Introduction to categorification and link homology

Added: 2026-05-31

[00:00:00]Can can you hear me? Yes. It's a great pleasure to visit Rick for the first time and give a talk here and I see this a very fancy talk and I'd like to thank Petto Sana and like to thank the staff at Ricken for organizing my talk. Um and the talk is about linkology theory. So and I'll target an audience of mathematicians and physicists. So please stop me and ask questions at any time. So, so um so the talk is about link tomology.

[00:00:36]Um and um let let me go back in time and just remind us that when you have um topological space X space such as manifold or CW complex then um one of the first invariants that people discovered of sufficient nice topological spaces is the oiler characteristic the sky I for instance in case of a surface. If you have a surface, you can triangulate it. You can break it into a union of points, intervals and um and triangles.

[00:01:15]And then you count the number of points minus the number of intervals plus the number of triangles and you get an invariant of a surface um x um and you can generalize to more general you can extend this to more general topological spaces and get one of the first invariants k of x and for noise spaces this is well defined and belongs to it's an integer invariant and then later and so so again long time ago people discovered that the or a characteristic is a shadow of a much deeper invariant of topological spaces called homology homology homology or sometimes cohomology um of topological spaces of x as h of x and this homology and I I hope that you are familiar with homology groups of topological spaces so it's it's um integrated so It's a sum of h subn of x where n is an integer and homology groups.

[00:02:21]Um and these are abidian groups groups.

[00:02:29]Um and you can recover the oristic from the sabian groups um um as k of x is sum of minus1 to the n rank h n of x as long as rank is finite and as long as we're only finding many non-zero terms this sum is finite and this is a way to um define real characteristic in a more intrinsic way for spaces that might not have a decomposition into points intervals um triangles etc. Uh so this is more intrinsic formulation of the or characteristic and it extends um so while this is only defined for some spaces homology is defined for for any space with understanding that this might be this might have infinite rank or infinite many groups can be non zero and discovering the discovery of homology of X was it gave the birth of to algebraic was the birth of algebraic topology that was led So it was I I would guess it maybe was about 100 years ago that Ponare discovered um other people with homology groups um and sort of it gives you much more information and leads to a much deeper theory and uh in particular I want to remind you that homology is funal oral invariant. So it's a funtor. H is a funtor functor from the category of topological spaces of topological spaces to the category of well what we have here is an abiden group for every answer. So we should say category of graded graded abidian groups and homorphisms between them and this factor to X it assigns homology groups H of X and to a continuous map from X to Y continuous map F it assigns a homorphism of groups from H of X to H of Y.

[00:04:57]F um lower um lower lower star. Um and uh so so you get information not only about topological spaces as in this example, but get information about continuous maps and you get a funtor from this kind of vague, more vague or more complicated uh geometric topological category to something very very explicit. And um so this has kind of greatly developed into algebraic topology where people generalize homology to other to general to more general homology theories such as K theory or coordinisms. Um they discover operations. So there is kind of huge there's a very deep deep giant theory uh that developed out of um this construction.

[00:05:50]uh um and again I think o often in in in math often one of the problems is to is to take some some some structure could be category structure which is not which is hard to deal with such as topological spaces and continuous maps or maybe metric spaces and maps that respect the metric to some degree and then pull out some algebraic pull out some more specific information out of it which often has to be algebraic.

[00:06:20]Um and sort of if you and so usually you want to map something more vague topological geometric into something which is easier to manipulate such as something algebraic such a number homology groups uh maybe or maybe homotopic groups or pi one is not a billion um but sort of a lot of mathematics and uh including algebraic topology is about taking um kind of linearizing things or building some formation of algebraic extracting invariance and information of algebraic origin from something which is harder otherwise to understand and I think this is also before even before algebraic topology before um um if you just look at the more basic applications of mathematics a lot of it is just linear algebra I mean applications ofatics in real life to a very to um economics uh um biology a lot of it is just linear algebra differential equation so it's it's just pulling out some algebraic information from something more vague and more topological um so it's um linearization so passing from a more complicated category to the category of say vector spaces and linear maps and in this example the category behind homology groups and homological invariance. So here is category of vector spaces and linear maps and here's a category of complexes complexes of vector spaces.

[00:08:03]So where you have a space vi a space in every degree and you have a map d is square= to zero.

[00:08:13]So D is a differential and you take homology groups on the D. So you take the kernel portion of the kernel by the image of the previous D. Um and so so I think a lot of applications of mathematics outside of mathematics is really it's about this category of vector spaces and in in this example we dealing with slightly more complicated category category of complexes and yeah >> the necessary smooth So there are various definitions of homology groups and there are several definitions that work for any topological space and any continuous map. So so there is >> it doesn't have to be smooth. For smooth spaces, for smooth manifolds, you can use the dam complex. You can use the complex of differential forms to write down the homology manifold X. You can use complex of differential forms on X with a der differential to write down the homology groups. But there are other constructions available for modal spaces or for other spaces.

[00:09:36]some one of the constructions is using the singular singular homology.

[00:09:46]So in some sense it's probably bigger complex than the run complex where you you look at all continuous maps from an n dimensional simplex. So delta n is n dimensional n dimensional standard simplex. You look at all continuous maps from your simplex into X very large number of typically there's a very large number of such maps X's that we encounter and um given given so each of this each of these maps um M from a simplex X gives you one generator of the N homology group. So there are as many generators as there are maps and then on on this generator we define the differential differential takes send send u restricts the syntax to its boundary it's oriented boundary and sends the sends m to the sum of the maps one for every boundary n minus one dimensional face of the syntax. So this is kind of this doesn't require smooth structure.

