 OK, so now without further ado, it's my pleasure to introduce the last speaker for this geometry festival, Peter Kronheimer from Harvard. He's going to talk about instanton homology for knots and webs. Thank you. This bangle. I'm very pleased to be here. I'm going to talk about some topics which go back quite a long way, but which have met and interacted rather more recently. Would you speak a little louder? No, that's for the taping. That's just taping. That's fine. I'll try to talk a bit louder. I will come over here, but then get back closer as well. So the two topics of the talk, it's really mostly the first of these that I'm going to be talking about. This is instanton homology as an invariant for knots. I'll also be talking, rather less, about Havana homology. And one of the reasons I'm interested in both of these right now is the way they interact. Instanton homology was originally thought of mostly in the context of the structure group SU2. The one can generalize it to SU where, and I'll talk a little bit about that. Havana homology also has a certain n equals 2 origin. And when one generalizes Havana homology, one obtains, amongst other things, Havana-Rozansky homology. So since both of the items on the first two lines have generalizations, it's natural to look for relationships between instanton homology for SUN and Havana-Rozansky homology. I'll be talking a bit about that too. And this is joint work with Tom Wufger at MIT. Before going on to the next slide, perhaps I wanted to say a tiny bit about instanton homology and how it first appeared in the work of Andreas Fleur. So the starting point here is a close-oriented smooth three manifold Y. And over it a vector bundle, which I think I might suppose is the trivial bundle Cn times Y with the standard Hermitian inner product on the fibers. And then one looks at unitary connections in the fixed vector bundle over Y. So all unitary connections, these form an affine space, infundential affine space on which there's the churn-Symons functional, CS for churn-Symons function on the space of connections with real values. My A is gradually turned into a U area. So the key properties of the churn-Symons functional are critical points, not exactly the flat connections. So the flat connections are rated the representations of the fundamental group in the unitary group. If you introduce a Romanian metric on the three manifold Y, then there's a L2 inner product on the tangent space, this affine space of connections. And then you can consider the formal gradient of the churn-Symons functional and the formal gradient flow for a path of connections A of T. The gradient flow equations, the formal gradient of churn-Symons is the matrix-valued one form, which is the hodge star of the two-form curvature of the connection A. So this is the formal gradient flow. And flow's construction of instant on homology rests on a key observation, which is the following. If I've got a one parameter family of connections on the three manifold Y, then you may look at it as defining a four-dimensional connection, a bundle, connection in the bundle over the cylinder R times Y. So if you think of A of T, this part of connections, instead of defining a single connection A on the cylinder, this equation, equation dagger, tells us that this connection A has an anti-self-dual curvature. So the curvature two-form in four dimensions. Being a two-form, you can split it up the way the two-forms on a four-manifold, manual four-manifold split up into the self-dual and anti-self-dual two-forms. And anti-self-dual curvature is the condition that the self-dual part, F plus, is 0. So the construction of this formal gradient flow and its relation to the representations of the fundamental group of Y is the germ of Flur's instanton Flur homology. And through this observation, one sees that it's related to something which is the anti-self-dual Yang-Mills equations, solutions of which, either on a four-manifold or on R4, often called instantons and this tied Flur's work closely with the then quite recent work of Simon Dahlsen in using instantons to study the topology of smooth four-manifolds. So oops, either 0 or 1 instant. Right, that's this time I haven't skipped. So here the situation is a three-manifold Y. I'm going to talk about an invariant of knots and links, which may lie in a three-manifold, but will usually, for me, lie in R3. This is a variant of the Flur homology, which goes back here to 1986, Flur's work. What is this instanton homology going to be? It's going to be a finitely generated convenient group for the unknotted circle in R3. It'll be Z plus Z. For the trefoil, I've written down the answer. It's four copies of Z and a Z mod 2. One key property that this instanton homology for knots is going to have is, here at the bottom, this unoriented skein relation. So you're supposed to imagine three different knots and links, which outside of a single ball are all the same, but inside that ball, one of them looks like this, like here, and the other two are obtained by this modification. So this is the unoriented skein relation between three knots. That's a bit more symmetric in three dimensions than it looks if you, instead of thinking of that two-dimensional picture there, you think of a three-dimensional ball and think of a tetrahedron there with its six edges. So there are three opposite pairs of edges. If you focus on this opposite pair of edges, or this one, or this one, you obtain three different pairs of arcs inside the free ball. That's another way of looking at what's going on here. So as