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 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.


The National Broadcasting Company presents the adventures of Sam Spade detective Is I sweetheart Sam. Oh, I was worried to death this office glass all over the floor holes in the wall That was just business going on as usual during all occasions F Well, what was it all about Sam? They tried if they just tried to pluck my feathers and cook my goose I'm Thanksgiving too. How could they? Oh, they were a mean lot. Are you all right? Hail and hearty every jiblet in place and on a feather ruffled. Did you reminisce Thanksgiving? Oh, it was heavenly. Mama had a turkey dinner, sage dressing, cranberry sauce, candied yams, hard siding A little. Come clean Effie. Well, I had two glasses Everyone was there. Cousin Gertie, Dwight, Mrs. Sloth. I was disappointed when you didn't show up, Sam Did you have Thanksgiving dinner? Sure. Where? At the helping hand rescue mission where there's plenty of free parking and never a cover charge For further details consult the report which I will presently be down to dictate on a tasty Chronicle of foul play that terrified turkey caper For NBC William Spear radio's outstanding producer director of mystery and crime drama Brings you the greatest private detective of them all in the adventures of Sam Spade. Oh You were waiting for having Thanksgiving dinner at a rescue mission where mama cooked a perfectly Thank your mother for me F tell her I'll be over to break wishbones with her tonight and to atone for my social indifferences Here's a little something I brought for you.  It's a gun our founding fathers used it in foraging for feathered food when they settle this abundant continent And it's mine to do with what you will Pencil boy Did you meet a founding father don't peak date November 24th 1950 to detective lieutenant I see Kelsey homicide details San Francisco police from Samuel Spade license number 137596 Subject Turkey dear Kelsey This was a big week for the cranberry pickers the butchers the sage makers and the stomach pill people But for private detectives it was strictly from hunger My office door opened only twice a day once to let me in and once to let me out and when on Wednesday I heard a knock on the door. I went into a paroxysm of delight come in come in Come in Entry boo under you stand when I ran out of languages I got up from behind the desk walked to the door and opened it Standing there was a small middle-aged man with a pink bald head His blue surge suit needed pressing and he was nervously fingering a strawberry birthmark under his left ear Mr. Samuel speed I am Make may I may I have a moment you may have several but not in the corridor. It's not in my lease I'll come in good good good Well, well you'll have to excuse me mr. Spade. I I've had so few dealings with private detectors I I find it hard to begin. Well, I Perhaps I shouldn't have come at all. Goodbye. No, no, no, wait a minute. Maybe I can help you Well, you see I oh What's the use you won't believe me nobody does I'd really better. Oh now wait wait I'll believe you all I ask is a chance now now. Let's start with your name. Oh my name. Yes. Yes Yes, my name To begin with you won't believe that but I can verify it. Yes. I can it's on the registration book of the old colony hotel In the 1943 phone book and on my old driver's license. Well, I have to know it before I can verify it Yes, yes, of course you will it's a It's Tom Well, that's not so hard to believe. Oh, you haven't heard the rest of it It's Tom Turkey You see I told you you wouldn't believe it. I'd better go. No. No, let me be the first to believe you now mr. Tom of what's your problem?   Yeah. Well, so long wait wait I'll tell you why I called I've had a pest in my office keeps coming back I think she's a turkey somebody wants to dress I brushed him But your name came up and I just wanted to warn you he might be in the sea. Yeah, I'm confused now I never knew you to turn your back on a buck. Oh, I don't want any of this one his buttons are loose My advice to you is to bounce it. Well, we've never traded advice before Coochal. Why now well after all We're in the same racket if we can't help each other sure Al sure I appreciate it. Give me a ring We've got to get together sometime. Yeah, when I get a free night will Jimmy parking meter. Yeah, we Yeah I turned back to the desk and what I saw in front of me was an empty chair Tom turkey had taken wing I got up and walked to the window and a minute later I saw him come out of the building downstairs and start to cross the street And then I saw something else a large four-ton truck was tearing down the street picking up speed Instinctively I shot at a warning and at the last second Tom turkey scrambled from in front of the truck and disappeared into The alleyway a truck roared up the street. Not its side was printed in gold letters Hey, and you drive it. There was nothing to say. It wasn't coincidence this near myth mishap But somehow I found myself intrigued wanting to hear more of a little guy's story He said the old colony hotel on the way. I stopped at the library found an old 1943 phone book and looked He was listed Thomas turkey it said out of curiosity. I rang the number. Hello. I wonder if you can help me I'm inquiring about a mr. Turkey turkey. This ain't his number no more. I know I haven't had any calls for him for years Screwing me. Yeah, I know I know a new woman named rabbit one Mrs. Rabbit about turkey. Could you remember what he looked like? I don't hey, Manny. What turkey look like? Yeah, yeah, small man around 50 nice fellow man. He's a strawberry under his left ear Yeah, strawberry under his left ear. Well, thank you madam for your information Thanks to Manny. Well, you're welcome, but I don't know what you're gonna do with it. Oh, man.  I continued on to the old colony hotel room 75 cents it said Tom's room was 114 Who is it Sam spade? I mean mr. Spade, I'm I'm sorry I ran away. I didn't think you really believed well I'm not sure I do yet. Tell me was that truck an accident? Oh, I don't think so.  