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1/3 Introduction to knot contact homology

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Automatisierte Medienanalyse

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Hey if Hey you have to indeed the subjects of my lecture series is not confined to monitor and so this is some subjects that we studied for a number of years and that is something that can be viewed as implications of all this is simply the field theory package to no theory more generally use the of differential topology and that the series will have 3 ports so today I'm going to give some kind of introduction defining not contact the momentum particularly starting by talking about the shot this honoring monitor which is part of this is simply the field theory package and then tried to get to the point where you can actually get the feeling for Hollywood compute these things and more concretely not just showing that the series exist but that the computer and then there's the moral and material tomorrow but this is 2nd lecture I would talk about probably continue with that afternoon talk about a relation to strained apology and finally in the 3rd lecture I'm going to talk about relations to more physical series like Simonson and and the topological string theory there was a more recent developments of the subjects over the last few years so but anyway so that let me start by giving and taking care some introductory remarks about the relation between differential topology and conflicting jails contact your so there is a lot of these basic this is saying that if you have a smooth manifold it's a content bond which is simply the manifold vector
Westerman is also expected manifold and 1 may now ask the question how much of this movie topology of adjusted biology 1st and that is simply John of the history the remember so that end and the 1st so I'm not going to explain these results and just going to statements of this era what we're going to move be the see if you have assumed that the morphism between the stock and and some the storms along the manifold and then if you if you look at 0 section in here and this season there is something wrong cement manifold in the image of that would exactly wrong manifold in the store and there's which has some history but some of these 4 version is due to the global and it said that if this is an exact origins manifold in the then there is the simple homotopy jubilant it's been in the 0 so I think and that is the home type of the year's station is remembered by its infected topology in this process goes by firm 1 win with local coefficients is there some history to this theory that I want to tell you about but there certainly were not contributions a compact so sorry to to an animal or plant false mindful of the incident compact so so the next question is does the sort of gold Beyond that would replace John feel anything more than the home computer and the answer a big game is and there may be cases starting point for such serious historical and when am in fact sort has no home of the series would mean that there is a hallmark of his land this is the biggest on the 1st day spirit prostitute this is not the same thing about this theory to the paper by myself and the and it says the following day that if signal and the prime for all dimensions N and if contended bundle of signal is simply more off include contended bundles of prime land same is an underclass of single-minded in the in the classical Tobias fears that equal a module bounded license this year so in this dimension To came from the sir when you look at this as some kind of contesting the differential topology which I won't talk much about but but if you have a home serious they founded the they form the sage Morrison group and inside there there's a subgroup the bounds Valley paralyzing manifolds and and we know that they are that the contended Monday and can be equally felonious treatment model the standard but so to thank you which corresponds to changing the orientation of things this sort of anyway so and so these there and tells you that in fact isn't likely don't you have gotten burned predecessor and I would say that the PRI that's not going to talk about but goes beyond the world for the series and assisting in the like it's kind of different version where modernize based on morph occurred not just as the 2 to get some fundamental sight of something like that but it's actually is Kayseri classes or you know something like that so some races raising fears you're using the modular space more than using things derived from its alleviates some very interesting way of using modernize bases but here it's a little bit of hope used so it's very hard use directly and so encourage everyone is thinking hard about this and see if you can find 10 all so that the sharpest expectation that this is nearby a conditional status and expected Rongen sitting inside here but he's Hambletonian I suppose seriously I'm not so sure about this a marriage of the qualities and that is the leading this year I think we could have had but there idea don't know forming on so for example yeah I really mentioned something 1 2nd inspections so now what we're going to talk about is not the only going to talk about the relative version of that so if you have some manifold came inside metaphors again closed everything is closed then we can associated simply geometric object was that the natural thing is to think that the Rongen ,comma inside the contended Monday that and what are going to do is related to the appearance of of catalyst on the incomes of refineries in England's inside the unit goaltending by land uses a complex manifold the London calling if this points in the bonds accused along the corridor along the basin peace accords 25 appointment so that she please and in the