On the semi-threading of knot diagrams with minimal overpasses

Given a knot diagram $D$, we construct a semi-threading circle for it which can be an axis of $D$ as a closed braid depending on knot diagrams. In particular, we consider semi-threading circles for minimal diagrams of a knot with respect to overpasses which give us some information related to the braid index. By this notion, we show that, for every nontrivial knot $K$, the braid index $b(K)$ of $K$ is not less than the minimum number $l(K)$ of overpasses of diagrams. Moreover, they are the same for a torus knot.