Okay, okay, so perhaps after the talk or between the two talks or after Jake's talk, which is the last one There's a sign-up sheet for people going to the airports either in Trieste or in Venice So that would be you know to organize taxis and so So very happy to introduce Claudius Dubrobius who's going to tell us Yeah, thank you very much. Thank you to the organizers for inviting me. It's an exceptional honor to be Able to have the opportunity to present my research at this awesome conference I want to tell you a story about two multi-curve invariants one multi-curve invariant for Not flow homology and another multi-curve invariant in the context of Kavanaugh homology Both of these multi-curve invariants are structurally identical First of all, they're both invariants of conway tangles. Here's an example of a conway tangle Conway tangle consists of two pieces of string entangled inside Three ball case such that the four endpoints lie on four fixed points on the boundary sphere Okay, so we're looking at proper embeddings of two intervals into the closed three ball okay, and the second thing about these multi-curve invariants is that they're both one-dimensional objects on the boundary three ball Containing the tangle Okay, so the one-dimensional objects on the two-dimensional boundary of our three-dimensional object of our tangles Okay, let me give you an example. Okay, so this is the two twist tangle. Okay, so it's As this diagram here Okay, and let's look at first at the figure flow multi-curve invariant Okay, which for the purpose of this talk, I will denote by gamma Hfk of the tangled T and With this tangle this multi-curve invariant associates well one-dimensional object on on a four punctured C on this four punctured C here And that invariant looks like looks like this is this Consist of a single component which is embedded Let me also give you another example What these look like what these what this multi-curve invariant looks like more generally so here's a more general Conway tangle. I should hold it like this Okay, so there's a non-rational tangle. Okay, it can't be Twisted to to the trivial tangle. Okay, and for this particular tangle Let me also draw the diagram for this tangle Let's see. Okay. No, I wanted to hold it like this. I think okay Okay, so it's this time tangle diagram and the multi-curve invariant on the Here go for the side Hfk of this tangle consists of three components Okay, one component looks Like this a wraps around these two tangle ends, right? I remind you You should think of this four punctured sphere really as being identified with The boundary of the three ball containing the tangle. Okay, so these are really natural invariants so the second component of This tangle invariant looks almost identical except that it now wraps around These two Tangle ends Okay, and then there's one third component Okay, which looks as follows. It's it looks almost Identical to that one. It's also embedded and it looks like this So that's the invariant on the noffler homology side Now on the commander for mology side There we have another invariant there we have The invariant gamma kh that's my notation for the purpose of this talk again and this Well, there's what one one one input that it work where the structure is perhaps slightly different We also need to fix fix the base point We need to Distinguish one tangle end from all the others in my little models here. I can do this by attaching this little felt piece to this tangle end, so now we have a pointed tangle and Once we've done this then well should also Distinguish this tangle end from the others from the others and now I can draw The multi-curve invariant for this one again. We only have a single component and It almost looks The same as this one except that it's now a figure eight curve. Okay, it has a self intersection point Okay, let's Okay, what about this other tangle? Okay, we're now in the Kavanaugh setting, so I should Fix my base point and now The They're now two invariant two components of the invariant cage one of them Looks well almost like this one again. So this green component. So that looks like So again figure eight curve. Hey, which is Again the same same as this one except there's now a twist here And then there's one other component instead of these these two And that looks a bit more complicated Some curve that wraps around These two tangle ends and again, I remind you This four punctured sphere should be identified with this With the boundary of the tangle So this is identified with a distinguished tangle end. Okay, and now let me finish this curve Connect this one two examples of what these multi-curve Invariants look like. Okay, so In summary with a four with a four in the tangle with a concave tangle in both cases We associate a collection of curves on A four punctured sphere namely the four punctured sphere that is the boundary of the three ball containing the tangle minus the four Tangle ends Now what makes this invariant really useful is the fact that both? Yeah, you have bounding co-chains I haven't said anything about focaya categories yet, but I'll come to that later. Yeah Okay other questions Okay, so what makes it makes these invariants really useful is the fact that they both Satisfy gluing theorems which allow you to recover not flow homology and on this side covenants of mology. How does this work? Well, I can take take two con rate angles and glue them together. Okay, so for instance I could I could close up this this con this convey tangle to get the trefoil knot. Okay, how can I do this for instance? I could do This okay, then I have the crossing here and then look that would be Trefoil not I can also do this with my models Okay, let me see if I can figure this out. So connect opposite tangle ends. Okay, and I have to be careful how I connect these two Does that work? No No, what? Okay, I think it works like this Okay, so again, we have we now have the trefoil knot okay, and so we glued to convey tangles together and Well, we know what the invariant looks like For the blue tangle now what need to tell you what the invariant looks like for the red tangle and that's that's very simple It's very simple. Namely the invariant looks like this. It's this red curve So again an embedded curve on this side and on the Kovanov side. It also looks yeah almost identical so there again, we introduce This point of self intersection