[00:10:55]This is more general at the cost of at the cost of having even in some sense even bigger space than special differential forms >> also doesn't have to be random.

[00:11:11]>> Yeah. Yeah. So so that that two. So here here um x is any topological space y is any topological space f is any continuous map and then yeah whenever you have a map of a simplex into x you can compose with f to get an induced map from a simplex into y and this uses a map of complexes from complex of x to the complex of y and it produces map in homology but then you have to then you have to do some work you have to So you get some giant complex but in this it's hard to compute in this complex. So people find people show that for nice X there's a a much much smaller complex of finite rank of homologist groupology group of finite rank um so if X is built out of points intervals two dimensional cells freed and so on.

[00:12:09]So if it's built if it's glued from standard pieces which are cells of various dimension. So X might be a manifold, X might be a singular space.

[00:12:19]But if it's glued in a standard way in a nice way from the pieces of dimension 012, then you can show that this very very large complex and this homology can be reduced to a very small complex with a basis in this complex given by zero cells, one cells, intervals etc. So you can show that there is a permatorial explicit commal model for um for a complex giving you the same homology but which is much easier to understand and manipulate.

[00:12:58]So um so I'll tell you about doing something similar where we replace topological space x replace it by a not or link or link L in the free space um where so to not L we'll assign homology groups H of L um of course there are sort of there are boring homology groups because if you have a link L in R3 you can take its complement the complement is a certain topological space. You can look at R3 minus L.

[00:13:44]That's an example of a three manifold.

[00:13:49]And three manifolds have a deep theory assigned kind of related to them which goes back at least to the work of properopolis in the 1950s.

[00:14:00]um and you get a very deep you get very deep theory has been studied over many decades. So somehow low dimensional manifolds are special.

[00:14:11]So three manifolds and four manifolds they're special in several ways. And so when people study algebraic topology they may study manifolds and spaces in all dimensions.

[00:14:25]And that's one theory. But then we discover that in low dimensions topology has very special different behavior that is unique to these dimensions. And some people speculate that that's because uh microscopically at least on large scale spacetime is four dimensional. Um but so who knows? But so for instance free manifolds they on one hand you can build lots of them on the other they have very rigid restrictions on the fundamental group of three manifolds and you can think of link links as toy models sort of give you toy models toy models of three manifolds and the three manifold corresponding to a link is the complement or 3 minus L. So let me draw an example of a this is a not not not but it's a link with one component not um not L and let me give you a picture of a of a link with two components L and so so again over the past decades people discovered that this simple looking pictures I mean not soft equivalent um link supplements of this simple looking pictures and the corresponding free manifolds they they have incredibly deep structures behind them. Um and usually people just apply they they somehow found ways to take some other theory maybe differential geometry or representation theory or mathematical physics and apply it to find structures new structures that involve nodes. Um so it's part of this was motivated by classification problems but to me it's sort of not even the point. The point is that if you study this threedimensional structures there's incredible amount of mathematics and mathematical physics related to it for for some reason. So just you get very very deep rich structure behind it and most of the time you need to apply something else to to nodes and three manifolds. You don't do it by just studying topologically not. You somehow take some other theory and it miraculously relevant to the study of not three manifolds and this is specific to three dimensions and into four dimensions in some cases. Um and um and again very very naively get a lot of information about not by just taking the fundamental group of 3 minus L you already get huge amount of information. Of course, these are very special groups. Not any group can have this form and it carries already a lot of information about notes. In some sense, you can you're almost done by working with this group.

[00:17:23]So, in some sense, you don't need all the fancy theory, but just structurally you get um and we'll see an example. You get lots a lot of things out of it. But >> you can Yeah. So, so you can also study four manifolds that's that's related to free manifolds. Again, it's a also some sense you get a special behavior here versus highdimensional story in high dimensional manifolds.

[00:17:58]Yeah, they're closely related but I think this is much better understood. This is maybe gauge theory.

[00:18:04]I think the main results are one of the main constructions of Donaldson theory flory and um for four manifolds and related to three manifolds.

[00:18:17]Um so let me before we get to this so what I want to do is tell you about this kind of more exotic strange theory this kind of exotic homology groups for for knots and links. But before this let me tell you um um um um um kind of before this. So first of all is the endlock of the oil characteristic in this case and uh I mentioned the John's polomial John's polomial of nodes and links and that's an example of a so-called quantum invariant which is related to chair Simons theory on the side of mathematical physics physics Um and mathematically it's related to representation theory of so-called quantum groups discovered by Jim and Ginfield.

[00:19:29]Um and and this is was a pretty fascinating theory where you start with a simple algebra simple algebra g a standard I mean the simplest example is this too and you study representation theory representation theory of simple algebra has a beautiful it's it's deep beautiful theory you can classify represent find representations of G so it's absolutely canonical um in its beauty and its application to combinatorics to geometry and physics.

[00:20:01]Um, and when you study representations, you usually you can rephrase them as modules over the universal developing algebra of G.

[00:20:11]And in this sphere of quantum groups, this is an example of a half algebra.

[00:20:16]And in the field of final groups, you deform this to to a more complicated more subtle half algebbras called the quantum group UKG. And you study its representations representations and this gives rise to the quantum invariance invariance of links.

[00:20:42]Um and one of the first one of the earliest quantum invariance was the John's polinomial which can be defined.

[00:20:54][snorts] So it it can be defined as follows. So if you have if you have a link L so you have a link and for convenience we we keep track of orientation. It's a pick an orientation for every component for every component of a link and to L there's an invariant uh assigned to it which has let me call it JF the J of L and it takes values in Z QQ inverse in the ring of LAN polinomials in Q within the Gefficients and um again you can define it in various ways but a downto- earthth way So conceptually you can think of it as coming from representation theory of this hop algebra conceptually for instance but you can also define it in a down to earth way you say you want the following condition that if you have three links that are almost the same everywhere in the free space but they differ just slightly they differ very slightly inside the threedimensional ball and this is one link this is another link so you see that they differ by crossing inside a small bowl and the condition and the third link is going to be different.

