 Thank you very much for inviting me to this beautiful conference and to make sure it happens despite the conditions. Yeah so well I want to tell you about some surprising at least in my opinion interaction between low-dimensional topology and the theory of motives. As a disclaimer I have to say that I'm not by no means an expert in motives so it could be that I maybe that I forget to name some people or misattribute some results. This happens don't hesitate to intervene and correct me. All right so well let me start by defining the first word in my title not. So here's the definition I fix a linear embedding of r into r3 and I'm going to study a space of long knots so it's limited like this embedding sub c from r into r3 so the space of embeddings from r to r3 that coincide with my fixed linear embedding j outside of a compact subset of r and you equip this with compact open topology so you well the topology not very relevant whatever topology has is subset by zero is really the set of knots that we know and more generally we can do this well there is no reason to restrict to r3 as target you can do it you can do the same construction with target rd for any g that is at least three the only difference that for what's special about r3 is that the space has many interesting connected components that the field of like not theories about studying the set of components of that space when d is at least four it's a connected space but it still it still has higher homotopy groups so it can be interesting as a space okay so here's a picture of a typical element in the space of long knots in r3 and okay so let me state the main theorem in a slightly and precise form and I'll I'll give a more a more precise version as soon as I define the as soon as I make more definitions so the theorem is that for d yeah for d at least four the space of embeddings of r into rd has a motivic structure and in the case of knots the space of embeddings of r into r3 it's not quite that space that will have a motivic structure it's it's what I I do you know t infinity of that space and I will define this in the talk so at the moment you can just view this as an approximation to the space of knots and in both cases the motives are over q and so what do I mean by a motivic structure so I said the space has a motivic structure there is a motivic homotopy type whose materialization is x and I'm gonna define what I mean by motivic homotopy type maybe naively you could imagine that a motivic homotopy type would be an object in the in the unstable motivic homotopy category it's not quite what I mean so let me let me let me define this now so uh I'm gonna fix uh once and for all a number field k uh an embedding of k into the complex numbers sorry we have a question uh c by materialization oh he just he just answered it sorry he just answered okay good um um right so once I fix that embedding I have a better realization so the most basic version of this is yeah when I have an algebraic variety or smooth that says smooth algebraic variety over k I can basically change it to a complex number along that embedding and then I have a complex algebraic variety and that's an underlying homotopy type and then from this construction I can I also have a better realization for any category of motifs over k and I'm going to denote this better realization function whatever the sources I'm going to denote it by b and it's always going to be relative so always there will be an embedding signal that is fixed um okay so I'm going to denote by d a k comma lambda the infinity category category of motifs over k always coefficient in the commutative ring lambda so I think it's standard notation so d a uh it's like without transfers uh and I I use the etal topology I think if I use the etal topology then matter if I have transfers or not um okay and now I can define what I mean by simply connected so I'm going to define first for the simply connected motif commutative type is and then well there is a slightly more involved version for if you want to remove the word simply connected um um so simply connected motif commutative type over k is a data of a commutative algebra in the in in my category of motifs with coefficients in q and it requires that this commutative algebra uh it's better realization so it's going to be a commutative algebra now in the in the derived category of q so it's uh like what people call it uh the cdga commutative differential graded algebra uh I require that that materialization is simply connected so it doesn't have cohomology in degree uh zero and one while in degree zero you just have the unit and nothing degree one and the finite type so finite dimensional the cohomology is finite dimension degree um yeah so that would that's you should think of this as the rational part of the omutative type then for h prime p I give myself a simply connected p complete space x of p again of finite type so finite dimensional cohomology I think we lost connection Karim do you hear us hello yes I hear you okay it's not it's not an issue with the IHS zoom room no no no Jeff do you hear us but Jeffrey use linux computer with the royal s network I think okay okay yeah okay okay yeah I'm sorry let's continue it's all right okay sorry you can still see my screen uh I see your pick uh I see your pick uh I see you but so don't see your file share my screen your beam of file okay sorry about that screen yes okay so I was giving the