 push you to be a chair so anyway so I'll can just continue from where we left off so there are several I mean I promise to give a more general definition of this 3d tqft is that hat and it includes one crucial component which will be the main point of the second lecture namely what's that hat associates to not complement namely that's this serious fk for not k as function of x and q so that will play the crucial role today and the way I want to do it is I'll give you a definition later in the lecture at least kind of hypothetical definition but it will be hard to compute and it will be based on modular spaces which are extremely hard to analyze so instead what I want to do is to list you lots of properties of both of the subjects which will together be will be over constraining so altogether they'll provide a lot of ways to compute them and then you can take each of these properties or some of their collection as a definition and view the others as theorems so this is the approach we use for example in Donaldson invariance nobody really computes Donaldson invariance directly but we usually write structural properties of whereas gluing formulae and so on and then compute Donaldson invariance using those properties used as essentially definitions so this will be perspective that I'll try to take today so some of the properties already mentioned and for example there was a question about relation or one class of properties to written recitation for Ivan variance at the level of this object that had which depends only on q the point was that if you take q to be root of unity then you get WRT invariance at that root of unity for a for closed three manifold so there is analogous version so this is closed M3 this object for not compliment depends on additional variable so this is picture for not compliment it depends on additional variable x and you should think of x as eigenvalue of the hallonymy of the meridian sl2c hallonymy on a boundary so it's of the form x x inverse something zero and you have additional variable x which is being integrated in the type of formulae that I introduced earlier and as far as connection to WRT goes an actual question is what about this F which is analog of Z hat for not compliments analogous statement is also of the same type you have something labeled by not but now it has two variables x and q and if you send q to be root of unity 2 pi i over k and if you take x to be q to the n then this we also has connection to classical invariance so we're still exploring the region which is boundary of the unit circle in this notation and this becomes k times call n color john's polynomial of the not k at q which is precisely this value so that's also WRT that's that's one of the properties meridian is the one which goes around so for example this is picture of the trefoil that's the meridian connecting it and halonymy should be of the form here so by halonymy I mean so halonymy of some gauge connection a is path integral path ordered exponential of a around your loop so in this case the loop is precisely the meridian yeah yeah halonymy of a connection so I mean I'm not gonna say too much about this modulate spaces of flat connections but I just want to point out that variables acts that we introduced before that came out of the blue they were just complex variables they actually have differential geometric meaning in analysis on a three-manifold so if I were to talk about modulate spaces that's that's yes complex theory yeah that's that's part of the reason no k times john's polynomial this is capital k and q is exponential of 2 pi i over k of q yeah I mean color john's polynomial depends on a not and k so there are two each statement that I'm going to give in the following we'll have two versions for closed manifolds and for not complements so this is statement or class of statements that you have for closed three manifold this is for not complements and as we discussed earlier today if you understand not complements well enough from generalization two links is very straightforward you can build any three manifold based surgery so this is class of properties about relation to wrt so then there are also relations to other things that I briefly mentioned so now let's let's discuss for instance relations to to rife torsion and alexander polynomial the rife torsion is going to begin closed version for closely manifold and alexander polynomial is going to be a version for nots for not complements so physics wise the subject is connected to twisted indices or this kind of statements come from twisted indices of 3d and equals two theories called t of m3 that I already mentioned and if I have time I'll say more about mtc of m3 this is a modular tensor category assigned to three manifold that on the one hand is describing modular transformations of those log DOA characters that appeared earlier and will appear more and also encodes the information about the twisted indices or how to build them in the theory t of m3 so if I have more time later I'll say how the statements I'm going to write on the board follow from from this physics but it's okay if I don't get to physics so closed version of this it will be very similar to what appears on this side of the board here we take something that depends on q or possibly x and q and take a limit so here we'll do the same thing we take for example for not complement this function fk of x and q and we'll take a limit another specialization where now q will be set equal to 1 we'll take a strict limit where q goes to 1 claim is that in this limit this function will become x to the half minus x to the minus half divided by Alexander polynomial of the not k the three manifold version of this statement is that if so