 OK. It's a pleasure to introduce Josh Wang from Harvard, who will be telling us about colored SLN homology and SUN representations of the trefoil. Great. Thank you very much, and it's really nice to be here. Thank you for the invitation to speak. So today, I'm going to tell you about a project that I've been working on concerning colored SLN homology and SUN representations. And the plan for the talk is that I'll first talk about a relationship between Kovanov homology and SU2 representations. And then I'll turn to a generalization of the connection to SLN homology and SUN representations. And then finally, I'll turn to a further generalization colored SLN homology. So the starting point is a nice sort of coincidence between the Kovanov homology of certain simple knots and links and the co-homology of a certain space naturally associated to the link. So here's the space. So given a link, we're going to look at the set of homomorphisms from the fundamental group of the link into the group SU2. And we're only going to look at the representations that send each meridian to a traceless matrix. And we're not going to quotient out by conjugation. We're just going to look at this set. Now, if you've got a link diagram, then there's a very concrete way to understand what this space is. So I'll describe that. So let's say you've got a diagram. And what we want to do is we want to associate to every arc of that diagram a point in the two-spheres, which we're thinking of as just the round two-spheres inside of R3. And then for every crossing of that diagram, we're going to have a constraint. We're going to have a requirement. And it's that if the crossing, the over-strand is labeled A and the under-strand has arcs B and C, then we want that rotation about the axis that passes through the point associated to A. That 180-degree rotation should swap the two points associated to B and C. So every arc, we've got a point. And every crossing, we've got a constraint. And one such configuration you can show is essentially the information of a representation in this space R2 of L. And so the whole space is just the set of all of these sorts of configurations of points. So as an example, if you take this diagram of the Hopf link, there are two arcs. So we're looking for pairs of points on the sphere. But the crossings tell you is that if you rotate about one of the points, it should fix the other point. So there's exactly two possibilities. So either the two points are at the same location or the two points are antipodal. So the representation space has two connected components. And both components are both copies of the sphere. For this diagram of the trefoil, there are three arcs. So we're looking for triples of points on the sphere. And the crossings tell you that if you rotate about any one of the three points, it should swap the other two. It turns out there are two possible configurations. So either the three points are at the same location or the three points lie on a great circle and are equidistant on that great circle. So there are again just two connected components. One is a copy of the two sphere and you can convince yourself the other one. You can identify with SO3. Okay, so it's a very concrete space. And the observation of Kroneimer and Rufka and also Jacobson and Rubinstein is that if you take a two end torus nodder link and the covanophomology of that link coincides, it's isomorphic to the coamology of this representation space. Now you can say something about gradings but for simplicity I'm just gonna ignore gradings entirely for this talk. So you can just think of covanophomology as just a single abelian group. You just take the direct sum over all gradings and then similarly with coamology just ignore the grading. And in fact this actually also extends to two bridge nods and links as well. But it certainly is not true for all links. There are even alternating counter examples. Okay, so why is there this coincidence? Well it fits into the story of the connection between covanophomology and SU2 instantanophomology. So Kronauer and Rufka defined an SU2 instantanophomology for links, i-sharp, and they defined a spectral sequence from covanophomology to this invariant. And very briefly, i-sharp is defined by a version of Morse homology for a function called the Tern Simons Functional or a version of the Tern Simons Functional. And if you look at the set of critical points of this function, you can identify that space with this representation space R2 of L. So in particular, I mean R2 of L is not a discrete set of points, so this function is not a Morse function. So really you need to take up perturbation of this function and then do more somology to that perturbation. But if the Tern Simons Functional is more spot in a suitable sense, so if this critical point set, this representation space, if it's smooth manifold and if the function is non-degenerate in normal directions in a suitable sense, then what you'll get is a spectral sequence from the cohomology of the critical set to the instantonophomology. So there's sometimes a spectral