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 l negative crossings, 0 ≤ l ≤ n, we determine the first n− l+ 1 terms of the Jones polynomial VL(t). We show that VL(t) satisfies a braid index constraint, which is a gap of length at least n − l between the first two nonzero coefficients of (1−t)VL(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.