Topological complexity of some planar polygon spaces

TL;DR

This paper determines the topological complexity of certain planar polygon spaces, showing it is either 2n-5 or 2n-6 depending on the side lengths, based on cohomology ring properties.

Contribution

It establishes exact topological complexity values for planar polygon spaces with specific side length conditions using cohomology ring analysis.

Findings

Topological complexity is either 2n-5 or 2n-6.

Values depend on whether n-r is an odd integer.

Results apply to spaces with one side length r and others of length 1.

Abstract

Using known results about their mod-2 cohomology ring, we prove that the topological complexity of the space of isometry classes of n-gons in the plane with one side of length r and all others of length 1 equals either 2n-5 or 2n-6, provided that n-r is not an odd integer.