Dual bases in Temperley-Lieb algebras, quantum groups, and a question of Jones

We derive a Laurent series expansion for the structure coefficients appearing in the dual basis corresponding to the Kauffman diagram basis of the Temperley Lieb algebra TLk(d), converging for all complex loop parameters d with |d| > 2 cos (pi/k+1). In particular, this yields a new formula for the structure coefficients of the Jones Wenzl projection in TLk(d). The coefficients appearing in each Laurent expansion are shown to have a natural combinatorial interpretation in terms of a certain graph structure we place on non -crossing pairings, and these coefficients turn out to have the remarkable property that they either always positive integers or always negative integers. As an application, we answer affirmatively a question of Vaughan Jones, asking whether every Temperley Lieb diagram appears with non -zero coefficient in the expansion of each dual basis element in TLk(d), when d is an element of R\[ - 2 cos (pi/k+1), 2 cos (pi/k+1)]. Specializing to Jones Wenzl projections, this result gives a new proof of a result of Ocneanu [27], stating that every Temperley Lieb diagram appears with non -zero coefficient in a Jones Wenzl projection. Our methods establish a connection with the Weingarten calculus on free quantum groups, and yield as a byproduct improved asymptotics for the free orthogonal Weingarten function.