definition of a simply connected motivic commutative type so there is a rational part which is a commutative algebra in the in the category in the category of motives with q coefficients for each prime p I have a p complete space xp which is simply connected finite type and has an action of the absolute of my field and finally I have a compatibility data between these two things so for h prime p I have an equivalence of commutative algebra between the commutative algebra of quotients over xp with coefficients in qp and the commutative the betualization of the algebra a tensor qp so maybe a few words of explanations here so the left hand side maybe is not quite a commutative algebra but merely an infinity algebra but that's not that's not very that's not very problematic you can either rectify it to a strictly commutative algebra or since we're doing we're working in in 50 categorical languages so there's no difference between these two motions the right hand side yeah the left hand side has a an action of gamma k just because it's a functor xp by by assumption of gamma k the right hand side has an action of gamma k and that's because the because the isomorphism between betualization and etal realizations so instead of instead of working with betualization you could work with etal realization and you get something which is equivalent and has has an action of the opposite field jaffa we can first give the question to you what is a commutative algebra in dkq a commutative monoid object yes but uh yeah maybe I should emphasize uh this notation dkq maybe usually is used for the the triangulated category of motifs here i'm reworking at the infinity categorical level so a commutative algebra is more data than just a commutative algebra in the in the triangulated category okay so that's that's a definition of a simply connected motivic commutative type um so the way you define it formally is by the the full input back diagram so well you have this category of commutative algebras in dakq take the opposite category because it's going to be the cohomology of my homotopy type here I have the the product of real primes of the category of p complete spaces with a gamma k action and at the bottom right corner I have this product of these the category of commutative algebras in the direct category of qp with an action of gamma k and again I take the opposite category here so again where by d of qp I don't quite mean the I don't mean the triangulated category but really the the infinity category of chain complexes with qp coefficients um so uh well this factor here is taking uh taking code chains uh on each factor that with qp coefficients um and this factor here um is uh well uh given a given a motive over k I have its ethyl realization which will be a chain complex uh over qp with an action of gamma k and then this factor is symmetric in rados it puts the commutative algebra to commutative algebra and I can do this for hp and this gives me a map like this so I take this pull back and maybe I should add some words everywhere so here it's not it's like a full subcategory of that where I restrict to spaces which seem to connect it and find a type and similarly here and here but that's that's how you uniformly define this this category there is a question I see no I don't see any questions to you at the moment but I see Marko Balov has raised his hand well uh yeah I have a question if I see it okay what is what is the question sorry uh q and a okay questions in q and a okay good um uh what was I okay so I have this pullback square and I can I can compare it to another pullback square which uh actually defines the category of simply connected homotopy type and this second pullback square is essentially due to salivan so how can you construct a homotopy type well you have a rational part so again by homotopy type I mean simply connected and find type so well salivan has shown has showed that um a rational homotopy type is the same data as a commutative algebra in in chain complexes over q so that's that's on this uh upper right corner then I have for h prime p a p complete part so I have a space with a p complete space for h prime p and I have a compatibility data which is well I require that when I extend scalars from q to qp the commutative algebra I get is identified with co-chains over the the component indexed by p with qp coefficients so so this map is taking uh co-chains with qp coefficients um so so yeah so that's uh what uh salivan called the the arithmetic square you can reconstruct a homotopy type from a rational data a p complete data for h prime and a compatibility between between these two things um and actually you you can compare these two pullback squares so there is uh the bottom pullback square is sort of there is a forgetful map from the top pullback square to the bottom pullback square so well to a to a space with a gamma reaction you can forget the gamma reaction and it gives you a space um for we have a question so you're not closure of qp uh algebra closure you mean yeah no it's it's really qp this is this is a coefficients not the not the base okay thank you sorry yeah so I have a forgetful map from this category this product to this product I have a better realization map from this category of commutative algebra to commutative algebras in the derived category of q and well these two I also have a frontier from here to here that forgets the gamma reaction so in fact the the first