here we take q to root of unity and here we take q to very special root of unity namely identity and the result is still function of of x for a closed three manifold this invariance that had was labeled by choice of label B and it's all only depends on q there is no no x variable but you can still take the same limit take q to be identity and in this case what you get is one over what replaces Alexander polynomial for closed three manifold and that's known as tori of torsion so we get t as a function of B and that's that's a very similar statement so this is tori of torsion or rather inverse of the tori of torsion so the second version is slightly more delicate because it requires you to talk a little bit more about nature of this indices B so in our previous discussion B started live at least for those manifolds that constructed by graphs as core kernel of the matrix q but from the viewpoint of a three manifold it's nothing but h1 of a corresponding three manifold so in general when I talk about surgeries and so on this index B takes values in h1 of a three manifold but the better way to think about this as a choice of spin C structure on a three manifold if it's closed and tori of torsion is defined precisely for choices of spin C structures yes I don't think there are scheme formulae for fk that would be nice to have I don't think they exist I mean it would be really nice that would be much easier way to compute things so yeah these are again two analogous versions to to this relations you take some limit either for not or for not complement sorry for a closed manifold or not complement and you get similar invariance in the limit so let me not go into the second one because it's a little bit more subtle it will require me introducing a bit more machinery but I just want to mention that it's similar and goes very much in parallel to the first version but I'll talk a little bit about this f because again in second lecture it will be a star of the show my goal is to tell you more properties about f because then by surgeries you can compute it for any three manifold that's where really technology or machinery comes very handy so first if you haven't seen Alexander polynomials of knots let me make a simple list or table so let's do some simple experiments and play with them so I'm going to make a table over here at least not K in the first column and in the second column I'm going to list to rise torsion also known as Alexander polynomial of a not K Alexander polynomial denoted delta K of X K capital K is the knot X is just a variable so let's see what they are so first and on trivial note of course you can have a lot that's a little to trivial in this case Alexander polynomial is just one the first no trivial not as a trefuel which I mentioned earlier that's the only not I can draw that's this guy and for this Alexander polynomial is X inverse minus one plus X next one is figure eight for one for this Alexander polynomial is minus X inverse plus three plus X rather simple I'll just give next one five two in part because all these knots listed here belong to a family of so-called double twist knots and all statements that I'm going to make today are theorems for all double twist knots and conjectures for more general knots so part of the reason I decide to give you five two is that it's a natural next knot and the sequence of so-called double twist knots again if you haven't seen this before it's not important anyway that's Alexander polynomial it's very easy and now let's let's do some simple experiments play a little bit after all the statement says that inverse of f if you kill the Q variable should be X to the half minus X minus half divided by this polynomial so it's a rational function but you should think of it as a double expansion in variable X going to plus infinity and minus infinity in terms of powers of X so I learned recently from Don that this is called hyperfunction it's basically some of serious expansions in those two limits and I just want to illustrate for you what this is because of everything on the board so I can open Mathematica and just try to take this and then see what happens so basically I want to evaluate the right-hand side of this formula for say one of this knots okay so it's very simple and again I'll do this experiment in real time for you so you trust me that this is easily computable and fun to play with so let's open Mathematica so I'll start new calculation so I'll ask to compute a serious expansion of this ratio as a function of X so numerator is X to the half minus X to the minus half divided by let's take the trefuel for example this this first non-trivial case so it's X plus 1 over X minus 1 I put Alexander polynomial in so now I want to do serious expansion in X around zero up to some order let's say 15 and I want to take symmetries it and do analogous expansion around infinity and and see what happens okay so first observation is that what happens is that we get a sum of the form sum over m some coefficients I'll call them f little m that depend of course on the choice of not but this notation I may forget if I write it later and X to the power m over 2 minus X to the minus m over 2 so that's the first thing you notice by looking at the formula on the screen okay yeah I'm just symmetrizing yeah I'm writing this rational function as yeah serious in X and X inverse around both points kind of in the paper with cheaper and we call the symmetric expansion as abbreviating as SE so symmetrized expansion meaning adding pieces at infinity and zero in X yeah okay so in that case it's not a hyper function that's I'll call it symmetrized expansion if you see a rational function I'll define SE as we do in the paper to be half sum of