Oh, who were they and why would they want to kill you? I don't know. I just don't know look Let's tack tack tack I dialed your old phone number and the people who answered said you're dead Oh, a lot of people think I'm dead. Yeah, look, do you still want me to work for you? Oh, yes Yes, please well, you'll have to tell me more than I can't I I guess I'd better tell you everything. Oh It's it's hard to talk about mr. Spade It's not easy to admit to someone You've been a foolish man. You see I just turned 50. I was quite tired of the life I'd let proper dull and unfruitful except in money My business was wearing and so was my wife, Henriette. This has a traditional ring anyway to make it short I decided to run away one day. I drove to work I parked my car in the middle of the bay bridge where the suicide note left it and disappeared Where did you go?  I took a job on a boat I did on a boat and then I settled in San Paolo, Brazil under another name now your back Why maybe I got lonely maybe I got wiser maybe Maybe I felt I paid enough for my mistakes. Let's just say I'm back I want to be with Henrietta. Have you seen her I checked into this hotel and wrote her a letter saying I wasn't dead I was back in San Francisco and I I wanted to come back to her if she still would have me But I told her I wouldn't bother her unless she wanted to see me that she could contact me here That was a week ago and you haven't heard from her no no and almost right away these attempts on my life began I see. All right. What's her address 3118 Monroe.  Come on. Let's go. No.  That's right Oh, thank you Thank you. Mr. Spade you you do believe me I think I'm really ready to face the world again now I deposited Tom in my apartment with instructions to open the door for no one but me and then I proceeded to 3118 Monroe in the high rent district I was ushered through a comely portico by a japanese maid who told me to wait in a study heavy with mahogany In a moment two people came in the woman wore a black dress Silver pendants flat shoes and a complexion the color of apple meat She was miss. Henrietta black and or mrs. Tom turkey The man turned out to be liander loose the lady's attorney business manager and canasta partner You say you have something important to discuss with me mr. Spade. I do I hope you don't mind my asking mr. Loose to be here not at all Well, mrs. Turkey. I just talked to your husband Tom mr. Spade if you please I say something a rather feeble attempt at comedy mr. Spade. Oh, I wasn't trying for laughs. You are mrs.  Spade someone was always going to stuff Tom based him dress him slice him this season They're going to kill him. They are not going to kill him. He's already dead. He's not dead mrs. Turkey and you should know it I should yes. He sent you a letter saying he was back in san francisco and wanted to see you mr. Spade This has gone absolutely far enough not quite What about the letter? I know of no such letter I see well, thank you for your time. I'm sorry. I bothered you. You used bad judgment and coming in the first place Yes, maybe you're right There was falsehood in this someplace lieutenant and it stuck out like a fat girl in slacks The only thing to do was to go back to my apartment get Tom turkey and confront mrs. T with her husband in the flesh When I got back to my apartment building. I spotted in rapid succession one an ambulance To a police car and upstairs outside my half open apartment door. I spotted three you I've been expecting you what's going on kelsey. Yes serious sam serious Who's that bald headed man moving around the apartment? That's McCracken the new medical examiner checking a stiff on your rug I stepped around you lieutenant and pushed the door all the way open I saw McCracken kneeling over the body and a couple of men from homicide taking photos I moved into the room feeling nothing good A little guy had given me a job and while I was jacking with his wife somebody got to him and in my apartment where I'd stashed him McCracken stood up and I looked down at the body Then I looked again Who I saw wasn't tom turkey at all. It was the late private eye al kuchel You are listening to the weekly adventure of radio's most famous detective sam spade You friday fans of sam spade there's mystery on saturday evening two on nbc Tomorrow the man called x sets out on another mission of danger and intrigue in some far off corner of the earth Herbert marshall stars as the man called x a man without a name who travels the world over Protecting his country's interests. He lives by his wits and his business is danger He is the man called x tomorrow over most nbc stations For top sunday listening. It's another broadcast of the big show on nbc This sunday your stars include fred allen jack carson mindy carson ed archie gardener ed wind and many many more And tulula is your mc as usual this sunday. It's the big show on nbc And now back to the terrified turkey caper tonight's adventure with sam spade While the men from homicide were taking pictures etc You and i lie lieutenants were going round and round on the question If i didn't kill the man found in my room Who did and you were sufficiently impressed with my insults kelsey not to hold me for the murder We bowed to each other and i left Thinking back to the truck that had almost run turkey down. I went to the hains you drive truck rental garage I'm a detective, could you give me a list of names for everybody who rented a truck from you during the past few days? Sure. He handed me a big registration book, and I read every name for the past week. For the first five days, they all seemed to be nice, normal, abnormal names, and then under the rentals for the day before was the name of John Smith. John had given his address as 7,200 Kearney, and I happen to know that Kearney only goes up to 2,200. The dispatcher said that Smith had returned the truck about three hours before, and he remembered him as an ugly, heavy-set, and rough-voice character who looked like an ex-longshoreman. They had already washed the truck so the fingerprints were all outstuck. Mr. Spade again. Look, I'd like to speak with Mrs. Turk, uh, Miss Black, if you don't mind. Come in. Come in. Thank you. Into the dip. Right. Well, I was sure you'd look into this affair a little more and realize that it was just a blind alley, a hoax of some kind. Where's Miss Black? Oh, she's upstairs lying down. The whole affair is upset her, and she asked not to be disturbed. I think the wisest cause of action for you, Mr. Spade, is just to let the matter drop.  