 All right, so thanks again for coming. So if you were at my previous talk, you might have asked like, why do we care about all this? So there was a lot of combinatorics. There was a lot of kind of links and link invariance. And then there was a lot of kind of form of logical algebra. But if you're actually interested in algebraic geometry, like why do you care? Why do you care about all this? And I think one of the goals for me today is to explain like why you might care and why all these abstract problems about chain complexes of bimodules actually make sense and compute something that you might be interested in. And so I will start really elementary. So today's topic is braid varieties. And I will start with the matrix where I have ones everywhere except this two by two block at some position, i and i plus one where I have zero, one, one z. So this is n by n matrix depending on one parameter z. And I would call this a braid matrix. And so an exercise, there is some kind of echo. Andrei, maybe you need to mute yourself. Anyway, so we want to associate these matrices to crossings in a braid and so we need to check a braid relation. And so to check the braid relation like one thing which we need to check is bi plus one bi. And this is not satisfied on the nose. It's satisfied up to change of variables. So specifically bi of z one times bi plus one of z two times bi of z three. So we have three different variables z one, z two, z three is equal to the other product. Bi plus one of z three, bi of z two minus z one z three, bi plus one of z one. And these are just computation for three by three matrices which you can do as an exercise. And so as a result we have this product of B is corresponding to i, i plus one and i. And we have another product of B is corresponding to i plus one, i and i plus one. And we can compare them by this explicit change of variables. And so if I have a positive braid beta, I write it as a product of crossing sigma i one up to sigma i r and then I just replace each crossing by this matrix by one of these matrices with all with different parameters. So if I have a braid with r crossings I will get r different variables z one through z r. And I will get a giant matrix depending on all these variables. And that's it. And so this braid relation or kind of braid relation tells us that in fact the giant matrix up to change of variables up to re-parameterization actually doesn't depend on the way how we write the braid. So as a function of z up to change of variables it is all the same. And so it makes sense to define this braid varieties. So in the simplest form we just look at the locus of z's. So z one through z r where this giant matrix b sub beta of z one through z r is upper triangle. So as explicit as it could be. So we just it is a locus in a fine space where z one and z r are coordinates. And we have n choose two equations which require that this giant matrix is upper triangle. Very, very explicit our final braid variety in z r. And so as I said by this equation star varieties for the equivalent braids braids are related by braid relations are as a more and again maybe let me repeat. So all this is very specific of course for positive braids. There is no, I mean you can talk about what you can do for negative braids but it will be much more complicated. And like so far everything is very positive but still braid relations are fine. And so let me give you a concrete example. So the braid is crossing to the power four. So this will be I don't want to draw this. So I'll have four crossings. It is a two-strand braid so there will be two by two matrices and I have all different parameters for all the crossings. So I have z one, z two, z three, z four. And so I get zero one, one z one, zero one, one z two and so on. I multiply all these matrices. I get some stuff here, some stuff here and some stuff here. I get z one plus z three plus z one, z two, z three. That's an explicit computation. And so my variety is cut out in z four with coordinates z one, z two, z three, z four by one equation that z one plus z three plus z one, z two, z three is equal to zero. And it's kind of curious and we'll see a little bit of this in general that this does not depend on z four. Somehow z four will be in other entries of the product but it won't appear in this lower left corner. But it's fine. It just doesn't depend on z four. And then we can actually solve this equation by factoring out z three. So this will be z one plus z three times one plus z one, z two and this whole thing is equal to zero. And then we have two cases either one plus z one, z two is equal to zero in this case z one is equal to zero from the equation but then this is a contradiction because one plus z one, z two is equal to one. So this is not possible and so in the second case we get one plus z one, z two is not equal to zero and then we just solve for z three. And so our variety is actually just a complement to hyperbola. So this is the equation in equality that one plus z one, z two is not equal to zero and this is an open subset of C is two of the plane and then z three is completely determined by z one and z two and z four is a free variable. It doesn't appear in the equation. So we have just this x of beta times a fine space a fine line with coordinate z four. And so in particular it might be not completely obvious from this equation, from the initial equation but it's completely obvious from this description down below that this variety is smooth because it's open subset of a fine space it's three-dimensional and obviously it's not compact. So it's not projective variety, it's quite projective and it's complicated but we can study it very, very explicitly. So maybe let me pause and ask for questions like is it clear what's the definition of x of beta in general? Is it clear how to multiply the matrices? Any questions about this? Okay and so this variety x of beta I mean and this particular one is the main object of today's lecture, like how to associate something interesting to a positive rate. And so note for people who have seen this and I'm sure there are people in the audience who have seen this so this 1 plus z1 z2 is not equal