Topological interpretation of color exchange invariants: hexagonal lattice on a torus

Abstract

We explain a correspondence between some invariants in the dynamics of color exchange in the coloring problem of a 2d regular hexagonal lattice, which are polynomials of winding numbers, and linking numbers in 3d. One invariant is visualized as linking of lines on a special surface with Arf-Kervaire invariant one, and is interpreted as resulting from an obstruction to transform the surface into its chiral image with special continuous deformations. We also consider additional constraints on the dynamics and see how the surface is modified.