The police wouldn't hear of it. Huh? Murder? Ooh. An unfrocked private detective named Al Kuchel. Well, what does this have to do with Henrietta Black? Al Kuchel called me earlier today and said that Tom Turkey was a crackpot, a little man with delusions. He tried to top me off taking his case. He sounds like a perceiving man. Well, he didn't perceive ending up in my apartment with a bullet in his head. Well, that's too bad. But I still don't see... I left Tom Turkey in my apartment for safekeeping, and when I returned, he was gone and Kuchel was dead. Well, that explains itself obviously. This detective knew that Tom Turkey was a phony, and Turkey killed him. It can figure that way, and a number of other ways. Mr. Spade, I have no desire to sit here trading subtleties with you. As yet, no one has demonstrated that the real Tom Turkey actually exists. Alive. Now, until you do have something more concrete and less mythological, Miss Black requests that you do not come around opening up old wounds. You've made an eloquent point. Just tell me one thing. If I can? When did Tom Turkey disappear? I mean, what month? What day? It was, uh... Oh yes, 1943, uh, November. But I'm not sure of the exact day. I think it was in the third week. Could it have been on Thanksgiving? Very possibly, very possibly. I returned thoughtfully to my office and did a little rapid mental arithmetic and came up with a number seven. From November 23rd, 1943 to November 23rd, 1950, was seven years to the day, and I pondered this. What did the number seven mean to the life or death of Tom Turkey? I had just hit upon the answer and was crying Eureka when my office door opened, unknocked, and a visitor came in unannounced. He was ugly, heavy set, and looked like an ex-long showman. I waited to see if the voice checked. You played? Who shall I say is calling? Captain John Smith. And here's my calling card. The first... The first bullet grazed my shoulder and tore the padding out of my coat. The second bullet hit the water cooler and it crashed over water and all on top of me. Where the third bullet hit, I wasn't sure at the time because darkness came rushing through my head like a freight train. When I opened my eyes again, I expected to see St. Peter checking my ID card. But all I saw were the dust balls onto my desk and a fly bathing himself in a pool of water spreading slowly over the floor. There was blood on my hand, but it came from a glass cut. I was in shambles but alive. Captain John Smith had shoved off, obviously thinking his bullets had done their work. Homicide, Lieutenant Kelsey. Sam, Kelsey, have you found anything more about Tom Turkey? Nothing, Sam. Frankly, I'm beginning to wonder if there is such a guy. Well, clever, Kelsey. A few minutes ago, a gorilla by the name, believe it or not, of Captain John Smith just tried to kill me in my office. Oh, go on, Sam. I find it hard to think. You find it hard to think, period. Really, Sam? Did you get him? My office is a wreck, and there's a whole blast into my wall, big enough to put a basket ball in. Well, what did he use, a bazooka?  Dumb, dumb? Well, that's illegal, ain't it? Kelsey, doesn't it strike you as significant that every attempt on Turkey's life has been vicious as if someone not only wanted to kill him but also mutilate him? Yeah. Yeah, now that you mention it, somebody probably wanted to make identification difficult. Even dead. They didn't want anybody to know who he was. Now, listen carefully, Kelsey. This is real deep. Some Turkey disappeared on Thanksgiving of 1943. A person has to be missing seven years before he can be legally dead and his insurance collected. Now if someone had Turkey insured, they could collect the day after this Thanksgiving, if Turkey didn't show up before. You mean somebody's trying to kill him for the insurance? I would say so, Kelsey. I would say so. Now, hurry up and find him. When I put down the phone, I heard a heavy pounding. For a minute, I thought it was in my head until I turned to face the door and standing there was a small pilgrim with bandy legs in black stockings, pantaloons, white collared coat and still piped hat. Hallelujah. He wore silver buckles and what he was pounding on the floor was an 18th century blunderbuss. Hallelujah. Have I got the right place? Full offhand, I'd say so. If you're looking for Captain John Smith, he just left. Pocahontas is expected any minute. Don't you go trying to confuse me. I'm too thirsty. What's on your mind? Well, I'm looking for a fellow named Dan. I'm so thirsty, I forgot. Sam Spade? That's it, that's it, that's it.  Yeah. Good, good. That stuff poisoned anyway. Hallelujah, hallelujah. Hey, say, do you happen to have any hard cider around? Fresh out. I'm kind of thirsty, you know. Any type of corn squeezes? Here, try this, Dad. Good, huh? You like that, huh? Hey, follow me. But before we go, you suppose we could have a little something for the road? It's busy cold. I gave him a little but not too much because I didn't want him to lose his way. He walked me right down Market Street so he could look in the liquor store windows. He said it gave him a comfortable feeling to know there was so much good in the world. And then we turned right a few blocks until we came to the helping hand mission. Across its gray front, a banner promised special holiday food and comfort to the unfortunate. And on the street in front of it, there was a brass band sending out signals to the fraternity that any minute the great feast of Thanksgiving would begin. The band members and other volunteer workers were all dressed as pilgrims. The quaint conceit. My pilgrim led me to a dark corner of the club room and sitting there unhappily was none other than Tom Turkey. Hello, Mr. Spade. Hello, Tom. What happened in my apartment? Why did you run away? Oh, I was afraid. You told me not to answer the door until you came back. Well, somebody knocked on the door and said it was you, so I opened it and two men came in. Tell me, was one of the male Coochle? Yes, the detective. The other man was a big, ugly-looking fellow. And when they saw I was alone, they started arguing. About what?  