Georgia Institute of Technology College of Sciences

The four color theorem states that each bridgeless trivalent planar graph has a proper 4-face coloring. It can be generalized to certain types of CW complexes of any closed surface for any number of colors, i.e., one looks for a coloring of the 2-cells (faces) of the complex with m colors so that whenever two 2-cells are adjacent to a 1-cell (edge), they are labeled different colors.

In this talk, I show how to categorify the m-color polynomial of a surface with a CW complex. This polynomial is based upon Roger Penrose’s seminal 1971 paper on abstract tensor systems and can be thought of as the Jones polynomial’’ for CW complexes. The homology theory that results from this categorification is called the bigraded m-color homology and is based upon a topological quantum field theory (that will be suppressed from this talk due to time). The construction of this homology shares some similar features to the construction of Khovanov homology—it has a hypercube of states, multiplication and comultiplication maps, etc. Most importantly, the homology is the $E_1$-page of a spectral sequence whose $E_\infty$-page has a basis that can be identified with proper m-face colorings, that is, each successive page of the sequence provides better approximations of m-face colorings than the last. Since it can be shown that the $E_1$-page is never zero, it is safe to say that a non-computer-based proof of the four color theorem resides in studying this spectral sequence! (This is joint work with Ben McCarty.)