sequence from the cohomology of the representation space to i-sharp as well. But then the coincidental isomorphism between covanophomology and the cohomology of this representation space, you can interpret it as a pair of coincidences that both of these two spectral sequences immediately degenerate in some strong sense. In which case all three of the groups here, covanophomology, i-sharp and the cohomology of the representation space are all isomorphic. So now I'm gonna move on to SLN homology, where we're looking for a similar, motivated by a picture similar to this one. Okay, so first of all there's a link polynomial called the SLN link polynomial, I'm denoting it PN, and it's characterized by this skein relation and it's a specialization of the Humphley polynomial. And in the case that N is two, the SL2 link polynomial is just the Jones polynomial. And SLN link homology, also called Kovanov-Rozansky homology, is a categorification of this polynomial. It's a bi-graded homological invariant obtained as the homology of a chain complex associated to a diagram. And in the case that N is two, SL2 link homology, categorifying the Jones polynomial is isomorphic to Kovanov homology. So you can think of it as a generalization of Kovanov. And this was first defined by Kovanov and Rozansky using matrix factorizations over a field of characteristic zero. And it was only more recently that there have been definitions over the integers. And the more recent constructions are, I think more closely analogous to Kovanov's original definition of Kovanov homology. So I implicitly work with the more recent constructions. Okay, and to tell you a little bit about how this invariant is constructed, first gonna mention that the polynomial invariant, it in fact extends to certain trivalent graphs in the plane. And the class of such graphs are the MOY graphs. And this extension of PN allows the polynomial to satisfy these different looking skein relations. It satisfies an oriented skein relation up at the top, but then it also satisfies this different looking skein relation. So given a crossing of a diagram, there are two different planar resolutions of that crossing in this context. So one of them is the oriented resolution. And the other one is almost the unoriented resolution. You take the unoriented resolution, but then you add an extra edge that connects the two strands. And then you label that edge too. So these trivalent graphs are, the edges are labeled and they're oriented. The labels are either one or two. The addition of this edge here takes you from the world of just complete resolutions like circles in the plane into the world of trivalent graphs in the plane. Okay, and then the construction of SL and link homology is obtained by sort of categorifying these last two skein relations. So the first thing that you do is for every MOY graph, you associate what's called a state space, the state space of that MOY graph. And what this is is just a free abelian group with a Z-grading. And with respect to that Z-grading, its graded rank is just the polynomial invariant. So abstractly, as a group, it contains no more information than just the polynomial. It's just same information as the polynomial. But what's new and interesting about it is that it's functorial with respect to a notion of cubortism between MOY graphs. And those cubortisms are called foams. So the abstract groups themselves are determined by the polynomial, but the maps are more interesting. Okay, and then the complex that you associate to a link diagram is obtained by the cube or resolutions used to construct Havana homology. So for every vertex of the cube, you have a complete resolution of the diagram, which in this case is an MOY graph in the plane. You associate to that vertex the state space of that MOY graph. And then for every edge of the cube, there's a foam that goes between the two MOY graphs and that foam is given locally by these pictures here. And then you take the induced map on state spaces and then add all those maps up together with some signs and that's the differential of the complex. And you'll notice that these foams, if you ignore the gray disc, they look like mergin, they look like saddles. And these particular local foams are called the zip and unzipped foams. Okay, so that's Havana-Vrasansky homology. The relevant space of SUN representations is as follows. So we look at again just the set of homomorphisms from the fundamental group of the link into the group SUN. And now we're again going to restrict where the meridians go. So we require that all the meridians go to a particular conjugacy class inside of SUN. And it's the conjugacy class of this particular matrix, this particular diagonal matrix. So we take first the diagonal matrix where you've got a single negative one as the first entry and then the rest of the diagonal entries are one and we multiply it by a root of unity and the resulting thing actually lives in SUN. And we can actually, we can understand this conjugacy class very explicitly. So if you take a matrix A inside of this conjugacy class, then