Ab-initio calculations are performed on single-wall carbon nanotube (SWCNT) caps of different chiralities. The nascent caps are the minimum structures for which the chirality can be determined; we hypothesize that their early formation is related to the abundance of certain types of chiralities during the catalytic of growth of nanotubes, in particular on cobalt nanoparticles. Small differences are found in the stability of certain nascent caps, the most interesting difference is between same diameter (9,1) and (6,5) nanotubes, whose chiralities are uniquely defined, with the (6,5) being more abundant than the (9,1) in some synthesis processes. The interaction energies of the most stable caps for the chiralities (6,5), (9,1), (7,5), (6,6) and (10,2) with an optimized 9-atom cobalt cluster are calculated as well. This interaction energy is expected to be favorable based on the analysis of the cluster-cap electrostatic potential map found in our calculations. The most favorable interaction is found for the (6,5) cap, which has an adsorption energy 2eV more stable than that of the (9,1) cap. The way the cluster interacts with the cap depends on the chirality, the polarization of the composite systems is found to correlate with the SWCNT chiral angle. The reorganization of the geometry of the cluster also changes for different caps, revealing some interesting features of the active sites of the caps and about the variation of the strength of the carbon-cobalt interaction as the chiral angle changes. Further, the growth of the nanotubes out of the nascent cap is studied by progressively adding C₂, this reaction mechanism is chosen based upon the compatibility of the two unpaired electrons of the cap with the two free electrons offered by the radical. It is found that the energetic of the successive addition favors the (6,5) chirality. In addition, it is observed that the closer the cap is to the armchair chirality, the higher the number of favorable sites for a successful C₂ addition leading to the completion of one hexagon. This fact would explain why the most abundant chiralities in the production of carbon nanotubes for most CVD processes are those lying to close to armchair chiralities. All the aforementioned factors show the complexity of the process leading to the selection of a specific chirality. It seems that complex and synergistic interactions of kinetic and thermodynamic factors determine the selectivity of the process of nanotube growth.