 Jones-Wenzl idempotent

The Jones-Wenzl idempotent has identity-diagram coefficient one, satisfies $f_n^2=f_n$, and annihilates each adjacent cap-cup generator on either side. With $\delta =-A^2-A^{-2}$ and ${\Delta }_0=1$, ${\Delta }_1=\delta$, ${\Delta }_n=\delta {\Delta }_{n-1}-{\Delta }_{n-2}$, it is constructed recursively by $f_n=F-({\Delta }_{n-2}/{\Delta }_{n-1})F{e}_{n-1}F$, where $F=f_{n-1}\otimes 1$. The required denominators must be nonzero. Its closed bracket trace is ${\Delta }_n=(-1{)}^n[n+1]$.