 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