pullback diagram maps to the second pullback diagram and uh well the the map I have the the induced map on the upper right upper left corner sorry I called a better realization of motivic commutative so motivic commutative type has a realization that is a homotopy type so again remember everything is simply connected um okay so how do we construct motivic commutative type well in particular smooth ultramaric bioteas over over uh over k uh will give me motivic commutative types so this uh this uh superscript sc means uh simply connected so what I mean by that is algebraic varieties whose better realization is simply connected so how do I construct this frontier well since this uh this category of motivic commutative types is is a pullback I just have to map to each of the three corners uh so the rational part well to an algebraic variety over k uh I can take its motive which is an object in the in the category of motives g a k q and I can take a dual the inner dual of that um and that thing will be a commutative algebra in the category of motives so the diagonal of x induces a commutative algebra on this on this object the p complete part is given by uh by the et al homotopy type so given a given a smooth algebraic variety over k I can construct its et al the et al homotopy type of its of its base change uh to to k bar uh and then I I p complete that thing uh and what I get is a is a p complete space with a with an action of the absolute garaboo as a field uh yeah and I didn't write it but these two these two pieces of data are compatible so what I defined here is really uh a frontier to motivic commutative types we have a question what goes wrong if you allow non-simply connected spaces um yeah some tree the definition is there's the same for simply uh non-simply connected I'm going to say word in a minute but uh um yeah the only difference is you have to be a bit more careful about what what you mean by uh by a p complete space maybe uh but uh I'm going to give an example in a second tree um um okay so what can we what can we say about comal uh but uh motivic commutative types well we have this uh theorem which says that the the the comology groups of uh motivic commutative types the the comogy groups of a motivic commutative type are naturally uh non-remotives and this structure is compatible with all the natural operations that you have in comogy group for example coproducts but also uh stiny rile operations maybe and you have a kind of uh Ackman Hilton dual of this theorem about the homotopy groups so the homotopy groups have a pointed motivic commutative type so motivic commutative type is the the data of a base point um can be given a structure of a non-remotive uh yeah maybe when I say comogy group oh I did it in the first year but not in a second so what I mean by the homotopy groups is really the homotopy groups of the of the petty realization and uh and this structure that you have an homotopy groups is compatible with all the natural operations uh for instance for example white type products um so maybe I'll explain a little bit uh how you can uh yeah I'll try to explain how you can prove these theorems uh it's not really hard once you have the right definition of the remotives um so I want to say a few words about that um so essentially this follows a work of ayub iwanari I misspelled that the first a and n I should be swapped uh shuduri and galawar um so how did it work so you we we have the the battery realization from the the category of motifs with z coefficients to the derived category of z so again everything is is uh infinity categorical so that's a left adjoint on the right adjoint below or star this counter preserves filtered coordinates and uh yeah moreover uh this adjunction is z in here so both infinity categories are naturally enriched uh over um over the derived category of z both the left and the right adjunct are compatible with the structure so so from these two observations uh we said so we have a as for any adjunction we have a common ad um so in that case we have a common ad on gz and because of the fact that the right adjunct preserve filtered coordinates um uh so it since uh it also preserves uh of course homotopy coordinates because uh gz is a stable infinity category so so in fact this uh this common ad preserves all coordinates um and also since it's z linear uh I mean you can you can uh it's it's not very hard to prove that any z linear uh uh limit preserving function from gz to itself is of the form uh c maps to c tensor with something and in that case since our function is is a common ad uh this h a guy is uh is in fact a co-algebra in gz so this h a is just what you get when you apply the common ad to z so you start uh you start with z here uh so the channel complex z concentrated in degree zero apply b star and apply b and and you get some co-algebra um but there is a little bit more uh structure than that uh the point is that b is a is a symmetric monoidal function and by abstract one sense b star is lax monoidal so so this means that our common ad is actually lax monoidal uh and this implies that h upper a is in fact a commutative hot algebra so z is is a unit of gz so when I apply b star I get some commutative algebra in motifs uh and then when I bet you realize I get some commutative algebra in gz so this h upper a is uh what people call uh i use motivic algebra group um and so by by abstract nonsense we have a factorization of the