Topological complexity of planar polygon spaces with small genetic code

TL;DR

This paper establishes tight bounds on the topological complexity of planar polygon spaces, especially for spaces with small genetic codes, advancing understanding of their motion planning properties.

Contribution

It provides new lower bounds for the topological complexity of planar polygon spaces with small genetic codes, nearly matching known upper bounds for most cases.

Findings

Lower bounds closely match upper bounds in most cases

Results apply to n-gon spaces with small genetic codes

Analysis covers a wide range of polygon spaces

Abstract

We determine lower bounds for the topological complexity of many planar polygon spaces mod isometry. With very few exceptions, the upper and lower bounds given by dimension and cohomology considerations differ by 1. This is true for 130 of the 134 generic 7-gon spaces. Our results apply to spaces of n-gons for all n, but primarily for those whose genetic codes, in the sense of Hausmann and Rodriguez, are moderately small.