We show that, at least for an alternating fibered link or
2
2
-bridge link
L
L
, there is an exact formula which expresses the braid index
b
(
L
)
{\mathbf {b}}(L)
of
L
L
as a function of the
2
2
-variable generalization
P
L
(
l
,
m
)
{P_L}(l,m)
of the Jones polynomial.