The ugly man pulled the gun and they started to fight. Oh, dear, I slipped out the door. And when I was halfway downstairs, I heard a shot and kept on running. Well, how Coochle is dead. Oh, my. I thought so. This was the only place I could think of to hide. Oh, when Henrietta finds out I've been mixed up in a murder, she'll never take me back. Henrietta, hey, tell me, did your wife ever have any insurance on you? Oh, before I ran away, she did. A $50,000 policy, but, oh, that would have lapsed by now. Maybe, maybe. Did it have a suicide clause in it? Suicide? Yes. Well, no. No, it didn't. I remember. Yeah. Yeah, you'd like to talk to Henrietta, wouldn't you? All right, here's your phone number. Call her up and tell her where you are. Oh, dear, I, I don't think I could. I'm too frightened. You've got to do something to help yourself. If you don't, by midnight you might be a cold turkey. Oh. I'm sorry. It just slipped out. All right. I'll do it. Well, he went and made the call. When he returned, he said that a man had answered who said Henrietta would come down and pick Tom up. He didn't want to wait, but I sat on him. The pilgrim brought us a dish of turkey dinner saying he couldn't stand food himself, and we munched a spell. In a little while, a limousine pulled up in front of the mission with someone in back whom I couldn't see. The chauffeur stepped out and came in inquiring for Tom Turkey. It was Captain John Smith himself. When he saw me, a look of shocked surprise came over his unhandsome face. Hoping to catch him off balance, I told Adam, it was the liveliest thing that has happened at the Helping Hand Mission in years, and we had a good house too. Money was even changing hands. When I heard the odds starting to go against me, I realized I'd better come up with something. He didn't use this, pardon me. And I did. The bandy leg pilgrim shoved his blunt of us right in my hand. And I swung. Smith dropped like pheasant on the wing. I looked up. The passenger from the limousine was just coming in. Yeah, what's the meaning of this? It means, Leander Luce, that you're not going to carve Tom Turkey up for your Thanksgiving insurance policy. Hallelujah. Drumstick, anyone? End of report. Sam. Well, it's as plain as the cranberry stain on your dress, Sam. Luce says Henrietta's business manager had her power of attorney, and secretly he kept making the payments on Tom Turkey's insurance policy. Oh, and then he'd collect for Henrietta and keep the money himself. Effing? Sometimes your lightning mind frightens me. I'll go type that up. Three chimes mean good times on NBC. There's fun and laughs with the chimes later tonight when Ed Gardner stars in Duffy's tavern. As usual, Duffy won't be there. But Archie, the manager, will definitely be on hand to serve his blue plate special of grilled English language. This Sunday, the big show comes your way again. Tallulah will be your hostess. And the stars include Fred Allen, Jack Carson, Edwin, Meredith Wilson, and many, many more. It's The Big Show, Sunday on NBC.  Was his name really Captain John Smith? No, Wiffy. Could we have a Thanksgiving caper without a Captain John Smith? It wouldn't be right. It was a coincidence, wasn't it? Well, if you promise not to tell any. I won't. His real name was Michael Giuseppe Yablonsky Smith. I call them John, for sure. You're so kind. Are we going over to your mother's for cold turkey snacks? Well, all right, but I don't think there'll be much left. Oh? You see, my cousin Gertie couldn't find a little boy. And mother phoned and said they just found him. He was inside the turkey, eating his way out. Effie, is there no way to curb that tongue of yours? Yes, there's one way. Well, come here. Good night, sweetheart. The adventures of Sam Spade are produced, edited, and directed by William Spear. Sam Spade was played by Stephen Dunn. Loreen Tuttle is Effie. Script for tonight's adventure by Larry Roman and John Michael Hayes. Musical scoring by Ludbluston, conducted by Robert Armbruster. Join us again next week, same time for another adventure with Sam Spade. Hear the magnificent Montague, then visit Duffy's Tavern on NBC.


As part of measures to boost welfare and productivity of Lagos civil servants, Governor Babaji Desaulu has promised to increase the minimum wage before the end of his administration. The governor announced this during an interaction with workers at the state secretary of Laosa Ikeja. Plus TV News correspondent Love Ikukoyedeku has details in this report. It is a working visit by Governor Babaji Desaulu. He is here to interact with the workers to know how they are faring, their challenges, and possible solutions. The governor arrives to arousing welcome from the workers who have gathered in front of the office of the head of service. Straight to the business of the day, he expresses appreciation to the workers for supporting his administration and enjoys them to do more to save all residents. Air of Excitement envelopes the workers as Governor Sao Wunu announces a planned increment in their wages. I instructed at a cabinet meeting, I instructed the head of service office and the minister of staff training and pension to start work on how we're going to increase the entire salary of the public service.  That is not all. He also announces that 100 directors have been penciled down for brand new cars before the end of this month. You know, that's why we started by also identifying the best in class, the best civil servants, the best teachers, the best public officers, and we've done all of that. And we've seen that data started. But now we need to make sure that as much as possible, we can make it go round. The head of service and myself have put heads together. And before the end of this month, we'll start the first set of 100 fleets of directors' vehicles. We'll start with a brand new 100 vehicles for our directors before the end of this month. Earlier, the head of service, Murio Konla, ruled out some of the achievements of the governor. Mr. Governor, in the health sector, which is a very, very crucial sector, you delegated the responsibility to me. Mr. Governor, that responsibility, with all sense of humility, has benefited legutions generally because of the automatic replacement of our exited health workers. Mr. Governor, this is also something that is magnanimous and we are benefiting from it. We ask you that you extend the same to our teaching sector as well, sir. He thanks the governor for keeping to his promises he made to the workers when he first visited in 2019. Love you, Kuku Uyedukum, plus TV news. Hello, hope you enjoyed the news. Please do subscribe to our YouTube channel and don't forget to hit the notification button so you get notified about fresh news updates.


