  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.
