The Bubble Skein Element and Applications

We study a certain skein element in the relative Kauffman bracket skein module of the disk with some marked points, and expand this element in terms linearly independent elements of this module. This expansion is used to compute and study the head and the tail of the colored Jones polynomial and in particular we give a simple $q-$series for the tail of the knot $8_5$. Furthermore, we use this expansion to obtain an easy determination of the theta coefficients.