 Thanks. Great, so I'll be telling you about Hager flow homology. So Hager flow homology, it's a package of invariants for three manifolds and knots inside of them. So to three manifold y, so let's say that it's closed and oriented. We'll associate a chain complex. So this is say either a vector space or a module endowed with a differential that squares to zero. The chain complex itself depends on your choice of description for y, but the chain homotopy type of this chain complex is an invariant of y. So in particular, you can take the homology, so take the homology of this chain complex and that's an invariant of your three manifold. And so this is defined by, it's too close. Okay, can people still hear me? Okay, great, thanks. This is defined by Ausroth and Zabo. And in general, it's pretty difficult to compute this from the definition, but for certain special families of manifolds, this may be tricks about how you can compute this. And now also, there's a knot invariant, so we'll be interested in saying knots in S3. We associate a chain complex, we'll call CFK. And again, the chain complex itself is gonna depend on how you describe your knot, but the chain homotopy type is gonna be an invariant. So if you take the homology, that's gonna be a knot invariant. So this is defined by Ausroth and Zabo. And independently, Jake Rasmussen, you heard from last week, great. And so what's special about this knot invariant? So there's different flavors of it, but if you look at the simplest flavor of it, we'll call that HFK hat. So this is a bi-graded vector space. Take the graded Euler characteristic of this vector space, where you get the Alexander polynomial. So in particular, we say that this categorifies the Alexander polynomial. And so I guess last week, you learned some properties of the Alexander polynomial. In particular, you saw that the Alexander polynomial gives you a bound in the genus of the knot. I guess one half the degree of the Alexander polynomial is a lower bound for the genus of K. And it turns out, well, knot flow homology actually dramatically improves this. Knot flow homology actually detects the genus. This is due to Ausroth and Zabo. And so since the knot is the only knot with genus zero, well, that tells you that knot flow homology actually can detect the unknot. So basically that says that if some knot has the same knot flow homology as the unknot, well, it has to actually be the unknot. I guess you also saw that the Alexander polynomial can obstruct fibrousness in the sense that if K is fibrous, then the Alexander polynomial is monic. And again, knot flow homology improves this in the sense that knot flow homology detects fibrousness. So knot flow homology can tell you if your knot is fibrous. This is due to Kijini and me. One other question you're wondering is you might be wondering, well, is there a relationship between the knot invariant and the three manifold invariant? And there is. So in particular, the invariant CFK, this actually determines the Hager flow homology for any surgery on your knot. So this is due to Ashfa and Zappa. So by Sn of K, what I mean is I mean N surgery on K. So that means remove a tubular neighborhood of K and glue it back in such that an N-frame longitude bounds a disk. Great. So today, I'm going to tell you about how we're going to describe our three manifolds and our knots. And that's going to be the input into this machine that's going to spit out a chain complex. Tomorrow, I'm going to tell you about the three manifold invariant. Thursday, I'll tell you about the knot invariant. And then on Friday, I'll talk about this relationship between the knot invariant and the Hager flow homology of surgery on a knot. What I'm going to tell you now is how we're going to describe our three manifolds. Question, so how do we describe our three manifold Y? And then the answer is going to be something called Hager diagrams. And that's where the Hager part of the term Hager flow comes from. So I'm going to begin with a definition. G is G handle body, a neighborhood of a wedge of G circles. You should think of them as N in R3. So let me draw a picture. So here is a wedge of three circles. And then here's a genus three handle body. Equivalently, you can think about it as a three ball together with handles attached. So these are homeomorphic. A Hager splitting of a three manifold Y is what's the decomposition of Y as a union of two handle bodies. So it is decomposition. So Y is going to be the union of two handle bodies. So it's H1 glued together with H2 via F. So here H1 and H2 are handle bodies. And F is going to be, let's say, an orientation reversing diffeomorphism, the boundary of H1 to the boundary of H2. Great. And then the genus of the Hager splitting is just the genus of H1 or H2. They have to have the same genus since the boundary is the homeomorphic. So let's look at some examples of Hager splitting. Great. So if you think about S3 as R3 together with a point of infinity, well, you can think about S3 as a union of two three balls. I guess that's a Hager splitting of genus zero. S3 also has a Hager splitting of genus one. Great. So if you think about S3 as R3 together with a point of infinity, well, let's take one of them. So you can take the z-axis together with a point of infinity. It's going to be a circle. If you take a neighborhood of that, well, that's going to be a solid torus. And then if you think about the complement of that, well, that's also going to be a solid torus. So