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

TL;DR

This paper explores a topological interpretation of color exchange invariants in a 2D lattice problem, linking them to 3D linking numbers and topological obstructions, revealing new geometric insights.

Contribution

It introduces a novel topological framework connecting 2D color exchange invariants to 3D linking numbers and surface invariants, expanding understanding of their geometric nature.

Findings

Invariants correspond to linking numbers in 3D.

Visualization of invariants as linking of lines on special surfaces.

Identification of topological obstructions related to the Arf-Kervaire invariant.

Abstract

We explain a correspondence between some invariants in the dynamics of color exchange in a 2d coloring problem, 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.