it determines an orthogonal decomposition of CN, it's eigenspace decomposition. And one of the eigenspaces is a line, lambda A. And that's the line sort of associated to the negative one block. And then it's orthogonal complement is a N minus one plane eigenspace associated to the block of ones. And then this correspondence between the matrix and the line, just one of the two eigenspaces is a bijection, it's an identification. So if you give me a line, then I can give you the matrix A just by defining it by the formulas for the eigenvalues. So just as for R2 of L, we can think of RN of L quite concretely from a diagram of the link. What we need is a line in CN for every arc of the diagram. And then for every crossing, there's a constraint. So here are some examples of the spaces that you get, the representation spaces. So if you have an unknot, you just need, you get one free choice of a line, so you get just complex projective space. For the two component unlink, you get two free choices of a line. Or the hop link, we think about it as the diagram with two arcs, we're looking for pairs of lines. So it turns out that the components are just where the pairs of lines are, the two lines are the same line, or the two lines are orthogonal. So one component is CPN minus one, and the other component is this partial flag manifold consisting of pairs of lines. That are orthogonal. Okay, and then for the trefoil, it turns out to also just have two components. One's a copy of projective space, and the other turns out to be the unit tangent bundle of CPN minus one. And you can check that when N is two, the unit tangent bundle of Cp one is SO three. And I'll note that if you look at the two end torus knots and links, the representation spaces are just built out of the pieces we've already seen. So they always have a copy of CPN minus one. If it's a link, it's also got a copy of this partial flag manifold, and then the rest of the components are just copies of X. Okay, so R2 of L, this space of meridian traceless SU2 representations, was first studied by Xiaosang Lin. And then this generalization to SUN was introduced by Kronheim and Rufka. And I'll mention that Lab and Zettner, and also Grant, they studied the analog of this representation space for MOY graphs in connection to the polynomial variant. And, well, what's the expected picture? So what you would expect is that there's an SUN instanton homology for links defined by a version of Morse theory for a function whose critical set is this representation space we've been talking about. And you'd expect a spectral sequence from Kavana-Rosansky homology to this instanton homology. And then furthermore, just as before, you'll sometimes get a spectral sequence from the homology of this representation space to the instanton homology. Now, there actually is a version of SUN instanton homology for links. Kronheim and Rufka defined it. But unfortunately, they later showed that its rank doesn't change under crossing changes. So, for example, for a knot, its rank is just N, the rank of the invariant for the unknot. And so, although it is indeed an SUN instanton homology for links defined by a version of Morse theory for a function whose critical set is this representation space, I don't think it really sits in this picture in this way. So, for example, the spectral sequence from Kavana-Rosansky homology to this instanton homology, it won't be degenerate for the truffle if it existed. But maybe it does fit in a similar sort of picture or a more general picture, so it might be related to a lead type deformation of Kavana-Rosansky homology. But I expect that you should be able to modify the construction so that you indeed get this picture and for say, two bridge knots and links, both spectral sequences exist and immediately degenerate. Okay, but before attempting to construct that and also construct the spectral sequences, I mean, the first thing you should check is just, well, does Kavana-Rosansky homology for simple knots and links match the homology of this representation space? That also will tell you if they do match that you've got the right representation space. I mean, you can define versions of instanton homology where the relevant representation space is different. Okay, and indeed, if you look at the two end torus knots and links, then in fact, the Kavana-Rosansky homology is isomorphic to the homology of this representation space. And I call it an observation because you can just calculate both sides explicitly and check that the answers are the same. Everything is simple and explicit enough that you can just do that calculation. But I want to explain a slightly better proof, something that is a little more clever than just calculate both sides explicitly. And the reason is that where I'm going, so if I'm, the next thing I want to talk about is colored acetylene homology, it's no longer tenable to just explicitly calculate both sides. The groups get way too complicated. You'd need some more indirect approach. So now I'll explain a proof of this observation for the Hoflingke and the