better realization uh through the category of co-modules over this uh commutative algebra function so I reward this here uh so maybe I call the first function uh b tilde so it's a it's an enhancement of uh of better realization and the second function u is just the forgetful function you you forget uh commutative structure um so an observation that you can make is see if the function b was conservative uh the better realization uh the first function would be an equivalence this this is the content of the monoidicity theorem so the conservativity is the only thing that is missing the rest of the uh hypothesis that I'm under the city theorem are are satisfied um so it would be cool that we'd express motifs as uh co-modules over some hot algebra unfortunately the function b is not conservative so that's uh something I learned from a paper by iub but the lack of conservativity uh like to construct uh which is conservative you have to use motifs that are very that are not geometric so there is a conjecture that is to open that uh called the conservative conservativity conjecture that the function b is conservative when restricted to geometric motifs so motifs that are that come from uh algebraic varieties uh so if this conjecture is true it would mean that the category of geometric motifs uh embeds fully in co-modules over h a uh maybe a remark uh the conservative conjecture is it's a purely rational question uh if it's true uh it's true is z co-efficiency if and if it's true is q co-efficient and the point is um with yeah in order to show this uh to go from q to z uh the problem just come might just come from torsion and and you have a theorem called susan rigidity that tells you that um with torsion coefficients this category d a uh it's just um um yeah essentially uh an element of d a k with torsion coefficient is uh a chain complex with an action of the absolute guarantee of the of k so the the materialization will be will be conservative um okay so what can you do is hope for the right space uh recall it's a hope for the right in uh chain complexes over over q essentially uh sorry over z so it's a priori it has a co-module g uh it could have co-module g in lots of different degrees so if it was concentrated in degree zero then the category of co-modules um would be the derived category of the abelian category of co-modules over well i denoted pi zero of h a or you could also write h zero so it would be that would be great uh and conjecturally this is a case so it's uh that's related to the the the conjecture of the existence of a t structure in the category of motives so in we know so of one half of this conjecture we know that the the hope for the right shade and half homogene negative degrees uh so it's a theorem of iuba and in fact iuba has an explicit chain complex uh that uh represents the self algebraic a and you think uh by staring at it you'd be able to decide if it has co-module gene positive degrees or not but uh it's not so easy um but in any case uh what's known it's a theorem of should we and gala where the if you take pi zero of h a or h zero the the zero's homology group of that thing you find the motivic category group of the category of no-remotives so that if you don't know what a no-remotive is you can take this as a definition so the category of no-remotives is a category of uh co-modules over this um so now it's a half algebra over like a it's a discrete half algebra it's really in the category of a billion groups it's actually a flat over over over z so so yeah if you want it's uh it's an affine uh an affine group scheme over z and you look at representations of that um so unconditionally uh we actually have factored her betualization through first the category of co-modules over h a so h a is something which is potentially derived and and then uh well since h a maps to pi zero of h a because h a is uh is connected it doesn't have homogene in negative degree you have a sort of a forgetful map to um if you have a co-module over h a it has an underlying co-module over h n uh namely an object in the derived category of no-remotives and then you have the forgetful frontier u2 uh to the derived category mc so so that's unconditional uh and the conservativeity conjecture would say that the b tilde is is uh uh maybe that's quite an equivalence but when you restrict to geometry motifs it would be very faithful and the t structure conjecture would be uh saying that u1 is an equivalence um so if these two conjecture are true you've written the category of motifs as the derived category of some a billion category um okay so now I can give the proof of the uh of the theorem I mentioned earlier but uh homotopy groups of motifs and motopy type so from the from the construction of no-remotives you can you can realize that a no-remotive is data of uh a rational part which is um uh so it's a q vector space with a co-action of this hop filter rubber in a q and for h prime p I give myself a finitely generated cp module m sub d with a continuous section of the uh sub garo group as a field and some compatibility between the two uh the two structures so I have an isomorphism between when I extend scalars to qp uh sorry I wrote h would be m so that thing sorry the left hand side has a gamma k action and the right hand side uh uh it also has a gamma k action because uh well you can show that there is a any uh any no-remotive over over q has a I don't know you're lying uh yeah when