Newton, Klein, Kauffman, Cayley and the Four Color Problem

Paul C. Kainen

Knots in Washington XI
December 7-9, 2000
George Washington University
Washington, DC, USA

Organizers
Dubravko Ivansic, Jozef H. Przytycki, Yongwu Rong, Dan Silver, Akira Yasuhara

Newton, Klein, Kauffman, Cayley and the Four Color Problem
by
Paul C. Kainen
Department of Mathematics, Georgetown University

An argument is given to show that the problem of map coloring in the plane, well-known to be equivalent to a problem involving pairs of rooted, cubic, plane trees, may be regarded as a generalization of Newtonian mechanics. Coloring is equivalent to the propagation of torque. The latter problem was treated by Felix Klein in his book analyzing the motion of a top. Klein introduces four complex parameters to describe a model. On the other hand, Kauffman noted that if the inputs are regarded as quaternions, rather than axial vectors, then associativity prevents any discrepancy. We have pointed out that the relationship between trees can be considered asymmetrically, with one tree regarded as generating the set of admissable colorings via states (clockwise or counterclockwise) at each vertex. By the now-standard technique of complexification, replace the state at a vertex by an element in the two-dimensional vector space over the complex numbers. Normalizing, the state is now a qubit. Thus, a gedanken quantum computer experiment would always register a solution for any suitable pair of such trees. A complete theory of quantum mechanics must, therefore, contain a solution to the Four Color Problem. We propose an approach based on the unification of geometry and arithmetic proposed by Klein and actually embodied in the octonions. Since it was Cayley who first announced the Four Color Problem to the mathematical community, it is ironic that the ``numbers'' he invented (with Graves) may be required for a natural solution.

Date received: December 1, 2000