expansion around X equals zero and X equals infinity so that's what this is that's what just mathematical produce so let's look at this coefficients f m right so first of all you agree that we have this structure so FM have the following form so so that we are asking about FM for the travel not right and they have the property that so just by looking at what's on the screen you see that when X is equal to one half coefficient is minus so we know that this is minus one when m is equal to one then there is nothing multiplying X to the with m equals two there is nothing with m equals three because that will be X to the three halves and there is nothing like that but there is something at five half right so when m was equal to five we see plus one so then when m is equal to six there is nothing but when m is equal to seven there is again plus one that's the coefficient so then indeed there is a little bit of a gap once we go to eleven and I can say that a coefficient is minus one so I can say that it's 11 as is in this list but then if we have 13 half and 17 half you can see that story starts repeating itself modular 12 so what we learn is that set of coefficients have this form that you have coefficient either plus one or minus one depending on whether m is one eleven five seven modular 12 and just like in music you see this pattern of signs going from minus plus plus minus minus plus plus minus and so on and so forth they keep repeating modular 12 so the way I got it is by just using the data of Alexander polynomial for trefoil not you can play same game very easily for figure 8 and for other knots oh it is a type of yeah thank you so much it should be symmetric in X and X inverse yeah I didn't mean anti-symmetric yeah it should be always symmetric yes correct right so we got we got this and net natural next question is how to restore q dependence so this was a statement about relation to Alexander polynomial to rife torsion and again there are several more statements that I'll give you if I have time but they already allow you to get the rough structure of what this f is as starting point so claim is that if you introduce q dependence if you don't take this limit where q goes to one what happens is that f of x and q is always of this form so without taking the limit you always have fk of x and q of this form and the question is how to find q dependence of this coefficients okay that's why I'm trying to walk you through actually efficient computational techniques not so much the definition to which I'll try to get later so to find q dependence we have to consider some limits or some properties which unlike the previous ones don't set q to something special for example setting q to one was a good idea but it kills two dependents so we have to use some of the properties that have q in the game and that's going to be the next property any questions about this so far no for four four four one not it's going to be more interesting if I had more time I would actually do it in a real time for you yeah I for five two it's actually going to be even more interesting because the Alexander polynomial is not money so a lot more things are happening but anyway that you get the idea so I want to illustrate with Mathematica just simple things and I can do in real time so I mentioned relations to WRT and the John's polynomial I mentioned the relation to Toriv torsion and Alexander polynomial so next class of properties we'll have to do with SL2 C to Rensheim's theory and a polynomial so that's that's another class of relations that I advertised in the first part so relations to SL2 C to Rensheim's and a polynomial in particular these are kind of relations that will help us to restore two dependents of these coefficients which so far seem to form again for the trefoil not complement this repeating structure of plus minus ones so to explain the relation to complex trance hymens and a polynomial let me mention in other perspective on the story which is very useful so it may help shed light on integrality of these numbers and why this is interesting so I mentioned that we're talking about this invariant zb for three manifold say closed one it has the form of q to the d labeled by b 0 plus q 1 q plus and so on and this coefficients are integer so that was the crucial property another natural limit that you can consider is limit where q is near one just like we did before but not necessarily equal to one so I'll write it as q to the small number I'll call it h bar and we'll ask what becomes of this function in the limit where q is is is near near identity in that case what happens is this whole series has the form of exponential of typical perturbative expansion of trance hymens theory by trance hymens theory again with complex gauge group so it will start as one over h bar as a zero labeled by this choice of b plus s 1 labeled by choice of b plus h bar times s 2 labeled by b where this terms is zero s 1 s 2 are classical one loop and two loop and so on higher loop contributions called either finite type invariance or vacillia invariance generalizations of tsuki series in three manifold topology so this invariance perturbative expansion in fact even in the context of complex trance hymens theory was discussed in mid 90s by many people involving Vafa that's how he's naturally involved in this more recent developments and his work with Gopal Kumar Aguri and others where they interpreted this exponential of generating series in h bar as exponential of generating series for Gromov written invariance in a certain system that's naturally associated to either three manifold or not complement I'm going to try to write it in general by saying that you get the same type of series or the coefficients