Abstract A method is proposed for constructing equilibrium shapes and studying the energetics of perfect single-wall carbon nanotubes in terms of five geometrical parameters including the twist, radial expansion, axial stretch, and two components of the inner displacement of the interwoven Bravais lattices defining the unrolled, planar, hexagonal graphene sheet. Values of these quintuplets at equilibrium are computed over an extensive range of chiralities by solving an unconstrained energy minimization problem using Brenner’s interatomic potential. An analogous formulation for computing the parameters defining the deformed tube shape under the auspices of molecular membrane mechanics is discussed. Three fundamental modes of deformation that preserve the cylindrical shape are identified describing extension that allows for expansion and twist, and their two cyclic permutations. The tube response is studied in each case by solving constrained energy minimization problems in four scalar variables. The results demonstrate the coupling between the elementary modes of deformation, and reveal topological transitions when the deformation exceeds certain thresholds. Under these critical conditions, a change in neighbors occurs signaling reconstruction and plastic deformation. Graphene rolling and tube deformation are shown to have a significant effect on the electronic band gap.