  It is well known that the braid index of a link equals the minimum number of
Seifert circles among all link diagrams representing it. For a link with a
reduced alternating diagram $D$, $s(D)$, the number of Seifert circles in $D$,
equals the braid index $\textbf{b}(D)$ of $D$ if $D$ contains no {\em lone
crossings} (a crossing in $D$ is called a {\em lone crossing} if it is the only
crossing between two Seifert circles in $D$). If $D$ contains lone crossings,
then $\textbf{b}(D)$ is strictly less than $s(D)$. However in general it is not
known how $s(D)$ is related to $\textbf{b}(D)$. In this paper, we derive
explicit formulas for many alternating links based on any minimum projections
of these links. As an application of our results, we are able to determine the
braid index for any alternating Montesinos link explicitly (which include all
rational links and all alternating pretzel links).
