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.

中文翻译：

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颜色交换不变式的拓扑解释：圆环上的六边形格子

我们解释了二维着色问题中颜色交换动力学中的一些不变量之间的对应关系，这些变量是缠绕数的多项式和3d中的链接数。一个不变量被可视化为特殊表面上的线与Arf-Kervaire不变量的链接，并且被解释为是由于将表面转化为具有特殊连续变形的手性图像而产生的障碍。我们还考虑了对动力学的其他约束，并查看如何修改曲面。