[00:22:16]So what we want is a linear relation saying that no matter what's outside of this region as long as this feelings are related in this way inside the threedimensional ball um then there there must be a linear relation on a linear relation on um um on on on the three John square of these feelings.

[00:22:45]Um and this is already this is an infinite this is a system of infinitely many linear relations because you're saying that no matter what's outside this should be the relation on the free links. Um and you you can check that this almost determines this almost determines the this infinite set of conditions. It almost uh determines the John's polinomial. You just need additional you observe that from this condition by closing up on the side you can check that the John's polomial of a link L union with unlink is going to be the John's polinomial of L* Q + Q inverse.

[00:23:28]So that's that's easy that's easy to see and then you introduce the condition of the John's polomial of the empty link when there is nothing is one. So with the John's polinomial of a circle is Q plus Q inverse.

[00:23:47]So this is our normalization for the John's polomial.

[00:23:52]Um and um this is one way to define it. Um but if you want to see where it comes from then you study quantum SL2 and you study representation theory of quantum cell 2 and then in this representation field interpretation then you have a crossing.

[00:24:13]So you you use two dimensional representation of your quantum sol that you pull at every end point. That's V. V is isomeorphic to C2 with an action of the quantum group and you put an copy of V at every end point of a diagram and then to to your crossing you assign a linear map from V tensor V to V tensor V which respects the action of quantum which commutes to the action of quantum and there are not that many maps in fact if you just have a solution.

[00:24:51]Then V t which is V square deco composes as the second symmetric power of V plus second exterior power of V.

[00:25:02]>> Yes.

[00:25:10]>> Yes. Yes. Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z QQ inverse It's these are polomials here. Q um in our interpretation Q is a formal variable you know expressing Q formal variable in the S science theory it's a root of unity for us it's a formal variable >> formal variables like X like variable X just just take a variable and use its inverse >> yeah it's like yeah >> just yeah just just a variable >> and So they take the value of that value.

[00:25:59]So how many?

[00:26:09]>> Yeah. Yes. So what's so this condition it's encapsulates infinitely many relations because it says that no matter what's outside you have three you have three links this link this link and this link and their invariance should be related in a linear way. So, so it's not obvious beforehand that this is well defined but this is consistent. Um and um you have to remember that a link can have many diagrams. For instance, this diagram of a link this it defines the same link as this diagram. So topologically these two are the same. So there are something on the outside but topologically it's the same link. So it's you know given for instance two diagrams it's not trivial to figure out do they represent the same link or not. So there are some subtle questions.

[00:26:59]So um but um so we're saying that one way to see that you get a well- definfined invariant is to interpret it using representation theory of this of this h algebra. So in half algebra if you study representations or vector spaces on this on which this h algebra acts you can take the tensor product. So the properties of of algebra is that you're able if you have two representations for instance if you have two representations of a group you can tensor them and you still have a representation of the group or if you have two representations of a V algebra you can tensor them and get representation of the V algebra and then you write >> SL2 somehow it's the first non trivial simple V algebra SL2 is the smallest smallest simple the algebra and it appears everywhere and this is the first trivial example of this theory. So there are just more technical examples for other for other the algebraas beyond the so but here sort of conceptually [snorts] I mean where does this I mean from the representation theory side where does this relation comes from comes from let me just do when Q is one you just have the usual SL2 and you have the usual SL2 you have this C2 with an action of SL2 you can take C2 tensor C2 so V tensor It deco composes into sum of two irreducible representations as two of three the three dimensional and one dimensional presentation and so the space of homes the space of intertwiners homes from v square to v square is two dimensional because um it has to take a st of to v and has to be a multiple of the identity minus two of V and it's a multiple of the identity on lambda 2 of V. So you get two multiples of the identity. So you have two dimensional space of homes from V² to V square. Um and the same in the quantum case when you Q deform when you add Q you still have two dimensional space of homes one but here we have three maps.

[00:29:23]You have this map this map and this map.

[00:29:28]So they have to be linearly dependent and when we do the computation you can check that out these are the coefficients. So this is a conceptual explanation why we have this soal scan relation because just the space of homes is two dimensional from the square to itself.

[00:29:48]>> Yes.

[00:30:02]This is this is simple simpler than three manifolds. It applies to this is specific for links. This is specific for links in R3.

[00:30:17]>> Okay. So for so for free manifolds you need to you need to do um I ch Simon's theory where Q Q must be a root of unity and then on the level of groups you need to make Q root of unity so you need to do additional manipulations this is known as the witten witten derive theory so it's more complicated so for free manifolds things are more complicated so links put somehow simpler case plus for links you can use generic Q you can use Q formal parameter okay okay but so there's also a this is conceptual there's a more combinatorial so our idea is to going to replace John's polinomial we can say that it's it behaves like the poly characteristic of some homology ology theory. So we're going to replace it by homology theory H of L and say that the John's binomial is the oil characteristic of this homology theory.

[00:31:25]But to motivate this emoji theory, I want to give you another derivation of this formula due to Louis Kaufman which also goes back to the early days of a John's polinomial and it quickly followed the original John's approach to this polomial. The idea is to simplify even further because if you look here then these two diagrams they have roughly the same complexity because they have the same number of crossings if something happens outside them these two diagrams have the same number of crossings so it's not clear how to which one is simpler this but Kman pointed out there's a way to to have a a kind of a iterative simplification which is immediate for these diagrams and so Kman's construction was to let me I'll I'll do a modification of Kman construction which is slightly less symmetric but which is more convenient for us. So calfman. So he takes he takes link L. He takes diagram D diagram D. So plane a diagram plane a diagram such as this. And at first he forgets about orientations. Forget forget orientations.