this figure eight curve. Okay now we can State the gluing theorem in both cases Which says that? Can I draw here and everyone read? Yeah, okay, so it says that the knot flow homology of the union of two tangles and we call them T prime and T so we take the union of these two tangles T prime and T Insert with the two-dimensional vector space, so I'm working over characteristic to here This is isomorphic to the Lagrangian flow homology of the two curves namely the curve rather than the mirror of the curve for Okay, sorry, perfect T prime and then the curve the curve for T so this This HF here. This is Lagrangian flow homology of these two curves Okay, so now let me say something about the fukairi category So from a symplectic point of view what we're doing we associate with these concave tangles we associate objects in the fukairi category of the four-punk shape sphere and The morphine spaces in this fukairi category They compute knot flow homology or on the Kovanov side They compute command of homology and Statement here is structurally exactly the same So if you want to compute the command of the reduced command of homology of the union of two tangles T prime and T and then We can just compute the Lagrangian flow homology Lagrangian flow homology of the two curves Now we take the Kovanov curves, of course any questions up to this point. No, this could also be a link and This could also be a link Yeah, and then in the case okay in the case of the link You would drop this factor then then we could link the flow homology. Yeah, good question Yes, you can you can make this work without with gradings, but I'm going to not talk about gradings But there are by gradings. Yeah, I can make this work with by grading So with relative by gradings on the non-flow homology side and with absolute right by gratings on the command of side Okay, maybe I should say a few words about the history of these invariants. So this is an invariant that grew out of my PhD thesis and it is basically a geometric interpretation of a bordered sutured invariant or alternatively you can also view this as a geometric interpretation of multi-pointed Higa flow homology Okay, so I should mention a couple of other names here. So this uses one for that Higa flow homology. So Lipschitz Robert Lipschitz. Is it here? Peter Oshradd and Dylan Thurston and I'm using a version called bordered sutured Higa flow homology and that's due to Roman Zareff from around 2009 and this is earlier I think maybe 06 Okay, so that's that's on the Higa flow side and on the covenants side this invariant in the form that I'm going to tell you about this is due to myself and my co-authors Artyom Kretelsky and Liam Watson Which we defined a couple of years ago and this uses there should also mention a couple of other names because this is essentially just a geometric interpretation of Algebraic again algebraic invariants that you can define in Kovanov homology. So there we're using work of Von Atten his general framework, I think it's 04 and This this project was inspired by work of Heiden Harold Chris Harold Nathogenkamp and Paul Kirk from 2018 Okay, so they defined a functor from Banat and Skobort in category to the Foucaille category and We then yeah simplified and extended that construction and also importantly proved this gluing theorem Which allows you to which makes this theory you really useful. So what why is this? Yeah This is reduced Kovanov homology. Yes. Yeah You can also get un-reduced Kovanov homology and there's also a version that I will mention later That recovers reduced Banat and homology So why why would you be interested in studying these Conway tangles? Well, for example, we could now we can now understand thanks to these two theories We can now understand how they behave under crossing changes. So for example, I could do something like this Okay, and Well, all I have to do. Okay, so I'm doing a crossing change. So here's a Basic crossing. Let me do this in the Kovanov sitting. So here's a basic crossing So what we had before I think was this this crossing now to get to the other crossing Well, I just rotate this by 90 do 180 degrees around this axis Okay, so well because these curves are natural with respect to them mapping class group action They really live on the boundary of These three balls. All I have to do is well change these red curves. So here I had six intersection points Okay, and now I rotate this Maybe I use yellow for this for the new curve And now rotate this and I obtain this curve here Which only has these two intersection points Okay, maybe I'm not a mod of the of the unnot Stabilized ones I can do exactly the same on this side. So there I rotate This curve and again, I only get two intersection points. Okay, and I get The Kovanov model you see are not one stabilized. Yes. Oh No question. Okay questions Okay, so, um, okay, so in principle we can now understand so how these How these two theories change under for instance a crossing change or other tango replacements But to really, yeah, um understand what's going on We need to better understand what these multi-curve invariants look like. Okay properties of these invariants and For this it's often useful to consider these curves not in these four punctured spheres But rather to consider lifts to a certain covering space. So we now lift These curves Same question as before. Um, yeah, and then there are no bounding co-chains in in here Ah, sorry, the question was are there any bounding co-chains in this picture and the answer is no Is there a deep reason for that? I'm not sure. I mean it says that so there's a classification result that tells you that So the the algebraic objects That these theories correspond to they're classified by these by these objects in the fukai category and The worst that can happen is that they you get local systems Okay, but I'll address that that issue later, but otherwise you just get curves and that's it They're bounding co-chains Okay, so now we look at lift these curves to certain covering space We lift these these curves along along the map Which this is to to a planar space So this is this covering map which you can think of as the two-fold branch cover of the four punctured sphere Franch of the four points that gives you a torus with four marked points And then you take the universal cover of the of the torus and that gives you this checkerboard coloring and So