 I almost got lost in the park. I thought I would be late, but I made it. And yes, I'm glad this didn't start without me. All right, I'll be talking about Schaehn-Lazania modules, which are some proposals for extending Havana homology to three manifolds. So let me first say a few words about Havana homology. If we have a link in S3, then there's this bi-graded theory whose Euler characteristic is q plus q inverse times the Jones polynomial of the link. And well, it's very similar to fleur homology in many ways. So it's factorial undercobordisms. I have some surface in, let's say, R3 times i between two links. And then I get a map between the Havana homologies. There's some ambiguity in science, but it can be fixed. And so this is similar to not fleur homology. And it's also related to fleur homologies by spectral sequences. So Peter and Tom showed that there's a spectral sequence from Havana homology to the instanton fleur homology of the knot. And Dowling showed that there's a spectral sequence from Havana homology to not fleur homology. Right, so now an important problem is to see if we can extend this theory to knots in some other three manifolds rather than R3. And well, just like not fleur homology works for other three null homologous knots in three manifolds. So there are different approaches to doing this. There's maybe the best known one is categorification of roots of unity, work of Havana, Vensacan, and Elias, and Chi, and other people. There's approaches from physics, YouTube Witten, and Gukov, Pei, Putrov, Vafa, and others. But the thing that I want to talk about today is the third one, which are these Cain-Lazanya modules. And well, to motivate them, let me first explain the analog at the decategorified level. So you can ask for the Jones polynomial. So how do we extend Jones polynomial of links to links in other three manifolds? Well, categorification of roots of unity has been done for these polynomials in the result, I'm sorry, without categorification. Just looking at the roots of unity and taking cables of the link, you can get the Witten-Rechetikin to arrive in variants of three manifolds. So that's the usual way. But there's also a simpler way, which is the Cain module. And well, what is that? It has a purely formal definition. So the Jones polynomial in R3 can be defined in terms of the Kaufman bracket. So it's minus a to the third times minus of the right, the Kaufman bracket of the link with some change of variables. Well, the Kaufman bracket is just it's some invariant of, let's think of this as a framed link. So that were a diagram of the link. And then you define an invariant of these diagrams by the Cain relations. OK, and that's the Jones polynomial. And these relations completely determine the polynomial. Oh, yes, thank you. OK, so now in an arbitrary three manifold, well, you can, so Brzezinski and to arrive, define the Cain module, the Kaufman bracket Cain module of Y to be the space of all framed links in Y, module of the same relations. So yes, in R3, it's easy to show that this will determine a polynomial. In general, well, you can make a vector space or a module over ZAA inverse. And just define this and see what happens. So, right, so then if you have a link, then of course you get some element. So for example, for S3 is just ZAA inverse. And that's why the Jones polynomial is a polynomial, is a Laurent polynomial. But yeah, but in general, where does your invariant live? It lives in this Cain module, which is easy to define, but things that are easy to define are hard to compute. So for, well, this started like around 1990. And for the last 30 years, this was kind of going under the radar. And people were computing it for various things, lens spaces, and so on. But it turns out to have an interesting interpretation. So recently, Gunningham, Jordan, and Sopronov in 2019 saw that KBSM is finite dimensional over, let's say, the field of fractions in A. So this is non-trivial. I mean, if you make a definition like this, you might get something infinite dimensional. That's completely uncomputable. But well, it turns out to be finite dimension. In fact, you can say what it is. So there's some work in progress. They announced it at some conferences by Gunningham and Sopronov, proving some conjecture of various people that the Cain module, again, once you're tensored with this field of rational fractions, is the equivalent SL2C blur homology of y, again, over the same field of fractions. So this is some version of what I defined with Mohamed Abu Zaid a few years ago. It's not quite the one in our paper. It's a covariant and just a degree 0 part. But you can think of it as roughly as a count of representations from pi 1 of y to SL2C. So yeah, so even though it looks like some formal definition, that's maybe just some complicated thing. It turns out to be related to representations from the fundamental group to SL2C. All right, so great. So this is for the Jones polynomial. So again, there's a formal definition, and then there are some mathematics that allows you to compute it. So now what can we do at a categorified level? Pardon? Oh, no, no, no. This