[00:32:53]Forget orientation.

[00:32:55]um and he and to without orientation he want to assign a coffman bracket of D which also lives in ZQQ inverse as follows just a scaled version of the John square but it's um it's as follows so you require that um so if you have a circle then the invariant is as before it's Q plus Q inverse and if you have many circles It's Q + Q inverse to the K where K is the number of circles and then if you have a crossing you it it's it's again it's a version of a scan relation. So it's a layer combination. You say that the CN bracket is given by combination. simplifying this way to Q two arr minus Q inverse simplifying this way to two arcs again I'm slightly I'm breaking the symmetry of his definition just to make it kind of easier for for us later so he has he has he uses Q square root of Q and then you can make things more symmetrical he uses overall minus sign here but let us do this kind of this role so um this role You can check check that that if you if you take this diagram like this di you can you know you can simplify this crossing this crossing the but things will cancel out. So you can check that you get the same answer as for this diagram which is a good sign because we want to define the same link. So you want this diagram together.

[00:34:49]>> Excuse me.

[00:34:51]>> Question.

[00:34:52]>> Yeah. Perhaps for the vertical resolution there should be Q uh in the Kaufman bracket >> here >> uh for the vertical resolution. So in Kman uses so his um Kman's normalization is that minus u plus Q inverse and uh crossing is Q2 plus Q minus 12.

[00:35:22]So I'm just slightly changing to make it easier for me later.

[00:35:25]>> Okay, thank you very much. This is more symmetric.

[00:35:31]But so if you can simplify, you can check that this is equal to this, which is what we want. Of course, we want the invariant under moves that don't change the topological type of a link. And you can check that for the so-called randomized three move the uh the invariant is the same. And not much that this is well defined because for each crossing you're reducing to something simpler for something which doesn't have this crossing. So these diagrams are intrinsically simpler. They have one less crossing compared to this diagram.

[00:36:05]>> Yeah. This is just readmeister smooth type three. Right.

[00:36:10]So, so these two so so these two fives two and three move then there's also you can check that is four yeah I mean so if you if you if you have uh so this is going to be plus or minus q so so you can check with this up to some simple scaling you you get the same invariant as here. Um and um so so the point is that we sort of ignoring orientations, but we we're also in the neighborhood of this statement because if you if you put here a V and V. So each of these maps is again an intertwiner from V tensor V to V tensor V.

[00:37:03]we just kind of we sort of need to identify B if it's dual and there are some later there's some kind of issue how you identify and so that's why this is this is slightly more slightly more less intrinsic than this description but conceptually it's very similar you're also saying that there are the space of intertwiners is two dimensional so there's going to be a relation on the three intertwiners a linear relation on three intertwiners it's very similar but it's just kind of simpler some sense And um so you can check that the John's polomial of of L is given by taking this of decompose all the crossings in every you'll have many terms but every term is a union of circles. So just count the number of circles and and convert it to this exponent. You can check that the Jones polinomial is the calman invariant calman bget of a diagram d times um minus one to some power and time two to some power that's easy to compute from the diagram and orientations which I'm going to skip but it's just this way you it's easy to check invariance under so-called randomized moves moves of diagonals that don't change topology ical type of the link and it's it's well defined because every time you reduce the complexity in a in a in a in a in a kind of strong way. Um so let us maybe questions >> J. So here so usually people start at first they start with with a link or not. So so it's closed it's closed. But if you want you can work you can look at links with boundaries. So link with boundaries I can say well so I I I have a say plane.

[00:39:12]So I I'm going to look at a link which has boundary.

[00:39:19]So you can study links with boundary.

[00:39:21]They're called tangles. And then the invariant of a tangle, it's not just a number. It doesn't live here. It lives in some vector space.

[00:39:31]And some this vector space. And this vector space relates to representation theory of SO2 and quantum SO2 because this this tangle this tangle gives you a map from a trivial representation C to representation to V tensor and where n is the number of boundary points.

[00:39:54]So the invariant of a link is going to be a morphism from a trivial representation to this representation.

[00:40:01]And this morphisms constitute the space of invariance what people call invariance of v10.

[00:40:07]So the analog of the john's polomial is an element of this vector space.

[00:40:18]Yes.

[00:40:28]>> How can we know the number of >> Yeah. Yes. So I think if you just set Q equal one >> Yeah. number of closed loops you can the the number of components you set Q equal to one >> and Q equal to one you get so for a circle you get Q here so guess J see so J of L it's going to be Q to the number of components number of components of L Yes.

[00:41:19]>> Yes. Yeah. This is kind of this first the simplest specialization Q= 1 gives you a number of components.

[00:41:40]Yeah. Yeah.

[00:41:42]>> Yeah. Yes.

[00:41:54]>> Yeah. Yeah. But for instance for if you take the trivial link versus the linking link with not linking number you will also see the difference in the John's polinomial. I think you should just expand Q. If you look at Q near one look at the first term of the expansion you should get the linking number.

[00:42:17]anywhere.

[00:42:22]>> Yeah, people study perturbative expansion of JFL of the John when Q is near one in powers of Q minus one. And this is the so-called um the theory of finite type invariance of CDF invarian.

[00:42:35]So it's been studied in in a lot of detail. So you get invariants you get invarian which are repackaged in a different way. you get you get this Jones phenomenal but rewritten in the perturbative expansion around one.

[00:42:51]So those are called funny type invariants.