to 0 is not completely random thing. So it's not a completely random open subset of c2, in fact there's a cluster variety of type A1 with one cluster variable and one frozen variable and so the frozen variable will be 1 plus z1 z2 and then z1 and z2 are two clusters corresponding to this type A1. So if you've seen cluster varieties you might recognize this and you might have seen this in different way and I'll talk about this a bit later. But there is already some interesting structure here and maybe one kind of geometric consequence of being a cluster variety is that we can we have two torii inside this. So we have this kind of complement to hyperbola this is this variety and so I can also remove the vertical line and I can say that I have a chart u1 which is where z1 is not equal to 0 and 1 plus z1 z2 is not equal to 0 and this is just a torus and I have another chart u2 where z2 is not equal to 0 and 1 plus z1 z2 is not equal to 0 and again it's a torus in both cases because in u2 I can express z1 as a function z2 and 1 plus z1 z2 and here I can express z2 as a function of z1 and 1 plus z1 z2 and then like probably speaking being a cluster variety means that transition function between these charts is very very specific so that's something interesting and another interesting thing which is not so relevant here but will be very relevant in a second is that there is a torus action which scales by t and z2 by t inverse so it doesn't change this equation in fact it scales these three if you want by t again and maybe it scales z4 as well but we don't care about z4 so there is a non-trivial torus action on this locus we have one fixed point but again it's non-compact and you have lots of subtleties but this equation or which we remove from the plane is a torus equation ok and so what do we know in general so in general it was studied under different names for a while I think like one concrete result which we proved recently with Roger Casaus, my brother and Susanne Misha somewhere in the audience is that so for reasonable class of rates this variety is very nice so we assume that our half twist so half twist is the positive lift of a permutation where just every element goes to n minus so i delta y n minus i this is a permutation and plus 1 and you take a positive lift of that that's a delta and we require that our braid contains this delta this is suffix so there is some positive braid gamma and then i have delta in the n so things are slightly easier if we contain a half twist so for those trends the half twist is just one copy of sigma so this certainly contains the half twist ok and so the first statement is that this variety that we're talking about it could be empty actually so if beta is just equal to delta then the variety will be empty and so this is really really easy to see because for example on those trends you will have this metric 0 1 1 z it's not never up a triangular because you have one in the corner so the variety will be just empty and in fact we have a criteria and when it is not empty so this is not empty whenever gamma contains another copy of delta as a sub word so not necessary as a suffix not necessarily prefix but some letters some generators inside gamma form another copy of delta and in this case the variety is actually smooth and it's smooth of expected dimension the lens of beta minus n choose 2 or the equivalent of the lens of gamma so this is of course expected dimension because we have L of beta variables we have one of these z variables per each crossing and you always have fixed number and choose two equations which are just lower triangular part of your matrix and so in this case the variety is nice and smooth as we see in the above example and another thing which is much more complicated but I want to say very clearly that this variety itself is an invariant of a link so wish link so it's an invariant of the lead of the closure of gamma delta inverse so gamma is a positive rate gamma delta inverse is not necessary positive rate but we can still talk about the closure of this non-tensory positive rate and then the claim is that the variety of beta is an invariant of the closure of that rate under conjugation and positive stabilization so you can think of it as delta inverse maybe slightly easier ways to say that this is beta delta to the power minus 2 so you just take a beta you remove a full twist from beta and if I have a different varieties which close to the same thing I actually get the same different rates such that beta delta minus 2 close to the same link then the varieties is the same provided that two links are related by conjugation and positive stabilization so we're not allowing negative stabilization and in fact they're not exactly isomorphic but there is morphic up to c to some power and c star to some power so my x so if I know I don't know anything because I can't raise but if I know that beta delta minus 2 is equivalent to beta prime delta to the minus 2 and there could be a different number of strands then I know that my x of beta times c to some power times c star to some power is equal to x of beta prime times c to some power times c star to some power and we actually know all these powers I just don't want to write them so this is very very explicit and somehow it's much more interesting than just talking about the homology or anything the actual algebraic variety is an invariant of a link with some restrictions on operations but that's it and maybe if I have time I'll explain an idea why this might be true but maybe I won't but I'll give some examples for sure and another thing which might be more interesting for you is that x of beta has a smooth compactification so the variety is smooth but it's very very non-compact you remove all these devices but depending on the braid words for beta whenever this variety is non-empty you can always write a very natural in some sense canonical compactification which will depend on the braid words for beta the complement to x of beta in this compactification will be a nice normal crossing divisor and the components or the strata in this compactification correspond to sub words of gamma containing the full twist and so very