Problem of the Week
Hilton Memorial Lecture
The Four-Color Theorem was first proved by Appel and Haken in 1977 with the aid of a computer. Later, a second proof was given by Robertson, Sanders, Seymour, and Thomas. While the proof was simplified, it still relies on a computer in a significant way.
In 1990, Kauffman proved that the Four-Color Theorem is equivalent to the ability to find a non-trivial assignment of the 3-dimensional unit vectors i, j, and k to the variables of two associations of the multiple cross product v1 × v2 × ··· × vn, such that both associations have the same evaluation. (An assignment is trivial if it evaluates to zero.) The associations are determined by the map being colored.
Since elements of Thompson's group F represent instances of the associative law, one can prove that the Four-Color Theorem is equivalent to every element of F's having a non-trivial assignment of the vectors i, j, and k for which that element's instance of associativity holds. I will prove that every positive element of F has such an assignment.
We call such elements colorable. I will consider several operations that preserve colorability of elements of F.