you when you extend scalars to qp you have a have a gamma k action so so once you know this uh the proof of the theorem is is easy so yeah recall the theorem is I want to show the homotopy groups of a pointed motivic homotopy type are no-remotives so let's take uh so we call a pointed motivic homotopy type it's like I have this which the commutative algebra in in motifs with q coefficients and I have this xp for p p complete spaces with gamma k actions so on the one hand on the rational part so I can apply this filter uh u and b so this is from the previous slide remember so u1, b so it's this filter that goes from motifs to the direct category of no-remotives so that gives me a commutative algebra in the direct category of no-remotives and by abstract yeah by abstract reasoning uh I mean by it's it's a very standard argument that so if you look at how salivan construct a rational homotopy type from a commutative algebra so that's what I denote by these brackets here so yeah this notation is the the rational homotopy type associated to this commutative algebra in in in the direct category of q the point is that the homotopy groups of that will have the structure of no-remotives with q coefficients is that is that the same sorry is that the same as the primitive homology of the um algebra yes yeah if you you can also yeah thanks right and also for h1p uh let's straightforward the homotopy groups of xp I will have an action again okay and uh well you have a compatibility uh you have a compatibility between the two pieces of data okay so what happens in the non-simple connected case uh I'm not going to do a general theory but uh let me give an example um and I think well this uh this goes back to to work of uh Dean and Koncharov so we'd like to say if you have an algebraic writing of k we'd like to say something like the the fundamental group is a motive but of course I didn't make a lot of sense because the fundamental group is is in general is in general not a billion um so how can you make sense of that um well how do you approximate a group by a billion uh groups you have something called the lower central series so we call uh the lower central series of a group I have uh it's defined ductively gamma 0 of g is the group g itself and then gamma a different gamma sub i plus one of g to be the things that can be written as a commutator of an element in g and an element in gamma gamma i of g um so it's a sequence of smaller and smaller uh normal subgroups of g um and I can consider the questions and the the organize themselves uh in into a tower uh indexed by the integers um so I can view this tower as a as a pro object in the category of groups um so that's what I call the new potent completion of my group um and the theorem you can prove so an example which is a motivating example for what I'm going to talk about uh the example of the purebred group so purebred group uh if you consider the hopf algebra of continuous map from p and nil into z so when i'm continuous in the sense that yeah pro group has uh has a has a topology so you you give it the inverse limit topology where you give each uh each term in the tower that is cryptopology so I can look at continuous functions from that group into z that's a hopf algebra over z uh and the point is that this hopf algebra uh can be given the structure of a hopf algebra in the remote keys over q and maybe yeah I should have said that well the example what do you expect that what do you expect this to be true the point is that the purebred group is a fundamental group of the of the the spatial configurations of n points in the complex plane so you can view this as the materialization of uh an algebraic variety which is the kind of a q you're going to prove so so in that sense well this is a particular uh a particular case of the general question that I have at the top of this slide uh it's not quite pi one of x that has the structure of a motif but it's this hopf algebra which is um sort of the best nilpotent approximation to to my group um all right all right so so that was that was it for that's all I wanted to say about motifs um and we're going back to knots uh so yeah recall now we have a more precise uh idea of what I want to do so we call my main theorem was about giving the space of knots a motif structure so I want to say first a few words about manifold calculus because if you remember in my main theorem I had this mysterious t infinity thing um so manifold calculus theory that was uh introduced by guli and vice so the idea is we'd like to understand the space of embeddings between from a smooth manifold m to a smooth manifold m m as dimension m and m has dimension m so that's hard to draw that's easy to do when when the source is a disk then the homotopy type of space of embedding is uh is what I denote f r sub m the tension bundle of n so this is um this is the bundle of m frames in the tension bundle of n so that's um so that's a fiber bundle over n whose fiber over a point is the space of space of linearly independent family families of m vectors in the tension space at that point so essentially what this is saying is up to homotopy embedding a disk in n it's just you can just shrink this disk to a very uh like you can just look locally at the at the origin of the disk and then all you all you have left is the data the derivative of that embedding which is which is a linear map a linear map from from r m into into the tangent