what, remind you what annular Kevon ophthalmology is on the next slide and I'll tell you what the map F plus is on the next slide or the slide after. Just before that just some words about convention. So I'll say this again on the next slide but my convention is that Kevon ophthalmology, the differential goes from the one resolution to the zero resolution and almost everything is with F2 coefficients. So like the reason I can tensor over F2 is because that was secretly Kevon ophthalmology with F2 coefficients. Okay, yeah, I mean you mean geometrically. Not as far as I know, but that's a good question. So I will tell you what the map F plus is and then at the end I'll ask what this means. Yeah, good question. So I should repeat the question. The question was what does the cone mean? What is this cone in terms of other more familiar things? Let me tell you what it is, well let me remind you what annular Kevon ophthalmology doesn't tell you what the cone is just sort of literally. It means, well stay tuned for two slides. So just a reminder about Kevon ophthalmology itself. Though most of the audience knows this, you start with a Kevon ophthalmology, you start with a knot diagram. Okay, there's one of my favorite knots. You look at all of the ways of resolving the crossings, you can do a zero or one resolution to each crossing that gives you a cube of resolutions. So here are two crossings, two dimensional cube. Each edge in the cube is associated to a kabordism which either two circles merge or one circle splits into two. This is a cube of kabordisms, one plus one dimensional kabordisms. Now there's a TQFT associated with this Frobenius algebra which is Z-adron X one X squared and the co-unit is that. So what that means, so I'm gonna compose the, we had a map from the cube, so the cube is a map from two to the end, that's just a, just mean the category that's shaped a cube to a kabordism and you can compose the TQFT, you got a cube of Boolean groups. What that means is you replace each circle by a copy of this ring A and any time two circles merged, you applied the multiplication, any time a circle split, you apply the co-multiplication and you take a total complex of that and you take homology. So you collapse it that way and you take homology. Okay. Annual equivalent homology is the same except with an extra, you take, there's an extra filtration, you take the associated graded. So suppose that I started with this annular knot on the left, so that's got a, the annulus is right in the middle of the left circle. Okay and I look at this cube of resolutions, well there's a hole is inside one of the circles in each place. Now assign a winding number to each of the circles, either plus one or minus one or zero, zero if it doesn't go around the hole and one if it goes around counter, sorry one if it's labeled one and X if it's, sorry and minus one is labeled X. So here this circle goes around and is labeled one so it has winding number one goes around positively, this one's labeled X, it's got winding number minus one goes around negatively. It's not hard to see or hard to show that that gives a filtration on the Kovanov cube. So every differential either preserves or decreases the winding number. Every term of the differential either preserves or decreases this winding number. So in this example, the arrow in black preserves the winding number and the two in red decrease the winding number and you throw away all of the differentials that decrease it. You take the associated graded and then you take, so that means throw away all of the differentials that decrease it and then you take homology. That's the annular Kovanov homology. Are they're not oriented? Good question. So the story is actually not oriented So what's up with the orientations here? I orient it counterclockwise if it's labeled one and clockwise if it's labeled X. So it's the Kovanov generator that's determining the orientation. Thanks, John. More questions? Yeah, so you can do it with more dots. There's a recent paper working out sort of the details of that theory by a student of Petkova's, whose name I'm blocking. But it works out the way you expect. So you have a filtration by Z to the N say and it works out how you expect. Well, you have a, I mean, once you have, so if you put in more dots, you get a filtration from each dot and that gives you Z to the N and you can collapse it just to a single filtration if you want and, well, it'll be an invariant of the, so what's an invariant? It's an invariant of the not in the N times punctured plane. So, and that's actually, I mean, mostly that's a special case. The original constructions I just didn't give an attribution. So this originally appeared in a work of Asaita, Przhtitsky and Sikora and they looked at knots in thickened surfaces, any eye bundle over a surface so that the total space is orientable. And this is the case of the surface, it's just the annulus, okay, but you could take the surface to be a, N times punctured plane and the construction goes through. They, their filtration is a little bit more complicated in that case than the one I just said, but it's close. Yeah, more questions. I didn't attribute Kovanov's knowledge either, it's