[00:43:04]>> Yes. Yes.

[00:43:08]>> Yeah. These are called when you have two circles, these are called two component links, component links and one component links are called knots.

[00:43:22]But the number of components is a very basic invariant of a link. It's like the simplest invariant of a link. And beyond this invariant, you can look for instance at linking numbers.

[00:43:33]Look at linking numbers. You can bound you can draw a surface area the surface which bounds one component and count the geometric number of intersections with the other component and that's the kind of max level invariant of the next level but before the John's polinomial people studied more elementary variants such as the Alexander polomial Alexander penal is more in the spirit of linking number and its higher level interpretations which can be recovered in a simpler from pi one from the fundamental group of a link complement but with the John Spano is a kind of more more subtle invarian that don't come immediately from group theory yes so um so so the sort of so this is uh due to calman and it's just another way to describe the John's binomial and see that it's well defined and so our so I promised you to lift the Jones binomial to the homology theory so that the idea is that you have you have a scan relation with on three terms and you can just ignore the skew.

[00:44:56]Then the idea is that this return relation can be lifted to a long as a sequence of homology groups. The point is to point is to say that we're going to replace the scan relation written there into we'll have a long exact sequence of homology groups. So, so to to a link L we now assign homology groups and the scan relation is going to be replaced by a long exact sequence on this homology groups.

[00:45:39]So there's going to be something additional about Q. So Q so J of L is in Z Q inverse.

[00:45:54]So it turns out that you can and recall that when we lifted the oilistic of Listic of the topological space X to homology of XMology of X this has parameter integer parameter H M of X.

[00:46:14]So here we already have parameter from the start powers of Q. So it turns out that to link L you can assign homology groups H of L which are now bgraded not a single grading but a B grading. So h of l is a direct sum h i j of l these are integers such that three johns polomial is the sum minus one to the i u to the j r rank h i j of l.

[00:46:54]So, so, so the J binomial um it gives you an integer for every power of Q. For every power of Q, you can look at the corresponding uh coefficient. So, it turns out that for each so you fix the fix J very very I fix J I and so you you're just taking you're taking homology with a fixed J to recall the corresponding coefficient of the of a run polomial. So you can sort of imagine that you have a plane and in the plane with this homog hij i + one j and h i j + 1 of l h i + 1 j + 1 of l and they extend in two directions only find many of them are non zero and then so you take the oil characteristic along in the horizontal direction to recover coefficient of a John's polomial.

[00:48:02]So, so, so if you write the Jones polomial if you write it as sum a subj U to the J rank rank what? Rank H I rank HI J of L.

[00:48:29]Um a rank. Yeah. Yeah. This is the rank.

[00:48:34]Yeah. Rank rank of a billion group.

[00:48:38]So it's like dimension. It's it's similar. It's similar to the dimension of a vector space. Yes. So other words we recall the coefficient say a subj by looking at the groups on the JF level on the JF level and taking the oil characteristic along this horizontal line. and then you get J + one you get a J + one as the so you can sort of bgrade it also similar to Here in some sense also by groups um and so the idea is that this scan relation on three terms it it becomes the it should become the long sequence of homology groups because then if you have long sequence Then then k of this minus k of this is going to be equal to k of this except that there's additionally q which I'm ignoring there is a q and q becomes additional grading um and sort of it turns out that it works maybe I'll sketch I'll sketch how it works Yes. So we need to construct. So for instance, if you want to inductively construct homology of the more complicated diagram from homology of simple diagrams, we need to know the homorphism.

[00:50:22]So you know as you simplify and simplify um so for instance for if you have let's look at this diagram with a single crossing right so we should it resolves into this diagram and into this diagram.

[00:50:40]So we need homology of this diagram homology let me move this a little bit.

[00:50:46]So we need homology of this diagram and we need a map between them and then if you have a map we can we can extend we can we can add the third term that then we can almost understand the third term if you have a map. So we do need this connecting well well at least this map. So let me adjust this maps to inactively reconstruct the homology of the diagram with more crossings.

[00:51:20]Let me so let me explain this.

[00:51:26]Yeah. Yeah. But yes so you get connecting but the idea is to just build this two build this two. Build the map between them and put here the corner put here the corner of this map inductively.

[00:51:39]Just put here the corner of this map. So you needed this list two and you need a map between them and then you iterate on the number of crossings and so how how do you how do you build this? So when you have no crossings you just have a circle or several circles and it's invariant is Q plus Q inverse.

[00:52:03]So Q plus Q inverse you need to have a copy of Z in degree minus one and one.

[00:52:12]If you have a copy of Z in degree minus one and one and if you take the and see take all the characteristic let's say you inological degree zero you just have this Z gives you one with Q inverse and this Z gives you one with Q1. So you kind of guess that to a circle you want to assign just two copies of Z in this degrees in degree in degree 0 minus one and in degree 01.

[00:52:42]Okay. But we also need some then to several circles. So when you have let me call let me call the sub group A. Well the sub group A with two Z's.

[00:52:54]So when you have more than one circle, you want to assign to this 10^ of a square because that corresponds to the invariant of this is going to be q plus q inverse squ.

[00:53:09]So that corresponds to the graded dimension of of a squ.

[00:53:15]So you can have guess that for k circles you assign a tensor k.

[00:53:21]But you also need some interactions between the 10 powers of a. So for instance here you have for the first term you have a square and for this term you have a.

[00:53:35]So you need a map between them map which is your in this map. Let me call this map. It's my pen because it looks like multiplication because you go from two copies of A to one copy of A.

[00:53:55]And um if you think about it, I mean there are not that many possible multiplications on an Aiden group of rank two, right? But so it turns out that you can what you do you think about let me just call a generator of the Z1.