trefoil, which doesn't involve actually just explicitly calculating these groups. Yeah, well, so it's kind of migrating and then analogous to the base point action on Kavana homology, which gives you the structure of a module over like the z-adjoin x mod x squared. Now a base point gives you the structure of a module over z-adjoin x mod x to the n. It's also a reduced variant, like reduced Kavana homology. Yes, there is. So this representation space actually is a fiber bundle over Cpn minus one. And if you take the induced map on cohomology, that makes the cohomology of the representation space into a module over the cohomology of Cpn minus one, which is z-adjoin x mod x to the n, and they match up. If you take a fiber of that fiber bundle, you take the cohomology of that fiber, you get something whose cohomology is isomorphic to reduced Kavana-Rusansky homology. So we take the full twist on two strands. So just these two crossings. And by definition, the chain complex that's associated to it has four terms. So the two crossings each have two different resolutions. But it turns out there's a way to simplify this complex. And I believe it's the first version of this is due to Krasner. And I won't explain it, but the upshot is that you can simplify it. So it's homotopy equivalent to a complex, which just has three terms. So the first map in this complex that I've denoted z is just the zip foam that showed up in the definition of what you associate to a crossing. And then the second map is a, it's a difference of dot maps. So I haven't really discussed dot maps, but there are dot maps in Kavana homology. There are also dot maps in Kavana-Rusansky homology. And then if you close up this diagram, so if you connect up the two endpoints on the left and the endpoints on the right, and you get a diagram of the hopflink, and if you close up the simplified complex as well, you'll get a complex homotopy equivalent to the complex for the hopflink. So if we close these graphs up, then this is what we'll get. But the important thing to notice is that the map given by the difference of dots, once you've closed it up, the two dots lie on the exact same arc. So the two dot maps are the same and so their difference is zero. So we find that this complex actually splits as the direct sum of two complexes. One has a non-trivial differential and the other has no differential. But then we notice that this first complex is actually just literally the complex associated to the diagram of an unknot with a twist in it. And so by right of Meister invariance, we can just homotop that complex to the state space of just a planar loop. And we're left with no differential. So that's the homology. So the Kovon-Avrazansky homology of the hopflink is just the direct sum of these two state spaces, the state space of an unknot and the state space of this theta graph. Okay, and as I mentioned before, the state space as a group is completely determined by the polynomial invariant. So it's just free abelian and its graded rank is the polynomial invariant and we can just easily calculate that polynomial invariant. And what you'll find is that the state space is abstractly isomorphic to the coamal... The state space of the planar loop is abstractly isomorphic to the coamalgy of Cpn minus one. And the state space of this theta graph is abstractly isomorphic to the coamalgy of this partial flag manifold, F11n, which matches with the coamalgy of the representation space of the hopflink. So this argument, as I've stated it, is basically just calculate both sides and see that the results are the same. But we won't do that for the trough oil. But before going to the trough oil, I wanna mention one particular thing. So the state space of the theta graph and the coamalgy of this partial flag manifold, although we know abstractly that they're isomorphic, they're both actual abelian groups that you can define an isomorphism between. And you can define an explicit isomorphism that plays nicely with certain extra structure on both sides. Oh, let me describe one nice feature of it. So over this partial flag manifold are two tautological line bundles. So one of them, the fiber over a pair of orthogonal lines, lambda A and lambda B, is just the line lambda A. Then the other one, it's a line bundle and the fiber over the pair lambda A lambda B is the line lambda B. So these are two tautological line bundles and they're complex line bundles. And their first turn classes turn out to form a basis for the second coamalgy of this partial flag manifold. So now you can define an isomorphism between the state space of that theta graph and the coamalgy of this partial flag manifold so that the isomorphism intertwines cup product with the first turn class of one of the bundles with the dot map on one of the two edges labeled one. And so that cup product with the other first turn class gets intertwined with the dot map on the other edge labeled one. So you can define an isomorphism that intertwines just natural endomorphisms on both sides. So now for the trefoil, there's a similar simplification for a three