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Biomedical researchers regularly stretch and twist single DNA molecules in magnetic tweezer experiments. By making the molecule writhe into a plectoneme (ply) and plotting its load-extension curves, key DNA parameters, such as effective radius, can be estimated. Adding untangling enzymes (topoisomerases) to the DNA's environment, their individual cuts are detected as jumps in extension. Sufficient information is now known about the topoisomerases for us to make good idealizations about their kinematics and mechanics. The novelty of this paper is to study their actions in the context of accurate ply solutions from the theory of elastic rods. To do this, we define an extended rod-plus-tension system that allows us to determine the stored energies from areas in the conventional link versus writhe plane. After a cut, the molecule relaxes dynamically to a new equilibrium state, and often there will be two or more alternative stable configurations onto which it might settle. Knowing the energy levels allows us to identify which states can and cannot be reached over the unstable mountain passes, and which of the accessible states offer the greatest energy relaxation. Strict energy bounds on behaviour are established. This knowledge has medical value because topoisomerase inhibitors, lethal for cells, are used as antibiotics and in chemotherapy for cancer.