 really great conference that you put together. So let's have a, so I'll introduce the last speaker who's Jake Rasmussen is going to tell us about toroidal three-manifolds curves and sutures. Well, thanks very much for the invitation. It's a real honor to speak at this conference for Tom. I mean, Tom has been an inspiration for everyone working in fluorohomology and gauge theory of course, but it's actually always amazing to me kind of the breadth of subjects that he reaches across. If you look at his students, they seem to do everything. So yeah, he's just amazing. So happy birthday, Tommy. All right, so, yeah, I wanted to talk about an extension of work that I've done a while ago with Jonathan Hanselman and Liam Watson and maybe let me motivate it first just by talking a little bit about satellite knots. Work in progress. So let's say we have a companion knot C inside of S3 like Juanita was talking about yesterday and then a pattern knot P inside of S1 times D2. Then of course I can write S3 as the exterior of C union S1 times D2. I just remove the neighborhood of the knot and sew it back in. And inside of here is P. And so then P sits inside of S3 in this way and then it's called the satellite with pattern P in companion C and we think about it as a knot in S3. Okay, and so maybe you're obligated to draw at least one picture. So here's some satellite of the trefoil like Paolo, I really can't draw a table of the trefoil. And it's known probably a theorem of Cypher that the Alexander polynomial has very simple behavior under the satellite operation. So if I take the Alexander polynomial of SPC, this is just the product of the Alexander polynomial of C evaluated at T to the winding number of P times the Alexander polynomial of P thought of as a knot inside of S3. And so a very natural question is what is an off-floor homology? Peter was talking about sort of soccer teams worth of thesis subjects. I don't think you could maybe make a soccer team of people who've written thesis. Thesis on this question, but you certainly have a respectable basketball team at least. And so maybe sort of starts with the work of Eftikari. So Eftikari considered white-head doubles and Heddon considered cables also white-head doubles. And what they found was that in these particular cases there were sort of formulas for the knot floor homology of the satellite in terms of what now we would recognize as the sutured floor homology of the knot complement, a complement of a companion knot, sutured along some slope. So I take two curves, two parallel curves with slope alpha on the boundary of this knot complement. And another way of thinking about this group is just that it's the not-floor homology of the knot inside of the dame filling of the complement with slope alpha. And then I take the core knot, which I'll just call K alpha. And of course, I mean a lot of the work that Eftikari and Heddon did was understand what these sutured floor homologies were in terms of the original knot floor homology of C. Okay. And so then sort of after this bordered floor homology, so this of course is due to Lipschitz-Thurston. And already in their very first paper on this subject, there's a wonderful example at the end that everyone should read if they haven't where they make some computations for a cable of the trefoil knot. And that's an inspiring example. And so then using this kind of technology, lots of people with theorems, kovah, avivine, just to name a few. And so the gluing here that you do in the bordered floor homology is along the surface of genus one. And in that case, there's this sort of curving variant find earlier with Liam and Jonathan. And let me just briefly recall how that looks. So if I give you M, boundary of M is a single torus, then what do I get? This gives me, let's just call it HF half of M, overload the symbol a little bit. What is this? This is a collection burst closed curves, possibly equipped with local systems, inside of a cover, the boundary of M minus a plane. And then there's a pairing theorem that says that if you take, say, HF half of a closed manifold obtained by taking two manifolds like this, M1 and M2, gluing them together by some map phi, this is sort of, I'll write this as hom, it's hom in the Foucaille category, but it's really just computed by pulling these curves tight and counting intersection points of phi of. An important thing about these curves is that for simple manifolds, you can actually compute what they are and write the answer down and it's simple. Okay, simple manifolds have simple invariants. So for example, manifolds say I have S1 times D2. So actually in this case, the cover is a punctured cylinder here. And the invariant is just a straight line between two points. I guess on the cylinder it's a circle, flat circle. On the other hand, if I had the exterior of the truffle oil, how does that look? Again, I'll draw the cylinder as I identify these vertical curves here, punctures and the invariant of the truffle oil looks like. Okay, so if I wanted to compute the Dane filling of truffle oil, maybe with plus one slope, truffle oil filled along slope plus one, well I pair these two invariants. This is just a straight line essentially. So I pair, I just draw a straight line of slope one here and count the intersections. And there's just one dimension of this. It's a dimension that we're working over field of characteristic two since we're doing bordered flow homology. So this is one dimensional, that's because this is the Poincare sphere. Well on the other hand, if I wanted to do minus one surgery, then I could just do kind of the opposite slope. And here I'd see three intersection points, minus one, that's going to be three. This is briskorn sphere sigma 237. Okay, another thing you can compute from this picture is non-flow homology. I'm going to stand up the phone and compute hf k hat of, let's say, k alpha contained in m of alpha. So I take my manifold