the tangent space at that point um and yeah by pitting a basis you can identify it with that space here that's for one disk what happened if you have many disks uh so if m is a disjunction of k disks then uh the summary zoning show that the the space of embedding will be homotopy equivalent to the uh uh m frame bundle of the tangent space of the space of computations of k points in n so for each disk you should remember where you where you send the center of the disk and the data of the the derivative uh the embedding at the center of the disk and that that captures uh that captures everything up to to a contractible choice okay so we know what to do for disk condition to know about disks so what we can do in general is try to approximate our space of embeddings by the by the by such embedding spaces so we can uh we can consider the the diagram which which sends to an object in it in the preset disk m i'm going to explain this in a second to you i assign the the space of embeddings of human to m so disk m is the preset of open subsets of m that are diffomorphic to a disjunction of disks and then I take the whole limit over over this post set and uh any embedding uh by restriction to to disks will give me uh so if I have a point here I have a point in each of these pieces and they assemble into a point in the homotopy limit so so that's the idea of of manifold calculus uh you try to compute the space of embeddings by uh studying instead this approximation which is a complicated whole limit whole complicated homotopy limit but of spaces are well understood and there is a theorem of Goudian Klein that says in some cases uh so the condition is that the co-dimension is at least three then this map is a weak equivalent uh but in general even if the co-dimension is not three you can denote uh the limit uh like the right hand side of this map you can denote it by G infinity and in fact you have a tower so for hk you can denote by tk of embedding mm the homotopy limit over the category of disks so what are you not by disc colors and are equal to k is a category of open sets that are diffomorphic to a disjunction union of at most k disks um so yeah everything these things are organized into uh they're organized themselves into a tower so you have a space of embeddings t infinity and then you have the tk for hk so if we do this for knots um uh then there is a there is a an important theorem which is due to Dwyer has some torching so maybe first of all this then quite fit in the framework that I explained the previous slide because you have this lower c thing so it's not quite embeddings but this compactly supported embeddings but you can adapt the theorem of manifold calculus to these um and you can uh yeah the theorem of Dwyer has some torching so they proved it sort of independently that you can understand uh these approximation this tk of embeddings in terms of uh this space here so omega two it means two full book space um um yeah but en is my notation for the for the little n disks operate lower than or equal to k means uh I truncate it up to rt k so this is the mapping space from the e1 truncated upright to ed truncated upright this is a mapping space from the e1 truncated at two upright to ed truncated at two upright then this map this map here is given by by restriction and I take the homotopy fiber of that map and then I take the two full book space this looks very this looks quite complicated and a bit crazy in fact if you think about it well I'm not going to explain the proof of the theorem it's quite involved but uh if you try to unpack what this uh right hand side is you will see um well omega two is some sort of homotopy limit whole fiber is also a homotopy limit if you pick a presentation of uh e1 this mapping space you can view also the homotopy limit of things that are going to be configurations of points in rg so at the end of the day if you if you unpack what this is this is a big complicated homotopy limit of things there are configurations of points in rg and if you recall the previous slide uh it's also what the what the tk thing was it was also some kind of homotopy limit of things that were configurations of points in the target and in our in this case the target is rg so that gives us a vague idea for why this theorem is true um so this theorem in particular implies a tk of the embedding space before the test and can I ask a silly question can I just roughly speaking and you're mapping the e1 the k to the two or you're just sort of gluing the little intervals and the e1 together along things is that the idea is that what the two means you're taking two intervals and then seeing how they glue together because you're you're you're trying to map a longer and longer piece of the of the r1 yeah I think it's it's it's off yeah it's not it's not quite uh it's not quite that uh okay sorry it's a bit complicated to explain sorry okay um yeah so the structure of a two-folded space is compatible with connected sum for knots so I think I have a picture yeah here's how connected sum works so you should have a long knot in rd and another long knot in rd but we can glue them together to get a third knot in rd that's uh that gives the space of embedding the structure of the loop space in fact um it's not it's not clear that it's a two-folded space but at least it's a it's a loop space and and recall that the the embedding space maps to each of the tk of the embedding space and the