maybe the picture looks something like this. So here's your z-axis together with a point of infinity that gives you a circle. And then a neighborhood of that is going to be a solid torus like this. And then the complement is the solid torus here. And that gives us S3. So that's a genus one Hager splitting of S3. You can also, there's also a genus two splitting, a Hager splitting of S3. So here's a genus two handle body. And then if you think about the complement of this, sort of just think of this as sitting in R3. And then while the complement of this take a neighborhood of the point of infinity, that's going to be a ball. And then imagine, I'm going to describe to you a genus two handle body. So you have this ball, it's a neighborhood of a point of a point of infinity, together with a handle that sort of goes through. You get sort of one hand, so you have this ball of infinity together with the handle going through here and the handle going through here. And that gives us a genus two Hager splitting of S3. And then maybe as an exercise, you could imagine a genus G Hager splitting for S3. Great. Then maybe another example, the Lenspace LPQ. I guess depending on your definition of a Lenspace, sorry, this is maybe your definition of a Lenspace is something that can be described as a union of two solitary. And so sort of exactly how you glue together the two solitary is telling you which Lenspace you're going to get. But why are Hager splitting is going to be useful to us? Well, they're going to be useful to us because every closed oriented three manifold admits a Hager splitting. So theorem, every closed oriented three manifold admits a Hager splitting. So let's prove this. We'll assume that all three manifolds can be triangulated. So let's take a triangulation of y. And now let H1 be a neighborhood of the one skeleton. The one skeleton is going to be a graph. So a neighborhood of a graph, well, that's a handle body. Well, if that's one of our handle bodies or the other handle bodies should be the complement of this. So the claim is that y minus H1 is also a handle body. And then the proof of the claim, well, y minus H1, this is just the neighborhood of the dual one skeleton. So what's the dual one skeleton? It's the one skeleton you get where the vertices are the centers of the tetrahedra. And then the edges, you get one edge for each face. Sort of perpendicular to each face. So maybe I'll draw a picture. So here's maybe a tetrahedron. And then dual to that, well, the dual one skeleton, you get a vertex in the center of the tetrahedron. And then you get an edge perpendicular to each face. And so that completes the proof. So every three manifold has a Hager splitting. And so a Hager splitting, well, it's two handle bodies together with a diffeomorphism between their boundaries. But we want a way to describe these Hager splittings. And so we're going to do something called a Hager diagram. Great, OK. So what's the Hager diagram? Before we define a Hager diagram, I'm going to give a way to describe a handle body. So definition, a set of attaching circles for a handle body H. So let's say the genus of a handle body is G. It's a set of G simple closed curves. So we'll call them gamma 1 through gamma G. Where do these curves live? They're going to live in the boundary of H, such that they're going to satisfy three conditions. So the first condition is that these curves are going to be a pairwise disjoint. So the gamma I, a pairwise disjoint. The second condition is that if we look at the complement of the gamma I, so sigma minus gamma 1 minus gamma G, we want this to be connected. Third condition is that each gamma I is going to be a pairwise disjoint. So that's a set of attaching circles. So let me give you some examples. So here's a handle body of genus 2. It's a neighborhood of a wedge of two circles. And well, this curve bounds a disk. And this curve also bounds a disk. They are both disjoint. The complement is connected, and each bounds a disk. This is a set of attaching circles. Maybe another example. Well, if I take this curve and this curve, well, these also both bound disks. So this is another set of attaching circles for this handle body. And in fact, well, if I just gave you this surface together with these two curves, that actually describes the handle body for you. And I can rebuild the handle body from just the data of the surface together with these two curves. So how do I do that? So we can build by, so first let's thicken sigma to sigma cross the interval. And now thicken disks along gamma i cross 0. So in terms of, say, this picture, right? So I have this surface. So thicken your surface. And now along surface cross 0. So we'll think about maybe that as the inside. Attach a thicken disk along each of these curves. So I'm attaching a thicken disk along each of these curves. And now it's an exercise that since these were attaching circles, in particular since the complement was connected, if you notice here, what's left on the inside? Well, what's left on the inside, that boundary, is homeomorphic to S2. So if you follow along here, you can see that that's homeomorphic to S2. So in particular, well, you can fill in the resulting S2 boundary with a 3-ball. So in particular, if you just fill in what's left here with a 3-ball, well, we've built our original handle body, for which these were attaching circles. Similarly, you can check that if you do the same procedure here, attach a disk here and attach a disk