Swap action on moduli spaces of polygonal linkages

TL;DR

This paper introduces a swap action on moduli spaces of polygonal linkages in 2D and 3D, preserving area functions and Morse critical points, with computational experiments and conjectures about the symmetry group.

Contribution

It defines a new swap action on moduli spaces of polygonal linkages that preserves key geometric functions and analyzes its properties.

Findings

The swap action preserves the area and vector area functions.

Critical points of the functions are well-described and preserved.

Computer experiments suggest symmetry properties and lead to conjectures.

Abstract

The basic object of the paper is the moduli space $M_{2,3}(L)$ of a closed polygonal linkage either in $\mathbb{R}^2$ or in $\mathbb{R}^3$. As was originally suggested by G. Khimshiashvili, the space $M_{2}(L)$ is equipped with the oriented area function $A$, whereas (as is suggested in the paper) $M_{3}(L)$ is equipped with the vector area function $S$. The latter are generically Morse functions, whose critical points have a nice description. In the preprint, we define a \textit{swap action} (that is, the action of some group generated by edge transpositions) on the space $M_{2,3}(L)$ which preserves the functions $A$ and $S$ and the Morse points. We prove that the commutant of the group acts trivially, present some computer experiments and formulate a conjecture.