An Almost-Proof of the Four Color Theorem

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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.

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One Response to “An Almost-Proof of the Four Color Theorem”

  1. Ars Mathematica » Blog Archive » Four Color Theorem and Lie Algebras Says:

    […] to Greg Muller, I’m looking at this paper by Dror Bar-Natan that reduces the Four Color Theorem to a […]

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