here. Well, again, because these were attaching circles, you can check that the boundary that was left is S2. And so there's a unique way to fill that in with a 3-ball. Your 3-ball's going to sort of live here. So the point is, I just handed an abstract surface together with the attaching circles. Well, it actually tells you, it allows you to sort of rebuild the handle body that you had. Yes. So the way I defined this, in this definition of a set of attaching circles, you had the handle body. And then you described the curves in the boundary. Oh. So the other way around is. So you're starting with the handle body by this procedure. So when you build the handle body, you're making it satisfy this condition because you put the disks in. That's right. Yeah. That's right. But sort of the third gets imposed by construction. Exactly. Yeah, I guess here I'm requiring this surface. Yeah, my surface, everything's connected. Everything's connected. Yeah, I guess I didn't say that. Other questions? All right. So now we're ready to define a Hager diagram. A Hager diagram for, so let's say we already have a Hager splitting of a manifold y. So this is, it's a triple. Surface comma alpha comma beta. So I'll tell you what each of these are. So sigma is a closed, connected, oriented surface of genus j is a g tuple of curves alpha 1 to alpha g. And what are they? So this is going to be a set of attaching circles for the handle body h1. And similarly, beta is going to be a set of attaching circles for h2. So let me give you some examples of Hager diagrams. Great. And then so there is a convention in the field that alpha curves are always red and beta curves are always blue. So we will stick to that convention. Great. OK. No. I'll give some examples, and maybe that will help. Great. So let's look at the, so we describe the genus 1 Hager splitting for S3 earlier. Great. And so we sort of saw that, we sort of saw, we had this solitorus, so the one you see sitting here. And then the complementary solitorus was a neighborhood of the z-axis together with a point i infinity. So for this solitorus that we sort of see here, well, OK, well this curve bounds the disk. And now for the complementary solitorus, this curve, this curve bounds the disk. So this is a Hager diagram for S3. I guess we also described genus 2 Hager splitting for S3, which was sort of this, sort of the handle body you see, sort of the natural one bounded by the surface. And then sort of the complementary one. And so, well, in this handle body here, these curves bound disks. And in the complementary one, well, these curves bound disks. Great. And so just like the attaching circle sort of told us how to build a handle body, well, a Hager diagram also tells us how to build a 3-manifold. And basically, by basically the same procedure. And if you want to build your 3-manifold, if you're given just the Hager diagram and you want to be able to build your 3-manifold, well, what should you do? So y can be sort of reconstructed from the Hager diagram as follows. So you first thicken your surface to sigma across i. Let's attach thickened disks along alpha across 0. So for example, in this picture here, I'll attach thickened disk along here. And now we'll do the same thing for the betas, except for those, we'll do it on the other side. So we'll attach thickened disks along the beta i cross 1. So in these pictures, maybe we think about you attach thickened disks along the alpha curves on the inside and then along the beta curves on the outside. And then just like analogous to before, well, now we're going to have two boundary components. And both of them are going to be S2. And we'll get an S2 on the inside and an S2 on the outside. So for each of those, you can cap those off with a 3-ball. So resulting boundary is S2, Distant Union S2. So fill each with a 3-ball. Great. So here, when you fill in the inside boundary of the 3-ball, you get back this inside solid torus. And then you fill in the outside. That's sort of giving you the 3-ball. That's the neighborhood of the point at infinity. Maybe let me give you one more example of a Hager diagram. Here's a Hager diagram for our p3. So as I said, this is our alpha curve. And then here's our beta curve. And so for technical reasons, we're going to need a base point. So we're just going to put a base point somewhere in the complement of alpha and beta circles. So maybe a base point here, w. So for technical reasons, a base point w that should live in the complement of the alpha and beta circles. And so I guess now our Hager diagrams are going to be a quadruple, sigma, our alpha circles, our beta circles, and our base point, w. You can have different Hager diagrams that describe the same 3-manifold. But a natural question to ask is, well, if you have two Hager diagrams described the same 3-manifold, is there some relationship between those Hager diagrams? So now I want to describe, so motivated by that question, now I want to describe for you Hager moves. Before we move on, are there any questions? Can I maybe elaborate why that's our p3? I guess it depends on how you describe our p3. So if you think of our p3 as L2, 1, and then if you think about length spaces as two solid toy glued together along with the appropriate identification of their tourist boundaries, then maybe you can see that this is our p3. So in order to describe these Hager moves, let's let gamma 1 through gamma g be a set of attaching circles for H. So a set of attaching circles for handle body H.