So I quickly earned enough for you today and to be honest, I wouldn't post this unless I knew it'd be worth your time. But remember, as with every guide first, make sure prices in your server are worth that grind. Although most take a grind, this one's real easy. How's it going people? My name is DPjen, if you enjoyed the video leave a like, really helps out. And if you like what you see and want to see more, be sure to subscribe. Okay so all you're right now is at an average of between 0.80 to 2.10 on my server, it just depends on the time of day. It's a material people don't want to farm but need for various things. And while we're at this farm today, we can earn thousands of literally half hour to an hour. The area route I planned out, which you will see in a second, will be at a point where you just do a circle back on yourself, ready for that first spot to be farmed again. Meaning you can just rotate this spot, rotate this farm until you're either full up or get bored. And worst case is that all yourselves with the lowest estimate on your server, on my server I just popped mine up for the lowest point of a 0.80 because these weren't like bin out farming, I just picked these up while I was running around doing other things. And they sold almost instantly. I could have probably gotten more, in fact I know I could have gotten more because the other ones for sale on my server, the closest to mine were £1.60 or £1.60. Now the farm is one I recorded ages ago but never posted it, but after a quick look it's still crazy good and barely anyone goes here still. So once really now this is where the farm starts and we this thing, go here, follow this route and earn that oil guys. Once you get this done this run will take you literally 15 minutes max and one run sees you easily earn around 400 oil. If you have any kind of boosters on, that would only increase massively. So let's say 500 per run, 15 minutes per run, 2,000 oil per hour, putting out for sale for let's say one coin each. You are getting more or less 2,000, probably a little under every hour. 2,000 gold coin per hour people is amazing. And I don't think that's a bad hour's work if you ask me guys. But remember before you do this and before you do anybody's farming when it comes to selling items, make sure the item on your server is worth your time. If not, scrap that farm, scrap this farm and carry on with your day. But on that note the end of the video has arrived. If you enjoyed it leaving a like really helps out if you're new around here and want to see more, be sure to subscribe. And if you never want to miss a video I upload, you can turn notifications on by hitting that bell button. But guys thanks as always for stopping by and hopefully I'll see you on that next one.


I don't even want to say an intro, I just want to get right to it man. I'm free, I was gone a week, now I was gone two weeks. Was one plus one, that's two, that's two times I was gone. And what happens on three strikes, are you catching my drift? I don't think it's going to get to that, but I need y'all to be, everybody to be on one accord man. If you're not following me on Instagram, if you're not following me on Twitch, if you're not following me on Patreon, you're going to get lost in the sauce. I was gone for two weeks just now, and every day on Twitch I was still recording.  If you follow me on Twitch you'll know exactly where I am. So here's my Twitch once again. If you don't see me somewhere, you'll always see me here. Also, Patreon, there's three tiers, there's four tiers on my Patreon. Three of them are paid, one is free. We also post to the free Patreon on here, so if you follow me on Patreon, there's a free tier on there that still gets videos, like Luther, you gotta be, you gotta participate in a membership, line of duty membership, top gear free, shameless membership, 24 hours police custody free, phone shop membership, Island with Bear Giles free. I'm watching all of this stuff for free, and y'all are not participating, because you just, I don't know why, I mean it is what it is if you don't, it's free content for you. So I'm dropping everywhere. I just want y'all to know that, and I want you to be clear, but more importantly I need you to follow me on other stuff, because at the end of the day, something catastrophic could possibly happen on one of these, and I won't be there no more, and you'll be lost in the sauce. Now an update, now what did I do on my little two week break, man, honestly I chilled. I got a lot of stuff done pertaining to my daughter and things of that nature. I re-entered the gym, I'm back in the gym, you can't tell right now, but I got a nice little gym routine going on, with weights, that's the main thing. And I do got a white board here, like goals of my couple of goals, one of my main goals is I'm trying to get into acting, so if there's any talent agencies out there, or you know somebody who's an agent, direct them my way, because I will come through, I am coming to the UK first and foremost, but I will come do a show there, and if the show needs to be take for a year, you guess where I'll be for a year, it's as simple as that. It is very simple, but like I said, I need y'all to lock in, if you're really rocking with me, you should be locked in to all of these places. I post on Instagram now too, so all the food videos that be on TikTok, that y'all don't like to go look on TikTok because y'all just don't, it's cool, they're on Instagram too and Facebook. The link to all of this is in the description below, I'll make sure it's in the description of this video. But this is the main thing you need to be following me on Twitch, because everything else is I'm going to mention it on here all the time. But I'm back, we're back to our regular schedule program, and thank you for being patient. You got me, I'm gone.