m, Dane fill it along slope alpha. And then there's a core knot k alpha inside of there. This is just given by taking the pairing, I'll just write pairing instead of hom, of Lagrangian L alpha And this Lagrangian L alpha is now, for example, for the meridional filling. So it's a non-compact per between two punctures with slope alpha. So for the meridional filling, it looks like doing this. So here I see that I have three intersection points. That's the non-flow homology of the trefoil and S3. You can see the Alexander gradings correspond to the different heights here. If I wanted to, for example, do the non-flow homology of the core knot inside of this Dane filling, I would instead use this slope. I find I had something of total rank five. Okay, so that's kind of the picture in the closed case. And so the question I want to talk about today is what happens if instead, we don't think about closed manifolds, but maybe instead think of links inside of manifolds with torus boundary. And so there's maybe a more natural general statement in terms of bordered suture and floor homology. But since trying to be a little bit simple here, I'm just going to restrict to this special case of links. So, yeah, suppose we have boundary of M is a torus, and then I give you a link L inside of M. And maybe I'm going to fix K a component. And now if I just go ahead and run the same machinery, really, what do I get? So state it this way. So theorem, there's, and again, this is really nothing more than the same kind of thing. It's already things I did with Liam and Jonathan applied to this particular case. So there's an invariant, let's call it little, H at L on L. Okay, so what is this invariant? What form does it take? I'm trying to be a little bit specific, because actually, be specific about spin-C structures here and what cover I'm thinking about, because both of those things are actually rather different from what happens in the closed case. So here I'm going to sum over a set of spin-C structures. Let me write this. Spin-C of exterior of L relative to the boundary of M. Of course, for each spin-C structure, I get a sum and all of S. This lives in partially wrapped, kayak category of covering space TM bar. Okay, long alone equivalence relation. Okay, and now what I'd like to do is maybe try and explain what each of these pieces are. Okay, so this set, spin-C of L boundary M, this is just in bijection with H lower one of EL modulo, the image of H one of the boundary, boundary M. Okay, that's the outer boundary, as it were. This covering space, so there's a covering map, high mapping from TM bar to boundary M minus C again. Okay, and the high one of the cover is the kernel of the map from high one of boundary M minus C to naturally goes to H one of M. Okay, so some comments about this kayak category and the equivalence relation. The first thing to say is that again, there's a structure theorem that says that, well, you can just think about objects here as being sort of a union arcs with ends punctures and again, closed curves, local systems. Okay, and here, the original paper I wrote with Jonathan Lee and we gave a direct proof of this. Here, we could also just be kind of lazy and use the theorem of Heiden. This case, I have been lazy and just referred to this theorem of Heiden-Katzarkov. I don't have a direct proof. Okay, and what's this equivalence relation? Roughly speaking, what this equivalence relation does is it divides out by the operation of gain twisting along the boundary components. Okay, so I forget about twisting pressures. Okay, and so I've admitted, I chose this component of L and then forgot to include it in the notation. Okay, so what was I doing when I chose this component? This is a sort of reduced version homology. There's also an un-reduced version. Right, so I define an equivalence relation on objects in the focaya category. And I think one way that you could define that equivalence relation is to say that two objects are equivalent if they're related by gain twists around maybe several of the punctures. So notice that once I've defined that equivalence relation, there's no longer a well-defined hom pairing on this quotient. Okay, so that's an important point. Maybe because I'm bad at algebra. I mean, that might equally, you know, this definition is mostly designed to draw pictures with. So yeah, this might not be the best thing to say. So remember we took the covering space of this thing, right, so this has a puncture. In the cover, I'll get probably a lot of punctures actually. Ah, well, that's because I didn't define the invariant, right, but you know, so if you know a little bordered sutured fluorophomology, what did I do here? I have a manifold with a bunch of boundary components. Yeah, I put little disc suture, you know, little sutured regions that look like discs and all, but one of them and on the other one I put a tube that goes out to the exterior. And the fact that, you know, when I put that tube there, I made a choice. This quotient here is designed to divide out by that, you know, give me something that doesn't depend on that choice. So there's also an un-reduced version. The un-reduced version look like, well, one way to think about it, I'll write it H, L, L, now it doesn't depend on a component. Well, this is the tube cutting bi-module applied to the un-reduced version, that's one way of thinking about it. In terms of what I just said to Marco, rather more concretely, like if I have just an arc between two punctures here, what should this do? Places this with figure eight. And on the other hand, if I had a closed component, I'd probably get two components, parallel components with opposite orientations. And this is very similar to an operation in Tangleflare homology studied by Kotelsky-Kirch-Harrell, similar operation. Okay, so then the pairing theorem says that if I wanna look at, say I have L1 inside of M1 and L2 inside of M2, and I have B mapping from