twist. It starts off with the same complex as before, a zip foam and then the same difference of dot maps. But then we've got an extra map and that extra map is also a difference of dot maps but the location of the dots are different. And if we close up this complex, then that middle map just as before will vanish. The first two terms are homotopy equivalent to just the state space of the unknot. But then we're left with an interesting complex and that interesting complex goes from the state space of the theta graph to itself and it maps by this difference of two dot maps. But now from the previous slide, we can identify all of these groups and all of the maps with these coamalgy groups. So the state space of the unknot is again just the coamalgy of Cpn minus one and then this other complex, it goes from the coamalgy of the partial flag manifold to itself and the map is given by cup product with this difference of first-turn classes. So this is the complex homotopy equivalent to the Kovano-Rosansky complex of the trefoil. And I'll remind you that the representation space of the trefoil has two components, one's a copy of Cpn minus one and the other component is a copy of the unit tangent bundle. So if we want to show that the homology, the Kovano-Rosansky homology of the trefoil matches the coamalgy of this representation space, suffices to show that the homology of this simple complex, going from the coamalgy of the partial flag manifold to itself, that the homology of this complex is isomorphic to the coamalgy of the unit tangent bundle of Cpn minus one. No, so I'm just literally taking the set of homomorphisms, not quotienting out by conjugation. Okay, so at this stage, it's possible to literally just calculate what the homology of this complex is, but you need to take a little care. It might not look like it, but it turns out that there's Z mod n torsion in the homology of this complex. But here's a better way of identifying the homology with the coamalgy of this space. So it turns out that this unit tangent bundle is actually a bundle in a different way. So it's actually a bundle over this partial flag manifold with S1 as a fiber. And as a circle bundle, it has an Euler class, and by characteristic class computation, it turns out that this Euler class is exactly this difference of first-turn classes. And then associated to a circle bundle is a geese in exact sequence, and it takes the following form. So two of the terms in the triangle are coamologies of the base, and the map between them is cut product with the Euler class. And then the third term in the triangle is just the coamalgy of the total space. So this triangle gives us a relationship between the coamalgy of X and the homology of that chain complex. And strictly speaking, you find that the coamalgy of X is an extension of the kernel of that map by the co-kernel of that map, but in fact, there are no non-trivial extensions. Just because the coamalgy of that partial flag manifold is supported in even degrees. So in fact, the coamalgy of this complex is indeed isomorphic to the coamalgy of this space. Okay. So I'll now turn to colored esolan homology. So colored esolan homology is a further generalization. So it's now not an invariant of just an oriented link, but the link has some extra data, and that extra data takes the form of a labeling. And what that is is just a number K for every component, and that number K is between zero and N. And ordinary esolan homology is just when all the labels are one. If the labels are zero or N, you get something trivial. If you're familiar with this story, then I'll just mention that K is supposed to indicate which exterior power of the defining representation of esolan that you're coloring the components with. Okay, so this is an invariant that was first constructed by Wu using matrix factorizations. And then in the same work that I was mentioning from before, there's a definition that's given over Z. It's quite analogous to how ordinary esolan homology is defined and how coamalgy is defined. And to just briefly say how the labeling sort of arise, I'll mention that basically instead of a cube of resolutions, you have a rectangular prism of resolutions. Though associated to a crossing between strands labeled say two and two, instead of there being just two planar resolutions and a map between the two, you now have three planar resolutions arranged in a row and you have maps that join adjacent terms. And now the planar resolutions, they're still trivalent graphs, but the labels are allowed to be bigger than two. Okay, and then for a crossing between strands labeled I and J, you'll get just a longer complex. And the length of that complex is just the smaller of the two numbers plus one. But here's what you would get if you are say trying to compute the homology of the Hopf link by definition the complex. So it's associated to it if the components of both labeled two will have nine terms arranged in this way. So the complexes get quite big very fast. Okay, now as for the representation space, so the way that the labels come in is