Symmetric Configuration spaces of linkages

A $configuration$ of a linkage $Γ$ is a possible positioning of $Γ$ in $\mathbb{R}^d$ and the collection of all such forms the configuration space $\mathcal{C}(Γ)$ of $Γ$. We here introduce the notion of the $symmetric configuration space$ of a linkage, in which we identify configurations which are geometrically indistinguishable. We show that the symmetric configuration space of a planar polygon has a regular cell structure, provide some principles for calculating this structure, and give a complete description of the symmetric configuration space of all quadrilaterals and of the equilateral pentagon.

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