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The rigidity properties of a framework whose vertices lie in a plane is governed by the 2-dimensional generic rigidity matroid, a counting matroid defined on the underlying graph of the framework. Minimally dependent sets in this matroid do not normally correspond to planar graphs and their embeddings in the plane may have edge crossings. We use the tool of x-replacement to obtain plane frameworks whose underlying graph is a cycle in the rigidity matroid and then use their non-zero resolvable stress to lift them into 3-space while leaving their boundary fixed.
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