[00:54:16]Let me call a generator of the ZX. So one in degree minus one X is in degree 1.

[00:54:23]If we shift this one degree up we get one in degree 0ero and X in degree 2.

[00:54:31]And this has a multiplication. You can you can think about Z1 plus ZX and identify this withology of a two sphereology of the ring of a two sphere. So it has a rank two with the generators and x square is zero zero and um so this turns off this is part of the structure. So you take a to b the economy of this two if basis one and x except that for for balance you from degrees 0 and two you shift the degrees down so that they symmetric around degree 0 you shift it to minus one and one.

[00:55:17]Then if you do this then if you look at the map from a square to a 1 goes to one one goes to one. So one now lives in degree in degree um minus2 and this one is in degree minus one.

[00:55:39]So you see that the map does not respect the degree.

[00:55:43]But notice what we have here. Q Q inverse Q minus one. Inverse that tells us that we need to additionally shift the degree down by one. So we need to shift down by one. And now the degrees match. So we we would we additionally shift the degree the degree down by one.

[00:56:04]And now the the same with differential is grading preserver.

[00:56:10]Okay. And we know what so what we want to put here is homology of this diagram.

[00:56:21]Uh but this diagram is the same. It's a trivial note the same as this diagram.

[00:56:28]So you know that maybe after world shift you should get the same answer as just a.

[00:56:33]And this works just fine because this map is subjective. If you look at the kernel kernel of m you get basis. So it's x t x goes to x² = 0 and x 10 1 - 1 x multiplication takes it to x - x that's zero. So these two elements give us a basis of the kernel kernel of n. So sort of size-wise it matches wise matches and then turns out that you don't need much more. So you just need thisology of the two sphere with this relations x² is zero and you also if you flip the if you reverse the type of the crossing.

[00:57:22]So if you look at this this diagram then you'll actually need a map from homology of this I need to homology of this.

[00:57:43]So you need the map in the opposite direction. You map from a into a tens 2.

[00:57:50]So it looks like multiplication delta.

[00:57:53]And that u inverse gives you additional shift.

[00:58:00]And that turns out it's not easy to construct. So not not hard to construct to construct this you you realize that a in some senses is aorphic to its dual aid group. A dual um and that's because this is what people call a fibinius algebra algebra.

[00:58:25]It's an algebra with a non-trivial trace. So it's an algebra that's isomorphic as a module over itself is isomeorphic to the dual mod.

[00:58:36]So I'm going to skip this details but I'll just say that a is isomeorphic to a star. So so when you have this multiplication from a square to a youize it to get a map back from a star to a* squalize it to get m And then this is aorphism allows you to identify a star va and identify this with a square compos likeorphism.

[00:59:11]All the composing this together gives you a map from a to a square which we call delta.

[00:59:20]And um you can also write this delta explicitly. You can compute delta of 1 is 1 x + x 1.

[00:59:29]Delta of x is x tanz x.

[00:59:36]Um and this turns out to be also part of a structure. So you put here delta and you shift the degrees as before. Um and and then turns out that this is already enough to iterate to more complicated diagrams.

[01:00:14]Um, you can think of it this way. You can Yeah, it's essentially um Yes. Yes. It's Yeah, because because it's a manifold. So, you can say it this way.

[01:00:45]Yes. Yes.

[01:00:57]>> Yes. So I mean what you're using. So if you have a manifold m of dimension n if you take h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h n of m let's say that is when hn of m is let me let me work over a field so over q this is q and the multiplication gives you a perfect pairing h of mq and h n minus i of mq it goes into H N M MQ isomeorphic per Q. So multiplicationology gives you a perfect pairing.

[01:01:39]Okay. And it allows you to identify this with the homology.

[01:01:56]>> Yes. Um yeah. So you just need to do well or you need to use pongraality. So you need to use pongraality but in the um so you so you probably you probably can uh yeah I would say it's part of this um I mean um yeah how to do using different operators. I mean that's yeah one would have to be one has to think about this but I think it's probably standard I would guess that it's standard standard but so you're using that h star of m for a manifold m is the fabinius algebrainous algebra and mean part of the Theory is that um this gives you a two-dimensional DQ oft.

[01:03:14]So more generally when you have a more complicated diagram, so you have a diagram of crossings, you can resolve every crossing in two ways.

[01:03:41]So you sort of you get and many ways to resolve your diagram into diagrams without crossings and then so combinatorally you can just assign you can build the corresponding tens of powers of a and define maps.

[01:04:04]which are on appropriate copies of a their m delta.

[01:04:11]Then you need to slightly tweak the structure add minus signs to make every square anti-commute.

[01:04:18]And so then you sort of collapses into a single complex collapse into a single complex by taking direct sums along the diagonals.

[01:04:33]So you get a complex. So from a diagram dagram you build a complex c of d which is sum of many powers of a powers of a one for every simplification one for every diagram without crossings. So that's going to be two co such diagrams where n is the number of crosses.

[01:05:00]Um and then you take the homology h over d and and this homology of a total complex is essentially the homology we assign to l h.

[01:05:21]This is a degenerate example because in this example the square computes for trivial reasons just because the diagram is very small but in general it's it's more complicated and the property that you need for this to compute to compute you need that a is exactly a commutative frainous algebra algebra. So a must be commutive algebra.

[01:05:47]So multiplication is commutative and frainous means that it has this trace trace similar to the top degree trace identification of top degree with Q making the pairing perfect.

[01:06:02]This is different algebas and such algebraas that are in a bjection with two dimensional TQFTs two dimensional topological quantum field theories biological field theories is a tar function from the category of comportisms in that dimension in dimension two to the category of vector spaces and they're kind of easy to write down the properties are easy to write down in dimension two turns out it's Exactly. A commutative frainous algebra. Fabinius algebra means there's a trace. There's a trace similar to the trace here which makes the algebra is amomorphic to the dual vector space.

