## An Almost-Proof of the Four Color Theorem

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 $L$ 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 $G$, lets write $W_L(G)$ for this complex number.

The general idea of Bar-Natan (and other people he quotes) is to figure out natural things that this number $W_L(G)$ is counting. The results are as follows:

Theorem. (Penrose) If $G$ has a planar embedding, then $|W_{sl(2)}(G)|$ is $2^{|G|/2-2}$ times the number of four-colorings of any embedding of $G$ in the plane.

Theorem. (Bar-Natan) Thought of as a function of $n$, $W_{sl(n)}(G)$ is a polynomial in $n$ of degree at most $|G|/2-2$. If $G$ is 2-connected, then the degree $|G|/2-2$ coefficent of this polynomial is the number of embeddings of $G$ in the plane.

Thus, if the vanishing of $W_{sl(2)}(G)$ implied that the polynomial $W_{sl(n)}(G)$ had degree strictly less than $|G|/2-2$, 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.