Moduli space of planar polygonal linkage: a combinatorial description

Abstract

We explicitly describe a structure of a regular cell complex $K(L)$ on the moduli space $M(L)$ of a planar polygonal linkage $L$. The combinatorics is very much related (but not equal) to the combinatorics of the permutahedron. In particular, the cells of maximal dimension are labeled by elements of the symmetric group. For example, if the moduli space $M$ is a sphere, the complex $K$ is dual to the boundary complex of the permutahedron. The dual complex $K^*$ is patched of Cartesian products of permutohedra and carries a natural PL-structure. It can be explicitly realized as a polyhedron in the Euclidean space via a surgery on the permutohedron.