Moduli Space of a Planar Polygonal Linkage: A Combinatorial Description

Abstract

Abstract. We describe and study an explicit 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 permutohedron. 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 permutohedron.The dual complex K * is patched of Cartesian products of permutohedra. It can be explicitly realized in the Euclidean space via a surgery on the permutohedron.

Manifolds associated to simple games

“…• As it was shown in [8], M (L) admits a structure of a regular cell complex. The combinatorics is very much related (but not equal) to the combinatorics of the permutahedron.…”

“…In Section 3 we associate with a quasilinkage Q a cell complex CW M (Q) by applying the rules from [8]. We prove that CW M (Q) is locally isomorphic to CW M (L) for some real linkage L (however, L depends on the location, and there may be no real linkage associated to the entire complex).…”

“…We refer the reader to literally the same proof of the analogous theorem for real linkages from [8]. In short, each cell is combinatorially equivalent to a Cartesian product of permutohedra.…”

“…The relationship almost automatically extends to quasilinkages. We stress that the below is a combination of the classical construction borrowed from [6] with the cell decomposition approach from [8].…”

“…The idea is borrowed from the cell decomposition of the moduli space of polygonal linkages (see [8]). This motivates us to treat a simple game Q as a quasilinkage since it provides a natural generalization of polygonal linkages.…”