I recently gave an Olivetti (our graduate colloquium) on chord diagrams, effectively covering my first two posts on the subject. In preparation for the talk, I read a little bit more about some cool things one can do with them, and I finally got around to reading a paper of Bar-Natan on connections with the Four Color Theorem. I figured I should write a post on it before everyone completely forgot what I’ve said already; that said, this post should be readable even if you didn’t read my other posts on chord diagrams.
In my last post, it was shown how to take a finite dimensional lie algebra equipped with an invariant inner product and combine it with a ‘generalized chord diagram’ to get a complex number. For the purposes of this post, we can let ‘generalized chord diagram’ mean a trivalent graph with a choice of cyclic ordering on each vertex, lets call these oriented trivalent graphs from now on. Given an oriented trivalent graph , lets write for this complex number.
The general idea of Bar-Natan (and other people he quotes) is to figure out natural things that this number is counting. The results are as follows:
Theorem. (Penrose) If has a planar embedding, then is times the number of four-colorings of any embedding of in the plane.
Theorem. (Bar-Natan) Thought of as a function of , is a polynomial in of degree at most . If is 2-connected, then the degree coefficent of this polynomial is the number of embeddings of in the plane.
Thus, if the vanishing of implied that the polynomial had degree strictly less than , the Four Color Theorem would follow. Of course, this isn’t yet known, and the four color theorem is proved; so this approach is mostly for simplifying our understanding of the Four Color theorem.