SCP-610. Staff Alert. Object Class Cater. SCP-610 is a biological anomaly under Triumvirate Containment. The Triumvirate is an alliance between the SCP Foundation, Global Occult Coalition, and the Horizon Initiative. SCP-610 has breached containment. Triumvirate forces are attempting to re-contain. This alert will explain the situation to all staff. SCP-610 is an infectious disease which, over the course of several hours, will convert any infected life form into a grossly physically mutated creature. These SCP-610 mutants will seek out any non-infected animals, including humans, and attempt to terminate or infect them, and convert them into SCP-610 mutants. All SCP-610 mutants have been contained in a small area around Lake Baikal. However, around 100 entities have recently breached this containment area. SCP Foundation, Global Occult Coalition, and Horizon Initiative forces have formed a perimeter to stop the SCP-610 mutants from moving into Europe and Asia. Triumvirate forces have been organized into Joint Task Force Alpha 3, Carnal Side. This Joint Task Force has anomalous weaponry, including Geo-C Ultra Heavy Engagement Chassis and Horizon Initiative Paladins. The SCP Foundation and its allies will resist SCP-610 at all costs. Please remain calm and perform your duties. You will be made aware if the situation changes. Secure. Contain. Protect.


As we have seen before that oxygen consumption during exercise can increase from 250 ml per minute at rest to approximately 4 to 5 litre per minute, so that is like 18 to 20 times the increase in oxygen consumption is occurring, how is that possible? That is because of increase in the ventilation, remember increase in ventilation where there is increase in the respiratory rate as well as increase in the depth of the ventilation. So as ventilation increases there is increase in oxygen consumption and if we draw a graph of oxygen consumption versus the ventilation we see a linear relationship as oxygen consumption is increasing ventilation also increases and linear relationship here also you see the slope oxygen consumption increases approximately 18 to 20 times and ventilation also increases approximately 18 to 20 times and what we see that the pulmonary ventilation at maximal exercise to be around 100 litres per minute ok, so that much it can increase to approximately 100 litres per minute, but remember that the maximum we can ventilate that is the maximum voluntary ventilation, what is this maximum voluntary ventilation? That is the maximum ventilation which can be achieved with the maximum rate and maximum depth of ventilation, this is approximately 150 to 170 litres per minute, understanding so at maximal exercise pulmonary ventilation is approximately 100 litres per minute, so still at that point of maximal exercise there is a reserve of approximately 50 litres per minute, why I am talking all this thing? This is because respiratory system is never the limiting factor for exercise in physiological condition, this is not the limiting factor because we have a huge reserve for respiratory system, so limiting factor actually is the cardiovascular system that we will see in a short while, coming back to respiratory system where we talked about oxygen consumption, where we talked about pulmonary ventilation, we also talked about the pulmonary reserve, what are the other changes occurring? Other change occurring is increase in the oxygen diffusing capacity that means more oxygen can diffuse, how is it possible? That is because at rest not all the pulmonary capillaries are open, there are some reserve capillaries which are like collapsed state, okay and whenever there is increase in the cardiac output, all the cardiac output goes from the right ventricle to lungs, right so lungs have to accommodate more cardiac output from the lungs, from the right ventricle, that time these capillaries open up and that is known as capillary recruitment, capillary recruitment, also the existing capillaries which are there, they also increase in diameter that is known as capillary distention, so capillary recruitment and capillary distention increases the surface area across which the oxygen can diffuse and that increases the oxygen diffusing capacity in lungs, then fifth change occurs in ODC curve, what happens in ODC curve? ODC curve actually shifts to right at the tissues, see at the level of the muscles which are exercising what will happen? There will be increase in temperature, there will be increase in the hydrogen ions, right, there will be increase in the carbon dioxide and if you remember all these factors cause the ODC curve to shift to right and hence at the level of the muscles there will be increased unloading of oxygen thus increasing the oxygen delivery to the tissues which require it most, so these are the various respiratory system changes which occur during exercise, one concept you should remember here that during exercise oxygen levels never go down and carbon dioxide levels never rise, understanding even though our body is requiring more oxygen and carbon dioxide is being produced at a more higher rate, still we don't see this, then you may wonder that how ventilation is increasing, if that is the case, it simply means that peripheral chemoreceptors are not the main reason why this ventilation is increasing, it is because from our higher centers from our cortex the information is coming and also from the exercising muscles that is the proprioceptors information is coming to the respiratory centers and there is stimulation of the respiration, so it is more like a feed forward mechanism preparing the body beforehand that exercise is happening rather waiting for oxygen levels to go down and carbon dioxide to rise and you may understand that it may not be very good for the exercise, now let's move on to the last part that is the cardiovascular adjustments during exercise, well during exercise obviously because of increase in the sympathetic activity there is increase in the cardiac output because of increase in heart rate and increase in the stroke volume both increase, now this cardiac output can increase up to five to six times during exercise and in marathon runners it has been seen that it can increase up to seven to eight times also, now people who are exercising for a long time what happens that there is hypertrophy of the heart, okay so muscle mass of the heart is increasing so this heart can pump with greater power so there will be increased stroke volume okay because it is pumping with a greater power, now at rest this increased stroke volume what it will lead to it will lead to unnecessary increase in cardiac output but we know that cardiac output is equal to heart rate into stroke volume, so if muscle mass has more power