[01:06:47]The algebra is amorphic to the dual vector space.

[01:06:52]Um and um and so you can um and then so what's interesting you you can so you get to this homology groups they're braded because you have you have one direction the direction um perpendicular to the direct sums but there is also internal direction coming from theology of degrees of S2 and its direct products to two directions. So you get a B credit theory by graded.

[01:07:31]So you need to do some work. You need to check that it's invariant under randomized under the moves that preserve the topological type of the link.

[01:07:44]Um and but when the theory so but you get some you get some invariance you get and then you can check that they have nice factorial properties that it's not just some random groups they fit they kind of extend to interactions between links and the interactions between links it turns out they given by cabortisms.

[01:08:08]So if you have you have a link L null have a link L1 you have a comportism S between them.

[01:08:20]So links are in R3. So this is L null is in R3 one is in L3 mass is in the space one dimension high.

[01:08:31]So it should be in R3 times the interval.

[01:08:36]But think of them as interactions between links one dimensionality. So such a cardboard is a mess induces a homorphism from you can check with this homorphism from homology the homology of L1 and you can check what it's that it's factorial that thing that the composition of uh composition of the state. Cabortisms correspond to composition of and so one subtle thing that was always that always requires a lot of work is to check with certain sign in this comparison the map is well defined and so among the many papers of Tettoana. So one very interesting work was to kind of a conceptually very simple explanation of how to find this science and why why it's well defined.

[01:09:44]It's a work from several years ago. Um but up to it's much easier to explain but sign is very subtle.

[01:09:55]uh but so so what you get is a two-dimensional fourdimensional theory get a 4D theory because now we get an invariant of something for dimensional of a surface embedded in the force space stretched out between two links so we get a fourdimensional invariant so it's not it's not invariant of for manifolds but it's a sort of some kind of intermediate model simpler model simpler than for manifolds but it's still four-dimensional What's Yeah. So if if you look at John's polomial this relates to finite dimensional representation of quantum group QSL2 it's some well-known theory it's some in particular it's semi simple representations are semi simple but when you categorify when you extend this to homology homology groups so this is in 3D so to homology which is essentially a fourdimensional theory due to the extension to coorms it's 4D there's relation but here you get into um um you get into much bigger even bigger chunk of representation theory relating to many things for like to highest weight categories of representations of sorry here you have the cell but once you left to homology already for the verification of genes you get high categories or representations of GLN for all N you don't need quantum you can do the usual but you have higher categories so the categories which contain modules like vermo modules modules of this kind need them for a length but you only need a particular slice you don't need the whole category you just need soal maximal maximal singular or parabolic blocks blocks so you just need some part of it and if you want to switch from SL2 to SK then the blocks here become less maximal so they become more complicated.

[01:12:35]Um we can also relate to um coherent shifts derive of coherent shifts shifts. So the categories I mean once you extend from if you extend from links to tangles and you look at the invariant of a tangle in the story. So these are nonomology groups and angles they're going to be objects of some triangulated categories and this categories. So they appear here coherent shaves on macajima on suitable macajima varieties and they appear the categories appear throughout representation theory for instance in modular representation theory various chunks of non semi-imple non semi-imple representation theory modular representation theory in many ways is modular representation feing algebbras algebra such as a algebbras algebbras algebraic he algebbras a lot of algebbras category categories so you get you see them throughout you see the categories that control this homology and that extend them to tangle they sort of sit everywhere throughout geometry and representation theory uh which is a very very very interesting feature um of this story. So you kind of go from this again also already beautiful representation theory of quantum groups finding dimensional semi simple representations you go to when you calcify you go to this non semi simple categories of high representations so you're dealing with infinite dimensional representations or you work with derived categories of shapes on varieties or this fine algebraic algebbras um secretly secretly in many of the constructions you're doing a categorical version of level rangularity secretly because you're switching from SL2 to certain pieces of GM representations for all N. So you're sort of doing categorical version of level rank duality but there's definitely lots and lots of structures but also it tells you that in this categories they control linkology theories but there are many factors coming from tangles coming from categorical quantum group actions but quantum group becomes categorical quantum group.

[01:15:27]uh s generators of common group they now become functors. So you get much more structure and you see things like um like canonical basis of koshovalistic it becomes basis of indosal projective modules over some algebraas. So you see so you see so this is all related to geometric representation theory and this was partially motivated partially motivated by geometric representation theory and again this kind of partially answers problems of cray frankl back in the mid '9s they asked about a clarification of quantum sol that question is still open. There are some kind of various technical issues with clarification of hop algebra. So their question is still open but lots and lots of things have been developed uh since then. Uh but you definitely get lots and lots of story on the representation theory side on the geometry side. It's a really nice study and you get lots of funs interpret various canonical functions within this categories as fun corresponding to tangles and functions corresponding to categorical actions of fun groups group becomes the fun group at the next level so that um ei become funers you can compose them transformation so there a new level of as well. Uh so so so but you see representations here on both on the 3D level as corresponding to say um finite representations of quantum groups and on this level where things are even even more complicated. But these are these are the kind of structures you see at the next level.

[01:17:35]But all the categories they have sort of common equivalent subcategories. So we also that there are lots and lots of factors relating those categories or making establishing various equivalences between their subcategories. Often we need to take derived category or homotopic category but that's so you see kind of immed you see connections between the categories seemingly which are seemingly of different origin. Yeah.

[01:18:00]You also see instead of coherent sheets you can replace it by fukay categories of lranions on the corresponding varieties with a rotated hyper structure. So all of that is present.

