Contributor info: Dr Olivier CEPAS

Olivier Cépas, Peter M. Akhmetiev

SciPost Phys. 10, 042 (2021) · published 18 February 2021 | Toggle abstract · pdf

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.

Olivier Cépas, Peter M. Akhmetiev

SciPost Phys. 7, 032 (2019) · published 12 September 2019 | Toggle abstract · pdf

We classify the sectors of configurations that result from the dynamics of 2d crossing flux lines, which are the simplest degrees of freedom of the 3-coloring lattice model. We show that the dynamical obstruction is the consequence of two effects: (i) conservation laws described by a set of invariants that are polynomials of the winding numbers of the loop configuration, (ii) steric obstruction that prevents paths between configurations, for lack of free space. We argue that the invariants fully classify the configurations in five, chiral and achiral, sectors and no further obstruction in the limit of low-winding numbers.