concretely again in this example so we have this one plus the ones we do which is not equal to zero in C2 how does it compactify so it compactifies to p1 cross p1 and the complement to p1 cross p1 to this thing in p1 cross p1 is what so I have this hyperbola which I have to put in back then I have this line at infins and that line at infins so each of them intersects the hyperbola at one point so if we come strata in this compactification we add hyperbola with two lines and we add three points which are intersections of hyperbola with the lines and intersection of two lines and in this case my beta was sigma to the 4 so my gamma was sigma q and there are three two letter sub words of gamma containing sigma which are essentially non-empty and they correspond to this three strata of one dimension one and then you have three points which correspond to letters of gamma which also contain them so there is some combinatorial way which might be not so important for many people here how to describe the strata in this and this will be relevant for us very very soon and for people who are interested in combinatorics so this notion of sub words containing gamma is related to so called sub word complex defined by Knudsen and Meyler and the collaborators studied a lot of interesting properties of this complex of sub words containing delta and so from our perspective and from the perspective of this talk this is just the CW complex which governs the strata of the compactification of X of beta ok and so this is the theorem and then the next theorem is asks like how this is related to not homology we discussed in the previous lecture and so this in some sense predates the notion of braid variety but I think it was phrased by many people in slightly different terms so it starts from I guess the work of Webster and Williamson and then it was rephrased more recently by Anton Meyler I mean Tom Trin and other people so you have this variety X of beta you always have an action of the torus which is C star to the power n minus one so n is the number of strands here and then we can look at equivariant cohomology of X of beta under the action of the torus so either equivariant or non-equivariant cohomology because this variety is non-compact it has very non-trivial weight filtration and so we can take associated the graded of equivariant cohomology with respect to weight filtration and we can take associated the graded of equivariant cohomology with respect to weight filtration and this is sum by graded vector space so we have homological grading here and we have the weight filtration and these are two different gradings and the claim that all these people say is that up to sum regrading this is the same as top we define yesterday so for grade again we can construct a complex of bimodules compute the homology of that complex and here we take just h h n of each term in this complex of bimodules and this recovers equivariant homology of this variety and there is a separate result which we proved with Matt Hogan-Cump Anton Meliton, Kate Ankaragana that this top degree piece h h n n of beta is actually the same as the bottom degree bottom Kavan-Frazansky lean homology of beta delta to the minus 2 so if we remove the full twist we exchange the top Kavan-Frazansky homology in the bottom Kavan-Frazansky homology and so this beta delta to the minus 2 is gamma delta inverse which we already saw and note yes later so this gamma delta inverse and this bottom homology as we discussed last time this is invariant under conjugation and positive stabilization and so we expect if we believe this theorem that at least co-homology of X of beta is invariant under conjugation of a braid and positive stabilization so then this part B of this theorem says that actually much more stronger statement is true that not only homology or a co-variant homology of this variety is true then the variety itself up to this factor is C to some power which doesn't take homology and C star to some power which changes the usual homology but doesn't change the co-variant homology is invariant and so because the variety is the same we see that the homology is the same and so as a conclusion this theorem too tells us that in fact we have you can read it in two different ways so one way is that you have explicit geometric interpretation of lean homology for positive braids or at least some part of lean homology for positive braids and this suggests homology of some very explicit algebraic writes you can also read it in the other direction and you can say we have very we don't care about like Zorgil-Beimodio so we get complexes or anything like this but what I really interested in this variety is where the product of matrices is upward triangle by some reason and then we can ask well when what is the homology of this variety and it turns out that the answer is given by Kavan-Frasansky homology and all these combinatorial techniques that I tried to outline last time actually helped to compute the homology of X of beta or its cousins and so for example for Torus notes as I explained last time as I will explain a bit later we can compute homology we can compute Kavan-Frasansky lean homology by now and so this gives us homology of very very non-trivial algebraic varieties which have different names and different incarnations but in some way you just take any positive braid which closes to Torus knot then this variety is essentially the same by the previous theorem and then its homology is the same and this homology is very very non-trivial and given by these combinatorics that I discussed last time so this is roughly the idea so I will explain the proof of this theorem in a second and maybe I want to ask for questions here so that's a good point to ask any questions before we go further okay and so a couple of remarks so one remark is that this variety is actually you can stratify by strata of the form c to some power and c star to some power like we saw before and so you can ask how does that behave but hodge filtration is actually easy and you always have just pp pieces of hodge filtration but the weight filtration is really really non-trivial and that's what is interesting and if some