is the place where the invariant, the de-categorified invariant takes values. So right, so a polynomial is an element of this module, and the generalization, still de-categorified, is an element of this vector space. No, no, no. This is a deep theorem. I mean, I don't know. If you know about the a-polynomial of a nut, that's also related to representations to SL2C. So it's kind of along the same lines, yes. And also, like if you know about Witten's proposal for Jones polynomial and Havana homology, that has to do with SL2C, well, Kapustin Witten equations, but in the limit, it's like flat SL2C connections. So kind of the set of values is, I don't know, presumably it's related to that. Okay, so for Havana homology, you try to do the same thing, but you define something formal, where you divide by the local relations in Havana homology. So what does this mean? So this is the same Lasagna modules. This word defined by Morrison, Walker, and Bedouin, 2019. And there are invariants of some link inside the three-manifold y, but it also has to specify some four-manifold with boundary y, the link is in del of w, which is y, and it's framed, everything is actually framed. So the scale Lasagna module, call it SWL, is the free-o-billion group on Lasagna fillings, modulus and equivalence relation. What's Lasagna filling? So this is w, this is the link, this is y. You have some interior balls. B1, B2, B3, so these are four-dimensional balls. You have some links on the boundaries, L1, L2, L3, and so on. And you have some co-boardism from these links to L, other components, this is a surface. And you also have some elements, B1, B2, B3. So Vi are elements in the Havana homology of Li. Right, so basically instead of links, now we have these surfaces. Okay, yes, I'll get to that at the end. Yes, so it should be in the chain complex, but we don't know how to do that in the moment, so it's only an approximation to the truth. Yes, doesn't it look like one? Here are some more sheets of the Lasagna, if the link has several components. Okay, I'm not an expert. Okay, is Paul Seidel here? Okay, that's true, that's true. Yes, anyway, it's due to them, the name. No, there are many. Is nobody telling me that Lasagna is Lasagna in Italian? Even if it's, no, even if it's more, if I have a noun that's a modifier, it has to be in the singular, it's Lasagna feelings. Okay, anyway, so these are the generators, let's do some math. The equivalence relation is given by, well, okay, so you want it to be multilinear in the VI's, and, okay, well isotope. The boundary, but more importantly, you can fill in one of the balls, so you can replace ball with, okay, so maybe you have one of these things, like L1 here, and you're gonna replace it with another sub-filling with some other links here, like LI and LJ, and some surface sigma i, and you have VI, VJ, but now you also have a V1 here. Maybe you have more things, and now you want, I mean, Havana homology has to come in at some point, so the map induced by this surface applied to VI, tensor, VJ, tensor, I don't know, if you have more things, it should give us V1, okay, so basically you're dividing by local relations in Havana homology, meaning co-boardism relations of this form. All right, there is something slightly interesting here, so this is in S3 minus some balls, and you wanna make sure that Havana homology is functorial under this, so more is on Walker and Webb, but when they defined this, they had to prove, like functoriality of Havana in S3 times i, rather than R3 times i, and this required checking one extra move from the usual functoriality of Havana homology, okay, so the most important example is that S of B4 and L is just Havana homology, so this is an extension of the usual theory, so well, why is this true? Basically, if my four manifold is a ball, and I have the link here, and I have these other balls and this sigma, well, I can always, we have colored chalk, yes, I can use this relation and replace and make like a large ball here, and just replace this, well, instead of having V1 and V2 here, I'm looking at the map induced by this co-boardism, and I get an element here, and this gives me something in, well, I get an actual element V in the, in the Havana homology of this link to L, because I can use the, yeah, because I can use the co-boardism maps inside B4, all right, so well, you can write down this formal definition of an extension of Havana homology to any three manifolds relative to some four manifold, and you can do various things, so for example, I mean, you can also do it for Kovanov-Rozansky homology, and this is just the, yeah, you can extend it to something called Blob homology, we won't get into that. Let me, oh, yes, let me say the main thing. This also has the advantage that functoriality is automatic, so if I have a co-boardism, if I have, well, first of all, W and Y, and I have a link, let's say Y0 and L0, and I have a surface sigma, I have a co-boardism to some other link, L1 in here, then I get the map, let's call it psi of, yeah, I mean, this can be something, let's call it Z, so psi of Z and sigma, it would go from the