Higher topological complexity of planar polygon spaces having small genetic codes

TL;DR

This paper investigates the higher topological complexity of planar polygon spaces with small genetic codes, extending known bounds and revealing that for certain dimensions, the complexity scales linearly with the number of segments.

Contribution

It extends existing bounds on topological complexity to higher topological complexity for planar polygon spaces with small genetic codes.

Findings

Higher topological complexity is either $km$ or $km+1$ when $m$ is a power of 2.

Topological complexity bounds are extended from known cases to higher complexity.

Results provide new insights into the homotopy invariants of polygon spaces.

Abstract

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 $km+1.$