Swap action on moduli spaces of polygonal linkages

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 $\mathit{\text{swap}}\mathit{\text{action}}$ (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.