Last time, we used knot theory as a way of motivating these funny things called chord diagrams, which were circles with a collection of chords. They came up as ways of writing singular isotopy classes of singular knots, but there are many other ways of thinking of them. They can also be thought of as fixed-point free involutions on elements (modulo cyclic permutations) or a trivalent graph with a distinguished Hamiltonian cycle. I list these other incarnations more as a way of appealing to a broader base, though the latter perspective will be relevant today.
The topic of the day is the 4T relation, which I mentioned at the end of the previous post. It, together with the 1T relation, determined which functions on the set of all chord diagrams of a given degree came from knot invariants. The 1T relation came from a fairly straight-forward observation, while I didn’t even attempt to defend the 4T relation.
One way of thinking of this relation is as a symptom of a larger structure. We will generalize chord diagrams to a slightly broader context, and impose a relation there that is more natural. From there it is obvious to see that these generalized chord diagrams can always be reduced in terms of the usual chord diagrams, and that the only evidence of this broader structure is the 4T relation.
To motivate the generalized chord diagrams, lets think about singular knots up to singular isotopy. The rule was we were allowed to pass the knot through itself, but we were never allowed to break the singularities. Hence, we were allowed to slide where a knot intersected itself along either of the involved curves, yet we weren’t allowed to move two singularities past each other (sliding each along a common curve).
This obstacle might seem reminiscent of the usual forbidden action in the study of non-singular knots, that we can’t move two pieces of the knot through each other. Well, the philosophy behind singular knots and Vassiliev invariants tells us something we can try. We can think of that one forbidden moment when the two singularities coincide as a new object, and study it. Furthermore, we can extend invariants to this new object by taking the difference of its values on the ‘before knot’ and the ‘after knot’.
So we want to think about singular knots, but now we want to allow singularities where three curves meet in one point. We don’t want things to be too bad, so we will require that in an infinitesmal neighborhood of such an intersection, that the curves are linearly independant (they don’t all lie in the same plane). We study these new objects, which we will call very singular knots, up to an analogous version of singular isotopy, where we can move the curve through itself, but we can’t break singularities or even split or combine singularities. Therefore, triple intersections stay triple intersections, but the precise points on the knot which intersect may change.
We will want to use something similar to chord diagrams to write down such very singular knots, but we need a way of writing down triple intersections. The answer is the obvious thing, we just draw a trivalent vertex in the middle of the circle and connect it to each point in the triple intersection.
One important but subtle point: at a triple intersection, the three involved curves are linearly independant. This gives them a cyclic order coming from the orientation of . However, they also have a natural cyclic order coming from the fact that they literally lie on an oriented circle. Our new diagrams need to keep track of this information. The usual way of doing this is to take advantage of the fact that we will always be drawing these in 2 dimensional space, and declare that the cyclic ordering coming from the orientation will be the clockwise order they come out of the interal vertex. Therefore, a chord diagram where the two above cyclic orderings disagree would be written in the following way:
Ok, so lets define the new chord diagrams. A generalized chord diagram will be a circle, a trivalent graph in its interior which may connect to the circle, and a prescribed ordering of the edges touching every internal vertex. The astute reader will notice that this definition allows a broader class of diagrams than will arise from very singular knots; this extra generality proves to be no problem. Also, since the external vertices come with an inherent ordering (clockwise), this definition can be simplified to a trivalent graph with orderings on each vertex, and a distinguished oriented subcycle.
The next thing to do would be to study knot invariants on very singular knots and generalized chord diagrams. However, this is a bit silly since every singular knot is also a very singular knot. Remember, an invariant evaluated on a knot with a triple singularity is the difference of the invariant on the two nearby knots. Pictorially, it can be written as:
The dotted parts represent connecting to the rest of the diagram which is the same for all three.
Using this relation, any generalized chord diagram can be written as a combination of chord diagrams, by repeatedly resolving the internal vertices. However, there is some ambiguity in this process. The relation above tell us how to resolve an internal vertex with respect to a point on the circle it connects to, but an internal vertex can be connected to several external points. These different methods of resolution must all given equivalent combinations of chord diagrams.
To see what ‘equivalent combinations of chord diagrams’ means here, just write down what it means to resolve an internal vertex with respect to one external point, and declare it to be equal to resolving it with respect to a different external point, ie:
Hey, its the 4T relation! Thus, we have demonstrated the original claim; that there was a natural generalization of chord diagrams, together with a natural relation, which simplified to the usual notion of chord diagrams and the less natural 4T relation.
I should point out that an actual proof of the 4T relation is quite a bit simpler than all this. One just writes out what it means to resolve each of the relevant singularities in terms of combinations of singular knots, and one gets eight terms on the left, and the same eight terms on the right. I just like this way of thinking about it much better, for the reasons mentioned here and some I will probably mention in a later post (its the Jacobi identity!).