which I'll call GW for Gromov and Witten take values in rationals and that's a typical property of Gromov written invariance they they not necessarily integers even though they count stable maps but stable maps in some Calabi out setting may have stabilizers that's why you get denominators so you typically get rational numbers so there is not a development in mid 90s which involves now just Gromov written variance and a fairly general setting where people notice that you can repackage Gromov written generating function usually written as exponential of such series in terms of something integer and that involves two steps so first of all it involves writing this generating series not inside the exponential but by expanding it so you write something as sum over n and then you also introduce a variable q which is exponential of each bar in terms of this two guys you get something which has the general structure coefficients times q to the n summed over n and now this coefficients are integers so this kind of developments which involve various people including Davesh Maulik, Nekrasov, Okonkov, Pandreapanda and so on were generated a lot of excitement in late 90s early 2000s when again there are two conceptual things happening I am emphasizing conceptual points here that first of all you replace exponentials by non-exponentials in two places the whole series comes out from exponential and gets written like this there is no x on the left-hand side but you write it in terms of q which is exponential of each bar so what happens here is in the q-plane if this is the picture of a unit circle this expansion in terms of each bar is defined for small values of each bar in other words it's defined near point q equals one right here so this is where q is exponential of each bar and now we're not sending this to one we're just keeping it near one but this series is naturally defined as a q series near the origin so what you want to do is to transfer the data from gram of written coefficients which control h bar expansion here to integer invariance which are called Donaldson Thomas Gopal Kumar Wafa and various generalizations in the q-series presentation so that's that's near q equals zero okay so the meaning of the equality that's a good question the meaning of the equality is that if q is near one asymptotically you should have this form so it's a statement about oh it's of the same type as here sorry it's it's equality of the same character that you want to take this function either on this side which is a formal series and write it as a q-series or so q-series here does converge all the way to one yes in fact the statement is that this horizontal relations are exactly this re-expansions which are very delicate from one region to another but vertical relations in both cases are just a quality yeah that's that's a good point yeah so I maybe I shouldn't do this but I just want to emphasize that naturally this series is defined near point q equals zero I'm only right yeah in fact in the whole open this exactly so some of these developments naturally relate to various enumerative invariants and therefore this question about integrality of coefficients that I mentioned earlier is not so mysterious you can say that these are just Donald and Thomas like invariants in whatever the setting involving five brains and and whatnot but of course that's very complicated way to answer a question about integrality but it teaches us something that we will do in a second and this concrete setting trying to introduce q dependence namely that first of all change of variables is such that q is exponential of h bar and we'll try to write exponential of this type of series in in the form that looks like this and what I want to emphasize is such a way of rewriting it is not terribly new again it has been seen 20 years ago in the context of relation between Donaldson Thomas Gromov written invariants and I don't want to present it as an accident because this whole story came as a spinoff of that development so this is not new at all of something so for this I refer you to g ppv so yeah I just want to mention that there is enumerative interpretation of even those coefficients is zero c1 and so on so there shouldn't be some mysterious in enumerative setting that that you can find for example here or just ask Pavel that's probably the easiest yes that's right so that that's not terribly critical for me here the reason I lost it is because that's entire function and I mean this analytic that's correct so I lost it yes yeah yeah exactly yes yes exactly so so yeah modular dt she eyes are precisely dt yeah so I just want to mention first of all that there is such interpretation and then secondly that this this is what we're instructed to do if you are if you witnessed developments in enumerative geometry in late 90s early 2000s so let's try to do this in part we can do this because this expansion in complex transiments theory is really easy it's it's something that's easy to write in particular it's completely controlled for a case of a not complement where I was trying to reconstruct the series f for every not k that depends on x and q it's completely controlled by a polynomial so what is a polynomial and now coming back to relation to transiments here so so on a not complement there's three minus neighborhood of some not it's complex transiments theory is completely determined by what you can call a spectral curve which is defined by equation in two complex variables x and y by a polynomial that depends on a choice of not so that polynomial is called a polynomial very naturally and here x and y belong to c star each so together they belong to c star cross c star which you also mod out by the two symmetry