well as defining this flow homology for three manifolds in 1986, Fleur also defined an instanton homology for knots, one of his last papers in about 1989. Our definition is slightly different from the one that Fleur used, but it actually gives the same answer for knots. So this really is an invariant of knots, which can be seen to go back quite a long way. In Fleur's work, he pioneered looking at exact sequences coming from skein relations in this context. They're not exactly this unoriented skein relation here. Although Fleur's work on Fleur homology has been very influential, many big developments, this particular aspect of it, this instanton homology for knots I think was not a lot developed. There's an exposition of some of those ideas from the first last papers by Peter Bram and Simon Donelson, which appeared in the Fleur Memorial volume in 1995. But very little else has been written based on this 1989 paper. So how do we define instanton-fleur homology for knots? So it starts with the knot group. I sort of define it in a rather backwards way compared to the sketch I gave here for the original instanton-fleur homology case. It starts with the knot group pi, the fundamental group with the knot complement, in which there's a distinguished element or distinguished conjugacy class, at least if it's a knot rather than a link. That's the original conjugacy class. You choose a base point for your pi1. Choose a path from the base point to the knot. Just go around that linking circle and go back. Different paths will give you a different element of the conjugacy class, but I'll write it as if it's a distinguished element here. And I want to look at homomorphisms from the knot group into, in this case, SU2. And I'm going to put the following constraint on these, that this written element or every element in that conjugacy class should map under row to this element of SU2 or something conjugate to it. That tilde sign means conjugate 2 for me here. And I want to write curly r of k as the set of all such homomorphisms, row. That's the starting point of what I want to do. So let's see what that actually does for the unknot k. So the knot group pi1 and the complement is a z for an unknoted circle k. The meridial curve is a generator for that z. So a homomorphism row from pi1 to SU2 is just determined by where m goes. And we've insisted that m go into the conjugacy class of i0, 0 minus i. So this is a little sketch of the 3-sphere, which is SU2. i0, 0 minus i lies on the equatorial 2-sphere. And its conjugacy class is that 2-sphere. So to specify a homomorphism is just to specify where m goes. It goes somewhere on this 2-sphere. So r of k is just a copy of that 2-sphere. So that's this representation variety for this unknot. For the trefoil, it has a non-Abelian knot group now. There are some Abelian representations, those that factor through the h1, which is z. That's, again, this 2-sphere. But there are also some irreducible representations, which are parameterized by a copy of SO3. If you want to think concretely about that, you can think of the presentation of the knot group by the wording of presentation, which is a standard way to write it. For the trefoil, the wording of presentation has three generators, x, y, z. You write it down. This and the three cyclic rotations of this relation are these are the relations of the knot group for the trefoil knot. And each of x, y, and z is an element of the original conjugacy class. So each of those has to go to some point in the 2-sphere. So if you're looking for elements of curly r, you're looking for three points in the 2-sphere. This relation says that if I reflect y through the point x, I'll get the point z. And either x, y, and z can all be the same point. There's a 2-sphere's worth of those. Or x, y, and z can lie on an equilateral triangle on a great circle. And there's a SO3's worth of those. So they're actually very easy to sort of see geometrically these representations. Normally when you talk about representation, the linear representation of a group, what you really mean in terms of homomorphisms is that that would be homomorphisms to the general linear group modulo conjugation by the linear group. But that's not what I'm doing right now. I was talking about just the raw set of homomorphisms not dividing by conjugation. Because if I'm looking at flat connections up to isomorphism, I more naturally get homomorphisms modulo conjugation by the linear group. The reason r of k is going to play a role for us rather than r of k divided by conjugation is the following study ad hoc looking, and indeed I think is rather ad hoc device. So given a not or link k, I'll write k-sharp for the following gadget, it's k, but then disjoint union is some remote ball, a hopflink h, the two-component link there. What's the horizontal bar? So the horizontal bar is an extra, an added extra for this hopflink. What I want, I'll talk more generally about trivalent graphs shortly, but this hopflink has actually been decorated as an extra bar, so it's actually becomes a trivalent graph embedded in three-space. And I want to look at representations of the complement of this augmented guy, which as well as having this condition on the meridian of k that they get mapped to something conjugate to this, I'll make the same insistence for the meridional curves on the hopflink. But I'll ask that this meridian, the meridian of the bar, mapped to this element minus one, zero, minus one, this non-trivial central element of SU2. The reason for choosing in that particular gadget is that there's a unique, essentially