due to Kovanov. Okay, I, there was one more piece of notation, the theorem I haven't told you what it is, so let me tell you about F plus, this map F plus. So my two different resolution, my two different quotients, they just differed by moving part of the knot over the base point. So I need a map that moves part of the knot over the base point. So here's how that works. It's a composition of two maps. You give birth to an unknot labeled one that goes around the, the axis, and then you merge that unknot with the, with the knot. So you do the annular Kovanov merge map. It's a composition of those two maps is this, is this map F plus. I claimed to have explained that. Questions about that, Danny? Oh, great, yeah, so this map is associated to moving, this F plus is associated to moving this strand like this little piece of the diagram across the base point. So there's a like mark point coming from which piece of the diagram moved across the base point. You didn't, yeah. There'd be, that's right. So F plus is depending on like, that's right, two different targets. That's right, that's right. Good question. So this is not the first time these maps appeared in the literature. There's some, there's a paper of Akhmachet and Kovanov from not that long ago, 2020-ish. That introduces a family of cabortisms for annular kovanovology that allows the cabortisms to move through the axis. But then they decorate, they keep track of the points where the cabortism intersects the axis and they labeled one or two. So each of those points is labeled one or two. And they have some of these, it's modular sum relations. And this, so what they were doing, so I was gonna say less, they were doing, they want a sort of foam style construction of annular kovanov homology and this was the input and they did it also for the SL3 case. And so this is, there's an obvious cabortism here which is you just, I mean going from there to there which is you move the knot across the base point and this is the map associated to it. And the reason it factors as this composition is because you can factor that cabortism as a cup like that and then a saddle, which is what we did here. So this doesn't really answer Tom's question sense of I haven't identified with something that was familiar to you unless you read the Akhmachev kovanov but it identifies at least with this construction there. Okay, so since we, I just thought I would put in the main theorem and also put in the gradings but since we have all the notation. So this is the theorem you saw before except that I stuck the way it affects the grading. So it does violence to the homological gradings so you just sum over all homological gradings but then the quantum grading gets mixed up with the annual, the k is the winding number around the axis and they get mixed up in a sort of controlled way. It doesn't matter, I mean it's gonna come up with one point later exactly what the gradings did but I would ignore it for a first pass. Great, you know this is the best I could get Microsoft PowerPoints equation editor to do with a big O plus with subscripts but that's a feature, not a bug. It incurs you not to type a bunch of mathematics on the slides, nobody wants to see slides with a bunch of mathematics. Yeah, sorry, say it louder. I still didn't hear you, why not humor? Beamer, oh, because beamer encourages you to type a lot of mathematics. In beamer it's really easy to type lots of formulas. Okay, so let me say something about the heritage of this result, where it comes from. So the first result of this kind is Deutas Eitel and Smith from 2006 and they studied the case of two periodic knots and they used symplectic Kovanov homology so this Fleur homological definition due to Zeital and Smith of Kovanov homology. They showed there's a similar spectral sequence in the two periodic case. That means here there's a symmetry around an axis which is disjoint from the knots. If you had the axis there in rotation by 180 degrees is a symmetry and there's the quotient. Great. So the way they prove this as well, you show that if there's a Z2 action on a symplectic manifold, preserving some Lagrangian set-wise, then there's a spectral sequence relating, I mean there should be these F2 brackets, theta theta theta inverse is fine. The equivariant Fleur homology in the total space and the ordinary Fleur homology in the fixed set. So this is a sort of symplectic or Lagrangian Fleur analog of something classical that I'll say a few words about later in the talk, also called Smith theory but due to P.A. Smith from the 1930s. Okay, they proved it under quite, I should have said they proved this under, this holds under quite restrictive hypotheses. So most simple Lagrangian intersection problems this doesn't hold for. And then this was used by Chris and Henry to give a number of results for Hagar Fleur homology including some results about periodic, not Fleur homology of periodic knots and Fleur homology of branch covers. Then much more recently, 2019-ish, Tim Large gave a fancier version, an improved version of this idle Smith theorem with weaker hypotheses and then some more applications to Hagar Fleur homology. And that uses ideas of Kronheim-Rufke so that's