downstairs Well, you have maybe The front the front is shaded. Okay, the back is still black I guess okay, and the front is lifted to these pieces and the back to these squares Okay, now now we did those curves and let's do this for for these particular examples So I'll claim that if you lift those two blue curves on the higafluor side you get This curve here, so that's one of them and the other one. Well, it's just translated Looks similar. Okay, and the green curve Green curve lifts to a straight line of slope two Okay, where do I draw this Straight line of slope two Okay That looks simple and we can do the same that looks certainly simpler than then this picture I hope and we can do the same thing on on this side and there Well, I've drawn this covering space just zoomed out a bit by a factor of two I said everything fits on the board and there the curve Well, we just have a single blue curve. Okay, and there the curve looks like this Okay, that's one of them and then there's also a Curve of slope of slope two That corresponds to this green green curve, but this one is more complicated it Days in the neighborhood of this curve of slope two, but it kind of winds around this curve Stop two now the classification result that I want to tell you about is the following first of the theorem that I proved 2019 about the the Higo flow multi-curve invariance and it says that Every component every component of Gamma h of k of t k for any tangle t and is one of the Following paths basically two types of curves and we see both types of curves in in this picture namely we see one type of curve That wraps around tangle ends and all these points are tangle ends downstairs Okay, so there's one one curve that wraps around these tangle ends And we call these a component special components. So this one Let's denote this by s to infinity. Okay, so infinity means the slope of this curve Is infinity right we can pull this straight okay to line of slope infinity and have length length to okay That's the index here And let's call this s plus and then similarly here. There's a curve Just looks just like the first But it wraps around the other two tangle ends. So let's call this s minus and we call the union of these two Just s s for we call this union of s to and One part of this classification theorem is that these These curves always come in pairs. Okay, and if you add the grading that always come in conjugate pairs The Alexander grading is is reversed Okay, so that's the first family of curves. So There we have some slope t over q in this case. It's infinity. Okay, but p p over q can be any slope There's some p over q in qp one And then we have then we have a length. Okay, and this length is always a multiple of Four so these front come in families. Okay, but we can stretch stretch those curves that's the first type of curve and the second type of curve is This curve here this which is this looks like a straight line Okay, and these are called so-called rational curves This one is a rational curve of length one and slope two Okay, so there is the slope this length and R stands for rational Yeah, what about the length? Yeah, just okay. So you could imagine Pulling sort of stretching these curves So instead of wrapping around these two tangle ends, they stay around these two tangle ends for longer on the right-hand side So maybe four four tangle ends of six Yes, yeah, that's what the length tells you and these rational curves. They always have length one Okay, so you might get the index Okay, and then there's rational curves First on some slow p of a q qp1 Okay, and here you might see some local systems Okay, so it's in local systems. You can think of them as matrices decorations of of these curves by matrices That are that are invertible Okay, but for the purpose of this talk you you can ignore these local systems Because one reason is because they have never shown up in this setting any other questions about this So you can you repeat the question the bi-gradings You can see some of the gradings more naturally in the covering space So the cover I was in in both cases the bi-gradings relative bi-gradings can be computed by looking at Domains of yeah between intersection points If you look at the pairing picture and then counting how many times did you cross the tangle ends? So it's essentially yeah, so the motto is you can do essentially he got flow homology, but on the four-pointed sphere That's that's the short motto and now the the theorem on the Kovanov side Looks almost identical So we proved this A year ago and yeah, it's almost identical so every component of this curve Now we look at the Kovanov reduce Kovanov curve Is one of the following okay, we have the special curves Okay, so this is this blue curve here is an instance of a special curve Special curve, but this is in the Kovanov setting. This should be h of k Again a curve of slope infinity and then some slope name some length namely length four In general we could also have shorter lengths. So here the length is only multiple of two some slope p of a q in qp1 and and in Some length okay, so that's the type of curve and then there's also again rational components And these can these rational components they can in principle have longer length again for P of a q some slope and Some length Okay, and this would be an instance of a rational curve so a Kovanov rational curve of sorry Of length one and again we see slope slope two so yeah Already I mean how many how many of you were surprised by this picture? Me okay, but I was surprised and there you should really really be I was really surprised to see This this phenomenon maybe we've already gotten used to this fact that if you see one interesting phenomenon on the not very modest side, then you should also go looking for it on the Kovanov side and vice versa, and yeah, maybe we've gotten used to this but I Would argue that we shouldn't at least I'd like to see a conceptual explanation for for why why you see the similarity Because the proofs of these I mean the construction of both invariants is really different and But also the proofs of these two classification results come from very different directions Okay, so one stays in turn or two you go flow homology and the other one uses homological mirror symmetry so really Very different Okay, any any questions Now I have a recipe to go from You mean from from Northland modality to to food