up to conjugation, a unique solution on that hopflink to that problem, which if you think of the Caternians i, j, k as elements of SU2, up to conjugation, this is m1 has to be i, m2 has to be j, the commutator is minus one, and that's this element around the bar. So because there's a unique irreducible solution here on this hopflink part, and because pi1 here is just the free product of these two guys, if I look at the original representation variety of k, without dividing by conjugation, that's the same as looking at this more complicated representation variety and dividing by conjugation. So I've enlarged my k to some more complicated gadget. The complicated gadget divided by conjugation is just the original guy without conjugation. So that's, I'll really just be talking about this guy, but the reason it's actually relevant is that we're really doing something a little bit more complicated. So I can think now as representation variety as something a little bit more natural is the space of representation's modular conjugation of this other gadget. So I can now think of it as a modular space of flat SU2 connections on the complement in S3 of this link k together with this hopflink gadget. Flat connections up to isomorphism. So I can now think of that as critical points of this Chern-Simons functional on a certain space of connections. If I take the connections modulo-automorphisms modulo-gauge, I'll write curly B for the connections modulo-gauge. The Chern-Simons functional is real valued on the space of connections, but on the connections modulo-gauge, it's not single value anymore, so it takes values should be normalized in R over Z. So now this representation variety curly R of k has begun to come into the realm of the setup used by Fleur. So I want to define now instant homology of the knot k, and I want to define it as the morse homology of this functional on the space of connections following Fleur's constructions. Technically to do this, after all, I'm looking at the complement of a knot in a three-manifold. A convenient way to do this technically is to view the three-sphere with this embedded knot or this trivalent graph, view it as an orbifold with cone angle pi, so that instead of two pi, the cone angle is pi along these edges, the edges big, but the knot k, the knot or link in S3. And I can view the space of connections as a space of orbifold connections in a bundle with structurally SU2 or SU2 over plus or minus one. So this is the picture of the knot k. It's in a three-dimensional space, but in the normal directions, it's a cone with cone angle pi. And the holonomy around this single locus for the SU2 connection I'm considering will be I00 minus I. If I pass then to the, locally to the smooth branch double cover branch along k, the holonomy will be minus one. So as a SU2 over plus or minus one connection as an SO3 connection, it's that there's no apparent singularity there. It's a sort of orbifold SO3 connection. Does that work for the singular point on them? It does. I'll actually talk about the trivalent graph again later. So now we're all set to sort of follow the outline of Fleur's constructions. We can think of the gradient of the Chern-Simons functional using an L2 in a product. Using orbifold, Romanian metric on this three-dimensional space. Well, here it is. So this guy, although it's a trivalent graph, this thing is also naturally an orbifold. If I take the three-dimensional ball with the three coordinate axes passing through it and divide by the client four group. So the group generated by these matrices, then the quotient is, again, homeomorphic to a ball. But it's a homeomorphic to a ball containing three rays. So that's the appropriate orbifold picture for the trivalent graph. So the critical points of the Chern-Simons functional are exactly our representation variety. And the gradient flow lines are sort of orbifold instantons, orbifold anti-self-tual connections on this cylinder. So I didn't talk much about, I began this talk, I talked about the Chern-Simons functional in the context of Fleur's original constructions. But the outline of what one is supposed to do after that, although the details will vary every time, but the plan is to define a homology theory as a Morse homology. So the standard steps are to perturb the Chern-Simons functional. So the Chern-Simons functional has as its critical points this representation of a R of K, which may be positive dimensional. But after perturbation, the critical points will be isolated and satisfy a Morse condition. I missed what you said when you explained what the tilde over R of K be. I didn't say actually, but nor does it say on the transparency. But the tilde R, I think, must be my notation for the perturbed representation variety. So R of K is the critical points of the Chern-Simons functional. The Chern-Simons functional is then perturbed a little bit, add a small perturbing term, which I won't really discuss. And then there's a new set of critical points, which is now a finite set as in Morse theory. And although there isn't a gradient flow defined by this functional, flow's idea is to construct a chain complex as one would in Morse theory, whose generators correspond to the critical points and whose boundary map counts is defined by counting flow lines, solutions of this equation, which are asymptotic one end to one critical point and at the other to a different critical point that defines a boundary map on this chain complex. Now a particular example that the definition of I sharp of K is that it's the homology of such a