  The moduli space of $n$ chains in the plane with generic side lengths that
terminate on a fixed line is a smooth, closed manifold of dimension $n-1$. This
manifold is also equipped with a locally standard action of
$\mathbb{Z}_2^{n-1}$. The orbit space of this action is a simple polytope
called the moment polytope. Interestingly, this manifold is also the fixed
point set of an involution on a toric manifold known as the abelian polygon
space. In this article we show that the moment polytope of the moduli space of
chains is completely characterized by the combinatorial data, called the
\emph{short code} of the length vector. We also classify aspherical chain
spaces using a result of Davis, Januszkiewicz and Scott.