skein module, sorry, the skein Lasagna module of W relative to L0 to the one of W union Z relative to L1, just by adding this sigma to whatever we're adding here. Right, so this is functorial under co-boardisms. What else? It also decomposes, so what is the structure? It's a direct sum over i, j in Z, and a relative homology class in, well, let me just say in H2 of WL according to the, yeah, the relative homology class of the surface and then S, i, j, WL alpha, so in i and j are the usual bygradings in Havana homology. All right, so what I want to talk about is how to compute this. Can we compute it for anything else? And, yeah, so basically, well, if you want to compute something for four manifolds, you have to understand how it changes when you add handles. So, this talk is to say how to express S, WL in terms of a handle body decomposition of W. Okay, and this is something I did for two handles. It's joint work with my student, Ekshu Naitalat from 2020, and then for one and three handles, it's joint work with Walker and Bedri. Okay, well, and for four handles, it's trivial. So, okay, four handles don't change S. For example, this is easy to see, like S of S4 is still, oh, by the way, when I don't mention, sorry, so in particular, I can do this invariant for closed for manifolds and then the link has to be empty because there's nothing in the boundary. Okay, so let me state the results, I won't prove them, let me start from the higher dimension to the lower in increasing order of complexity. So, three handles, theorem three, if I have a W and some link, and I attach a three handle, so that means that I have a sphere in here, an S2, and I attach some three handle, no, and so originally I had a Y and then now I have some new thing, Y prime. Right, then, well, so what happens? You can look at the equator of a sphere, let's call it J, and push the two hemisphere inside the four manifold, let's call them now delta plus and delta minus. So, the theorem says that SWL, sorry, SW prime L, so W prime is W and I attach this, the three handle is SWL, modulo the image of F, where F is psi of I times Y delta plus minus psi of I times Y delta minus, yes, yes. No, so well, if you're interested in W prime and it comes from something like this, then the co core of this is one dimensional and it doesn't intersect the link, generically. If you think of it from like W prime to W, yes, yes, yes, I mean, the goal is if you have a four manifold with some handles and you want to express it in terms of something easier, so you can just choose which way. You get different expressions, yeah, yeah, yeah. Yes, okay, so just to make sure, so these are disco board is a map corresponding to the upper hemisphere and the lower hemisphere and you take the difference and you get this relation. This is actually not too hard to prove, yes. Oh, separating sphere, yeah, I think it's fine. Okay, now the two handles are the main complication in four manifold, so this is more interesting. So again, I have a link and I have some K, K now it can, K is another link where I attach, well, let's say a knot and it can be linked with L and I attach the two handle. So again, I go from some W to some W prime. So now the theorem is SW prime L is the direct sum over all R one and R two natural numbers S of W, L union K, R one, R two, modulus and equivalence relation. Yes, what is, this is a K ball of K, so K is the attaching link for these two handles and again it can be linked with L where you have R one things in one direction and R two things in the other direction. So you have to understand kind of the ball. Let's say you attach things to a ball, then you have to understand the Havana homologies of all the cables of the attaching link and then you have some, well, some equivalence relation, maybe for reasons of time, I'll just say that this is given by some co-boardism maps between cables. Let me not write them down, but yes, basically you have to understand the scan lasagna for the, for W for L union cables of K. All right, and finally for one handles, let me just maybe explain the simplest case. If I have one one handle and I attach it to a ball, let's say I have S one times B three and I have some link in here. Okay, then I cut this along a B three. So the boundary is S two, let's say the link is in the boundary and it intersects it maybe in two B points. So I cut this open, I get just a B four and I have this tango. Yeah, let's call it R. So R is in, well, it's in S two times I. It's in S three minus two balls. So the scan lasagna of this thing, I'm just doing this example, but I mean you can generalize it by attaching one handles to anything. Well, is some Huxchild homology of some category associated category of tangles associated to R. Let me just put it like this. So this is some direct sum over all tangles T with delta of T, the boundary being two B points of Havana homology of T union R union T bar, modular sum equivalence relation. Get into that. But basically you have to look at all possible ways of closing this R with a tango T and it's mirror T bar. And yes, you get something like this. So in the end, in