and it is causing increase in stroke volume why not decrease the heart rate so that the cardiac output at rest it will become normal, so that is why trained athletes trained athletes have resting bradycardia okay resting bradycardia because the increase in cardiac output you see that will lead to increase in blood pressure, baroreflex will be active and there will be activation of the parasympathetic system that leads to decrease in heart rate, so resting bradycardia occurs because of increased parasympathetic activity what is known as increase in vagal tone, so that is one concept we should remember, now when we are exercising which of these factors heart rate or stroke volume contributes more to exercise, so let us see it in graph manner, so I will not write the values here but just for the concept I am telling you suppose on exercise axis it is cardiac output in liters per minute and on y axis it is the stroke volume and also heart rate is there okay, now whenever we start to exercise stroke volume is the one which increases first maximally and later on it is the heart rate which contributes much to the increased cardiac output, so both have started increasing in the beginning try to understand this both have started increasing but you see the increase in the stroke volume it is much faster, so this reaches maximum value much much faster compared to the heart rate, so why this concept is important, see in trained athletes because of increased vagal tone there is resting bradycardia, so this portion it will be much lower right for the heart rate and that reserve increase in heart rate which can occur is much more because the maximum achievable heart rate will be the same for trained athletes and for the untrained people but the reserve of the heart rate is much more plus also the stroke volume will be more, so obviously the maximum of the stroke volume which is reached is much more, so that is how this resting bradycardia is very beneficial in case of trained athletes, now I was before referring to why that respiratory system is not the limiting factor in case of exercise it is cardiovascular system, see cardiac output is equal to heart rate into stroke volume, now during maximal exercise what happens is stroke volume and heart rate both reach up to 90% of maximum achievable okay, so obviously cardiac output also reaches to 90% of maximum which can be reached isn't it, was it the same case in case of respiration, no we had 50 litres of reserve actually respiratory system reaches only 65% of maximum okay, so the reserve in cardiovascular system is getting finished, so cardiovascular system is the limiting factor for the maximum exercise which can be attained and this depends on something known as VO2 max, what is the VO2 max that is the maximum amount of oxygen which can be utilized during aerobic exercise, so the limiting factor for VO2 max is actually cardiovascular system and not respiratory system in physiological foundations, now one question which is frequently asked is that what would be the changes in VO2 max with training, actually it has been found that VO2 max does not change much with training, even after three, five, seven weeks of training there was an increase in VO2 max of approximately only 10% and furthermore VO2 max could not be increased, so it has been thought that maybe that again this VO2 max is dependent on a particular athlete only, it cannot be changed much with exercise, so that was in brief about exercise we talk about the strength, the power, endurance we talked about the muscle metabolic systems and how they are important for different types of exercise, then we also covered oxygen depth and the recovery of the muscle glycogen and which diet is best for the recovery and finally we covered about the cardiovascular and respiratory changes occurring during the exercise, thanks for watching the video, if you liked it do press the like button share the video with others and don't forget to subscribe to the channel Physiology Open, thank you.


Hey everybody, what's going on? I am Greg Sosman joined the day by Megan Unia as a fandom is here break down the conference championships and give us her best bets for the weekend. What's going on Megan? Not a whole lot just trying to find some really good bets for the really good folks out there. Trying to make people money as always. We begin with the NFC Championship Tampa Bay and Green Bay. The box are three point underdogs in Green Bay. Which way do you lean? Okay, so this is a game that I kind of teetered with until I kind of sat down and looked at the whole picture. Obviously Tampa Bay had their way with Green Bay their first meeting, but Green Bay is a team that will make those well needed adjustments. I will say that I was way more impressed with Tampa Bay's win over the Saints than I was with Green Bay's win over the Rams. I mean that Tampa Bay secondary is young, but they were able to force three picks onto future Hall of Famer Drew Brees and pretty much embarrassed him into retirement. Brady used all of his tools in his arsenal. He got creative and he has his team find tuned and absolutely humming. I want to make it a point to say this is not a fate of the Packers. The Packers are phenomenal. Aaron Rodgers is playing like out of his box nut. So however, if you offer me a six time Super Bowl champion who his motivation is to prove that he could get to and possibly win another Super Bowl without his ex head coach as a dog. I'm taking it every time. Give me the box plus the three. A dream for the NFL getting Aaron Rodgers against Tom Brady here in the NFC Championship game. Brady and Megan teaming together here as Brady over his ex trying to get to another Super Bowl and win another one has to go into Green Bay to do it. Whether he does it or not, we'll see what he he in the box will cover the three against Green Bay on Sunday, according to Megan. Let's stay in Green Bay, Wisconsin for a moment and talk about the total in this game. Total sits here at 51. Neither team has a fantastic event. Well, they're pretty solid, both Green Bay and Tampa Bay, but even the offenses are just too powerful for these defenses here. So yeah, I like the over 51 here. I know we saw a low total the first time these teams met back in October, but I think we can all agree that these two teams look completely different than they did in October. That first game Green Bay came out completely flat and weren't ever able to get anything really generated on offense. But as I said, this game will be completely different. And I think they come out firing and the Bucks match it score for score. I know a lot of people are screaming the under because of the cold, but let's face it. It's Tom Brady and Aaron Rogers. They've done some