summer and fall campaign claimed his ticket to the defendants face of the if he was 0 2 in the 2nd normal normal vectors overcame a sort of person lying in the final actions is obviously going to some wonderful and the the father the just as came in 2nd with the unit content and want to women in metric and that's the strongest In an altar going to do is actually going to specialize quite quite a bit more so we are going to afford for this lecture series would they came to me and it's clean and which is equal to or 3 that sometime perhaps history doesn't make much difference so we would consist of the very special a special case of this but before he go to the special case of defined on watercolor theories that and we are going to use it's worth mentioning it some resolve about this stuff and see how well it works With respect to things like this these things and then so much as saying that be again with recent reports about this this theory so when you morphing as if the package tours from Portland to the land the care that
it's quite successful so for example as we see the theory dictates the I'm not so that the release of explaining this to be associated an open the studio looks like that of the on not exactly only when it is done from northern so and it is a pretty pretty nice Walton variant that knows about things a polynomial The Norton such-and-such sober but but but let's 1st considered some of his counterpart using higher dimensions where there was some observations recently made by this through said and and another student of mine in recent months and and if the following so if you start being green or seek then In our seats you can imagine 3 spheres of this standard through here in the North Sea area here in fact is pretty so if you if European topologies was sincere apologies and each topologies and again from each of the point is that there is only gonna 1 note :colon but there is an worth of seasonal and the invariant can be seen as follows you take this is betting on the screen or northeastern and then you'd find a four-dimensional Cypher surface embedded undertakings signature of the stages some kind of multiple holds 16 and you can realize all this so there are these not 3 Renault's invites space retraction differentially note that but please stand so 1 might ask what the slammed the cases the city so what about Islam the which is the call normal Western and topologically science and it is the boundary line boundary to bring neighbors S 2 times 3 does this the no for the but the boundaries would be interested rights navigate if you wanted to strike at the time and the Treaty on Monday and the answer is no because in effect it does not and why is that so the alterations that actually but that it was not until to name None of neighborhood the call can also be done is clearly isotopic to a columnist with the outward 1 of this boundary over to the neighborhood it doesn't matter if you if you if you leave light on the North Korea fattened up and among others things and the the dead to the boundary to the neighborhood of the norm and the standard 1 another not or not isotopic the boundaries of their neighborhoods or isotopic so this is the kind of just consequence of which could worsen the current cancelations of two-year can you can you find a functions more functional if put their seats in the Boston so you have a minimum here and in the industry have appeared and so that they can complement of if you you can cancel the critical point so that means that a compliment looks like an times of D 4 so somehow make the complementary tracks interest in seeking and yesterday was completely free in Aussie resistant to so so that means that you can Isa told this to neighborhood to that neighborhood and therefore you can I support the Katyn lambasts straight on the other hand and this is something of interesting open so the secret of this is that the business we're not there have not exactly leaves here somehow he had to own the boundary to name and the 1 from for example the question that if you pick 1 of these the phone focus anyone would have to take could take a section of this kind of news about the intersection of the other 1 probably I would guess the answer is no the statement also about this but what it does the privatization question it's in the best that the stronger and I suppose we have to really messed up the position look at the top no no it doesn't break really anything but for the nearby running conditioning great question is if this is in court seeks and and if you think about it the store it's 3 also 6 dimensions and a moralist the counterparts of these is a serious action that would have counterpart of 0 section and so on and I think it's unknown if if there are represented by Lagrangian but I think this is reasonable questions can you find the growing industry In the isotopic class of a half-interest in the that and that would serve break the question is not the basis of the way OK so that this was ended and the introduction almost all but 1 of many say 1 of the things that need to be gone wrong years of good local dimension is to be so when wrinkle dimension to then certainly it is not called entomology exactly as in the case of in Austria we pretty powerful because it feels that all the properties of the fundamental purpose of the compliment so but that's enough for all of us introduction to the general area so we now studying is not convertible on the other you know I mean the thing this year in the about what I suspect is that is that land the kid not that some wonderful but a parameterize 74 as a primer tries the traumas of muffled those rumors we will remind them that those accosted to promote the use of the he has
