We prove that, in all dimensions d ≥ 4, every simple open polygonal chain and every tree may be straightened, and every simple closed polygonal chain may be convexified. These reconfigurations can be achieved by algorithms that use polynomial time in the number of vertices, and result in a polynomial number of “moves.” These results contrast to those known for d = 2, where trees can “lock,” and for d = 3, where open and closed chains can lock.
Polygonal chain, Bar framework, Knots, Locked chains
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Cocan, Roxana and O'Rourke, Joseph, "Polygonal Chains Cannot Lock in 4D" (2001). Computer Science: Faculty Publications, Smith College, Northampton, MA.