Well, its been quite awhile since the last post. The real problem is that this blog can’t be a higher priority for me than, you know, important stuff like learning, teaching and research. So, when time and energy get tight, its one of the first things to go. It also doesn’t help that the more I bend my mind towards research, the less time I spend thinking about things that would actually make good posts (I have started and abandoned several posts on uninteresting research-type things in the last couple months).
I’ll try to get back into the swing of things by talking about some stuff I really enjoy, but is far from my research: knots. I’ve always had a soft spot for knot theory, since its like a poor man’s number theory; a source of simple problems which require techniques from advanced math to solve. A good example of this are Tait’s conjectures, three basic conjectures from the 19th century that resisted proof until the discovery of the Jones invariant using techniques from analysis and representation theory (and secretly physics).
Today, though, I’d like to talk about ‘chord diagrams’, a type of object subtly related to that of knots, and whose study can yield some interesting new knot invariants. They also come up in a number of different areas (solving Feynman diagrams, the representation theory of lie algebras) that I know very little about. If anyone reading is more familiar with some of the places these come up, please send me a reference.
Just to review: an (oriented) knot is a smooth inclusion of the loop into , modulo ambient isotopy, which says that we can move the knot around, but we can’t let it pass through itself (ie, it must remain an inclusion throughout the isotopy). The main problem in the study of knots is to figure out when two knots are definitely the same, or when they are definitely different. The main tool in doing this is to discover knot invariants, numbers which can be assigned to each knot in a computable way, such that two knots which are the same get assigned the same number.
The first thing we will do is broaden the concept of a knot. An (oriented) singular knot is a smooth map from into , with only transverse self-intersections (and finitely many of them). So a singular knot is like a regular knot, except where the loop can intersect itself in a simple manner. The new notion of ambient isotopy is basically the same, except that we can’t break a self-intersection; two pieces that intersect stay intersected. To be more precise, we are allowed only those isotopies between knots which are restrictions of isotopies of the ambient space (hence the name).
Now we could try to study this broader class of knots by coming up with new and interesting knot invariants, but a far lazier solution is to try to turn an older knot invariant into an invariant of singular knots. Fortunately, there is a canonical way to do this. Lets say I have a knot invariant for regular knots . Given a singular knot with only one singularity, there are two non-singular knots nearby, corresponding to the two ways of moving the two intersecting pieces of loop apart. Call these knots and . Then we define . There is the problem that we did not distinguish between the two ways of resolving the singularity, and so the invariant would seem to be only defined up to sign. Using the orientation we can make this choice precise, but it requires more knot theory than I am willing to introduce.
Now that we can define for knots with one singularity, we can extend it to knots with any number of singularities inductively. That is, we pick a singularity of , look at the two ways of splitting it apart, and look at the difference between evaluated on each of the simpler singular knots. Note that this the resulting number is independent which singularity we chose to split apart.
For a first look at why any of this is interesting, imagine we have a knot invariant which has the property that it is zero on any knot with at least one singularity. Now lets say we have two knots which differ by a single ‘pass through’, that is, we can get from one to the other by letting the loop pass through itself once. However, at that moment that the knot is passing through itself, it is a singular knot. Since of that singular knot is zero, the value of must be the same on each knot. But we can get from any knot to any other knot by a finite sequence of pass-throughs, and we just proved that the value of can’t change at any step along the way. Hence, can’t tell any knots apart.
Thus, any non-trivial knot invariant is also non-trivial on knots with at least one singularity. This begs a natural question: are there interesting knot invariants which are nontrivial on knots with at least n singularities? For , the answer is yes, and they are called the n-th Vassiliev invariants. They were first studied in 1989, and their collective power is yet unknown; for example, it is unknown if there are two distinct knots which have the same value for every Vassiliev invariant. There is an explicit (if daunting) method for computing all of them simultaneously on a given knot, called the Kontsevich integral.
Let be an n-th Vassiliev invariant which is non-zero on at least one knot with n-1 singularities. Now lets consider all of the knots with n-1 singularities. If two such knots differ by a sequence of ‘pass throughs’, then the difference in the value of on those knots will be a sum of evaluted on knots with n-singularities. Of course, these are all zero, so the value of is the same.
Therefore, the invariant only cares about knots with n-1 singularities up to singular isotopy, which would be a regular isotopy which doesn’t break any singularities. However, we can record all the relevant information of such a knot in a more convenient package. We look at the parametrization of the knot, and we draw a line connecting the two points involved in each singularity:
This circle with chords contains all the information about our singular knot (that cares about, anyway). It is clear that all that matters is the combinatorial data of the order in which the chords connect, and not things like the precise point on the circle they connect to. Such data is called a chord diagram, and its ‘degree’ is the number of chords.
Therefore, the invariant determines a function from the set of chord diagrams of degree n-1 to whatever group takes values in. Furthermore, this function determines up to a Vassiliev invariant of lower order.
Great, then the next question is: What such functions are possible? There is one immediate restriction we can notice, which is as follows. Imagine we have a singular knot with a singular point such that the complement of that point is two seperate pieces. When evaluating an invariant on that knot, we resolve that singularity in the two possible ways, and take the difference of the invariant on each. Trouble is, these are both the same knot. To see this, just resolve the singularity one way, and twist one of the two pieces 360 degrees to get the other possible resolution. Hence, the invariant will be zero on this singular knot.
What does this mean for chord diagrams? Well, its easy to see a chord diagram that corresponds a singular knot with such a point. It will have a chord which never crosses any other chord, and effectively divides the diagram into two pieces. Such a diagram is said to satisfy the ‘generalized 1T relation’, and we have just proved that any Vassiliev invariant must send it to zero.
What other relations are there? There is a more elaborate one, called the ‘4T relation’, which looks like:
The dots represent parts of the diagram which can be arbitrary.
The fun (and hard) theorem is that these two relations are it. Any function from the set of chord diagrams of degree n-1 to a field which satisfies these two relations comes from a Vassiliev invariant of degree n. This was proven by Kontsevich in 93 with the aforementioned Kontsevich integral. The proof is well beyond my capacity, but I would like to talk more in the future about chord diagrams and the wonderful things you can do with them (they multiply!).
I would be remiss if I didn’t mention the book I have gotten most of this out of: Knot Theory, by Manturov. Also, I got the pictures of chord diagrams from Bar-Natan’s Knot Atlas Wiki. There is a page there that goes much more into depth about Vassiliev invariants (called finite-type invariants there).