Density functional theory is used to show that the adhesion between single-walled carbon nanotubes (SWNTs) and the catalyst particles from which they grow needs to be strong to support nanotube growth. It is found that Fe, Co, and Ni, commonly used to catalyze SWNT growth, have larger adhesion strengths to SWNTs than Cu, Pd, and Au and are therefore likely to be more efficient for supporting growth. The calculations also show that to maintain an open end of the SWNT it is necessary that the SWNT adhesion strength to the metal particle is comparable to the cap formation energy of the SWNT end. This implies that the difference between continued and discontinued SWNT growth to a large extent depends on the carbon-metal binding strength, which we demonstrate by molecular dynamics (MD) simulations. The results highlight that first principles computations are vital for the understanding of the binding strength's role in the SWNT growth mechanism and are needed to get accurate force field parameters for MD.