Configuration spaces of convex and embedded polygons in the plane

TL;DR

This paper explores the topological structure of configuration spaces of convex and embedded polygons in the plane, showing they are homeomorphic to Euclidean spaces or balls, extending previous convexification results.

Contribution

It characterizes the topology of configuration spaces for polygons with fixed edge lengths, revealing they are homeomorphic to Euclidean spaces or balls, depending on convexity or embedding.

Findings

Convex configuration spaces are homeomorphic to closed Euclidean balls.

Embedded configuration spaces are homeomorphic to Euclidean spaces.

Results extend the convexification theorem of Connelly, Demaine, and Rote.

Abstract

This paper studies the configuration spaces of linkages whose underlying graph is a single cycle. Assume that the edge lengths are such that there are no configurations in which all the edges lie along a line. The main results are that, modulo translations and rotations, each component of the space of convex configurations is homeomorphic to a closed Euclidean ball and each component of the space of embedded configurations is homeomorphic to a Euclidean space. This represents an elaboration on the topological information that follows from the convexification theorem of Connelly, Demaine, and Rote.