also been years you wouldn't want to have a surface of what you would have a search is going to have to use your history and your strange not so they have for manifold interpolating between the 2 of you could take the signature of the answers that would be that would be the counterpart of the signature thing but but this is just the just assuming no alters and then you asking can actually find the bronze representative so that these waters and has treated signature and that's review can it the OK so what I'm going to do next is I'm going to describe for you sundry and contacts who monitored in this setting which is sufficient for for what we are going to die later but not completely in you know content what the but I was sort of made me more generally the wiser contact contract to unwind 1 24 and in the context of the Cold and inside this my we have an and and what is this disorder in DJ itself will be fine the In his portraits of so if you want to just before I'm going to depend on what the qualities the attempt is the answer to the following question you're trying to do some kind of on the face so is In his land and this is the path gamma taking landed itself you try to do the 4th of Florham order for the action function which sends spare no matter the actions and then you so you cannot quite different and and some of these as if the types of things that you see the forces you to use some of the other right structure than just the chain complex and that's that's what I'm going to have to explain so far as to say this is a lot of work and technical business going on so so that and the precise so we will we will restrict so we assume that there are no known plausible In in which is which is the case for the unit called and won 3 the name of the company s 3 of them and also that the 1st shot class contact Plainfield the series and I think that I would thing that is why I think they will opponents of the measure I'm not going to be enough to be included in there I want to some of the things we have to see what you've worked so and then and then I would also like to talk about this most of us without make the most of land of the music that the theory grade this please use it is just to make it easier to explain so they're not necessary and that some of them are slightly serious so these things to these effects the grading so you have to do some some something about the grating write to the meeting always the serious the year's assumption and what would happen is that defined for you algebra which just generated by courts and it would be another resource coefficient ring but if there are more its banned the Lashonda in algebra would be an algebra over the conduct of knowledge orbit rather not going to talk about and that war without ever certainly does require a lot more machinery of the nature that's being discussed here the pontiff also something of perturbation whereas what we're doing here we can actually get away without we get away with the classic with classical perturbation thank you so much probably forgetting have me some more assumption but hasn't so so what is this thing so what they're going to lose University under the reign of the impact of the problem security discussed his rights this summer you come on the fire formalizing whose equal to what saw the generators of we made it clear so there be a cost and points on them In the end it itself will be the following severe another over the brokering opening of the 2nd all the and wife land generated by say universities is the unilateral generated more cost and this these are the result commuter demand anyway so that right corner companies nothing could be century could could suspend and this is it's if you have handle on I guess so right now so for for the 1st time I wanted to say this is the kind of simpler version so I was 1st taken to come is not really necessary so you can soak this up and we will do that tomorrow but for now I will take the coefficient just community already it's not really necessary this could be like some kind of moderation could put these things in between that don't necessarily come and we will see geometric reason why everybody can do such things in world this is
the sort of redefining DJ and we justify a knowledge of the great the court said so what do we think it passed inside the on this is In In this is why I wasn't because we somehow extend reconfigured with these In on the boundary of these these we have attended space itself land the giving of Field of Blood Rongen planes in the contract paying and when you have the ban use of you can't take the time and space of land at the end .period and transported to the starting point with the new rice trade flow and you get that 1 of the close passes to close it up somehow by posting rotation and take the the necessary in the event of war or the condescendingly so that I'll explain examples of computers so away that the great all of this is equal to the center from the sea which is is mostly in the summer was talking about minus 1 side of the island just you have to think about this is somehow related to the home of a class of this the wrong and playing fields and this would only and disintegrating when your computer industry's so far so we have an idea about the grading at least what is the difference in the cost important piece of them so so far the things so far because it was done before here because it is the politicization of our times and here we think this effect from the heat of the offer this is what it was before I I think the most complex structure G which is translation translation and thanks to the contact benefited itself and somehow jailed the exiting the action is seeking a waiver from its this problem everybody you're probably talked about you know what I did not you have to conclude that there is a lot of the schools so that the official divided by conflict or you just want to know about the but anyway because it's OK but it was also generate