Chirality-selective synthesis of single-walled carbon nanotubes (SWNTs) has been a research goal for the last two decades and is still challenging due to the difficulty in controlling the atomic structure in the one-dimensional material. Here, we develop an optimized approach for controlling the chirality of species by tuning the oxidation degree of Co catalyst. Predominant synthesis of (6,4) SWNTs is realized for the first time. The detailed mechanism is investigated through a systematic experimental study combined with first-principles calculations, revealing that the independent control of tube diameter and chiral angle achieved by changing the binding energy between SWNTs (cap and tube edge) and catalyst causes a drastic transition of chirality of SWNTs from (6,5) to (6,4). Since our approach of independently controlling the diameter and chiral angle can be applied to other chirality species, our results can be useful in achieving the on-demand synthesis of specific-chirality SWNTs.