Foams and the Four-Color Theorem
Foams are cool mathematical objects studied by my brother, Mikhail Khovanov. I already wrote about them in my previous blog posts, Foams Made out of Felt and Tesseracts and Foams. Here, I would like to explain why foams are so cool, but first, I need to remind you of their definition. Foams are finite 2-dimensional CW-complexes, such that each point’s neighborhood must be homeomorphic to one of the three objects below.
- An open disc. Such points are called regular points.
- The product of a tripod and an open interval. Such points are called seam points.
- The cone over the 1-skeleton of a tetrahedron. Such points are called singular vertices.
Foams are cool: they are 2-dimensional CW-complexes embedded in 3-space, with singularities only of the most generic kind, which makes them relatively simple. Moreover, they are combinatorially defined, which makes them easier to work with than with many other geometric objects.
My two previous blog posts have some pictures, but now, I just want to discuss a generic planar cross-section of a foam, which is a planar graph. In the cross-section, seams become vertices, and faces (regular points) become edges. The tripod condition above implies that the resulting graph is trivalent: each vertex has degree 3.
The most interesting foams are tricolarble: foams where their faces can be colored in three colors, so that each face has its own color, and, at the seams, three faces of three different colors meet. The cross-section of such a foam makes a tricolorable trivalent graph. This coloring is called Tait coloring. The cool thing is the Tait’s theorem connects the Tait coloring to the 4-color theorem.
Tait’s theorem. The following two statements are equivalent.
- Every planar graph is 4-colorable.
- The edges of every planar bridgeless trivalent graph are 3-colorable.
I won’t discuss the proof here, but I will explain how to color the edges of a graph in three colors when the faces are colored in four, and vice versa.
Assume that the four colors of the faces form a group of four elements, called the Klein group. Let’s say that gray is the identity, and red, blue, and green are the rest. Then, the product of gray and x is x. The product of any two non-gray colors is the third non-gray color.
Given a trivalent graph G whose edges are colored in three colors, we can color the faces of that graph in the following manner. Color one of the faces a random color. Then, calculate the colors of the other faces so that each edge’s color is the product of the colors of neighboring faces.
Going back, if we have a planar trivalent graph with faces colored in four colors, we can assign an edge a color that is the product of the colors of neighboring faces. As neighboring faces have different colors, the product of those colors is never gray (the identity). Thus, the edges will be colored in three colors. I leave it to the reader to check that three edges incident to a vertex must be colored in different colors.
Kronheimer-Mrowka homology theory of graphs states that the Kronheimer-Mrowka homology of a trivalent graph is non-zero if and only if the graph has no bridge. If one can prove that the rank of the homology group is the number of 3-colorings of the edges (or at least that the non-zero homology implies the existence of the tricoloring of that graph), then the Four-color theorem would follow from Tait’s theorem.
Foams are cool by themselves, but there is hope that they might provide a conceptual proof of the Four-color theorem, making them awesome!
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