For a closed
n
n
–braid
L
L
with a full positive twist and with
ℓ
\ell
negative crossings,
0
≤
ℓ
≤
n
0\leq \ell \leq n
, we determine the first
n
−
ℓ
+
1
n-\ell +1
terms of the Jones polynomial
V
L
(
t
)
V_L(t)
. We show that
V
L
(
t
)
V_L(t)
satisfies a braid index constraint, which is a gap of length at least
n
−
ℓ
n-\ell
between the first two nonzero coefficients of
(
1
−
t
2
)
V
L
(
t
)
(1-t^2) V_L(t)
. For a closed positive
n
n
–braid with a full positive twist, we extend our results to the colored Jones polynomials. For
N
>
n
−
1
N>n-1
, we determine the first
n
(
N
−
1
)
+
1
n(N-1)+1
terms of the normalized
N
N
–th colored Jones polynomial.