nice for this talk. And in fact, you could ask, well is there a version in Hagar Fleur homology of the main theorem and the answer is yes and that follows from a paper with Kristen and Tai but using the result, the key input is the result of Tim Large. So there is an analog of the main theorem from this talk for Hagar Fleur homology because of this theorem of Tim's. I'm not gonna say state the version of the theorem precisely. The second sort of strain of thought that was important in getting this theorem due to Tylerman and Chipprion. So they showed for monopole Fleur homology that if you have a regular P-fold cover then there's a spectral sequence relating the monopole Fleur homology upstairs and monopole Fleur homology downstairs. This is the non-equivariate version of the monopole Fleur homology. So it's the analog of HF hat in Hagar Fleur homology. How do they do that? Well, I didn't say, maybe I'll say in a sec. Maybe I won't say in a sec. Let me say just in case I don't say in a sec. Sorry, so how do they do this? They constructed a space whose, I mean, Chipprion constructed a space whose homology is this monopole Fleur homology. Ordinary homology is monopole Fleur homology. And then they, Ty and Chipprion showed that this is a fixed set of an action. The space giving this homology is a fixed set of an action on the space giving that homology and then they applied classical Smith theory. So this went through using this Fleur stable homotopy type. Stoffer and Zhang applied the same idea, but for Kavanaugh homology. So they got, there's a result for, this is for periodic knots. So again, this sort of symmetry that just joined from the axis. And there's again a rank inequality. I have not, by the way, I have not listed. You might think that by this point of the slide, I'm listing all possible results of this form that I mean, oh no, no, it's okay. Got it, oh sad. Hang on, play from current slide, so much for having cues. Zoom, apparently, it's karma. Oh no, I have to click through the whole slide again. Okay, I haven't listed all. There are a bunch of other results of this form I didn't list, okay. So this was reproved from a different perspective by another group. Their theorem, or at least the key lemmas in proving the theorem and input to the theorem that I'm gonna give you. So this is not, I mean ours is not independent from theirs, ours uses their ideas. Oh the last thing I wanted to mention for context is that there are any work on Kavanaugh homology for strongly invertible links. In particular, Andrew Lobb and Liam Watson defined an equivalent Kavanaugh homology but using the other kind of diagram, not this kind of diagram, the other one. I'll come back to say something about that at the end. Okay, this by the way is the, if you type in proofs in PowerPoint, this is the image that PowerPoint comes up with. I don't know, I sort of like hopscotch, you know. Okay, sorry. Okay, I wanna try to convince you that the theorem is basically obvious, the main theorem, at least that it's almost basically obvious. So here's the main theorem again. How do we get it? The symmetry of K, that's this, we have this rotation around the points in the middle. It uses a map on Kavanaugh complexes. So I'll just call that also tau from just by the obvious thing. It takes a resolution to resolution and generators to generators. So now you can consider the by complex given by, well this, you take the Kavanaugh comp, infinitely many copies of the Kavanaugh complex, so there's a vertical differential here, the differential inside each of these terms and connect them by differential of form, identity plus tau. I think I learned this construction from Chipry on a long time ago. So this is one way of defining or computing the equivariant homology of a chain complex. If you have any chain complex with Z2 action, you can write down this complex and that's a version of the equivariant homology. If you start with the singular chains on a space, this gives you the equivariant, real equivariant homology after inverting theta. Yeah. Oh yeah, sorry. I should have attributed to myself that point here. I mean, so sorry. So this uses the Lipschitz-Sarkar-Kavanaugh stable homotopy type is how this theorem is proved and that's gonna come up next. Yeah, sorry. Sorry? Why do I, so why do I, why are I boring you with this? Bear with me. I'll come back to the other thing. I, you see, here's the point. I want to almost convince you that our theorem is a corollary of this theorem and I want to tell you why our theorem is not quite a corollary but almost of that theorem. So like, is it actually new? I don't know what that is as a spectrum. That's right. I mean, I do, but you haven't been told yet. I have written that up. I haven't written it up on this slide. I should have said this paper's on the archive. So like, if you don't, if you want any details, it's available. So, okay, we have this by-coms. I take the homology of each of the CKHs that gives me a spectral sequence and the, so that is filtered by how far along you are horizontally that gives you a spectral sequence. The E one page is the Kavanaugh homology tends to with F2 join theta theta inverse. That's the E one page that I