that we're still gonna look at homomorphisms from the fundamental group into SUN. But now what we'll require is that a meridian of a component labeled K will go into a conjugacy class depending on that label K. So CK is the conjugacy class of again just a particular matrix. It's you take the diagonal matrix with K negative ones and the rest of the terms one then multiply it by a root of unity, kth power of the previous root of unity. And just as before, we can explicitly understand and identify what this conjugacy class looks like by just sending a matrix to one of its two eigenspaces. And what you get is just that this, you can identify this conjugacy class with the complex gross monion of K planes inside of CN. Okay, and some examples of this space. So if you take the Hopf link where both if you take the Hopf link labeled 22, consider its representation space, then there will actually be three components. So one of them is a copy of the gross monion G2N and the other two are partial flag manifolds. So this first one is triples of lines that are pairwise orthogonal and then F22N is a pair of planes that are orthogonal. For the trefoil labeled to the representation space also has just three connected components. One of them is the gross monion G2N and then the other two, I don't really know simpler names for the spaces you get, but they're both homogeneous spaces for UN. So the space of say left cosets of a particular subgroup of, and you can say exactly what those subgroups are. It won't really matter for this talk, but they are just literally explicit sets of matrices. Okay, so now the first trouble that you encounter when attempting to say compare the comology of these representation spaces with colored Esalen homology is that there are very, very few computations of colored Esalen homology in the mathematical literature. Well, for the unnot labeled K, it's kind of baked into the theory what the colored Kavan-Prasansky homology is. It's just the comology of the complex gross monion GKN. So it matches up for that. For the Hopf link, where one of the components is labeled one and the other component is arbitrarily labeled, this was computed by Yonezawa. And I believe that's strictly speaking it for what's appeared in the mathematical literature for links where at least one component is labeled by a number bigger than one. There are computations in physics for the Hopf link labeled I and J. And there are also computations in the mathematical literature for closely related invariance, but not this one in particular. Okay, so my main result that I've been writing up and been working on is that colored Esalen homology is isomorphic to the comology of this representation space in the following cases. So they match when the link is the Hopf link with arbitrary labels. And they match when the link is the trefoil with an arbitrary label. And they also match when the link is a two-end torus knot or link and all the components are labeled two. So you could view this as a computation of colored Esalen homology, but at the end of the day, I show that these two things are isomorphic but I don't have an explicit formula for what the groups are. So for example, if you ask me what is, how much ZMOT7 torsion is there in the trefoil labeled 13 in its SL27 homology, I don't know. But from this work, you can actually do some explicit computations for small values and I'll show you some of them at the end. I'll also hopefully illustrate how large and complicated these invariants are. But first, I'll explain briefly how you prove this for the Hopf link and trefoil when the labels are all two. Okay, so from before, we found that the Hopf link labeled 11, its complex was homotop equivalent to the direct sum of two state spaces. For the Hopf link labeled 22, its complex can also be homotoped to a direct sum of three state spaces. And it's the state spaces of these three particular MOI graphs. And as I mentioned before, the representation space has three connected components. Just as before, you can define actual explicit isomorphisms between the state spaces of these MOI graphs and the co-amologies of these partial flag manifolds in such a way that they intertwine dot maps with churn classes. For the trefoil labeled one, this was what we got on a previous slide. It's the sum of two complexes, one of which is this complex going from the co-amology of the partial flag manifold to itself, given by cup product with the difference of first churn classes. Now, most of the work goes into simplifying the complexes for, say, the trefoil. And for the trefoil labeled two, this is what you get. So it's the direct sum of three chain complexes. The first has no differential, just the co-amology of the Grassmannian G2N. The second piece, it looks a lot like the piece we've already seen before. So we've got just the co-amology of a partial flag manifold mapping to itself by a difference of first churn classes. But then the third piece is new. It's got four terms, and you've got first and second churn classes involved. And I'll remind you that the representation space has