 the following link of un-nodes with framings A1, A2, A3 and so on. Then the result will be LPQ which is defined similarly and so the coolest relation now is modified a little. So for that one the phase shift is the same 2 pi divided by P and for Z2 it should be minus 2 pi i Q divided by P Z2 and so where P and Q are related to the framings A1, A2, A3 and so on by the following continuous fraction decomposition. Example 3. If I take a dense surgery on left-handed trifoil with framing 1 the result is a quotient of a 3 divided by gamma where gamma is what is called gamma is a certain subgroup of SU2 which is in turn subgroup of SU4 which acts on the standards S3 embedded in R4 and which is called binary Iqo binary Iqo sub-hedral and for those familiar with the classification of subgroup of finding subgroups in SU2 this is a which is given by EAD this is a subgroup which corresponds to E8 in AD classification and this manifold is also known as Poincare homology questions. Well there is nice correspondence between dinking diagrams and links and links can be well as I mentioned in the moment this representation by surgery is not unique but there is nice for example correspondence between dinking diagrams and certain links essentially you draw a link like with unnot which format is dinking. So I think this is left-handed so if I go this way it rotates sorry minus sorry minus well this yeah minus thanks of course you can see the right-hand side is plus one but this will be like opposite orientation thanks okay any other questions so now there are two important statements so we obtained the map from framed links from frame links to closed oriented manifolds and of course here we will probably want to consider this up to ambidezalopy and here we consider up to homomorphisms so the question of course if this map is subjective can we obtain all three manifolds in this way and if the other question is kind of I can understand in a sense a kernel of this map and there is a precise answer to both of these questions so first there is theorem by Likaish and Wolff so any closed oriented three manifolds can be obtained by the insertor is actually a stronger version of the statement so I can actually make only consider links of surface certain of some specific type it's it's you but let me not don't go into this so this is actually claims that this map is is subjective so it's kind of another different well and another theorem which is well kind of the original statement is due to Kirby and but I will use a version which is modified version by fan rook so s3 is the surgery on link L is homomorphic to a surgery on link some other link L prime if and only if L and L prime can be related by sequence of the following moves I have a link which locally looks like this have a bunch of strands coming through some link components and a particular link component let's say L and this will be something like li 1 li 2 and so on of course some of them may coincide because the link can go to like a few times so this thing and so if so the framing on these are not surrounding bunch of other link components is plus or minus one and the framings here let's say the AI 1 AI 2 and so on then this this link can be replaced by the following here I make a minus plus one full twist so for example the plus one twist means that I take the whole bunch and rotate it by 360 degrees in a positive direction and and then identify with it and the we also want to change the framings and the framings I changed according to the following rule so AI I will say n prime is AI n plus minus plus the linking number of the link component with index I n with the link lm is a link component lm squared okay so this gives exactly what is the equivalence relation on the links of the frame links which gives the same three month so in the problem said there will be example which had to do to relate for example to relate this this particular this particular surgery on this particular frame link to a different to a couple of other links using this moves any questions no there is no we remove we remove completely remove this thing so the point is that you if you have if you have a if you have a link component this framing plus or minus one we can remove it by doing this modification this modification affects the other link components which which not really linked so this so the link component you want to remove so let me give you some example for example if I was this so if I have something like let's say plus two plus three and I have here plus one I mean suppose this is a different different components so when I do this move well when I do this move I replace it by I want to go into negative this thing with framings plus one last well I take you can imagine so here I kind of free suppose locally I can I can always assume that those all components align the same plane and then I rotate this plane whenever on this on the same line then when I go in this direction I rotate this line by 360 degrees and then I glue so that here I I do the full rotation so this is an example any other questions very bad with with orientation but I think this is a plus one yeah I don't know okay so what what do we learn from here so one thing we can learn so for example the invariance of framed links which also invariant under so this move were usually called Kirby move or sometimes Fennroth moves or Kirby Fennroth moves so the original moves were raised by Kirby was a bit different in the key removes provide invariant of three monthfuls so this is a say and this is essentially the fact which is used into construction of WRT invariant so so let so I will use the same notation for the link as before so let and so I will denote so I can see the color Jones polynomial associated for link to link so in my normalization to be Laurent polynomial in Q and Q inverse sorry in Q one half and Q minus one half so that so again this labels mention the integers they mention the integers so in principle we want a color link by representation of reservation for SL 2 and these integer numbers correspond to dimension of representations of SL 2 and in my normalization I will use normalization so that the Jones polynomial the