chain complex. Some very simple examples, it can turn out that there are no relevant parts of differential, no flow lines between different components of the original representation variety. It may be in some situations sort of perfect Morse function situation, that what this homology group ends up computing is exactly the ordinary homology of this representation variety. That's exactly what happens for the unknot, for example, we saw that the representation variety for the unknot was a copy of the two-sphere. Well it turns out that the instant on homology of the unknot can be thought of as just the homology, ordinary homology of that two-sphere. It's a copy of Z plus Z. For the trefoil knot, the representation variety was a two-sphere and then a copy of SO3, these were the reducible, these are the irreducible representations. The ordinary homology of S2, this joint union SO3 is a Z4 and a Z mod 2. And that is indeed the instant on homology of the trefoil knot. I haven't talked about the gradings here, and of course the ordinary homology is a graded group. The instant on homology as we've defined it is not graded. There is a grading, a cyclic grading by Z mod 4, but not an integer grading. And this isomorphism here, you wouldn't expect it to preserve the grading, you'd expect some shifts, like as you would even in a familiar finite dimensional morse theory. So this is a slide you've seen before repeated here. We've talked about instant on homology. I haven't talked about why there is this unoriented skein relation. That's an interesting story. It's closely related to a surgery exact triangle which Fleur proved for the three-manifold instant on homology that he defined in 1986. But I'll explain briefly this invariant of knots and links, the fact that it's more or less by construction of finitely generated Abelian group, and we saw briefly why the unknot and the trefol give you these two. So Tom Moffko and I were playing around and we wrote this definition down quite a few years ago now and we computed these two examples. And then there's something which is quite striking. There's something else which is gonna feature in this talk now, which is Havana homology. It was defined at the turn of the millennium by Mikhail Kovanov. It's an invariant of knots and links in R3. Can't be generalized in any easy way to knots and links in general three-manifolders, R3 or S3. Like instant on homology, it's a finitely generated Abelian group. And it so happens that for the unknot, it's Z plus C and for the trefol, it's Z to the fourth plus Z mod two. And more or less from its definition and construction, I'm not gonna talk much about the definition of Havana homology, but this unoriented skein relation is satisfied by Havana homology. If you have three knots related by this unoriented skein relation, then there's a long exact sequence relating their Havana homologies. So this was sort of defined by Havana off in 1999 or so. And by the time Tom Rufka and I were trying to play around with instant on homology and do these calculations, the Havana homology of the trefoil was I think well known to people who knew anything about Havana homology that didn't include us at the time. The similarity is very striking and it's particularly striking in that these instant on homology and Havana off whoops homology would have such different origins. So Havana homology grew from the Jones polynomial which was defined in 1984. It comes from the world of quantum algebra. The definition of Havana homology is very algebraic. You can present it to smart undergraduates quite easily. Instant homology is defined at a similar time to the Jones polynomial. There's two big developments in the 1980s defined using gauge theory through connections. And although there were some terms in common, for example, the Jones polynomial was considered by Whitton as coming from the Chern-Simons functional. Instant homology and the Jones polynomial were really rather separate things for a long time. So the fact that they turn out to be so closely related is a relatively recent discovery. So the fact that they're the same for the trefoil and the unknot is a consequence of a particular relationship here. This is the existence of a spectral sequence. The spectral sequence which starts with Havana homology is its E2 term and which abuts to the instant homology that we've been discussing. The reason that they're the same for the trefoil is that nothing happens in the spectral sequence after the E2 term. I'm not gonna define Havana homology in detail, but given a planar diagram of a knot or link with n crossings, then you can smooth out these crossings, they're n of them, each one can be smoothed out in one of two ways, like so, like so. And you'll obtain two to the n different circles, smoothings of this knot diagram. So each smoothing will just be a bunch of unlinked circles in the plane. So for those of you that have sort of seen Havana homology either firsthand or presented in a talk before, what one does, a bit of algebra to compute Havana homology is you draw the n dimensional cube with its two to the n vertices here, n is three. At each vertex, you put a vector space determined in a simple way from this bunch of circles. And then there's a differential, you make this into a chain complex, you have a differential whose components run along the edges of the cube in my picture starting here and ending here. It turns out that you can compute instant homology from exactly a similar picture. It's just that the instant homology has some additional differentials which will run not