Coefficients of the one- and two-gap boxes in the Jones-Wenzl idempotent Sarah Reznikoff 46LTemperley Lieb algebrasubfactorBurau representationJones-Wenzl idempotent The first $n-1$ projections forming the Jones tower of a II$_{1}$ subfactor generate a semisimple quotient, $\mathcal{TL}_{n}(\delta)$, of the Temperley-Lieb Algebra. This algebra can be represented pictorially by planar diagrams on $n$ strings in a box, and these diagrams can be classified according to the number of non-through strings, or "gaps" they have. The Jones-Wenzl Idempotent is the complement in $\mathcal{TL}_{n}(\delta)$ of the supremum of the projections generating the Jones tower. We prove Ocneanu's formula for the coefficients of the one- and two-gap boxes in an explicit expression of this element. Indiana University Mathematics Journal 2007 text pdf 10.1512/iumj.2007.56.3140 10.1512/iumj.2007.56.3140 en Indiana Univ. Math. J. 56 (2007) 3129 - 3150 state-of-the-art mathematics http://iumj.org/access/