three connected components. Though the first component is accounted for already in the chain complex. And then the second component, this thing turns out to be a circle bundle over the partial flag manifold F111N. And you can use a similar argument from before to identify its co-amology with the homology of this complex. But you need something new for the last complex and the last component of the representation space. And at the end of the day, what I use to identify the homology of this complex with the co-amology of this component of the representation space is this theorem of Guggenheim in May, which concerns the co-amology of homogeneous spaces. So in our case, our space in question is a homogeneous space. It's UN, MOD, a particular subgroup. And Guggenheim in May show that the co-amology of this space is isomorphic to a torsion product. So it's Tor of Z and the co-amology of the classifying space BK as modules over the classifying space BG. And I'll remind you that Tor is defined by taking a projective resolution of say, co-amology of BK as a co-amology of BG module. Then you tensor it with Z and you'll get an actual chain complex. Then you take the homology of that chain complex to get the Tor group. And the key idea in how I apply this is that I actually choose a projective resolution so that the resulting chain complex you get is literally the complex that arises in the colored Esselen complex. So you get an identification at the chain level. And then it's a separate matter to actually take the homology of that complex if you want. Okay, and so I'll end with some explicit computations. So here's the SL4 homology of the Truff oil labeled two. So I didn't talk about bi-gradings but these are bi-graded invariants and the techniques I'm using allow you to calculate bi-gradings as well. So on the bottom is the homological grading and along the left is the Q-grading. And you'll notice that there's Z mod two torsion, there's Z mod four torsion. And if you take the graded order characteristic of these groups, you'll get this polynomial, this big polynomial and you'll find this polynomial at the bottom of the not atlas page for the Truff oil. I've always wondered what these polynomials mean and now somehow they're at least related to the homology of these representation spaces. Okay, and then just for fun, here's the SL6 homology of the Truff oil labeled three. The biggest complex that I've ever taken the homology of. There's Z mod two torsion, Z mod three torsion and Z mod four torsion. That's it, thanks very much. Well, somebody, I think, somebody, yeah, somebody could write a computer program to tell you the answer. The question was when I mentioned that I don't know how much seven torsion there is in the 17 labeled 25 or whatever, is that just that I don't remember it off the top of my head or is there actually, do I actually not know like an algorithm to figure it out? But I do know an algorithm to figure it out. I mean, in a certain sense, like I have an explicit, pretty minimal chain complex and I know what all the differentials are. So yeah, so just as there, the question is I mentioned the base point stuff, I mentioned reduced. I also, well, Claudie suggested I mentioned equivalent, but maybe, okay, I mentioned it at some point. Yeah, so what can be said about the other versions or equivalent versions of SLN homology? And right, so there's a version of Kovanov homology called U2-equivariant Kovanov homology and the complex, the U2-equivariant complex determines all the different versions of Kovanov homology. So Lee homology, Barnetang homology and so on and so forth. And analogously, there's a UN-equivariant Kovanov-Rosansky homology. And yes, you can show that the UN-equivariant Kovanov-Rosansky homology of the knots and links that I mentioned, so say the trefoil, is isomorphic to the UN-equivariant, UN-Borel-equivariant homology of this representation space. Yeah, I mean, my hope is that in the near future that maybe the two end torus knots and links are possible, but I predict that the statement is true for two bridge knots and links. But I don't really know how to prove it. Well, at the very least, I think that this tells you that we've got the right representation space. So I mean, there are versions of instantonomology where the relevant representation space is different, but whatever version of instantonomology should this representation space should be the one that's relevant. I guess I have some ideas for how to approach constructing that spectral sequence and invariant. We can talk about it later if you want. Say that again. Yeah, those are all, so, so Lob and Zentner and also Grant, they were studying that question of looking at the representation spaces of the planar MOI graphs. And there's a class of simple ones for which the cohomology of that representation space matches the state space, but they don't in general match. And in particular, actually, the representation space might not be smooth and its cohomology isn't symmetric, like it doesn't satisfy point-created duality, but the polynomial always does.