just along the edges, but also along the diagonals and great diagonals. So from that point of view, Havana off, excuse me, differential on the same complex as some D one and for the instant on case, we've got a D one. In fact, it's only the odd diagonals and the next one in that picture would be the great diagonal D three and then there'd be a five dimensional great diagonal D five and so on. So there's a larger differential on the same complex which computes instant on homology. So a consequence of the existence of this spectral sequence is as an application that Havana homology detects the unknot. That is to say, you can tell whether or not it is modded by computing the Havana homology. If you get Z plus Z, it's the unknot and if you don't, it's not. This is an attractive application because Havana homology is such a sort of basic simple thing to compute. I mean, it's a very ungeometric environment. It's an open question whether the Jones polynomial detects the unknot. Some sense, this is a related question, but I don't expect that it's a useful step in the direction of that unsolved problem. The existence of this spectral sequence and do that cube picture, it's really, it's an idea related to a similar idea of Osvat and Sabo who considered the Hagar homology of branched double covers and really what's going on there, but the core, mathematical core of it is the fact that instant homology satisfies this skein exact sequence. It's like generalization, that picture's really what makes that work. So why does that imply that the Havana homology detects the unknot? So from the spectral sequence it immediately follows that the rank of the Havana homology is at least as big as the rank of the instant homology. An earlier work shows that the instant homology detects the unknot. The rank is at least two always and strictly greater than two for non-trivial knots. That in turn comes down to really an interesting state about instant homology. It's, you can think of it as, it's a corollary really, that if you have a non-trivial knot then the representation variety is not gonna just be this two sphere. It must contain some other stuff. There must be some irreducible representations. The reason one can prove something like this for instant homology is because it's really a three-dimensional thing. You can put it in an arbitrary three-manifold. You can cut your three-manifold along incompressible surfaces. You can glue your three-manifolds. You have all this flexibility of three-dimensional topology. In particular, a notion of sutured manifold decompositions which are introduced to topology by Gabai and were used in the context of Haigard for homology first by Kajini and then Ni and Yuhaj. So it's this extra flexibility of instant homology that allows you to do this. We defined our representation variety using SU2 connections and we asked that the marital element be mapped to the equatorial two-sphere, the conditional class of I00 minus I in SU2. I could have defined it and it would have been more equivalent to ask that the marital element map to this element of U2. So multiplied by I. Think of the representation variety defined in a slightly different but equivalent way. Having done that, you can see one of many possible ways to generalize this story. Suppose I want to replace N equals two by larger N. I think of this as the essentially only interesting involution in U2. Let's look at the involutions in UN. So that's a bunch of minus ones, say K of them and a bunch of plus ones. So let's write sigma sub K for that involution in UN. It's a reflection in a N minus K plane. So now I need a not K in R3 and a choice of weight lowcase K, somewhere between one and N minus one. And let's look for representations of the not group in UN mapping the marital element to this chosen involution. So we've got a choice of K there. Previously that would have been N equals two and K equals one. So there's a representation variety we can consider here just as we did in the previous case. It turns out it's profitable to generalize this picture a little bit to trivalent graphs rather than embedded curves, which would be the not K. So suppose I have a trivalent graph and a weight on each edge. So here there's a single weight on the not K. We've got now a graph with edges and a different weight K for each edge. So then you're looking at a representation variety which was also recently considered by Andrew Lobb and Raphael Zetner. I've drawn a sort of example here. So this is a trivalent graph. I've associated a weight to every edge. Not arbitrarily, at every trivalent vertex here there are three weights K1, K2, K3. And in some ordering I haven't written down the condition. In some ordering I've always arranged that K1 plus K2 is equal to K3. So, although there may not be a global orientation if you look at it locally as a sort of flow that the flow is preserved. If the K1 and K2 are incoming the total flow K3 is outgoing. So one plus one is two here and two plus three is five at the top vertex there. And let's look at representations of pi one of the complement. I wonder if I didn't miss a, I think I've lost a transparency at some point. I want to look at representations of the complement pi again into UN such that each meridional curve there's a meridian for each edge, m sub e for the edge e should map to an involution whose negative eigenspace has got dimension K sub e. And if you look at these conditions K1 plus K2 is equal to K3. I've got a block of size K1. If I put the block of size K2 with its minus ones here and multiply those then I'll get