that originates from the while symmetry of fl2c okay so simple example of a polynomial if I can find it since I gave you for example trifle not and so for trifle not this polynomial has a simple form it describes it has factor y minus 1 times y plus x to the 6 and the first factor is due to reducible flat connections on a not complement the second factor is due to irreducible flat connections on not complement and if you're a physicist the first factor in this vanishing locus is Coulomb branch of a theory T of m3 the second factor is the Higgs branch in any case for every not you have a polynomial in two variables and not only it's a nice algebraic equation defining this kind of spectral curve but as often happens in matrix models and other instances where a spectral curve appears in fact in the context of this enumerative invariance you look at a symplectic form dx over x wedge dy over y which is very typical when you have something valued on c star this is typical spectral curve for say trigonometric integrable system and what you do you quantize your symplectic phase space with this symplectic form by replacing x and y by corresponding operators that no longer commute in other words you consider a non commutative algebra which in this case carries a name of quantum torus or exponentiated while algebra is generated by x and y such that x y is q times y x so sometimes you can add hats in order to emphasize that you passed from commutative algebra to non commutative algebra as a result every element of this algebra in particular any polynomial in x and y gets replaced by some operator or some combination in terms of these x hats and y hats and the statement that's easy to show in complex trans diamonds theory is that this perturbative partition function is in fact annihilated by this operator so if I apply a hat of x hat and y hat to this expression you should get zero so exponential of 1 over h sum over n as b of x times h bar to the n should be zero well oh yeah sorry sorry yes that's right so is it correct oh yeah I yes I was trying to over correct myself yes thank you so much yeah so in particular it allows you to say that in this expression the leading term is going that multiplies this one over h bar is going to be of the form integral of log y dx over x and determining recursively all their sub leading terms for for for this h bar expansion so this is again sometimes called perturbative expansion of complex trans diamonds theory so this is SL2C trans diamonds on not complement as 3 minus the knot and now as I mentioned earlier since we're talking about manifold which is non-compact in particular it has a boundary so it has this boundary torus you have to fix boundary condition and here boundary condition is fixed by saying that halonymy of the meridian is equal to matrix in terms of x so this variable x that appears here is on which this this partition function depends is eigenvalue of the halonymy around meridian so using this recursive relation yeah yeah so in this story in general it's a good question if you have random equation that doesn't come from anything I don't know the answer actually that's a good question here there are many techniques one which I really like and which is most efficient is topological recursion so in this case it basically determines the ordering and at least gives the right one so that that's that's a good question exactly yeah so what's important I should have put it in orange because that Q is the same Q as we use all the time and in fact another thing that I put should put in big letters is that here Q is always exponential of h bar so since now I keep writing things on a blackboard that depends on both h bar which are perturbative and Q which is actual variable inside unit disk I should always emphasize this relation I think that's that's important okay so let's see if I can get back the screen okay very good I don't need this so now determining this as sms for not complements is a fairly straightforward in fact when I was 20 years younger we did this project with Don and using this data of a polynomial which is very explicit you can fit it in solve recursion relations and that's what we did so this is the paper called exact results but only in perturbative terms simons meaning that you can determine this as ends to any order you want but only perturbatively meaning that you get just h bar expansion for not complements yeah there was a question what's the correct but this is this is first term this is semi classical as I'm writing this as zero so this is a zero term what I'm trying to say is that it's one over h bar term in this expansion but then I'm not writing as zero I mean as one as two and so on so this oh why in terms of x you for this particular integral what you do you integrate over some contour gamma which belongs to the curve a of x y equals zero meaning that you substitute you solve y as a function of x so this I should write this that's a perhaps good question I write y as a function of x dx over x but then I have to choose a root of the solution I have to choose a branch so that's precisely where a choice of this be that I suppressed for most of the discussion comes in for instance here you can see that for trefoil not I have two branches I have one choice of y and another choice of why for figure eight for example I'll have three branches and in general you may have lots of them so the picture will look like a ramified cover of the x plane where you have bunch of branches so you'll have some kind of picture which looks like this involving say this abelian or reducible branch and then various branches called geometric conjugate geometric because it corresponds to hyperbolic geometry where I sell to see connection