boundary M2 to boundary M1, then I could glue them up and get a link inside a closed manifold and HfL L1 union L2 sits inside of F of M2. It's going to be hung, I'll just write pairing again, of say little HfL L1, I'll reduce along some component, and big HfL L2. And one interesting case that you might consider is where one of these links is empty. M2 has no link inside of it at all. Well, then I obviously can't reduce, there's no component to pick, but I can still consider the un-reduced version, and that's just the empty set inside of M. All right, so this is awfully dry. Not a good question, right? So notice that this un-reduced homology, this is a compact object. Okay, and there's not a well-defined pairing between two non-compact objects once I divide it out by this equivalence relation, but there is a well-defined pairing between the compact and a non-compact. So that's, yeah, it's very important that one of these two things is compact. Okay, so let's consider a couple of examples. First, a very trivial example, say I just have this knot inside the solid torus. So then the first thing to understand what's the covering space. So here, TMR actually just looks like R2 minus a lattice of points. Okay, so maybe I'll just draw the punctures as thoughts. This is canonically identified, for example, with H lower one. So R2 is H lower one at boundary M. So I should choose some coordinates. Maybe I'll call this direction A and this direction B, and this A, and this is B. And there's only one spin C structure in this case. And so the invariant here just looks like a little arc in the direction given by A, okay? And for example, this has to be the answer, okay? Because one thing that this pairing theorem says is that if I take and glue this solid torus into a knot complement, such a way that I get the original manifold back, then this should just give me the knot floor homology of that knot, okay? And that's exactly the description of the knot floor homology in terms of this picture over here that we already knew. So this is, the yellow is say little HFL. On the other hand, maybe I could draw an orange. Big HFL, that looks like this. Okay, so for example, if I wanted to compute the regular link floor homology of the Hopfling, okay? Well, I could get that by sticking two of these tori together. One of them I use the un-reduced invariant. The other one I use the reduced invariant. And I see maybe, so the pairing here, four different possible pairings arranged in this pattern, which if you're familiar with the link floor homology of the Hopfling, well, that's what it is. Okay, so that's a very easy example. A next example, well, let's just consider the next easy example, which is the knot, a pattern that gives us the two-one cable. Looks like this inside of the solid torus. Okay, so here it's again true that TM bar is C2. Okay, if I'm looking at sort of knots in the solid torus, this is actually always true unless the winding number is zero. But now the set of spin C structures has two elements in it instead of one. Okay. And two spin C structures, S0 and S1. And what does little link chat now look like? It's like this. So I just have a pair of little arcs. And what I want to say, a good exercise is to figure out how to compute the link floor homology of this link. I get by gluing in the half length from this picture. I won't do that, but instead I'll talk about sort of what the Euler characteristic means in this case. Okay, so the general rule of thumb is that, right, if I have knot floor homology or link floor homology, the Euler characteristic is supposed to have something to do with the Alexander polynomial or the Terai of torsion. And so how do I see the Euler characteristic in this picture? Well, it's just the homology class given by the ends of these arcs. Homology class is a boundary of H of L. L, let's just say this determines, gives prior torsion of the exterior of L times the product one minus MI over all the link components. Li, that's the usual formula for the Euler characteristic of link floor homology. And what do I mean that it gives it? Well, notice that this thing here naturally lives in something that looks like Z of H one, a boundary M that sits inside of Z of M. But this is, you know, it's not a surjection here. This, in this case, for example, this is one in M two and this is Z of joint M one and M two squared. And so you have to combine these two pieces together to get the full Euler characteristic that looks like. So, draw this now on a lattice where this is two. Here's one of those two pieces. And here is the other. Okay, and so the Euler characteristic here looks like one minus M one plus two M one minus one, something like one minus M two times one plus M one and two. And this is the Euler's inner polynomial of that link component. Okay, so coming back again to satellites. Well, maybe first we could consider groups. We compute the groups, Hfk hat of our satellite. Well, this pairing theorem just tells us, right? It says that I could write this as the exterior of a companion paired with Hfl of patterns. And again, let's just consider an example of how that looks. Maybe I'll pair, I'll compute the two one cable of the trefoil. Okay, so I have to draw a curve for the trefoil complement, which looks so. And then I will pair it with translates of Hfl of p. Okay, there's a convenient way of sort of organizing the way to do that. I'll just draw translates in here, and then I'll also draw translates of this S one. This, and so this, for example, gives me the non-floor homology in Alexander grading two. There's one generator. Here I have Alexander grading one. Let's see these two generators. Here I get a equal to zero. This will give me a is minus one. So one thing to notice here is that it's kind of immediately obvious why Heddon found a formula for the non-floor homology of the cable in terms of sutured floor homologies. Okay, because here I'm pairing with