from Kovanov to Yeah, so that would be amazing. So that would be a first start of to look a lot Give a local construction of Nathan Darwin's vector sequence. I would be great. I don't have that so Okay, so now we understand these components of these multi-curve invariants So if we now want to understand how well how these two invariants behave under Say crossing change we really need to understand So how in what combinations these components can show up as the as invariants of actual tangles Okay, so we need to find out some some properties of these curves Okay, but maybe that's a that's a really I guess that's a really hard question So maybe let's first look at the at the components themselves. Do they have any meaning? Right, I mean for example, we see these rational components these rational components are identical to the invariants of rational tangles Okay, and we know that So one one can show that every tangle. So it's not a coincidence that this tangle here In both settings contains a component that looks like the invariant of a rational tangle Okay, that's that's true in general. So that's that's kind of remarkable So what what is the meaning of this rational component? For instance, um? They are so in the example that I raised for this tangle and there is this bijection Okay, but it can fail for more complicated examples. So the examples where yeah, the rational components have different slope for example Or where the number of? Special components is different in both settings. So, yeah, there's not an obvious bijection I mean that would be that would be amazing, but no, that's not true. Okay, I'm so and many 30 minutes or so I want to study one particular simple class of Convay tangles that you construct very easily. So here we the construction starts with a knot. Okay, and we cut it open to get a 1-1 tangle and then we take a second piece of string and we tie it into this into this blue Tangle strand certain such a way that the two strands stay parallel throughout Okay, so we just double and this tangle this time these tangles have very nice properties For instance, if I close off like this if I take these closures, what do I get? Yeah, I'm not okay. I can yeah, okay. Don't have to prove this. Okay, so here. It's the I'm not Okay, so Well, let's draw a diagram of this Okay, so we have So that's the The one one tangle that I start with And now I tie in a second component Okay, which stays stays parallel throughout and I can correct the framing I can choose So this tangle now depends on the kind of tangle diagram Okay, but so we can compensate that by adding some some number of twists up here. Let's do that six twists Okay, and we can call this this tangle the double tangle tk for This instance the trefoil Okay, that gives us a construction for any not in the not table we get some conmate tangle and we can ask Okay, well, what is the multi-curve invariant for this? and this corollary Proposition that you can easily prove from this property that you have this this unlawed closure is that That these that the multi-curve invariance for these tangles have a very simple form Okay, there's exactly one rational component and all the other components are special components Hey, so for any not K in S3 and Star in either H of K or Kvanov So gamma gamma star of Tk Consane contains exactly one rational component of slope one and then some integer slope M Okay, so there exists one some integer M Okay, which is actually always even and then all the other components are special components So maybe s I one and all of those special components have slope infinity So that follows from is next easy exercise from the fact that you have this this unlawed closure You can still say this This is just general so for cap zero tang of cap trivial tangles As Liam likes to call them Yeah, there's not special to this particular family, but yeah other questions So this is this true in both settings and now we can try to understand. What is what is the slope? What is the slope here and of this rational component? Okay, can we so now get our hands on this and the answer is yes, at least in the Here I've learned setting and it's called this a proposition Proposition Which I should have proved years ago because there's a really easy computation. So this says that and for any So gamma Tk H of K Is equal to? So we have this rational component are one and it has slope four times tau of K and then special components Okay, and this this proposition it really is a Just a computation of a certain border tutored by module That allows you to extract to extract this tau invariant From a certain third multicart invariant that I want to mention here. So sketch proof. So this uses Uses a multi curve invariant gamma hf for three manifolds With With boundary Equal to the torus and this this multicart invariant is due to Handsome and Jake Rasmussen and beam what can this is a multicart invariant and that sits again like like These examples we have a three manifold with torus boundary and the invariant is a one-dimensional object on the two-dimensional boundary on this torus In fact once punctured torus of this three manifold Okay, and this computation tells you exactly how to extract this invariant here for Tk from the invariant of the not complement Okay, a very simple exercise So now the next question is what about and that's all I'm going to say about the proof. Sorry So now the next question is what about what about the Kovanov sitting? Okay, and there Lucas Leibach and I The following theorem which says that Gamma Tk in the Kovanov sitting is Equal to well, okay, we can have this rational component Recording to our proposition over there And here the slope is two times theta two times theta of K k plus special curves and this theta 2 of K is a Concordance homomorphism or induces a concordance homomorphism That is independent is linearly independent of The Rasmussen invariant. Yes, and I'm again working over characteristic to here Okay, how do we prove this really sketch proof and so there we use a fourth multi-curve invariant In the the one for Manhattan homology, which associates with the conveyor tangle again Curve on the four punctured sphere, and it is very closely related to this Kovanov curve invariant And this is again due to my co-authors and I And this this what this allows you to do is it allows you to relate the slope of this rational component to The