  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.