as identified with geometric hyperbolic structure if you're not complement as hyperbolic and so on so I mean you have lots of choices and that's where the choice of be is going to be important we'll talk about this in a second but in general what you do is you solve this equation if x y equals zero in as a function for y in terms of x so this is y of x well yeah so it'll come to this so be always morally is what you should remember be tries to label whether we're working on a closed manifold or not complement be labels the choice of flat connection but among flat connections some are really special and I'm gonna to come come to this in a second namely abelian once that we already saw earlier that are parameterized by h1 by homology they play a special role exactly in this picture so far it looks like be can be anything that's a very good point so it looks like be can can be abelian non abelian anything you want yes absolutely so at this point there is no connection to spin c structures but it will emerge in a second any any other questions so what I want to do is to determine so the goal is to determine this function fk of x and q by effectively doing the same thing as relates gromo-fritten invariance and Donaldson thomas invariance that we discussed earlier by writing this h bar expansion as a q expansion so let's let's try to play this game but first I want to say that this h bar expansion is easy to determine so here in very ancient work that don't be try to determine it for figure eight not complement very explicitly so this is a picture of figure eight not and also the five to not for which I was writing Alexander polynomial earlier so we go through so let's see where is where is the computation yeah this is the same expansion and slightly different notations so the condition is that it's annihilated by this a hat and then for the case of figure eight where we have these three branches as an illustrated on a blackboard you can solve for s s n's you is so m m is x so or rather I think x is equal m square in my notation but it doesn't matter so you can see you can determine it up to six loops or maybe even further I think seven eight eight loops so at lunch we were discussing whether one can do computations and QCD up to two loops and that's already a challenge so this is eight loops and anyway there is nothing stopping you from doing this up to 80 loops and so on so anyway you get something but the point is we were very proud of this calculation back then in old days but we didn't know what to do with this so it looks like okay it was a cool formula but so what it's very impressive that you can go to to this high water but it doesn't have anything nice in particular it doesn't have integer coefficients nothing interesting because see you have all this kind of fractions involving denominators and since we're talking about this series where you should think about this as an as gram of written variance of course it's not surprising that the rational I mean in this numerative problem they are gonna fit and variance but now we know what to do we should take the same series and re expand that and try to write the answer in terms of q rather than h bar so this is basically what we try to do with cheaply and we took this kind of expansion and if you have expansion for a function in terms of h bar and x first thing you can do you can just exponentiate it open the exponent so this is what the answer looks like where I try to do this first for the trefoil knot with which we played a little bit earlier and what I'm writing here for you is a result of taking function that we're looking for for the trefoil knot this will be three one function of x and q which we know is of the form fm of q x to the m over two minus x to the minus m over two and I'm trying to re express this as a function involving h bar expansion so now this is a series when q is exponential of h bar this is a series involving something some other coefficient cm of h bar times this x m over two minus minus m over two nothing happens to x okay so therefore in order to reconstruct q dependence what we want to do is to look at a given power of x because nothing happened to x so let's do that let's look at the given power of x in this formula for example picks x equals to one half and what do we see about h bar expansion multiplying x to the one half so we have a term x to the one half h bar times x to the one half h square over 2 x to the one half h cube over 6 x to the one half and so on does it remind you of anything exponential very good so term multiplying x to the one half is basically e to the h bar but e to the h bar is q so what we determine is that in this expansion from by looking at power x to the one half which is m equals 1 f 1 of q is is just q itself okay what about the next one so we don't have any power x to the three halves anywhere right or for that matter just x itself so therefore we learn that f 2 is equal to 0 f 3 is equal to 0 and next non trivial one in fact comes a 4 is also equal to 0 and next non trivial one comes when m is equal to 5 doesn't remind you anything we did discuss Alexander polynomial before and somehow Alexander polynomial knows about the structure already we already start seeing 1 5 next look 7 is going to be next choice so next question is what is f 5 as a function of q this is a little trick here huh well I believe q squared is the correct answer the only question is with which sign I'm working I'm always I mean what's the coefficient and yeah it's minus yeah and so on you can easily continue it doesn't take too much work to see that in general what we're discovering is basically the general formula that in this case f m of q is equal to epsilon m where epsilon