behavior of the Rasmussen invariant under satellite operations of winding number zero and wrapping number two And then using that connection you can you can show that it is a concordance homomorphism This slope any questions up to this point Say again Yes, yeah, so there's so you can set this up over any characteristic the only sort of the only thing that we don't know at the moment is Whether these are also concordance homomorphisms We expect that they they are But at the moment we only get this result of a characteristic two because these multi-curve invariants are best understood of a characteristic two Sorry, look, sorry. I should have repeated the question The question was whether this theta invariant also works about other characteristics. No Okay, sorry No, the Rasmussen invariant is over f mode Zmod two coefficients in characteristic two Yeah, but I mean the Rasmussen invariance you can define them over any characteristic. Sorry. Yes So in both cases, we're yeah work over characteristic two Yeah, that's an open question. We don't know that we'd like to but We don't know whether that should be expected. Yeah That's an interesting question. Yeah, I Don't know we haven't thought about this So you'd be thinking about the What would be the the not for the homology? What would be the K in this case? Okay, maybe we should talk about this. So maybe I should wrap up So I want to end with with three open questions. So the first question is What about tangles with more endpoints, right? So here I've only considered these tangles with with four endpoints Which is the simplest non-trivial case Okay, and I try to to make there that the theory look as simple as possible Okay. So now the question is well, what about tangles with more endpoints? Can we perhaps also? Prove some classification results like these right? I mean both settings both not for homology and kavan of homology We have algebraic invariance, right? We have on the not for homology side. We have these boarded sutured theories We have the algebraic invariant algebraic not for the homology So can can we yeah also prove These similar classification results in that setting as well and similarly here on the banatan Kavan of side specifically, I'd be interested if there is an analog of rational Components for these tangles with more endpoints because this would really have some fantastic applications That's the first question second question. What about so I talked about I mentioned this banatan invariant at the very end Which is the a covariant version of this kavan of homology, okay? So there we have both a hat flavor and a minus flavor What about the higafluor side right there? We have we have a tangle invariant that That works for that recovers in the hat flavor of not for homology. What about a minus h of k? Multi-curve invariant for tangles, okay, and then the third curve the third question which is The the biggest question of Yeah, most most important the one that I care most about is why I'm still surprised by this picture Why why do these two invariants look so similar? So if someone has a conceptual explanation for this I'd be really interested To know that thank you. Do I'm pillowcase? Yeah, okay. So the question is do I know? how these two multi-curve invariants are related to pillowcase homology and The short answer is no there are some interesting computations that Poor Kirk has has done in the pillowcase case and These computer in there are certain similarities especially between the instant sorry the pillowcase curve and especially this banatan So it seems that there's yeah, if you I mean In examples you can always kind of tweak the curves and maybe resolve the crossing here And then then you get the pillowcase curve, okay, but we don't have any Conceptual I mean again you would be looking for kind of spectro sequence analog But no at the moment we don't have that Okay, but mentioning Paul Kirk so he he recently Two weeks ago was in Regensburg and he suggested one particular family of tangles Now I should compute these multi curves on namely some some tour is not a cut open at some point and that was really really interesting because That gave us the first Example of a multi-curve invariant that has a higher dimensional non-trivial local system on the banatan in the banatan setting so this banatan curve can be as complicated as you We observe all kinds of Pathological behavior that you could could imagine which is in stark contrast to these classification results for the other two Curve invariance I Have not thought about this. I haven't been bold enough to prepare. Yeah. Okay. Maybe I should have been I don't know Jake And The not this invariant But the banatan invariant so sorry I should repeat the question the question was can I compute the S invariant of a Cable of two strand cable Using these multi-curve invariance and the answer is yes from that multi-curve invariant for for Banatan in the banatan setting but not for the for the Curve I maybe I should give an example so what the curve looks like for for example this This tangled here the curve looks as follows you you You have Something like this and then you goes off at so this this part here has a slope two times theta theta of K and Yeah, and then then Let's say it ends up here. So let's say theta of this thing is equal to I know for I guess And then then if you pair this with with some other with some pattern tango you look at Well, you look at compute that curve and it also ends somewhere It does something something funny and then it also ends at this point Okay, and then well to In the banatan setting you have to do wrap the gargantuan flow homology So in that case you well it comes maybe from down here and then you have to wrap around these tangle ends and that gives you an infinite series of intersection points and That gives you the tower the free component in banatan homology and the first intersection point its quantum grading computes the DS invariant That that curve does not fit into that classification. That's what I so this this curve here This is the banatan Curve for these tangles and and this is really yeah for this one. I don't expect to have any Classification at least not as simple. I mean as I said this can even have non higher dimensional non trivial local systems, which Yeah, it's it's really awesome. I don't know. I don't know Yeah, sorry Can I say anything about maps? Board isn't maps. I Guess it would be easier to say things in the Kavanagh. I'd be feel more comfortable saying something about this here But no not really not anything


