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

Abstract:

For a closed $n$–braid $L$ with a full positive twist and with $\ell$ negative crossings, $0\leq \ell \leq n$, we determine the first $n-\ell +1$ 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 $n-\ell$ 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.