is that choice of plus minus sign that we saw before depending on what 12 multiplying q to the power m square minus 1 over 24 for example if m is equal to 1 this this gives 0 or maybe plus 1 everything times times q I guess if m is 5 this is 25 minus 1 over 24 that works and you can check that it works in general oh that's that's a good point so now let me come back to this so here in doing this exercise I actually did something interesting I chose this a billion branch you can try to do this game play play this exercise of converting this expansion for for every branch so in this previous paper that I showed you we have geometric branch then goes a billion branch somewhere that's the one I am working with and there is a conjugate one so you can try to do this for any choice of b which so far was generic choice but something special happens when the branch is a billion claim is that this magic that I just showed you happen will own only when you had choose a billion branch and for that guy the leading term just finishes so why it happens only for a billion branch I'm not going to discuss in detail but the reasons are in work with Marino and put proof that's that's explained there but interesting point is that this magic does happen and you can see how you can reconstruct this q dependence very easily by by by this kind of analysis it's easy in this case and for all other tourist knots for more general double twist knots you need to use slightly more general machinery of so-called resurgent analysis but it basically does exactly what what we just try to do so now let me give you an example so using this technology I showed you several properties of the series for not compliments how it's related to Alexander polynomial as you can see that allows us to get a very good approximation to what these numbers are in fact if you send q equals to one in this formula you can easily see that we obtain the previous answer as we should if you do the same thing for figure eight not it's kind of messier but somewhat similar you can go through this machinery and obtain same coefficients FM for for figure eight not and now comes the key point so I have to tell you so so this is analogous H bar expansion for figure eight in this this one is for figure eight yeah it's still a billion branch so for example some of the things are easy to see for instance coefficient of x to the half is just identity because there is no x to the half multiplied by powers of each bar so it's coefficient just a constant independent of q but then you get analogous structure q multiplying higher power starting with five two five halfs and so on and so on so you're not gonna get q series with integer powers into your coefficients you can do it in fact you can do this Borel reclamation that's behind the scene but what you're gonna get is a function of q but in for example for conjugate branch is going to give you a tecumular tq of t but it's not gonna and that can be computed independently in many other ways but it's not a q series with integer coefficients so that that part is gonna fail so there is gonna be nothing enumerative about well since you came late I actually talked a little bit about it but but so in particular I mentioned that since Fafa is involved to him it's part of the same package but it's slightly different form of integrality it's it's in particular it's it's independent not as far as I know I think there are in the same general theme but but independent as far as I can see okay yes yes in this case it does in fact I can give you a first few coefficients so for figure eight since Don asked so I'll erase the answer for well actually here is the general answer for Truffwell you can save that but next example is for example figure eight capital case for one in this case what do you get do I have the answer you have f1 equals 1 so that's kind of clear from the formula f I'll write just non-trivial ones f3 is equal to 2 that's probably also but next one f5 already starts being interesting it's Q inverse plus 3 plus Q and I encourage you to play with expansion of inverse Alexander polynomial and verify from the previous property that I gave you that you get coefficients one to next one is going to be five so this looks like Fibonacci numbers that's seven well some there is relation to Fibonacci numbers which in view of time I'll probably suppress but anyway you can determine this guys it's it's very straightforward very easy so the basic point is that if you want to play with this there are many structural properties which allow you to quickly bootstrap what the answer is first look at Alexander polynomial that will give you classical version then if you want to restore Q coefficients look at something like this Q difference equation involving a polynomial or several other relations that exist for for this guy yeah these are earlier characteristics of some graded homology spaces associated to three manifolds I don't think the refinement will be topological invariant in fact question big question in this field that's why integrality is important is to even find this vector spaces first of all we build the homology so cheaper in motivation is actually in category fine this this whole thing so that's why integrality is important and that that's morally the space of BPS states that that's the home yeah exactly but this homology will be the homology of that space of BPS no no no that's that's what no that's in fact even in the previous case oh sorry that's right so yeah no no that it's true they're all positive yeah but anyway still even though this is true I don't want to you to be so excited about it because it's supposed to be an index so there is indeed homology or Hilbert space of BPS states