[01:18:21]So maybe I'll stop here. Thank you.

[01:18:41]What's steam group?

[01:19:00]spinning.

[01:19:01]>> Yeah.

[01:19:05]>> Spinning. Um I mean there are for instance there are various equarian versions of the story where you take S2 you take S2 but instead of taking themology of S2 you take various equivalent for the group G which is a subgroup of U2 I don't know I don't know if that answers your if I don't know if that's related to your question but there are some modifications of the story where you change theology of a student a homology but I don't know if it is spinning but you can there's equivalent version And then you use If we consider others different Yeah. So let me let me I mean to partially answer your question, let me point out that I mean for instance the in story v um for any irreducible representation v lambda we have the corresponding quantum invariant um from colored for notes labeled by lambda. Um and I just want to point out that in this story you can you can get various various types of linkology theories and there's been many developments many people discovered various theories the theories with the best behavior on the coortism on the coortisms there they're known to be for GL let me say gln and when you color components of a link by fundamental representation lambda lambda A of V is C N. So these are kind of the best understood um cases and in this theory the homology of the trivial not labeled by the corresponding representation let me say of quantum group UQD4 and lambda QV the corresponding homology um group for the unnot. So this is the simplest is actually ends up being of the complex grass manion of a dimensional planes in c in c.

[01:22:30]This is this n um this is the so already for just for the unnot groups are rather complicated of the grassman the dimension of the kagmologist spaces n a which matches the dimension of the axial power of v. Um but this is the example where you get the best behavior of gmology groups and there are other problems. There is a work of Ben Webster again also from a while ago where he griifies [snorts] um for any simple algebra and any lambda but in his construction and in probably related constructions by other people homology doesn't cannot have cannot be factorial for some trivial reasons. So there are various constructions which are not functoral which and it's an open problem to modify them to make them functoral which can probably be also related to the fact that like in the related story for a fine grain the sort of corresponding spaces that you see in that man are going to be singular unlike this space which is a smooth manifold. But sort of the but you get very complex very complex theory once you go beyond the cell. So in in this case already for the unnot you get a very complicated spaceology of money. So it's manifold we have fian structure. So that's all part of the story but technically it gets very very hard very complicated.

[01:24:23]What is the underlying way >> I I think you can trace I mean various big chunks of genomeic representation theory can be restated can be reported from this language but I think it's it's part of the package that's you know that you get some very very deep beautiful theory with many facets which connects lots of mathematics algebra geometry mathematical physics to me it's sort of it's part of the package which is sort of of course not fully understood but to me the important is structural to me I mean not so fine but it's just that this is a way to to to uncover new structures which has many connections to many brain many areas of mathematics in mathematical physics. So to me it's the structural it's the structural beauty of the theory that's kind of of worth studying connection to many many fields and the structures they're kind of there then they're kind of precious they are rigid and it's always good to find them means so we're just uncovering new structures which are some sense I mean very special very very precious like I can sort of it reminds me for people it's a bit different context people been studying um groups recently and related to other categories and the work of uh Andrew Snowden um Nate Harmon Sophie Chris and Sophie in one of her papers she cited the the cited pier who said that this these categories they're like isolated diamonds they're very hard to find but they're intrinsically beautiful and there are not many of them and they deter a kind of special study and so this is not exactly the example he had in mind but it's sort of similar you get very kind of special region which connects to many things and it's worth studying for for that connections and connection to n is just one part of a story connections to n to 3D and 4D topology is one part of a story but there also connections to representation theory geometry mathematical physics and all of that so for me this structural richness that's worth studying.

[01:26:54]>> Maybe the question is whether there's a universal construction that works for any lever.

[01:26:59]>> Yeah. Yeah. So it's an open problem to if you look at Webster Webster construction webster construction of verification of derive invariance for any g and any lambda question is it's an open question. Find a modification of this fraction which is funal under the linkortisms.

[01:27:35]So um you made an algebra a and uh out of crossed or uh link links and uh it it turns out that uh it has a commutative provenous algebra structure. It's natural because um c commutative proven algebra corresponds to closed uh topological quantum view theory into person to 2D but um um I think from TFT bound view um open string sector is more fundamental and it's related to symmetric for venge I think so the question is are there um open string version of the homology theory and if yes um is it related to symmetric streak for Ben Sarra.

[01:28:23]So, so my understanding that at least in this kind of the most naive interpretation if you look at I mean if you look at the so so these algebraas are not so simple and I think in TQF people often prefer semi- simple algebas and semi simple algebra they're not hard to classify very few of them I mean up to isomorphism in the 2D case very few semi- simple algebas and you can do open case but it's not by itself it's not that interesting here. But what you could do, you could you could extend you can extend homology to angles in a more subtle way where things are non-unitary not so much simple just it's more subtle extension but it's it's not super hard it's doable and then you see connection to all the categories and geometry so there's an extension to the open case whereas the links you work with tangles but it's not exactly I think it's not exactly open it's not exactly open string theory I think it's topology I think it's it's kind of something else but it also it gives immediately very rich very rich story I don't know if that's answering question but very very if you just naively naively look at semi-imple the most simple 2D open costic UFS you do not get anything interesting and already here things are not semi-imple this algebra is not semi- simple soology of S2 is not that's again that answers your question.

[01:29:54]>> I think but but you can definitely there's one way to get into things with boundary and then it leads you to very rich mathematics.

[01:30:06]>> So I think I think there are many more questions but I think it's a good time to wrap up here. So uh let's thank the speaker again. Sure.

[01:30:21]So we'll have a casual reception after this. So I think uh professor Koponov will be happy to answer questions. So yeah, let's move on to the reception. So thank you very much for coming today.