$p$-Jones-Wenzl idempotents

For a prime number $p$ and any natural number $n$ we introduce, by giving an explicit recursive formula, the $p$-Jones-Wenzl projector ${}^p\operatorname{JW}_n$, an element of the Temperley-Lieb algebra $TL_n(2)$ with coefficients in ${\mathbb F}_p$. We prove that these projectors give the indecomposable objects in the $\tilde{A}_1$-Hecke category over ${\mathbb F}_p$, or equivalently, they give the projector in $\mathrm{End}_{\mathrm{SL}_2(\overline{{\mathbb F}_p})}(({\mathbb F}_p^2)^{\otimes n})$ to the top tilting module. The way in which we find these projectors is by categorifying the fractal appearing in the expression of the $p$-canonical basis in terms of the Kazhdan-Lusztig basis for $\tilde{A}_1$.