two the two loop space fractures are are compatible okay and so this uh this good good advice uh this good good advice tower uh so this many focal costower is related to the three of five type invariant for knots so let me let me explain quickly what this is this is a definition that was due to Vasiliev and Gusarov so what is so the definition is what is a an additive invariant of degree at most k for knots so it will be a map from the set of knots by zero of the space of embeddings to an obedient group a which is a monoid homomorphism so this has a monoid structure given by connected some and which is invariant to their infection by pure braids lying in the k plus one term in the lower central series of the pure braid group so that's that's how the this lower central series that appeared in the in the theory of motives here also appears in the in the theory of knots so what do i mean by infection by pure braid here's a picture so here's uh here's a knot uh an inside is not well inside is box so it's like a three ball in my uh in my r3 and the intersection of my knot with this three ball is a trivial braid with three strands and i can uh now i can pick any other uh tour braid with three strands and replace what's in that box by my pure braid and this gives me a different knot so that's what i call infection of a knot by a pure braid and so yeah an invariant of degree at most k will not see the difference between this knot and this knot if the braid that you used is in this term of the lower central series so the higher k is the more the finer invariant is um and there's a conjecture uh which is in uh i think in this precise form is a first print a paper by balmy conant co-chef and senha the conjecture is that the map from pi zero of the embedding space to the k plus first term the good device tower is the universal additive invariant of degree uh at most k the universal means it's the initial object in the category of uh additive invariance any other additive invariant was factored through this one um so what's known of this conjecture that it's true after answering with q uh and that's essentially to concevich it's a it's a construction called concevich interval um and uh danica kosanovich and her thesis has shown that it's um that this map is subjective which means um this implies that maybe it's not the universal additive invariant but at least it's a it's a quotient of the universal additive invariant of degree at most k okay so now i can give a precise uh precise formulation of my uh the main theorem i had at the beginning so now i had this um so d is uh relation equal to three k is any integer uh at least k is at least two and it could be infinity and i can look at this homotopy fiber so this mapping space of truncated operat to this so recall from the previous slide if i take the two full loops on this this gives me the case term in the goodwill device tower uh that approximates the space of knots so yeah so that's that's a two-fold looping of the case case stage of the tower um so that's actually a simply connected space uh and the theorem is that this has a the structure of a non-trivial number to become a type so now since it's simply connected i don't have to worry about um by one so this is really the the definition i've given at the beginning um so in particular the homotopy groups uh at this uh goodwill device approximation not just by zero but all homotopy groups uh are going to have the structure the structure of an derivative and and when d is at least four then we have this goodwill decline uh conversion theorem that says that the the limit of the goodwill device tower actually computes a space of embeddings so then we have a statement about the actual homotopy groups of the of the embedding space so yeah the important word in ethereum is a word non-trivial because you could there is you you can always give a sort of stupid mutilic structure to to any homotopy type where well essentially uh essentially a mutilic homotopy type you can think of as a homotopy type with an action of some complete headed group and you can you can give a trivial trivial action so here what's important is that it's non-trivial and we can actually deduce stuff from from this non-triviality um yeah this space isn't connected as i said i can give a very quick sketch of proof so we we can start with um so the input we need is the structure of a mutilic homotopy type on the operator e2 so here we have to actually use unsympathetically connected spaces because it's a pretty too uh the spaces that that appear in the separate e2 and are not simply connected so we have to do a detour through non-sympathetically connected spaces even though the statement of the theorem is is a pretty simple connected theorem um and then we use a result due to Jacob Dury called additivity for the little uh d-discs operator you can write the little d-discs operator as e2 times e d minus 2 uh and what you do is well you give e d minus 2 the stupid mutilic structure um and you give it to its uh its non-trivial one so i'm going to explain in a second where it comes from um and the point is that yeah the the map yeah well if you pick a map from e1 to eg that will also factor through the e d minus 2 part the one that is trivial um this will be uh well this space will have a motivic homotopy type structure and this this will preserve the base point and here as well uh so these two spaces will be uh pointed motivic homotopy