recognized this and then and then we have this notion that the problem of the cover of finite energy so we have the energy on all managers are area in the contact planes there has all these couple area in the end of the world of development London and some other people so so going into this other direction where you have to take a couple function and then that's probably will discuss the days in the case of warming we know that it finds itself so fine identity cards and thinking This is the presence of unintended cars at home watching enough far as Ray ports so so we don't abide by this requirement if you take that just race course said times or and this is all more strip with the better of the strips which counterparts attributed interest from the sport and now if you have any such final and beast split-second mapping and Leonard has a number of positive bunches number of negative bunches where it says it strongly Leslie hasn't altered to these these provided that all the race course on on the job look at it and the modernize places that we will use they are modular expenses of a specific kind of these with long positive and the related a toward war and 4 the underlying which they want to be we we consider them on MySpace so which is it I think there is the but "quotation mark on the
following these these kids want positive puncher at the end negative punches which appear when you go around the beast has been 1 of the 3 of them and finally be and that's that's determined these work and finally remember that we have these cap the so we can somehow make it to the skiers these here a year they we just shows that it's there for each quarter features such as these so when we feel by these capping that creature that more victories than the capping they defined the homeowner he inside-edged so that's that's what this is going to be the place we used in mansion the former dimensions and in our is the perfect rating only minor degrading B which is the sum of the ratings of the factors in so so the differential that we are going to define isn't defined by counting such things in one-dimensional modernize bases of the kind of minimal dimensions the interest of the people you heard about or and some of them the an interesting these cases 1 of differential From his itself satisfied life for and is leaner local visions of just go through and generators is defined as the deal is the sum over they might be the best 1 and the number of all our families in this module based the could hear each of these indicated in the group and then it's something definition of what the differential than we had expected the parents the Securities include era an aide let me say much of this is not the Commission that's lot of some so in the case that we needed so indicates when the ambient space is wonder manifold 1 spaces as we will see that the unit cost about 3 years he said there was actually sort of fully proved to be the work of myself and summer so so and that you can do without using absurd for donations for for some reason they maybe I'll see if I get to explain the license can their during discussions of what it says is that the ship did repeat this J. generic the squared equals 0 and also in form the thing you gets home when I wanted to say what is this home about in particular the particular knowledge of the things in "quotation mark "quotation mark the story is that this is that it is on the 1 hand the the to the union this is just a bit invitation for instance contains while and then I write some peaches for the same thing in Prince so I I think this is going to number of times this thing is integral to use just you can think it wanted to think about it it's just it OK so let's consider the simplest example of this steering it was also the 1st time that with discovered for the economy as a whole not fond of on so consider the benefit to the author the standard companies such of the mine is widely seen here Sonoran alters North which lies in the corner of this thing and in particular projected I should say 1st the raid the radius just cleared the vertical that deficiency directions to projected strong Renault In it's space than projects designed to take job that double points and the amount 0 area if you got to take that might get into college In the
course of very easy to spot the correspond to to double points in the 1st instance direction and now 1 can peaked at almost complex structures to be the pullback the content on the projects want to this plane so that nothing like this project so what demonstrates Pfizer morphism on display this open so that you can take the standard complex tertiary and pull it back to the company and therefore you can read all hormone-free costing the simply decision but rejecting them down and say that the projected growth in here the
fact it is possible to computing by the falling so we need to distinguish what is a positive sign terminated 2 the earliest that positive the whole of the car goes up the raid coordinated by choosing goes down so here such migration and that the differential counts what's more the quality of week corn corners so that some of the scenic scenic can mean that's requirement if you really we want positive instead of negative court so so we need to take a few more words to complete this description so here In this case this age to relative just page 1 of the all so that there will be kind of 1 variable in the grouping and so we think point on an altar and what more so the
