A self-stress on a framework of surfaces yields a monodromy-free piecewise-linear lifting whenever the self-stress space has dimension greater than 3 times the first Betti number of the surface.
Demaine, and G¨ unter Rote
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Liftings of surfaces in the plane
A self-stress on a framework of surfaces yields a monodromy-free piecewise-linear lifting whenever the self-stress space has dimension greater than 3 times the first Betti number of the surface.