a block of minus ones of size K3. So that's the model that's going to happen near each trivalent vertex that act on the fact that for some conjugate of sigma K2, sigma K1 times that conjugate of sigma K2 is sigma K3. I'm writing gamma here for such a decorated trivalent graph. K is a trivalent graph, lowercase K underlined as its assignment of weights. Again there's a gadget I can put out to this bit like the Hopfling of before. So I can think of actually r of gamma as representation's modulo conjugation on some slightly larger object. I can think of this again as a modulized base of flat SUN connections in some orbifold context as critical points of an orbifold of a Chern-Simons functional. And there's really very little difficulty just generalizing the previous constructions to this i sharp n of gamma. So more some ology again of the Chern-Simons functional. So trivalent graphs appeared in the generalization of Havanov homology earlier. They appeared in the work of Havanov and Rosansky on their Havanov-Rosansky homology of knots and links. They appeared, for example, in a version of the unoriented skein sequence. If I have three knots or links related in this unoriented skein way, for Havanov-Rosansky's generalization of Havanov homology, it's no longer true that there's a skein relation related, a long exact sequence relating the homologies of these guys. And if you assign weights one, one here and one, one there. There is, however, a long exact sequence relating three guys and knot or link here and knot or link here. But the last guy is a trivalent graph with a edge of multiple, a wedge of weight two there. So trivalent graphs were a natural thing to look for in the SUN situation. So then it's natural to try and compute these things in the very simplest cases for the unknot. The representation variety is going to be the conjugacy class of this involution, which is, for example, if K equals one, it's a CpN minus one. For larger K, it's going to be a grass manion. The instant homology will now turn out to be Z to the n. Previously, it was Z2, Z plus Z. What about the trough oil? The same way we looked at the representation variety of the trough oil before, we can analyze what the representation variety is, in this case, in a very similar way. If I give it weight one, I again get the abelian, the reducible representations, a copy of CpN minus one. The irreducibles, up to conjugation, it's again a unique irreducible representation, just coming from the two-dimensional representation. Moving it round by conjugation, though there's this, that S is the sphere bundle, the sphere bundle inside the tangent bundle to CpN minus one. So the unit tangent vectors to complex projective space. Well, what is the instant homology then for the trough oil? So our first guess and second and third guess are quite what we expect to see is the ordinary homology of this representation variety, as in the case n equals two. But there's a surprise which happens here. The instant homology, generalized away from n equals two, you expect to see the same patterns. But for n bigger than two, it turns out the instant non-homology of the trough oil is just z to the n, exactly the same as for the unnot. And in fact, for any not k, if I take gamma to be k decorated with weight one, the instant non-homology just turns out to be z to the n. So that's a puzzle and a potential disappointment, I'm going to return to shortly. I've mentioned Havana-Frozansky homology. We still expect I sharp of n to be related to Havana-Frozansky homology. SU2 led to the Jones polynomial, led to Havana homology. If SUn, in its n dimensional representation, there's a corresponding quantum invariant which leads to Havana-Frozansky homology. So it's a very natural thing to look for here. There are many similarities between these. There's, again, a skein sequence for both of them, a long exact sequence relating to the homologies of those three knots. Both are functorial for these interesting gadgets called foams. And the Havana-Frozansky homology, these trivalent graphs are called webs. Foams are certain singular co-boardisms between trivalent graphs. You can see, I tried to sketch an example of a foam here. The interesting point is there's one in the middle here where there are six two-dimensional facets meaning at that central vertex. I expect both to give the same answer for planar webs, that's trivalent graphs decorated with arbitrary integers by lying in the plane. But that's a conjectural statement still. Foams, by the way, are overfolds too, just in one dimension higher. If I have a foreball enacted on by the diagonal matrices of determinant plus one with minus ones and ones there, it's a group of order eight. The quotient is, again, topologically homomorphically a foreball, but it's got these strato of non-trivial stabilizers. And what that foreball is, it's a cone. Take the one skeleton of a tetrahedron in the free sphere and then take the cone on that in the foreball. So the tetrahedron has six edges. You have these six two-dimensional facets in the ball. For planar webs, I expect the answer to be related to, well, the answer, Havana-Ferozansky homology, which is a vector space whose dimension is what's the number of so-called moi states. There's a very simple combinatorial thing you can do. Looking at the representations of pi one of the complement is rather tricky in un. But if you just look at representations where everything maps to this group of diagonal elements with plus or minus ones on the diagonal, then it's just a simple combinatorial map problem of