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So, as we have discussed resonating structure contributors, so here we are going to discuss the relative stability of resonating structure and how do we define the relative stability of RS. So, for that we have certain set of rules and that we are going to discuss, the rule one is the structure having structure having more pi bonds is more stable. The first thing is we will just calculate the number of pi bonds, the structure which has more number of pi bonds is more stable, why is it so, because the formation of bond is always an exothermic process, formation of bond is exothermic image and hence more number of bonds, more energy releases and more will be the stability. So, you see some examples I write on here, this is A and this is B, if you calculate the number of pi bonds, number of pi bonds is more in A and the stability of A is more than to that of B, number of pi bonds in A is 2, but that of in B is 1, so stability of A is more than to that of B. This is A and this is B, here the number of pi bonds is more, so stability of A is less than to that of B. So, like this we just calculate, just we count the number of pi bonds and then according to that we will check the stability. So, the next rule is what happens if in some of the molecule we have equal number of pi bonds, then what happens for that we have rule 2, if the structure has equal pi bonds, stability decreases with chart separation, pi bonds then stability decreases with chart separation. Now, you see some examples on this, initially this is A, this is B, so number of pi bonds here is 3, here is also 3, but here we have more chart separation, so stability will be less, so A is more stable than B. Similarly, equal pi bonds more chart separation, A will be more stable than B, equal pi bonds more chart separation, A is more stable than B. Like this we can find out the priority or the order of stability of this resonating structure. So, these are the two rules we have discussed, the third rule is what happens if the molecule has equal pi bonds and equal chart separation, rule 3, if the molecule has equal pi bonds and equal pi bonds and chart separation. So, in this case what happens, the molecule having iron, molecule having more atom complete octave is more stable, the molecule and iron having more number of atoms, atoms with complete octave is more stable. For example, you see and this is B, so stability of A is more than, stability of B, the stability of A is less than to that of B, because B all the atoms having complete octave, go on as 6 electron here, here it has 8 electrons complete octave. And it is 2, B is 2, you can do this with the help of number of pi bonds also, this one is more stable because it has 1 pi bonds. So, if it is A, this is B, obviously B is more stable than A, this pi bonds also you can do or the help of octet rule also, go on as something like that. Here you see this carbon would not have the complete octet here, but here it has the complete octet, so stability of this B is more than to that of A. These are the 3 rules we have discussed according to the number of pi bonds, according to the chart separation and then the number of atoms having complete octet, okay. Now, what is rule 4? If rule 1, 2 and 3, if rule 1, 2 and 3 are fails, if 1, 2 and 3, rule 1, 2 and 3 fails, then what we can say that the negative charge more electro negative atom is more stable, positive charge on less electro negative atom is more stable, okay. Few examples I will write down here. Now, you can pause the video and solve this question I am explaining it now. First question you see, we have equal pi bonds and same charge separation, 1 pi, 1 pi, negative charge, negative charge, okay. And octet is also coming for all the atoms if you see, okay. So, you see we use rule 4 here which says negative charge on more electro negative atom is more stable. So, oxygen is carbon will have negative charge, oxygen is more electro negative. So, B is more stable than oxygen having negative charge. So, B is more stable than A, right. Here we have negative charge on nitrogen and carbon, this is A, B and C I am assuming, okay. So, first of all the two electro negative atom, the most the top two electro negative atom here we have oxygen first, then nitrogen and then carbon, right. So, which one is more electro negative oxygen, right. So, the negative charge on oxygen and nitrogen is more stable, B is the maximum. Then negative charge on nitrogen and carbon we have oxygen is more stable, then we have C and then it is, here you see negative charge on oxygen is more stable, B, A, B is more stable. There is we can find out the stability order of resonating structure. Rule 5 is positive charge is more stable when it is surrounded by more alkyne group and that is the cause of plus i, okay. And negative charge is more stable, more stable when it is surrounded by lesser number. So, here you see this carbocation is tertiary and this is primary. So, this one is more stable, this carbocation is secondary primary, secondary is more stable, this carbocation is secondary, this carbocation is tertiary, right. So, which one is more stable, secondary is more stable, okay. So, like this we can compare the stability again. Now, rule 7, rule 6, sorry, rule 6 is Fry's rule. What is Fry's rule? Molecule containing more number of benzenoid form is more stable. Benzenoid form is more stable. For example, now these two structures you compare, number of benzenoid form means what if you see this ring, it is a benzene ring, right. And this pi point we can count into this ring also, this is also a benzene ring, right. So, we have two benzenoid form here. Here you see we have only one benzene ring, this is not a benzene ring, right, because the pi point is here. So, it has only one benzenoid form. More number of benzenoid form, more stable, this is more stable than this. This compound we got as naphthalene. Number of benzenoid form here it is two, these two. Number of benzenoid form here it is only one. So, obviously two is more stable than the first one, sorry, this one is more stable than this. Now, rule 7, which is aromaticity, aromaticity, right. When the compound is aromatic it is the most