people in the audience have seen p is equal to w conjecture so this is kind of the w side of that conjecture in this particular setting so this is some weird version of a character variety and we're taking a homology of this weird version of character variety depending on the weight with the weight filtration and so hodge filtration is easy but you have interesting weight filtration and another remark which some people might ask is that very recently I think couple of weeks ago actually min tam trim posted a paper on archive where he found a way to compute all hodge homology using very similar techniques so you have an analog of this x of beta and then you roughly speak in tensor take a fiber product with spring resolution and do some formal springers here to recover all HHI and so you have all hodge homology, all kavan-frazanskylin homology from the construction of x of beta or really really similar constructions and so there are lots of interesting things here and he has some other construction related to unipotent matrices and things like this but I guess I won't talk about this and maybe yeah maybe before going to the proof let me actually compute the homology of this right so you have complement to the hyperbola in C2, how to compute its homology well so the easiest ways to use alexander duality so maybe I'll write it as a side remark so you have x of beta 1 plus z1, z2 is not equal to 0 cross C so we can use alexander duality and say that the homology of 1 plus z1, z2 is not equal to 0 is in fact the same thing as homology of 4 minus 1 minus blah, maybe with compact support where you have 1 plus z1, z2 is equal to 0 and this is just C star and so the hyperbola is isomorphic to C star, we know it's homology in two dimensional in two neighboring degrees and then we have by alexander duality we have two homologists there and you have one extra piece of homology because this is reduced homology and the same thing works equivalently I just don't want to write the answer but if you do this properly then for example non-equivalently we have h0 is equal to h1 is equal to h2 is equal to C and all other homology vanish 4x of beta and then equivalently you get slightly more interesting thing which I guess we'll discuss a bit later but in this case it's something very concrete in a slightly more general case computing homology and using alexander duality is much harder because this is very complicated algebraic and we'll see some examples of this variety okay so let me spend some time talking about the proof of this theorem too so we need to compare this variety with something that we saw last time so what did we actually see last time so last time we saw bimodules bi which are r tensor r over rsi so these were bimodules over r which are again very very formal. How do you think about them geometrically so geometrically they came from and in fact Zorgel invented this bimodule by thinking about both Samuelson varieties and these are called both Samuelson bimodules so what is a both Samuelson variety it's the variety of pairs of flags f and f prime where f and f prime are two complete flags which coincide except for one place so fj is equal to f prime j for j is not equal to i and then we can compute its homology so we have line bundles on one side which are fj mod fj minus one we have line bundles l prime lj prime which are fj prime mod fj minus one prime we have axes and x primes okay that's already something that we can see so this is axes are churn classes of line bundles on the left and x primes are churn classes of line bundles on the right all these line bundles are actually the same except for fj is equal to i and i plus one so this means that xj is also equal to xj prime for j is not equal to i and i plus one and then f i plus one mod fi minus one so the two we skip this f i which is not equal to i prime but this quote f i plus one is the same as if i minus one prime and then this quotient is the same and this quotient is filtered both by l i and l i plus one and by l i prime and l i plus one prime and so in particular the churn classes of this rank two bundle are symmetric functions in l i and l i plus one and symmetric functions in l i prime and l i plus one prime and so the symmetric functions in x i and x i plus one and x i prime and x i plus one prime agree and so these are just churn classes of f i plus one mod fi minus one and so in fact this bimodule bi is very closely related to the homology of this bsi and so then we had to tense a product so we tense a product to a term complex but let me tense first these bi's so on an algebraic side we have bi one, tensor bi two where we tense over r and geometrically this corresponds to this again both samuelson varieties more complicated both samuelson varieties where we have a sequence of flags f one, f two, f r plus one so we have one more flags here and for each pair of neighbors we have this btsamuelson condition over here so that two flags coincide except for one place and so the first two flags coincide except for the place i one the second pair of flags coincide except for i two and so on and so we have a sequence of flags abstractly you can think of it as some kind iterated fiber product of this simple but samuelson varieties but you can just say that this is a sequence of flags with this condition and so we can actually understand these bi's and you can compute the homology of this variety which were studied by many people in god and samuelson there are many others using this algebra using this fact and you can just explicitly compare the homology of this variety to the tensor product of this primonium so we were interested in ti so ti was and I didn't say this but the co-homology of bsi of course is generated by axis and x prime so it's another important result that axis and x prime actually generate bsi and these are essentially relations plus relations on complete flags that are always there and we were interested in two-term complexes ti which are the cone of bi to r and again it was very very formal what does it actually mean what is the map from bi to r so if you think that bi is a co-homology of this variety and r is just a co-homology of the flag