Moduli space of planar polygonal linkage: a combinatorial description

TL;DR

This paper provides a detailed combinatorial description of the moduli space of planar polygonal linkages, relating its cell complex structure to permutahedra and offering explicit geometric realizations.

Contribution

It introduces a regular cell complex structure on the moduli space, linking it to permutahedral combinatorics and providing explicit geometric constructions.

Findings

Cells of maximal dimension are labeled by permutations.

The complex is dual to the boundary of a permutahedron when the moduli space is a sphere.

The dual complex is assembled from Cartesian products of permutohedra with a natural PL-structure.

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.