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The basic object of this paper are the moduli spaces M2(L) andM3(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 M2(L) is equipped with the oriented area function A. In turn, we equip the space M3(L) with the vector area function S. The latter are generically Morse functions, whose critical points have a nice description. In this paper, we define a swap action (that is, the action of some group generated by edge transpositions) on the spaces M2(L) and M3(L) that 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.

Swap action on moduli spaces of polygonal linkages