stable compound, ok. The aromaticity provides extra stability to the compound, ok. What we can write aromatic compounds most stable. We will discuss this aromaticity in this resonance session only, ok. In the coming classes we will discuss this aromaticity. But one thing just you remember now, aromatic compounds are cyclic planner conjugated system. Conjugated system with n plus 2 for n plus 2 pi electrons where n is any natural number 0, 1, 2, 3 and so on, ok. Cyclic planner conjugated system with 4n plus 2 pi electrons, ok. For example, you see this molecule, right, draw the molecule here, ok. Or we can also draw this. This is b and this is c. One more thing I will tell you here. There are two more compounds we have. The stability of aromaticity is maximum then we have non-aromatic compound and in the last we have anti-aromatic compounds, ok. So, stability order is this. Non-aromatic compounds if it is non-planet or we have other condition also we will discuss all these things in detail in aromaticity when we discuss this separately, ok. Order is this. Anti-aromatic compounds all conditions are same but it only has 4n pi electron. The only difference is it has 4n pi electrons, ok. It is cyclic planner conjugated system with 4n pi electrons, ok. That is anti-aromaticity. Now if you see this compound here how many pi electrons it has? One lone pair, two, four. It has 4n pi electrons, right. So, for n is equals to 1 it satisfies this rule, right. This rule we call it as Huckel's rule for aromaticity. Huckel's rule for aromaticity. So, compound b is aromatic. Compound c has 4, 2 plus 2, 4 pi electrons, right. So, it is anti-aromatic and this compound is non-aromatic, ok. So, order of stability will be what? First aromatic, then non-aromatic and then anti-aromatic. It should be b most stable, then a and then c least stable, ok. So, with this resonance also there are chances of getting aromaticity. The compounds may become aromatic compounds, ok. According to the structure of the compound you have and the number of pi electrons it has. So, with aromaticity also the stability changes. Aromatic compounds actually are highly stable it provides extra stability to the compounds, ok. Non-aromatic compounds are the normal compounds like at the room temperature, right. Anti-aromatic is highly unstable, right. So, these are the 7 rules we have discussed to assign stability of resonating structure. On the basis of this you can easily say that which one is major contributors and which one is minor contributors. Major contributors are those molecules and ions which has more stability, right. So, if you compare these two, right if you compare this molecule the actual hybrid molecule of this, right will have the maximum property of the most stable compound which is nothing but the aromaticity, ok this b. So, this molecule will provide maximum property to the hybrid structure of this resonating structure, ok. In this two if you compare both gives you a hybrid structure, a resonance hybrid in which these two will provide property like we contributes into the property of the resonating structure. But since the second one is more stable it is major contributors this one is minor contributors, ok. So, this is what the discussion of contributing like the contributors we have is relative stability of contributors or resonating structure.


Question is from Andrew Beth. What is the ideal role that hit and list cardio play in a workout to maximize fat loss and retain muscle? Can you give an idea what a week would look like if incorporating those two forms of cardio? All right, so first is break those two down. So hit and list are acronyms. Hit stands for high intensity interval training. This type of cardiovascular activity is characterized by a short sprint followed by a slow relaxed portion and then a repeated attempt at the sprint. So if you're on a bike, you would go all out for 20 seconds and then maybe a minute and a half kind of slow, get your heart rate back down and then repeat it. List cardio or list stands for low intensity, steady state. This is the traditional form of cardio that most of us are familiar with where you just get on a elliptical, a bike or treadmill and you just cruise for 30 minutes. Now, what are their values? What are their detriments? Hit cardio, you're gonna get better performance. You might retain more muscle from doing it. You could do less of it and burn just as many calories so it's more time efficient. List cardio's benefits, it's easier on the joints, less likely to hurt yourself, less stressful for the body overall. So which one do you do and how much? Depends on who you are. If you are a cortisol junky stress person with low bad sleep, high stressed out job, you're already pushing your body to the limit with resistance training. Go list route. Yeah, go list, it's calming, it's relaxing, it's rejuvenating, go for a long walk, something like that. Do it that route.  Hit cardio, and if you're very timed, if you don't have much time, hit cardio can work very, very well. But it really depends on who you are because if you apply hit on the wrong person or list on the wrong person, you're gonna get terrible results. Yeah, I think too, like, you know, there's benefits and detriments. So it will put a little more wear and tear on your joints, like if you're doing hit cardio quite a bit, but you are stimulating that fast twitch muscle fiber response, which is, you know, simply you're not necessarily gonna get with list cardio. So, you know, just considering that in terms of like performance and overall function, like it's a good caddy to, and that's why you see a lot more athletes utilizing hit training. But it is, if you get a low impact type of equipment, like I use like an assault bike or something like that, where I can actually sprint where it's pretty relatively low impact on my joints, you know, you can actually like max sprint and you can sprint without actually getting that type of detrimental stress on the joints and, you know, it could be something that you can incorporate. But for the most part, it is something to consider, like how well are my joints, you know, reinforced and do I have that kind of stability to do it, you know, versus list will, you know, be another option again for somebody who's like always doing high intensity activities. So I'll give you an example, a generic example of how I use hit and list