Higher topological complexity of planar polygon spaces having small genetic codes

We study the higher (sequential) topological complexity, a numerical homotopy invariant for the planar polygon spaces. For these spaces with a small genetic codes and dimension $m$, Davis showed that their topological complexity is either $2m$ or $2m+1$. We extend these bounds to the setting of higher topological complexity. In particular, when $m$ is power of $2$, we show that the $k$-th higher topological complexity of these spaces is either $km$ or $km+1.$