principle, basically you can decompose a four manifold into handles. And once you do all this, you reduce it to just Havana homology or like the scan lasagna inside B four. You just cut it until you get to B four and then you get some Havana homologies over things that involve how you attach the one handles and the two handles and the three handles. Okay, so unfortunately, of course, the lower you go in dimension, the more complicated the description. Like here you just divided by some maps. Here you had to take the direct sum over all possible cables and here you had to detect the direct sum over all possible tangles and divide. So all right, so it's some way of expressing the invariant in terms of things, but in terms of Havana homologies of the ordinary Havana homologies, but it's a big direct sum modular sum equivalence relation. So still let's see, so what could we, what can we compute using these? We can do some computations. For example, S of S one times P three. So let's just look at one handles and the simplest thing I can think of is S one times two P points. So just two P circles going around. I have to work with the coefficients in a field for some technical reasons. Well, this turns out to be K. So I have, yeah, so this involves some calculation of partial homology. It's K when P equals zero. It's dimension four when P equals one and it's infinite dimensional when P is greater or equal to two. Okay, you can also look at S of S one times S three. This case you get something, again with the empty link in the boundary because there's no boundary, you get one dimensional. You can look at S of S two times D two. Let's say with the empty link. This corresponds to attaching the two handles. So now as an application of VRM two, let's just attach a zero frame to handle along the R naught. Then I can understand the cables of the R naught, the zero frame cables of the R naught. That's no problem and I can compute this and I get something. I get some polynomial ring in A zero and A zero inverse and A i, several i, sorry, just one i, one i, one a, one a one. Degree of A zero is zero zero and degree of A one is zero minus two. This is also infinite dimensional but it's, I mean it's like locally finite dimensional meaning in each ij alpha. So recall that this decomposes according to ij alpha and okay in this case, once you specify these things it's finite dimensional. This is kind of bad that we got k infinity here. This is not locally finite dimensional. I mean we like the places where our invariants take values to be finite dimensional like the scale module was. But okay, yes. Sorry, where, which one? Oh, because otherwise it doesn't bound anything. You want, yeah, you want it to bound something. Right, okay, well there's some calculation for CP two in homological degree zero only. I get that, well this is zero and whereas for CP two bar, I get that this is k equals zero and zero otherwise. So these are pretty different. This involves the two hand-off theorem applied to the R not framed one and that gives us when you take the cables you get some torus links and we use something we know about the Havana homology of torus links. You can also do P framed cables of the R not but kind of that's about it in terms of Havana homology. It would be really nice to understand Havana homology of cables for other knots if we want to do more calculations. All right, so well that's the theory. So let me some open problems. So what can we do with this and what should we try to do in the future? Well, okay, so what is it? Well, we don't know but one thing you can try is to define something similar for nut flour homology. So my student there in Chen he defined as k lasagna using H of k. H of k has slightly different functor reality properties than Havana homology but you can prove a similar to handle addition theorem but the challenge is how is this related to the actual H of k in. So for nut flour homology we have something in any three manifold and you can pretend we don't and just start with S three. Well and in simple examples they're the same but I don't know if they're gonna be the same in general. Here's something that would be interesting. So this is an invariant of four manifolds. It's based on let's say we'll just look at closed four manifolds with the empty link. So does S of W detect exotic smooth structures on W? Well, it comes from Havana homology which can be used to show that there are exotic structures on our four so maybe there's some hope. Okay, some other hope is that, so this is a TQFT and S of CP2 is different from S of CP2 bar. So this is something you definitely wanna have to have a chance of detecting exotic smooth structures. There's a theorem that if you have a unitary TQFT where the invariant of X is related to the one of X bar then you don't have any hope of detecting exotic smooth structures. At least they're different, which makes sense because Havana homology, the maps on on ATAR sensitive to orientation. But the real question is what happens for S two times S two? So, you know, most exotic smooth structures become trivial after you connect some with enough S