current Jones polynomial for unframed unframed not is just the quantum number n so here we can see that we can see the Jones polynomials of color Jones polynomial of framed link and but the dependence on the color so the dependence on framings is very simple so the here L is assumed is assumed the frame the frame links with framings as before a 1 and 2 a L and I can simply relate it to to to Jones polynomial of unframed link there will be just a following the product of following simple factors which correspond to each frame each framing and here I have the link with zero framings so L prime link with zero framings so where is this here it says framings times so there's a product time G the color Jones polynomials of L prime which is the same links with framings zero and so let me also introduce the what is called Lincoln matrix which is a Lincoln so for off diagonal elements it just it's just a Lincoln number between components L i and L j so this is L by L matrix and for diagonal the diagonal components are framings and so let me let me denote by V plus V minus and V 1 this is a number numbers of respectively positive negative zero eigenvalues of M okay now I'm ready to to say what is a WRT invariant so WRT invariant of three manifold which is homomorphic so for any three manifold I can find a dense surgery as they always exist L such as m3 is a dense surgery on L is a so you can be sort as a as a map from a set of positive integers well this is not most generic definition but let me define it so for just like this two complex numbers and it's given me specificity by the following formula so for each color I do a summation from 1 to k minus 1 then I take a product of from I to L of quantum numbers and I and I colors and then I multiply it by this color Jones polynomial the framed link and then there is a normalization so first well there are different ways to normalize it but I will use the following so first I do the same thing where I take so instead of my link L I take a link which is consists of B plus number of unnot this framings plus one and B minus number of unnot this framing minus one and I also want to introduce the following another normalization factor which will depend on B1 which is so I just take a sum of from from 1 to k minus 1 of a quantum number and squared and take it to the power of B1 divided by 2 yes yes so and I specify this whole thing so specified at q equals e to the i divided by my key so let me write it here and so the theorem just by Chetikhin to arrive and it's also worth mentioning to kind of a bit of alternative approach by Kirby Mellon is that so this formula star so it's well as the environment is well defined so star is environment under Kirby Kirby moves so I think yeah my time is out so I have to stop here questions comments you see in this final expression you are you have two things written for q so I didn't I didn't get the final thing what what is q supposed to be it's supposed to be a root of unity it's a case it's a particular case primitive root of unity e exponential of 2 pi divided by k okay and this specify that so this is a so the John's polynomial is a Laurent polynomial in q well in q half and so I need to say essentially what I want to so to be precise I want to specify q half to exponential to pi i divided by k well this is a didn't have time to mention this area wanted to mention this but in this lecture but maybe I'm much so the motivation is coming from the victims realization of the same environment so this was ill-defined but it was a what was defined as a path integral but here there should be some a particular there should be a kind of a ticketing structure the particular gluing properties so essentially what I mean this formula essentially realizes the surgery procedure which I mentioned above on the TFT level so we so we kind of we some over some of our states on on it on a toy which surround which surrounding components well well this is a this is a serum which I didn't prove but I mean the basic idea is that first you can see if you just introduce like I'm not without with not to link with anything this frame is passing minus one this is this corresponds to actually just surgery on this guy is just three sphere so this is just correspond to taken connected some with three sphere which doesn't modify multiple but now after that you can actually see if there are some other link you can try to see what happens if you can have slide it around and you see this result in some twisting but I didn't I didn't prove it yes well let me clarify a little bit so well this is a sum I mean this sum is there actually L what I mean here the L sums so for each say well let me let me clarify so again we have some sort of link some frame link maybe some disconnected components and so for each link component I introduce a color so and eyes are colors so they go from one to K minus one in this form and so I want to sum over colors colors so my car by allowing them only go from one to K minus one so this one is again in the number corresponds to the dimension of the representation of SL to which I color the component well this he means all all for yes but I mean it's not literally I mean there are some some subtleties in that case it's again a question about the formula you are in the denominator the first sum is going over I but you have their N and be I but we didn't define what is be I we have just three different bees B plus B minus and the B for zero in the first time in the denominator oh sorry no the first in the first bracket in the first so this is B1 yeah but you have there the sum over I and I don't understand why we have I sorry this is M yeah this is just and the single M thank you okay and yes so here this was B plus plus be mine so one part is essentially yeah one point easy one part of the meter you do essentially the same thing as here but you replace this the link with B plus I'm not with framing plus one and B minus I'm not with frame minus one and then there also this simple factor which is the sum of quantum numbers N squared from one to K minus one and power is B1 divided by two well we should probably be wrapping up so you can ask a little more question