want up here. Okay, that's my E one page that I wanted here. And so, if I could figure out what the E infinity page, make sure the infinity page was this, I would be done. Or if I knew what the infinity page was, I would be done. Okay, so if I take the homology first the other way, so filtered by the grading on each of the individual terms, so that the first differential is the horizontal maps. It's not hard to see that the E three page is the right hand side here. I'm gonna basically do that argument in a slightly disguised form in a couple slides. And so the question is, does this other spectral sequence, why don't you take the horizontal differential first as it collapse? Well, so I'm, you know, this is just repeating what we had at the end of last slide, I ran out of space. So the Kavanaugh complex is a mapping cone of a map from K one to K zero. That's just the skein, you know, the unoriented skein relation at the middle crossing. And, Stoffer and Chang showed that each of K zero and K one are periodic knots, so we already know that there's a spectral sequence of this form from Stoffer and Chang. So, okay, if we plug that in, speed up, you see, cause Tom was bored. If we plug that in to each vertex, I see that our by complex looks like this. And if I restrict to the top row or the bottom row, it follows from Stoffer and Chang that this spectral sequence that I wanted to collapse collapses. But I couldn't figure out, or we couldn't figure out how to rule out having some higher differential in the spectral sequence that relate to two rows, sort of going from the bottom to the top or something. So, this almost gets you approved, but doesn't quite at the last step because there could be some annoying differential coming from the fact that it was, you know, that intertwines the two rows. Too bad. Okay, so back to the drawing board. So I'm gonna tell you where the Stoffer and Chang spectral sequence came from. So, let's recall something classical. So this is going back to 1930s again, 30s, early 40s. So Stoffer and Chang deduced the localization theorem from classical Smith theory for ZP spaces. That's what Ty and Chipper and did also. So I need a word, a Z2 CW complex, a CW complex with a particularly nice kind of action. It's cellular, it takes cells to cells and on each open, so the fixed set is a union of open cells. So each cell is either fixed by tau or two different cells, or cells are commuted by tau, but you can't have just some piece of a cell fixed by the involution. Okay, so here's a Z2 space and there's a, so it's a sphere, right? And it has an action. Okay, the action is not looking that good. What looks fine on my computer, that's a zoom problem. Okay, so what am I supposed to do? I'm supposed to make it into a cell complex, so now my space has become a cell complex, I triangulated it, and the Z2 action takes cells to cells, right? I went around, it's happening twice here, but you went around, each cell is taken to a new cell. So given a finite Z2 CW complex, you can form this complex that I said before, this equivariant complex using the cellular chain complex. Let's think about what happens with the, what happens when I do the horizontal differential's first spectral sequence. So the kernel of the horizontal differential, the two kinds of things in it. If I had a cell that was fixed by tau, then id plus tau of it is zero. Again, everything's with F2 coefficients, right? So for example, the cells at the bottom and the top, this vertex is fixed by tau, so id plus tau of it is zero. If you have a pair of cells that are exchanged by tau, like the front and the back, then id plus tau of the sum is zero. The first kind, this ones are not in the image, so this cell is not in the image, it's plus tau, but that of course is id plus tau of the front triangle. And so I take homology of that, the E1 page, so I say the spectral sequence, is just the cellular chains on the fixed set. Because all I was left with was these cells that were fixed by tau. And you can see that the differential is the differential on the cellular chains of fixed set, that uses this GCW complex property. And fine, the fixed cells however span a sub complex, so there are no terms of the differential out of it. If you've got what is the differential in a, one of the next differentials in a, in the spectral sequence, they're like zigzags. But since there are no differentials that go from a fixed generator to a non-fixed generator by a property being GCW complex, C2CW complex, there are no zigzags. So there are no higher differentials, the spectral sequence collapses and you get the result. Okay, so there's the sort of classical story that shows that you get spectral sequence from the homology of X, the homology of the fixed set. Finite is important in order to know that the two spectral sequences converge to the same thing, otherwise I mean this is extremely, so exercise the statement is completely false if you drop finite from CW complex. Okay, the other ingredient we need is Kovanov space. Let me tell you something about the Kovanov space that Tom kindly reminded me exists. So, so given a link diagram, I can build for you a CW complex whose homology is Kovanov homology