Helically coiled carbon nanotubes are modeled using topological coordinate method which is based on the toroidal triply connected graphs, containing pentagons, hexagons, and heptagons. Their regular incorporation into the hexagonal carbon net induces transition from the straight to the helical geometry. Relaxation of the structural model is performed in two steps: Firstly, molecular dynamics based on the Brenner potential is applied and then the coil parameters are, once again, optimized within symmetry preserving density functional tight binding (DFTB) method. Model of smooth regularly helically coiled single-walled nanotube structure is obtained. Correlations between the helical angle, tubular and helical diameter are found. Cohesive energy of the coiled structure is calculated by DFTB method within symmetry based POLSym code. Its dependence on the diameter of the coil is shown. The calculated energies range from 7.5 to 8.0 eV/atom.