the number of minus ones has to equal the weight and at each vertex the minus ones have to join up correctly. So a combinatorial thing, you don't have to talk about un at all, but those combinatorial things are the moi states, certain assignments to the edges. So moi states are the fixed points of a torus action, diagonal matrices. So this representation value, it's Euler characteristic as the number of moi states. This is a result that Lob and Zettner proved independently. I will just spend a few moments at the end just talking about why something changes for n bigger than 2. Perhaps I'll only try and say this briefly. So if I told you that some proper Morse function on some non-compact space had as critical points, a single point and then a circle here and I asked you what you thought the Morse homology was going to be. So you wouldn't be able to tell me because you wouldn't know what the differentials were, what the gradient flow lines looked like. You might guess it's a perfect Morse function and you just get the homology of the circle and the homology of the point. But you might also imagine that there's a function of this shape with a single critical point and then also a circle's worth of critical points. In which case there'd be a flow line here and in fact the homology would just be Z. When you look at what I told you the representations of Vardy was for SUn and even for SU2, I told you the representation of Vardy was a copy of Cpn-1 and then a copy of the unit sphere bundle in the tangent space of Cpn-1. Now that's very much the picture you'd expect if you had a function on the tangent space of Cpn-1 which on the vector spaces, each tangent space at a point had this shape that had a critical point at the origin and then a critical set on the unit sphere in that vector space. So rather than get the homology of this guy plus the homology of that guy, it's quite natural just to get the homology of this guy alone as the Morse homology. So if you think that's what's going on then the question is not why for larger n is the homology so simple. The question more is why is it for n equals 2 that there's more than one? Why aren't we just getting Z plus Z in the n equals 2 case? So let's think about the n equals 2 case. This point here is supposed to think of that as being the 2-sphere. Think of this circle here as the SO3. It's the unit circle bundle in the tangent space of the 2-sphere. The whole space here is the tangent space of the 2-sphere and its homology is Z plus Z. So why is that not the answer for the instant homology for SU2? The answer is that for SU2 and SU2 alone this trajectory has a companion which you can't really see in this finite-dimensional analog but there are actually two cancelling trajectories here one of which is a rather non-trivial path of connections other which is a path of connections which you'd expect to see here in this finite-dimensional picture. What happens for n equals 2 which doesn't happen for n equals 3 is a symmetry. If I have a connection on the complement of the knot whose monodromy around this little circle is I0, 0 minus I I can multiply it by minus 1 here. So if I think of a flat bundle E over the complement of the knot I can change it to C tensor E with the original flat connection on E but on C, C is a line bundle and its homology is minus 1 on the link of the knot. This is a slightly non-trivial operation on flat SU2 connections of the sort we're considering. It doesn't apply for any larger n if I take minus 1, 1, 1, 1 and multiply it by minus 1 I haven't got something which is conjugate to this guy that these guys are playing a different role there's a symmetry here for n equals 2 which isn't present in the other cases. This was a little bit more about that situation. So I want to just end with this last thought. So the instant homology I've been talking about just comes out to be Z to the n is when the weight associated with the knot K is just 1, the simplest case. For other weights I think this guy might be something in some sense more interesting. The particular case which I think is interesting to look at is the case where p equals n over 2 which is somewhat much closer to the SU2 case from the point of this symmetry. Again, multiplying by minus 1 will map this guy to something conjugate to it at the size of the minus 1 and plus 1 blocks are the same. And that's the story of a relationship between instanton and Havana homology. Two things which trace their origins back to the 1980s and which came together about five years ago. What's the level of complexity in computing the Havana homology for a knot with n cross 6? Does it make sense to ask that? You mean in the computational sense. That's an interesting question because there's more than exponential in theory in the number of crossings n. There's the amount of time needed. But in practice, algorithms are actually quite quick and there's no trouble to compute the Havana homology of a 40 crossing knot. The algorithms are implemented by hand or by computer? No, they're computer, but there's a... If the knot is long and thin, then you can compute efficiently. For a knot which isn't long and thin, you can try and break it up into tangles which are long and thin and have sort of divide and conquer strategy. And you can you can put that... You can implement a little heuristic for that on the computer so you can compute quite fast. But still, as far as we know, the theoretical worst case is still just the naive more than exponential time algorithm. Yeah. Other questions? In that case, let's thank the speaker again.