and the stable homotopy type of the CW complex so that doesn't matter for this talk at all, but how does that construction work? So I need an intermediate, I mean the two versions of the construction but the one I want to give, I need an intermediate notion which is this category B, this is the Burnside category of the trivial group, Burnside two category of the trivial group, but let me tell you concretely what it is, the objects are sets, the morphisms are correspondences, so it's just a set mapping to X and Y, so this is a one morphism in the category, it's just a set mapping X and Y is a map from X to Y. It's like if you've seen Lagrangian correspondences, it's just like that except not Lagrangian. Whoops, sorry. And the two morphisms are rejections of correspondences, so maps from this to some other one that commute with the structure maps. It corresponds like a matrix of sets, so it's like an Y by X matrix of sets. Okay, so what we construct is I'm gonna construct a map from the cube to one plus one embedded cubortisms. It turns out that it uses the way these cubortisms were embedded in R three cross I, R two cross I, to the Burnside category and then from the Burnside category of spectra, so I need to explain, well this arrow I explained previously, give it a not diagram, I think I stuck it there, give it a not diagram, you get a cube of cubortisms, okay, there. I need to explain the other two, so let me do that, so this map, the Burnside category, I need to tell you what correspondence there is associated to each edge, so to each of the, or to each composition of edges, so to each map in this, each cubortism, I need to tell you what's the correspondence, okay? If the cubortism has genus zero, then the correspondence either is empty or has a single element and it's just recording whether there's a term of the Kavanaugh differential or not, okay? So for genus zero correspondences, it's either empty or has one element, then all one element sets are the same, so it doesn't matter what set it is there. If it's got genus bigger than one, there is only, there are no L, it's empty, the correspondence is empty. And if it's got genus one, the correspondence has two elements and they correspond to what? You do a checkerboard coloring of the complement of the, the checkerboard coloring of the, the complement of the correspondence, so here I've, my checkerboard is white and sparkly purple, okay? You look at the white region, H1 of the white region, well actually H1 of the white region is sort of big, like there's a loop there and loop there, but you look at H1 of the white region, mod H1 of the boundary, that's one, that's one dimensional, so there are two generators, that one and that one, and that's the two elements of the correspondence. So again, the correspondence for, for a genus one, the corbotism has two elements and they're the generators of H1 of the white region, mod H1 of the white region of the boundary. It doesn't matter so much for what I'm gonna say, but I thought I would give you something like a complete definition. Then I need to tell you how you compose, like if you compose two genus zero things, get a genus one thing, how do you, what element of the correspondence you get? I'm not gonna tell you, so there's something about the composition that I'm suppressing, but roughly that's the definition as functor B, so it's functor sorry, from corbotisms to B. Oh, notice it used the embedding because I'm looking at the complement. Just an aside so I can see Tom was already getting bored. If you wanted to do something like lehomology, which you would love to, you need some way of getting some sort, like if you wanna extend the story, you'd want some sort of correspondences or something associated to higher genus corbotisms. You need to get rid of this like, this is some sort of quotient that killed off some potentially interesting structure and we don't know how to do that, but it could imagine that something from gauge theory or something else would construct that, so it might be interesting that if you could see these correspondences in some other way that might give a hint of how to generalize this story. Okay, back to the main talk. So next I need to tell you, given one of these things, how do I get a map from this burns high category to spectra? Well, what I'm gonna do is, so to an object that's just a finite set, we're just gonna take a wedge sum of spheres. So a wedge sum over all elements of the set of the n-sphere. So in this case, the two elements, so it's just the wedge sum of two spheres. To a correspondence, I'm gonna produce a, so a correspondence has arrows this way, from the correspondence x and y, I'm gonna map from the correspondence to y, gives me a map from this wedge of spheres to that wedge of spheres. So this arrow you understand, it's a wedge sum over a and a to wedge sum of y and y, well there's a map of index sets, there's a map of wedge sums. The left one I need to invert this arrow to get an arrow going that way, and that's a version of the Pontiagin-Tom construction. So what do you do here on my spheres? So this is a, I've drawn it in a slightly funny way, but this is a two sphere and here's the other two sphere. So