of their best work in the cold. These guys are seasoned veterans. I'm going to go ahead and take the over here and think that we're going to get that big firework game that we all expected to see whenever we watch the Saints and the Bucks last week. Fireworks in Green Bay would be fun. It's going to be very cold in Wisconsin this weekend, but it shouldn't stop Aaron Rogers and Tom Brady from slinging it over 51 says Megan. Our last game of the day final conference championship. It's the AFC featuring the Buffalo Bills in Kansas City. The question of course is will Patrick Mahomes be out there in the concussion protocol at the moment, but Megan, you believe the Kansas City is going to get the job done. They're going to win by three here against Buffalo Kansas City. Of course, the defending Super Bowl champions Buffalo trying to back the Super Bowl for the first time in over 25 years. You believe in Casey. Yeah, I have believed whole heartedly all season long that this Kansas City team was going to be right in this spot exactly where they are. And so that's why I have to take them. I know that it's not the fun and the sexy pick like bills mafia, but I really feel like Kansas City is just better in almost every facet of the game. I know Mahomes has not cleared concussion protocol, but I have talked to some really, really reputable people that know what they're talking about and they have pretty much assured me that Patrick Mahomes will be out there and will be cleared and ready to play. And if that is the case, that minus three seems just too sweet. The bills are going to try to contain Patrick Mahomes. I don't think it's going to work. Josh Allen's going to end up getting into a shoot out. And if I'm going to be honest, I think this is where the cream just rises to the top and we're going to see the chiefs going to another Super Bowl. We'll see Andy Reed in a beautiful Hawaiian shirt enjoying that Florida Sun. Mahomes gets the job done and he's on the field according to Megan. We'll see what happens with the chiefs going back to the Super Bowl from big time match potentially against Tom Brady or Aaron Rodgers should be fun this weekend. That's it for us here at the FanDuel Hurry Up. Megan, we appreciate time. Good luck this weekend. Thank you. Good luck to you guys as always. More conference championship coverage comes your way tomorrow where Jim Sanos will join me and give us the top stacks here for the conference championship playoffs. For Megan and you guys, I'm Greg Salsman. Thanks so much for watching. Have a wonderful evening. I'll see you back here tomorrow for another edition of the FanDuel Hurry Up.


There are over 50 health conditions related to poor posture ranging from the easy ones like Technic and chronic back pain to the more challenging ones like erectile dysfunction in continents and sleep apnea Chronic disease Costs the US over $4.1 trillion Have to look my nose By contrast the better your posture is The more attractive you are which leads to the better opportunities for professional and social advancement The the bed the bed of your health is the stronger your bones are which makes you more resilient against disease and pain and You'll feel better about yourself and people will feel better about you as radical as it may sound Improving your posture is an act of risk management and that's why it's important to make sure that you have the best Opportunity to have the impact that you intend to have in the world Today let's delve into the deeper reason why we all need to take action to improve our posture. So Sit up straight and let's get started Throughout history having good posture always meant strength confidence and beauty and For 40 years there was even the miscorrect posture contest sponsored by the American chiropractic Association There and who wind the clock forward there was the the the lady named Twiggy. Anybody remember Twiggy? Yes Twiggy Somehow made slouching cute and the notion went viral Within a few years in 1969 the American chiropractic Association canceled the miscorrect posture contest wind the clock forward The computer age began people started sitting at their desk to work all day and now wind the clock forward again and today People have elevated slouching To an art form in some parts of the world but nobody is talking about the end result the the consequences of having a life of poor posture We sit to eat We sit to drive Many of us even sit to work out our technology driven sedentary lifestyles Give us so many problems. We can't even count them But that's not the problem The problem is we don't have a system to restore our uprightness and our good posture There's no medicine no surgery no gadget no brace that can truly fix our posture problems You actually have to do something now. Wouldn't it be great if the body just went into upright alignment easily and naturally My name is Aaron Lloyd Ulysses Parnell I served on the first Olympic sports massage team at the Los Angeles Olympics where I helped athletes optimize their performance. I Decided to make a career out of helping people get the best out of their bodies the solution to our poor posture problems lies in posture optimization and fascia focused fitness Oh my gosh Posture optimization is the art and science of being able to Being able to make the bones through the work of weight-bearing and fascia focused fitness leaves your body free and effortless to make it easy So to have great posture 100 years ago fitness pioneer Joe Pilates introduced the notion of core strength to the world and I believe that he changed the course of history and The thing that where I think he left off is with core flexibility Which leads to posture optimization now in a world where posture optimized Bodies are the new normal.  I love to dance music is in my bones and for every every day, there's some kind of music that is the soundtrack of my life and What happens for me is I'm here because there are people suffering because they can't do what they love to do and I see I'm running out of time So I'm gonna jump to the end and I say When I look at people who are suffering with the with the inability to do what they love to do I think to myself This is like a freight train that just ran into their life didn't anybody hear this freight train coming and I'm here to tell you today that there's a freight train of posture problems heading in our direction and This is this is a bad thing Because there's no surgery no drugs no brace that can fix our problems You actually have to do something and if this matters for you Please see me at the break. There's lots more This turned in from a 12 a 12 minute talk into a five minute talk And so here talk to me at the end and we'll we'll we'll explore what's possible for you