two times S twos. So if you do this here you can show that it's the tensor product of the two parts. So then you would like to compute this and that would involve, well, it would involve the hop flink and cables of the hop flink. So if someone could compute the Havana homology of cables of the hop flink, then yes, then I would know if I should think about this or not because if this is, you know, if this is zero or infinite dimensional, then there's some hope. If it's not, if it's finite dimensional, but non-zero, then there's no hope because, right. I mean, if it's zero or infinite dimensional, then once you know this invariant, you cannot tell the invariant for X. So then the invariant would be zero. No, yeah, I mean, I'm hoping that it doesn't stabilize. I mean, it does stabilize, but maybe if it stabilizes by zero, then it will fine. Yeah, it doesn't matter. I mean, it's, yeah, even, yeah, if you stabilize many times, then, you know, if this is zero, then I don't know, you get zero. Okay, we can, oh, sorry. Okay, let's talk later. So the final thing and maybe the most important thing is what Tom asked in the beginning. So this is not quite the right thing. So it would be good. So here's the challenge is to define S at the chain level and then take homology. So here we had elements in Kovanov homology, rather than in chain Kovanov complex. And that's why, for example, that's why I think we got something infinite dimensional there. So we can study this for let's say S one times V three and the link in the boundary, which is S one times S two. And then we know we saw that S times S one times V three L can be infinite dimensional. It's not what we want. Like flow of homology, for example, is finite dimensional. But the better version, the chain version, the expectation, I mean, you, so remember in this case, this was some Hawkshield homology of some category. So you can now do a Hawkshield homology of a different category, which is done at the chain level. And this is what Rosanski did. So this is, so Rosanski wrote a paper, you get Rosanski's Kovanov homology for links in S one times S two. And this is locally finite dimensional. In every degree, it's finite dimensional. It may be infinite dimensional overall, but that's okay. So this tells us that, and this is a good theory, it's factorial, there's a Rasmussen invariant for it, and so on. And that's what, yeah. So really, it would be good to, the correct theory would be to define it at the chain level. And then I think this would have like nice properties. And well, the conjecture is that it would always be locally finite dimensional. But why can't we do that? Oh, and by the way, if you can define it, then there would be analogs of the theorems I stated for handle additions. But the problem is that we need functoriality, well, and naturality to infinite order of the chain Kovanov complexes. And that is not known. Yeah, what do I mean by this? So if I have a surface, then, well, if I have a sequence of random master moves, okay, I get a map between the chain complexes, but I want this up to homotopies and higher homotopies and everything to be just kind of, depend on a contractible set of choices. Sorry, which one? No, I don't think it's, I mean, it's not known for CFK either. Oh, it should do the instant on theory. Okay, maybe, okay. All right, we'll think about that. Okay, well, okay, so there's, okay, so my collaborators like Paul, we have some ideas in this direction, so that's the challenge. So hopefully later we'll get to something, but this is just, yeah, some ideas. So I'll stop here. Can you repeat the question? How does it behave under stabilization with CP2 bar? It multiplies like this. I mean, it's still tensors like this. Yeah, yeah, connected sum is always like that. Yes, and that's, oh, sorry, by the way, which one, what am I doing here? This is... I forget. Yeah, I mean, I mean, I mean, you know, I don't expect it to detect all exotic smooth structures, just some, maybe. Like, so yeah, so connect sum with this, presumably, it's fine, connect sum with this would just give you something zero, so it wouldn't detect anything. It's like that, yeah. Oh, well, yeah, okay. Sorry, so Tom is saying that, you know, in inside but within theory or instanton, you use the mixed map like plus, minus rather than on hat. Well, sure, I mean, I don't know. The map on hat still detects exotic smooth structures on some manifolds with boundary. And, you know, you have Bernatin theory, you can do this kind of stuff, yeah, yeah, sure, sure. Yeah, yeah, yeah, yeah. This is just the simplest theory that we can compute at the moment, yeah. Oh, everything works. I mean, the theorems work. The computations are, I mean, I don't know how to compute it for CP2. Like there's no, yeah, we just don't know. Like the cable, the SLN homology of cables, but of course links that well. Sorry, say that again. Yes, yes, we're not there yet. Chain exact, oh, like a long exact triangle? Yeah, probably not. I mean, I don't know if it would satisfy the chain level either, but yeah, I don't know. Yes, yes, yes, you can get exact triangles like that, yeah.