behind it and that should be right invariant but in any case the point is what sorry that's the same question that's that's basically a same question so q is epsilon one equals minus epsilon two in this emulative story if you want to introduce t that's refinement or categorification that would be two epsilon's or two parameters so many people are now interested in thinking about it that's a great question it's perfect problem for your next research paper I have maybe five ten minutes since there are quite a few questions I'm slightly slower so I hope it's okay so one thing I want to tell you is kind of the punchline so where we heading is the following I told you in the first lecture that any three many fold can be built by surgeries so that's one way to build it and develop tqft where you can do cutting and gluing are all along total boundaries second thing I told you it's important to know the basic building blocks analogues of these that had for knots and links so I told you some of the properties related to to rife torsion Alexander polynomial and things like that so now I have to put it together and give you the final answer which I already said in words I want to put it on the blackboard the gluing formula or the surgery formula if you have this fk how do you get an variant of a closed manifold very roughly you substitute this fk inside the integral that we had earlier in the first part in the first lecture but if you do just a surgery on a single not on the link with a single component I can give you much more direct formula so you take this result for fk now this is could be general k and do this so so far we worked on many fold with open boundary and what we're trying to do now is to cap it off so glue back this solid torus that we removed when we're doing the surgery operation so now we're gonna put it back so next point and the last one for today is the surgery surgery formula in other words how to go from fk to to that had of a closed manifold and I'll give you a general formula but then illustrate it in one example which will be specific and based on two notes that we discussed at most trifle and figure 8 so if you want this for closed manifold which is obtained by p over r surgery on a not k not surprisingly you have to do is you have to do the integral and take your fk which depends on x and q and you had to get rid of x in the same way we had to get rid of x in the first place since the result just depends on q so you integrate on a unit circle that part is not surprising so you write here dx over 2 pi i x you write fk of x and q and you multiply it by theta function which I'm going to write explicitly in a second but you also multiply it by x to the actually I'll just give you expression in terms of the function so this will be some over values of n such that our times n is congruent to a mod p so this is the index a here and you write q to the r over p n squared x to the n so that's a complete closed form formula for surgery on a not k now using this formulae and given that we already worked out what fk is for the trifle and for figure 8 not you can do the following example so in fact formulae are here on the board you can say let me compute what this answer is for four four three manifold which is plus one surgery on figure eight not so p is equal to one and r is equal to one on that formula so you see that we already computed f n's and then you can in the paper which you printed give complete list but they're easy to work out so we already saw several of them and you go ahead you do your calculation and if I can find answer I'll give you the answer in this case there is only one choice of label because there is only one's can see structure on this manifold so I'm not even gonna bother with this label a which should be not not there be important but the answer you get is the following it's q to the one half sum over n great or equal than zero minus one to the n q to the n n plus one over two divided by kappa Heimer symbol of the form q to the n plus one q and okay so you get something so you substitute function that was function of x and q that we determined you integrate out x and naturally you get a q-series so it has all the desired properties as promised it starts with q to the one half which is this correction term in here floor homology then has integer coefficients if you expand it as a q-series and all the statements about relation to WRT other things are correct but now you can play the following game you can say aha same manifold actually can be obtained using Kirby moves as a surgery on a trifle not so remember that they're doing to QFT that means that we allow to build three manifolds and many different ways starting with different knots and links so here if you start with a trifle not it's actually minus one surgery and using similar formulae that apply here you actually get the same answer so that's the first and on trivial check that it's a topological invariant you do completely different calculations but they produce exactly the same answer of course as they should if this is a full TQFT so now there is a second surprise and then I'll stop that this q-series is actually character of the log of the OA and in the log of the OA there are several families which are constructed so far and their characters worked out one of the families is called 1, p series of singlet log the ways and it happens that this is precisely a character of so-called 1, p singlet for value of p which is 42 so when I gave the stock a couple of weeks ago in Bonnet the four manifold conference somebody said that 42 is the answer to this question about meaning of life and he swikers a guide to the galaxy so I'll stop here that's the answer to everything thank you