for somebody who's getting, trying to get shredded, right? So this is different than the recommendation for just general health, how I would utilize hit or list. Normally if I'm utilizing both hit and list with a client, we have a specific goal we're trying to shred down for a show and this is how I would use this. And again, this is generic, but give you an idea. So it's eight weeks left until showtime with a client that we had to present the best physique we possibly can. I've done my due diligence, I've built my client's metabolism up, they're eating a healthy amount of calories. They're stepping or walking moderately, which is probably six to 8,000 steps, maybe 10,000 steps depending on what kind of job they have is what was keeping them at this point. Now we have eight weeks left and we get to get ready for show. And up into this point, I have not used any cardio whatsoever. In fact, while I'm building a client's metabolism, I have them shut down all types of cardio. I don't want them going out of their way to move any extra. I'm trying to build muscle, get their calorie intake up as high as I can before I reverse them down. So now we get ready for the cut. So the very first thing that I'm going to do with that eight week mark is I'm actually going to increase their steps. And I typically increase about 2,000 steps a day, every day for the week.  Now, what that looks like is kind of like lists. I mean lists is low intensity, steady state cardio is pretty much walking around or hiking. It's pretty close to that, right? It's somewhere walking around or hiking outside is right between neat and probably less. And so I'm going to try and get them to just create more activity.  Now I'm going to keep increasing their steps week over week until they start getting to places like 12,000, 16,000 steps. Now they're starting to, now they're looking at clients normally around this point are telling me, okay, I am getting up an extra hour early and kind of walking. I walk the dog an extra time. My spouse and I walk after dinner this time and it's, I'm barely hitting my 14 to 16,000 steps you want me at. I go, okay, now let's introduce hit. So now what I want you to do is every day that you work out post-workout, I want you to do a 12 minute hit session. So after the theory and the idea behind that is that roughly 80% of their glycogen stores has been depleted when they've worked out. Now I'm going to completely deplete that for sure by doing 12 minutes of hit post.  And glycogen is the energy that your body utilizes from carbohydrates. And once you run out of that, then your body burns fat. Right. And so the theory and the idea here is that I make sure I deplete that completely from somebody. So now when they are driving home and preparing their next meal and then even consuming their next meal before that starts to get digested and then converted to glucose, their body is like metabolizing fat to get them there. So that's the idea. So then I introduce, and this is normally like hit doesn't come into play until about weeks, four or five, like say we have four or five weeks. So the first three weeks I'm just increasing steps. Then I use hit post-workout. I do that for probably two weeks. That's normally enough to kick up the, how many calories a day they're burning because we've now included in that. And now I'm going to start to add like hour bouts of like list cardio where I'd say, okay, this is our final four to five weeks before the show. Now you're doing your 12 minutes a hit post-workout. So we deplete the glycogen stores. Now, in addition to that, every other day on week, you know, four weeks out, I'm having you do an hour of lists. And then when it gets to three weeks out, I'm having you do it every day. And then that's kind of what it looks like leading up. You're ramping up to hit is the peak. Yes. So, and I start with hit before I do anything else because it's the less time demanding. It's only 12 minutes. I'm asking him to do that. And then I start to, the very last thing I start to recommend to client when trying to get shredded is the hour long sessions on the treadmill. And that's just because when I reverse them out the other direction, that's the first thing that goes. It's the most time consuming thing. And the rest is more lifestyle and short 12 minute bouts for me to reverse out of. It just seems to be the best strategy that I have found for competing clients and getting shredded. And that's kind of the model that I've used. Yeah. Now, no, here's the, here's how I like to look at cardio. I look at resistance training, lifting weights as the workout. That's when I'm training and pushing my body and working out. I look at cardio as a way to rejuvenate my body as an act is a form of active recovery. Okay. And this is for the average person that I would train. Now, when I would look at him that way, I almost never did hit because hit is more of a workout. It's more like resistance training than the list is than the steady state stuff. So if I have the, if I have the, if the person has the energy and the ability to push in a workout, I'm not going to do hit with them. I'll just do more resistance training. When I have them do the, you know, let's get recovery. Let's do active recovery. Let's rejuvenate the body. I just have them do less. And I like to have it in the form of walking outside or things that seem to feel rejuvenating. Now, if I do combine the two, because the last part of the question is, what would a week look like? I tend to do a two to one ratio. Two list workouts for everyone, hit workout.  Usually what it looks like for the average person who's already fit and doing everything else right is two days of list cardio, one day of hit cardio. So that two to one ratio, because again, I want to minimize that high intensity push because I want to save that for the resistance training. No, and I think you're right to point that. I would totally not like do a regular workout and then hit, even with athletes. Like it was something that we would focus on, now we're in metabolic conditioning. We're conditioning the athlete now to then build up their endurance and then, you know, like emulate it somewhat towards their sport. So if it's like a few seconds of like high intensity bursts and then, you know, slow kind of movement right after that, you know, that's what it looked like. So it wasn't a combination of both.