On the tail of Jones polynomials of closed braids with a full twist

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