other thing is the science so he signed is the following remember that these these words they have operating rights Canada and in this case I should say the grading of the remaining years the following you start at the top you don't know so I and not as maddening as you go along the what you end up here and you close it up with a positive rotation so degrading he is twice the last time the rotation the positive might well so In some ways this is the whole thing that they can be endorsed by the leaders of the cover themselves but they they need no branch but the 1st their ,comma and I'm kind of getting slightly confused by so now what you do in order to get researchers this is on the side of the pictures and some of the shame going to explain so you're saying is even in schools you're saying is that that if you use you half of the corners like these and then in the end to so what we would have to look at the papers to get things right but in this account number shaded corners with the miners wanted to something so we said before the I'm Not There is nothing is what is the differential In the old days this is the 1st of the what is degrading always agreed on is too if rotates among strike uses it to miners once that's 1 and then there are to be used to reduce this 1 and that 1 and 1 of them it does not have to be chosen based on that 1 and in the other 1 the non-traded forestry concern or even even ones that all the other 1 just passed through this base .period which means that we can use as going once around not 1 plus an old would new which is the quality classic so is something that that the danger of this what is sometimes the use of most often used action but if you think you minus 1 and is not seeking to us so that's new Nunavut is nearly is generated by age 1 or all on our land to the article the coefficient of of the year you want to know what's going on the team is sometimes used as property to take the lead in the form of each 1 of which is the year of the facility coefficient so I was kind of win the time flying near the notice so but anyway I was asked to give them exercises so that the the exercise of course computers for some of the world computers for the transfer it was created from this compare it's something wrong it was it should be the during the so computes the J for this and I think what is what is the significance here on the this of course record ?question mark while the 2nd 1 was 1 of the most special
thing here OK good so the next step so so we that this theory is actually very computable Computerworld in the simple case that we're in Australia or maybe the surface times or something like that can compute but we're actually aiming for something new to meet higher dimensions so we're aiming for for the unit the through which in fact is the 1 that's basically so the next thing you want to explain this thing can be done in 1 space 1 this is if you have a manifold and the longest space of time it's just put turn bungled thought of us as differential the Financial Times on which is the function and on here there is a nice contact structure which is at the same time the action PDQ because much of the case all on the plane this actually kind of special case when we take is to be oranges wanted space of the of all are so we can also hear use begins fine the almost complete structure 4 the almost complex structure on the storm and so in other words we take an almost complete version of the story and the contact planes that again isomorphic onto the 10 days of the start of the put it back to contact friends and extended in the usual way in the other direction and then we had never seen a description all all the more the cars so that the whole thing we In response 2 or more .period on the same nature but now we on the protection of land which is now in the Groningen summer inside the store so basically it's the same with the same thing as for naught but now no 1 would broaden its high-dimensional and in the last number of points and we have to find out how to count but the 1st thing to service where we went to here that news notion of positive negative quarters exactly the same as the crossing at the positive refined goes after a coordinated goes down so we have some exactly the same speech now the question is how long possibly companies including areas of we're lucky in the sense that this using remote mapping theorem of land and space as hard mentions there's no such thing easy way to finally overcome but maybe 1 can degenerate the the growing into a serious sections that we try to explain this so we have so projections land sitting inside the storm this is the following is here we have a serious action now in use and so on it is exactly wrong as a matter for the year year analogy want those who want to the state so we take that we have some almost complete version of scale so no you take you the new method to jail ,comma the and the rest the world is tending to
0 so what happens in support this thing is getting skinnier and skinnier and gets closer and closer to 0 and then 1 can prove that the homomorphic proves that the Basic case and this is very old and the from that the whole occurred when when you have a graphical what they correspondingly integrated phone lines on the function friends that this graphical thing defiance right so that we have more complicated things like here then you cannot say that they correspond to gradient flow lines but rather they would correspond to trade infiltrate clips locally have locally and the looks like a great influence than it did in its systems tree that is being being drawn so let me try to to explain what those sources and probably not convinced that the explanation to the American Stock said that some of the main next means of trying to exclude this called the infiltration the way the next time I can explain that placed the grading form so when and if the rotten inside the bond it's based on the projected it it's the 2 teams start with some of the important we can also projected for this year and yet space which is just and so on it looks like what happens when you predicted a stronger in the sense that you get from inside and so on this is what he is for example the final 2 