a bunch of little line segments here, which are just giving me the sutured floor homologies. So we could make that formal is linear if L is a direct sum of line segments. And I'll just write L-alphas here. And there might be several with the same slope. They might be translates of each other. That's totally fine. And then a corollary is that if inside of S1 times D2 is linear, there's a formula for the link-floor homology of a satellite. So non-floor homology of a satellite, HFK hat ESC is just a direct sum over, let's say, alpha in little HFL of the pattern. And then I just take SFH exterior C with slope YAML. So I say, ah, okay, good question, right? So there, yes, okay. This curve looks non-compact, but in fact, that's just because I've drawn this in a covering space of the space that I actually should have taken the pairing in. You'll notice that if I were to translate up by one, I would get exactly the same picture. So really, what I should be doing is I should be dividing out by that translation. And that's just because this curve that I drew for the trefoil is really some cover, the real invariant of the trefoil complement. Okay. So this kind of raises the question, well, what kind of knots are, what kind of patterns are linear? Um, and, well, some, but not all, particular sort of simple looking knots, maybe have linear patterns. So here's a theorem, says that, suppose maybe L is K1 union K2 inside of S3 is alternating and that K2 is an L-space knot, then I could look at K2 inside of the exterior of K2. One, this is linear. And in fact, L, K2 is determined by the multivariable Alexander polynomial and the signatures. So, for example, any two-bridge length. Okay, so lots of simple patterns are linear. Another interesting pattern that's linear that doesn't fit this criterion is if I take maybe ZN, this is S1 times N points inside of S1 times P2. It's an example of a linear pattern. But, for example, cables of two-bridge knots are not usually linear. Okay, so the reason there was a formula is that we got kind of lucky and we had a simple answer for link fluorophomology of the pattern. Yeah, that's, so, but if I'm just thinking about pattern knots, if I wanna, something that actually gives me a pattern, then I need to take K1 to be the unlock. Okay, another thing you might ask about, sort of, maybe I just don't wanna know the groups here, but I might also wanna know what are the differentials. And, well, if I wanna know what the differentials are, here, for example, differentials, S3, I can take sort of horizontal and vertical differentials, two different sets. And what does that correspond to? Well, geometrically, it corresponds to the fact that in the cylinder, and I'm gonna draw the cylinder, consider this non-compact object. That's just al-alpha. Well, the two different ways that I can kind of add a differential to that to get this curve at a differential going this way to get this curve, okay, this differential on it. And pushing those forward, here you have to make an appropriate assumption about that not being non-trivial in the right group. You'll find that there's some sort of differentials on this thing that go to a hat of two. And in this instance, the differentials are really easy to see. There's only one thing that they can be. So what they correspond to is kind of pushing off these corners. So there's one push-off that goes like this way, another push-off that goes the other way. And we'll finish with two observations. One is that if I push off either of these two ways, I can straighten the curve that I get out to just be a straight line curve in variant of S1 times D2. But they're in different positions. It's separated by kind of a height two. That's really just the fact that the linking number here is two telling us that height difference. And then so the second remark is that so nice paper of Wenzhou Chen that tells you, sort of doesn't say it in this way, but it tells you exactly how to find these differentials for one, let's say two-bridge compliments. And so in particular for two-bridge compliments, it's always the case that actually these differentials just correspond to gluing these pieces of string together, so you could ask whether that's true in general. And that's going to follow from something that Wenzhou Chen and Jonathan Hanselman. Okay, so I think I'm over time and we'll stop. What about? Corbettism maps, okay. So good question. I haven't thought much specifically in this case. Of course, you expect Corbettism maps to be given by counts of holomorphic triangles in this picture in general. And indeed, I asked Robert this a long time ago and he pointed out a place in the paper relating bordered fluorophomology to the Kovanov, to Kovanov to fluorophomology of the double branch cover spectral sequence that should come close to proving that. I mean, I think the main problem in saying something about Corbettism maps is getting the naturality, right? Right, so in order to prove a statement about Corbettism, you have kind of the right naturality for what these curves are, for example. Yeah, that's right. I mean, that's, yeah. Is that like a part of the curve? Yes, yeah, that's right. So now you can push this tube cutting operator around. Any chance of characterizing patterns that are linear? So, I mean, you might hope to prove, find some broad classes of patterns that are linear beyond the kind of examples that I gave. But I mean, like, it's like, you should think, this is a lot like having thin, not fluorophomology, or thin link fluorophomology. I mean, in fact, what's really being used here is just the fact that the link fluorophomology of this link is thin. Okay, and then it's a totally formal argument, just following an argument that's already in the original L and P paper. So, I think, for example, I think it's probably pretty hard to characterize all the thin links. I mean, characterizing all the linear patterns is probably similarly hard.