this is this wedge sum, SN wedge SN. And you embed the correspondence in it. So I embed A, B and C in it and thinking them up a little bit to be disks. Okay, so I've embedded A, B and C in it, so that A and B, which map to x, live in the sphere over x and C, which map to x prime, lives in the sphere over x prime. And now you do a collapse map. So now I've gotten to this middle one and then you do this fold map which takes you to the right-hand side. So there was the, there's the, there was first that map and then that map. Questions? Yeah, I think I can do that. Let's see. Maybe. Yeah, should we go back? So I embedded, I took this correspondence, A, B and C and I embedded it in this wedge sum of spheres in the wedge sum over x and x of SN. So each of A, B and C went to a point. A was that black point there, B is that black point there. Just, yeah, embed those points. Somehow. And now you say, but Robert, what if you chose different embedding? And you also say, but Robert, what was N? So N is some large number and the space of embeddings is like N minus two connected and minus one, N minus two connected, something like that. So you make N big enough that it doesn't matter. N at Q was just, there are no arrows of length bigger than the number of crossings and so if you chose capital N big enough then it doesn't matter to get everything to be coherent. Yeah, good question. More questions. I need to pick it up. Okay. I need to pick it up a lot. Okay, so this gives, so you do the whole thing equivariately. Instead of embedding them, you embed that you embed them equivariately, you put an action on the spheres and then the work is you have to identify the fixed set. So you identify the fixed set. So there's sort of some sub functor of this function. The Bernstein category that's fixed and you see that the fixed set on the space is the space associated with the fixed points of this functor and that's interesting and takes work and that was done by Stoffig and Zhang and we just used that. Everything works the same. So a strongly vertical case, it works the same with a new work as you have to identify the fixed set and if you look at it, that's where you see this map F plus. The end. I'm going to talk about applications and related results. So I thought I would put in Tom's thesis brothers and cousins and so on, but I just put in his students and it was already too big, almost too big to fit on a slide. So there's Tom's descendants. Let me tell you an application and then a couple of related things. So here's a theorem that's not due to us but I'll give you our proof. Here are two slice discs. You do this band after you do the band, you've got to unknot and you fill them in with two discs. Okay, so here's a slice disc. Here's another slice disc that's not 946 and these slices are not isotopic. Rail boundary. Proof, it's enough to see they induce different maps on Kevonophomology. The map associated to the right is the map associated to the left composed with the strong inversion tau. In the relevant grading, you can see that our spectral sequence converges to zero. You look at the quotient, the quotient is not too big, it's not too small either, but it's not too big and you can just compute that F plus was an isomorphism and so the spectral sequence converges to zero. Wonderful, there's only one, do you look at how the gradients behave? There's where the interesting part is, the relevant part is and that tells you there's no way for this stuff to die, if you pay attention to grading, except by dying by tau. It has to be that tau is non-trivial on this part. So for some basis, tau is that form. You mess around a little bit and you see that, okay, there's a boxed dean and you look at the kernel as one dimensional and you multiply x, that gives you some element there and that element is taken to the same non-zero element by both of the two disks because of some property about keyboardism. So there's no actual, there's no work here, there's just a little bit of messing around with formal properties. So that tells you that these two disks give you have different values in a tau. So the only work was you had to compute that F plus was an isomorphism two dimensional and F plus was an isomorphism. So that forced tau. Okay, I went a little fast there. I don't remember who proved this. First, these are not topologically isotopic. You can distinguish them from the using the Alexander module also. So this is a smooth proof of something that's actually topologically true. Okay, this was proved. This is not the first Kavanophonology proof for this pair. This was proved by direct computation by Sonberg's swan. As is in some sense easier because you don't have to find any cycles. You just compute some annual Kavanoph group and you see that it was zero and therefore the rest of the proof is formal. Let me tell you some nice related results before I run out of time. So there are exotic disks that are topologically isotopic but not smoothly isotopic and are distinguished by Kavanophonology. So this was proved by Hayden and Sonberg. This originally, so these two disks, again, you do that, there's two, I'm not, you fill it with disks, are topologically not smoothly isotopic