years before has the following 4 intersection .period send intersection .period do we have to reconstruct the American strike Q the is equal to that portion of the tape of say with respect that you are where locally you have a great afternoon asserting which have not talked much about now but at least you can see that the double points they correspond to points on the front and this isn't committed to good but anyway anyways .period on the floor and where the tenant planes or pair itself together as 1 to coordinate and now I want to give you the grading 4 so the grading such rare according a statement to say 1 more thing that the singularity managers of the strong they have the results of water describing local dimension lowest with the highest start of the singularities local cast engines they look like that it's a systemic cubicle costs of the squared sequel to next year he is not a good thing and just multiplied by on the other direction and those are all the singularities encoder mentioned once we find people past from top to bottom then I received only only these type singularities this coordination to anything you don't have to describe and the grinding away the number of downcast the main thing going down the casting his information my number up the officers go up in the direction class of the Morrison the the exteriors have a critical factor in the difference the top minds about whether critical point of some function and the Tennessee condition is that this is Moss function I think the number of negative eigenvalues of the Hessian and then subtract 1 2 for the I'm not hearing the features you start here go down 1 because that's 1 of them 1 for the moss in the system and then I subtract when I get my 1 back OK so that we can expand the scope of this but we can expand another the next exercise without any nations because computer has hired a personal goal is thing which is not being used it hopes here
and dimensions of its politics and because this is the front of the season and so on cost and so computer the area and compute because you don't know about the only thing about trains anything at all to compute the Jerry's so I would stop at this and continue on Wednesday and will continue 1st the trees and some trades come will yes of course it's a general here so I when when you're at the point of the base where you have to go to and parents if you look at the function which is the difference of the opera graph analog rather than you have a financial just agreed .period the point that the moss a critical point take the Moss index of that With the positive function difference with top of the this is the right moment of my life the French people think you should go on logical thinking convention this thing is over the problem and the more "quotation mark mention 1 of the you say so if you multiply by this 1 the theory changes especially 1 alignment exercise supported by the beginning of the year changes so so you have to basically over this as 1 year when you so when you just want to play together from Canberra to generating significant as far as 1 found itself patrol the ways to get 1 something sitting on the minimum something to the maximum or minimum you have exactly a copy of what you had before and all the maximum you get so this is the year so go generated have corresponding generated and when I want to come here and and have that again got the so what is the deal this includes some someone obviously wanted to up to me end and and the dynamic of the long after Sutherland forget about science is doing the following being 1 of the 2 the plan has been widely believed to have been the last and so it goes I just moved ahead what is likely isn't it homotopy operates huge that's what what's but what it does not the when you multiplied by are you kind of have to chose whether you want to live here you will live here so it's not so clear I have to say what you think
Nachbarschaft <Mathematik>
Gewichtete Summe
Gerichteter Graph
Gebundener Zustand
Dynamisches System
Deskriptive Statistik
Negative Zahl
Vorzeichen <Mathematik>
Gruppe <Mathematik>
Translation <Mathematik>
Kontraktion <Mathematik>
Vervollständigung <Mathematik>
Kategorie <Mathematik>
Güte der Anpassung
Kritischer Punkt
Sortierte Logik
Rechter Winkel
Ordnung <Mathematik>
Explosion <Stochastik>
Orientierung <Mathematik>
Klasse <Mathematik>
Mathematische Logik
Überlagerung <Mathematik>
Weg <Topologie>
Algebraische Struktur
Endogene Variable
Inhalt <Mathematik>
Topologische Mannigfaltigkeit
Offene Menge
Dimension 4
Prozess <Physik>
Natürliche Zahl
Familie <Mathematik>
Fortsetzung <Mathematik>
Komplex <Algebra>
Einheit <Mathematik>
Stützpunkt <Mathematik>
Differenzierbare Mannigfaltigkeit
Zentrische Streckung
Nichtlinearer Operator
Lineares Funktional
Endlich erzeugte Gruppe
Kommutator <Quantentheorie>
Arithmetisches Mittel
Körper <Physik>
Projektive Ebene
Physikalische Theorie
Fundamentalsatz der Algebra
Orbit <Mathematik>
Komplexe Mannigfaltigkeit
Physikalisches System
Objekt <Kategorie>
Singularität <Mathematik>
Quadratischer Raum
Numerisches Modell


Formale Metadaten

Titel 1/3 Introduction to knot contact homology
Serientitel 2015 Summer School on Moduli Problems in Symplectic Geometry
Anzahl der Teile 36
Autor Ekholm, Tobias
Lizenz CC-Namensnennung 3.0 Unported:
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DOI 10.5446/16318
Herausgeber Institut des Hautes Études Scientifiques (IHÉS)
Erscheinungsjahr 2015
Sprache Englisch

Inhaltliche Metadaten

Fachgebiet Mathematik
Abstract We define knot contact homology as the Legendrian differential graded algebra of the unit conormal lift of a knot. We show how to compute it for knots in the three sphere using flow trees and discuss some of its basic properties. We also introduce the augmentation variety.

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