variety again it's not exactly but it's very very close and then you have a map from the flag variety to bsi so maybe let me mark this so this corresponds to the map or diagonal map where you have just the space of complete flags and you embed it to this btsi and then you have a map in co-homology which goes backwards and this corresponds to again up to some subtleties which I don't want to discuss to the map from bi to r and so we take the cone of this map what does it mean that we have a cone of a map between two algebraic varieties well so this is a closed embedding we can just take the complement and so we're saying that f and f prime are in position si if fj is equal to fj prime for j not equal to i so this is the usual samuelson condition which we had before and we remove the diagonal by this kind of argument so we require that f sub i is not equal to f sub i prime and so we can say that this is an open btsi variety or there are many different names for this variety and again if now the tensor product of this cheese what happens is that you have a sequence of flags fi f1 f2 dot and f r plus 1 and the sequence of flags satisfies the following conditions the first two are in position si1 meaning that all sub spaces are the same except at one place and at that place they must be different so because we remove the diagonal if we compare f2 and f3 again there should be the same except for one place si2 where they must be different and that is this condition and so on and so this is kind of open btsi variety that we are talking about and this variety actually appeared long before all this discussion so this variety of flags appeared in many many works in geometric representation theory most notably in the work of Delyne Michel and there is a beautiful paper of Delyne on break group actions and categories and many other people and it plays a very important role in Delyne-Lustic theory because they consider similar sequences of flags so if you are interested in geometric representation theory or in Delyne-Lustic theory you might have seen this space in some way and so you can define this space for any break but actually it's a very useful exercise which for example you can check and it was done by in that paper of Delyne that braid relations are satisfied so maybe let me put it as a remark Delyne said that this variety of bsb is invariant on the braid moves so the same braid but for different presentations of the braid as product of generators you get isomorphic varieties and in fact canonically isomorphic in some sophisticated sense although these Bots-Samuelson varieties are definitely not invariant and so you can think of this as very similar to what we had before and like in some very precise sense that Bots-Samuelson variety is a projective variety because they have closed conditions for the product of flag varieties but this open subset or bs is something smaller and in fact it's an open subset in bsb and so you can think of this Bots-Samuelson variety as a compactification of this variety of bs or whatever Delyne and Bray-Michel called it and so you have a variety which is a braid invariant but this simplification is definitely not invariant of a braid and this is very similar to what we had before for braid varieties and so just to summarize right so now if you have relation between the complex of Bs and the complex of Ts how to relate the homology so roughly speaking what happens is that the complex of Ts so the product of Ts as a product of Bs as a complex of products of Bs is this some kind of inclusion exclusion formula for open Bots-Samuelson variety inside close Bots-Samuelson variety so for example here we can say well so we either remove diagonal in the first place or we don't remove the diagonal in the first place in the second place we either remove diagonal or we don't remove the diagonal and so you can imagine that you have a giant complex of close Bots-Samuelson varieties which resolves in some sense this open Bots-Samuelson variety meaning that for example the complement to open Bots-Samuelson variety in the close Bots-Samuelson variety is a game paved by close Bots-Samuelson varieties and all the intersection of Bots-Samuelson varieties and so inductively we understand the structure of compactification and that's a very rough sketch of the idea and then you can ask well okay so we know all this how this is related to this variety X of beta and a very useful lemma says that X of beta is actually subset of this open Bots-Samuelson variety where we require that the first and the last flags are standard so there are many things that you can do here and this could be really confusing as it is so there are different versions of what you can do now which all corresponds to different notions of closing the braid you can require that for example the first flag is equal to the last or the first is equal to the last up to some kind of twist and the induced variety would correspond to the fact that the first flag is equal to the last times the action of the prebenoesotomorphism but here we require that not only the first is equal to the last but also they're both standard and then basically parameterizing Briocells gives you these matrices and it gives you a relation between X of beta and OBS and so then embedding as I said of OBS beta to OBS beta correspond to the compactification of X of beta above and so you can think of this compactification over here as an embedding so this is some piece of open Bots-Samuelson variety where we require that the first and the last flag is standard and this is P1 cross P1 adjust the close Bots-Samuelson variety where again we require that the first and the last are standard and then how do we compute homology with this inclusion-exclusion formulas? We're just saying that to compute the homology of the complement of hyperbole instead of using some tools like Alexander-Duality we use this stratification so we take homology of P1 cross P1 then we have homology of these three strata which are hyperbole two lines at infinity and three points maybe I should write it down so I have three points and let's take a co-homology so here you have co-homology of hyperbole and you have