so they're exotic. A key ingredient to know that the topologically isotopic is a theorem of Conway and Powell that under some conditions guarantees things that disks are topologically isotopic or servers are topologically isotopic. You can also, I don't even want to say that, you can get something like RP2, a case of RP2 is there's a beautiful theorem from this year, you can even do it with Cyford services. So there are a pair of Cyford services, here's one of them for the same not, which are topologically isotopic and before but not smoothly isotopic. So it's a beautiful theory of Hayden, Kim, Miller, Park and Sonberg from a couple of months ago. It's also a beautiful picture of Hayden, Kim, Miller, Park and Sonberg from a couple of months ago. And in all cases, these maps, these are distinguished by the maps on Kavanophonology. So computer maps on Kavanophonology is interesting. Another question you can ask, so is how many, if you take two services you can attach one handles to make them isotopic and you can ask, do you ever need more than one one handler? How many one handles do you need to make your services isotopic? So two versions of that, maybe for time I won't say the difference but in a topological category it was shown that you can, you sometimes need more than one, you need an origin number of one handles. Okay, by these people. But you can ask, okay, what about smoothly? Are there ones that are topologically isotopic but need a large number of one handles? Well, one way of approaching this, there are lots of examples that seem to have a strong inversion. So you can use a strong inversion to say something about this. So using a sort of similar strategy to what I did in Kavanophonology before us, at least before you write anything down, I'm going to use Hagrid for homology to give some abstractions to disks being the same and bounds on the stabilization distance. So again, it's like the argument two slides ago that I rushed through. And a very recent theory, theorem of Gary Gooth is that for any end, there are knots which are topologically isotopic but they need end stabilizations to become smoothly isotopic. So that's a relative version of a very old question of walls that is whether any pair of homomorphic form manifolds become defumomorphic after a single connection with S2 cross S2. So this is the version for surfaces. And so we don't know the question of walls is still open. Yeah. No, let me rush it. Yeah. The boundary is important. So it's for closed things, it's still open. That's right. Yeah, I think it's still open for closed things. I agree, the boundary place, absolutely. Everything I'm saying is for boundary. That's right, for open things it's closed but for closed things it's open. Yeah. Last quick one, there's a concordance group of these things and the concordance group used to being abelian. This is where the butterfly legs came up by the way but I'm a little out of time. But for the invertible ones, the concordance group is non-abelian. So here are two different connected sums of these two knots at the bottom. And as a recent theory I want to preach that who I think is an Italian graduate student who I haven't met unfortunately that these, huh, undergrad, Italian undergrad, very good. That these, this group is non-commutative. So, and that uses this notion of butterfly legs. Okay, I had some questions but I won't tell them and then I have an apology for having too many pictures. So, thanks very much for listening. Oh, so the question was can I go back one slide? The answer is yes. Yeah, so, yeah, fine, you guys can read. So I say it again. Yeah, that's a good question. So there is some work distinguishing strongly of vertical knots by, we didn't try to do it. I mean, I'm sure that it distinguishes, that it distinguishes some. But, Lob and Watson did that using the other kind of diagram. So using this sort of diagram. One thing that's nice about this sort of diagram is it also works for links. If you have a link where the intersects the axis in more than two points it's obvious what to do here that's similar to I am. Here you'd have a whole bunch of lines going all through that middle point. You'd have some sort of complicated bigger number of crossings, like the higher crossing. Anyway, yeah, so they give some very simple examples in their paper distinguishes some strong inversions. But I don't have any sense of how good these invariants are. And also I don't know if theirs isn't anywhere related to ours. So I'm sure it does but I haven't actually complicated to see that. Say it again. Yeah, so Tom's comment I guess is isn't this current the definition of the analytic monofomology of a knot that hits the axis? And the answer is maybe. I mean, yeah, you can define them to be any, like. But yeah, it seems, I agree. This is how to sort of saying it's like the difference between pushing it off one side and pushing it off the other side. So what's happening? The picture, I agree. Yeah, I agree. So in some sense, this is the analytic monofomology. In fact, we in the paper introduced notation, we call this AKH of, I don't know, K where K is the knot that intersects the thing in one axis. So in one point, well, more questions.