On the tail of Jones polynomials of closed braids with a full twist (Q2839326)

In this paper the authors determine the first \((n-l+1)\) terms of the Jones polynomial for a closed \(n\)-braid with the positive full twist \(\Delta^{2}_{n}\) and \( 0\leq l \leq n\) negative crossings. They also prove a similar result for the colored Jones polynomial for a closed positive braid with positive full twist.NEWLINENEWLINEThe proof is based on the following two observations: the first observation is that the Kauffman bracket of the \((n,n)\)-torus link, the closure of a positive full twist \(\Delta^{2}_{n}\), has particular lower degree terms which the authors call the gap block. The second observation, the main technical point in this paper, is that adding braids with at most \(n\) negative crossing to \(\Delta^{2}_{n}\) does not affect the gap block.