co-homology of P1 at infinity another co-homology of P1 at infinity and then you have co-homology of P1 cross P1 and so you have restriction maps given by these inclusions and you have restriction maps here and then the claim is that co-homology of my open space is actually sitting here in degree zero and it's given by homology of this complex and it's a cyclic everywhere else and again you can take homology or co-homology it will be dual to each other and you can do everything equivalently because the stratification is equivalent and everything makes sense and so this is a wrap idea and then the claim is that this kind of giant complex corresponding to the stratification precisely matches the complex computing the homology and again this is an idea which goes back to Webster and Williamson for sure and Zorger in some sense and many other people that you can compute homology in this way looks complicated, it is complicated but you can certainly do this so I think that's all what I want to say about the idea of the proof so you can think of X of beta and many people like to think about this X of beta instead of matrices and the product being triangular saying that in fact we have a bunch of flags in the first and the last understanding so again let me pause and ask for any questions okay and so then with all this let me actually talk about some examples because like what are the examples so I gave you just one like how does this variety actually look like and I think the most beautiful example was very recently shown in slightly different terms but essentially equivalent to what I will say it was shown by Galatian and Lawn so they observed that the two links correspond to open piezotroid varieties in the Grasminian so open piezotroid strata in Grasminian KN so let me explain what this thing is so Grasminian KN parameterizes K dimensional planes in n dimensional space so if you choose a basis of this plane we have K by n matrix of rank K up to row operations as usual and given such a matrix you can repeat it periodically you can repeat the columns periodically so here I have the columns v1 through vn and I require that vI plus n is equal to vI so I have infinite sequence infinite matrix if you want and then this piezotroid cell piezotroid strata corresponds to the condition that the determinant of K by K matrix vI vI plus 1 up to vI plus K minus 1 so this is a K by K consecutive minor or kind of cyclically consecutive minor is non-zero so all these minors should be non-zero and again we're quotient by row operations and this condition that minor is non-zero all these minors are unchanged or multiplied by non-zero scalars under row operations so this is a well-defined subset and it was started by many people starting by Knudsen, Lahn-Spar and many other people in the cluster algebra community and so the claim and the first claim is that up to some c to some power and c start to some power which I again don't want to write explicitly because that would take a while this braid variety for the torus node so this is just the torus link Kn is exactly the same as this piezotroid variety Kn which is an open subset in Grasminin up to c to some power and c start to some power and this actually depends on how you draw the torus link so you can draw it as a link on k strength you can draw it as a link on n strength you can draw it as a link on k plus n strength so by the theorem that I mentioned all these varieties are actually essentially isomorphic up to c to some power and c start to some power but you can also draw it as a braid on k plus n strength and that's I guess the closest to pkn so one way to draw this which is most relevant here is to say that we have k strength like this and n strength like this and this closes up to Kn and torus link and the braid variety for this thing is isomorphic to pkn without any c start but with some c anyway so concretely so we have our friend 1 plus z1 z2 is not equal to 0 what is it how it's related to any positroid so we have t24 so Grasminin 24 we have this open subset p2,4 and the claim is that this open subset is actually 1 plus z1 z2 is not equal to 0 times c start square and the braid variety we need to have an extra c here because we had an extra c in the braid variety and so in this sense the torus link t24 or torus yeah the braid variety for this link corresponds to positroid variety p2,4 and this is very very easy to see but maybe for the interest of time I don't want to do it it's one of the exercises so if you are interested please do this exercise and come to discussion session okay and so again like maybe the most important thing here is one of the most important points of this paper of Galatian alum is that they've been trying to compute the homology of positroid varieties for a long while using various methods using cluster algebras using various recursions provided by cluster algebras and that's just hard this is a genuinely very hard problem and so what they observed is that instead you can compare it to these braid varieties or their cousins and compute the homology and the querying homology using the machinery of link homology and in link homology there's recursions of Elias Fogh and Kampanmenut that I mentioned last time actually compute this thing for you and compare it to Q2 cutland numbers and other things and so if you're not interested in formal stuff that I discussed last time you might be interested in for example computing homology of this very very explicit algebraic varieties and it turns out that to compute it you might need all this abstract, combinatorial and homological algebra stuff and I think it's a beautiful and very interesting question to understand going around without using these recursions of Fogh and Kampanmenut and like doing it directly here in the setting of presetroid varieties or maybe braid varieties in general and so for example like one concrete application from last time which maybe I will write here so theorem so Galash and Val and it uses the results of last time anyone has any questions? iterated torus nodes correspond to presetroid varieties I don't know that's a good question what are iterated torus nodes in the setting I don't know