types and then the homotopy fiber will will have will inherit uh a pointed motivic homotopy type structure okay so the only thing i have to explain is where does the motivic homotopy type structure and e2 comes from but i realize i only have five minutes left so maybe let me explain this very quickly um uh so essentially as you as always uh there is a rational part and uh and the p p complete part which prime so rational part actually it comes from the fact that the rationalized little to discover that has an action of the groten a group called the groten dictation group and this groten dictation group receives the map from uh i use motivic galora group so to the rationalized little to this surprise has an action of uh of i use motivic galora group and that's how you can you can uh you can do the rational rational part of this uh this story uh the p complete part is uh similarly there is a there is a p a pro p groten dictation group and the p complete little to this surprise has an action of this uh this pro p groten dictation group uh so these two results are essentially due to drainfield and they were put in and so in the homotopy context the rational situation is due to press and in the p complete case it was a paper that i wrote um so we have this action of the pro p groten dictation group and the on the p completed little to this surprise and there is a map of profiled group from the observer group of q to this p completed uh groten dictation group to his p pro p groten dictation group so that's how you construct the galora action and these two pieces of data are compatible that's what gives motivic homotopy the structure of a motivic homotopy type and there is a second approach that is uh that uh comes from recent work of uh dimitri weintraub uh he actually has a an algebra geometric model for the little to this surprise uh using using log schemes i'm not going to explain this um okay uh so we have this motivic homotopy the structure of a motivic homotopy type on the on the space of knots or maybe on on this good device approximation in the case of knots in r3 so what can we use this for so one one theorem we can deduce from this uh is it's a partial answer to this conjecture so recall there was this conjecture that i mentioned that uh that the good device tower produces a universal additive invariant for knots and so it's trying to work with uh with bedroboevida we can prove that this is true so the map to a k plus first stage of the good device tower is the universal additive invariant of degree at most k after inverting prime numbers uh if you invert small prime numbers uh sorry this and here should be k if you invert small prime numbers with respect to k this is true so the larger k's the more prime numbers you have to invert um so that's a that's a theorem that we prove uh using this this motivic structure essentially the idea is um this good device tower so we have this good device tower that computes uh that computes uh the homotopy type of the space of knots uh and this this tower induces a spectral sequence that tries to compute the homotopy groups of this the limit of the tower and using the the point is since now this tower has this extra data of it's a motivic homotopy type the the spectral sequence has has some has some more algebraic data the differentials it puts out the restrictions and the differentials in the spectral sequence because they have to be compatible with this this data and this forces many differentials to be to be zero um and then we use uh we have to use uh the work in in Denitsa Denitsa-Kazanovich thesis where she she proves that uh uh if you know that this good device patrol sequence collapses this implies that that this implies a positive answer to this conjecture so the fact that the k k plus first state of the device tower is the universal ideal invariant of tp at most k and i think i'm out of time so maybe i'll just say yeah you can also compute higher homotopy groups in a range in a range of degree using this kind of uh data and that's it thank you okay many thanks in here any questions or comments yes so i have a question so uh uh is it possible that that that you could still lift this motivic homotopy type to homotopy type in say and i don't know sh of k or something like that or is it completely out of i think sh is fine um what's not uh yeah uh i think i mean i think that no one has really written the details of this but everything i've explained with uh ga you could do is uh sh instead but you you really want these two gluing parts some some periodic thing and some uh that's it i mean in some sense this is not a very good definition uh it looks a bit ad hoc yeah but okay uh yeah essentially the idea of motivic homotopy type is trying to capture as much as possible uh from like as much unstable data as possible from from stable information so that's that's um yeah that's that's the idea of of this motivic homotopy type so yeah and the galois action is is really used for example for the vis vanishing of differentials that you mentioned yeah yeah in fact you you don't you don't need the full really it this theorem is just a theorem about the p complete part of this data so it's just a theorem of the galo action i mean that's how we wrote it in our in our paper but i think it's cool to have a full yeah yeah structure other questions it seems that we don't have uh any other questions and let's thank jeffre again thank you very much