Coefficients of the One- and Two-gap Boxes in the Jones-Wenzl Idempotent

The first n – 1 projections forming the Jones tower of a II1 subfactor generate a semisimple quotient, JLn(δ), 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 JLn(δ) 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.