and there are lots of questions and it's just been explored so all this like connections and explicit equations and this work of Galash and Val it's all just last year essentially so there are lots of open things here so what I was saying is that so using these results of Fogh and Kampanmenut let me advertise them one more time so if GCD of K and N is equal to 1 then kohmology of this presetroid variety K and N is supported in even degrees and this is like as concrete as possible so you also have some explicit combinatorial formula for it but even explaining this thing without link kohmology I think this is an open problem because there is no like cell decomposition for this variety there is nothing like this it's just complicated open algebraic variety non-compact and it turns out that this kohmology is even under some assumptions so maybe it's retracting something with even paving anyway so the second example is also related to some known varieties in geometric representation theory so if you have a pair of permutations in SN and W is greater than U in Brear order you can form the fallen braid so you have the positive braid lift for W and you have the positive braid lift for U inverse W0 so W0 is the longest element in SN which is delta essentially and then you add an extra delta and anyway some explicit braid that you can cook up from W and U and this turns out to be this variety is open Richardson variety for W and U so this is an important subset of flag variety which was studied by many people and another thing which is I guess related to Richard's question W and U are in SN W is greater than U in Brear order again and in addition W satisfies some technical condition of being K-grasp mining then you get opposite droid varieties so Knudsen-Lavinsch pair defined like the whole specification of grasp mining where this pKaN was the top open stratum but we have lower order strata and all this lower strata actually correspond to some smaller braid varieties of this type and this actually is known that open Richardson varieties there is morphic to this posidroid varieties so this is also fine but it's interesting and so what we recently proved again with Roger Misch and Jose that you can realize the same posidroid variety in many different ways so up to C-to-sum power and C-to-sum power this posidroid variety PWU corresponds to four different braid varieties so we have four different braids some of them on N-strand some of them on K-strands and all of them are isomorphic up to C-to-sum power and C-to-sum power this is one of them but we have many more which look differently and so again I think it's an open question for example which Richardson varieties are isomorphic in general and maybe different number of strands and so this is one indication of why this might be interesting anyway so I think I try to convince you that there are lots of interesting varieties most notably posidroid varieties and lower posidroid varieties that are related to braid varieties so it might make sense to study all this and then the last thing which I want to mention is a theorem also from last year by Gaussian and Wen that X of beta under some assumption that this beta delta to the minus 2 is a positive braid has a structure of cluster variety or maybe I have to say upper cluster variety I'm not 100% sure and again up to C-to-sum power and C-to-sum power and so in fact all these varieties which we've seen here in particular posidroid varieties are cluster varieties by another work of Gaussian and Wen and these varieties one plus one to two as I said was cluster variety so there is a lot of interesting structure which I'm not an expert in and I don't fully understand for sure but you have cluster you have interesting charts in these varieties you have lots of interesting other structures and I think it's a very important question which I won't answer for sure but what does it tell about co-homology so what kind of structure do we have in co-homology with cluster variety what kind of structure does it tell us about the link homology which I for example interested in and so one example I will explain next time so I can just say one sentence that any cluster variety has a canonical two form and from this form we'll have an interesting class interesting operator on link homology which I will construct next time and really thinking about the construction of this form and different constructions of this form really helps to understand this operator and different other operations in the co-homology but again as I said there are lots of stuff to explore it's very active and interesting subject and I think I'll stop here for today anyone have any questions can you get us Eugene there is a question on the okay yeah co-homology but Samuelson variety is related to complexes of bimodules yeah you need to do equivalently so you need to work equivalently and adjust the Malaysian I don't want to explain equivalently but if you use proper equivalence and all this proper like actions of Brails and G and the torus on this flag but Samuelson's you can recover the bimodules and maybe I have to say so I didn't say it but some people might ask so you have this variety for positive braids can you do anything for negative braids and it was kind of a big breakthrough of Rukia who realized that you don't have a variety for negative braids but you can still construct find a complex of constructable sheaves on flag variety so this thing is some kind of correspondence between flag variety in itself a projection to the first flag a projection to the last flag and so these correspondence give you a constructable sheave on flags cross flags and so Rukia explained that like this formal stuff that we did yesterday you can think of some computations between constructable sheaves on flags cross flags and so in this sense you can do anything for any braid except that it won't be given by an explicit correspondence I don't know how to think about this geometrically and I'd love to know but like the best you can hope for is like at least for sure there is always an equivariant constructable sheave on flag variety and that's I think an important remark to me what if we consider equivariant K theory I don't know that's an awesome question never thought about it but that's an awesome question yeah the definition of class of variety play a new role I don't know so there's two form that I mentioned in the end is probably responsible for quantization but again I'm not an expert and maybe my brother who is in the audience can see more I don't know Yes, is there a definition of braid varieties for general cocceta groups I think there should be and maybe it's somewhere in the work of Leustic so you general cocceta groups I don't know but for general wild groups I think so and the reason is like you can think of this matrix as fallen so you have your bra cell which is something like B S I B you can just write it so S B on the left and then you have this braid matrix B sub I Z for some Z so the claim is that any B S I B you can write in this form and so this works in general so you can say that you have a simple reflection you have this two sided bra cell and then you can kind of move B to the left and see what's left so you always have some matrix and I've seen some work of Leustic to remember exact reference when this is discussed and then I think from general grounds you know that there is some kind of of this relation I don't think anyone has ever written the explicit change of variables aside from type A but I think it's possible and I think again there was some paper of Leustic where this is discussed that I mean it's always exists by considering just general machinery bra cells writing this equation explicitly for other types it's an excellent question but yeah I don't think it's done at least in this explicit way as I presented here the flag write is though so if you don't want to think about that thing so this so this open but same as the write this certainly makes sense for all types at least for wild groups and that was started by all these people in all types and like rewriting it as matrices requires some work but again that can be done probably I'm not sure if it is done already by Leustic and others is there anything corresponding to tangles so there is something corresponding to braids if you have a tangle with partial closure I don't know I don't know I don't know and like the definition of Kavan-Prazansky homology somehow very much depends on the braids you have to present your link as a closure for braid and then check markup relation so you can just draw arbitrary tangles somehow this Kavan-Prazansky machinery breaks down and I don't know if you can do anything for tangles and then Roman has a second question which I can answer why the variety only depends on the node so this is an excellent question so this is this is this part B of the theorem so let me say maybe one word for this about this if I can don't crash sorry I can put it back so why is it being converted and so the idea is the following so you need to consider braids with negative crossings as well and let me say this word so Legendre link possibly with negative crossings so to a Legendre link Chicano associated dg algebra and prove that like this dg algebra is roughly speaking a Legendre link invariant in particular the homology of this dg algebra is a Legendre link now so what we is that we found in explicit algebra geometric model for this algebra in terms of these braid varieties and building on the work of Kalman the specs of h0 of this dg algebra what it is is your x of beta roughly speaking if beta is a positive braid I'm lying a little bit and in fact the dg algebra you can think of it as the generators correspond to the crossings of a braid and then generators of the next degree correspond to the equations in the matrix so a roughly speaking is a causal complex the equations of x of beta and so if beta is not a positive braid but equivalent to positive then we can also have some algebraic model for spec of h0 of beta just by looking at the braids again some generators correspond to crossings differential counts some things and so we use this algebra geometric model to check that this is invariant under all braid relations and Markov's and so this is invariant then Markov's and maybe I don't want to say more because that's a lot of stuff but in some sense we reproved the work of Chukanov worked over z2 and we worked over complex numbers and we had to do some other improvements but we have some very explicit algebraic geometric model for replacing this work of Chukanov and computing h0 of this dga and proving that this is a link invariant this is kind of the rough idea and maybe one last thing that I want to say here is that this model is the following so you have some algebraic varieties plus a collection of vector fields which integrate to a free action p to some power and this vector fields are parameterized by negative crossings and so if the braid is positive you just have an algebraic variety which is nice and smooth and we understand everything if we start for example introducing negative crossings and like making right in my series 2 when we introduce sigma sigma inverse then we have to introduce this extra piece of data for each negative crossings we have a vector field all these vector fields can be used and integrate to a free nice action of c to some power and then the quotient by this action is a link invariant and that's what we check but again like we don't have a general insight like why it is except that she kind of tells us so and we check every move separately we check right in master move we check marker moves and this is in the end the genre link invariant and we use pretty subtle properties of this digital algebra which I'd love to understand better and maybe another remark is that I'd love to understand I've never seen this construction in kind of geometric representation in literature that you have a variety of vector fields and it would be very nice to understand how this construction where you have vector fields for negative crossings how does it correspond to this constructible sheaves in perspective whereas before it seems like varieties but that's also not done I would say but yeah that's the idea of how to prove that the variety is the same right everyone so let's thank Eugene again