Modules of the Temperley-Lieb algebra at zero

We explicitly describe the category of modules of the Temperley-Lieb algebra $\mathrm{TL}_n(\beta)$ under specialization $\beta=0$ for even $n$ in terms of a quiver algebra, analogous to a result of Berest-Etingof-Ginzburg. In particular, we explicitly construct an exact sequence of the standard modules of $\mathrm{TL}_n(0)$, which categorifies a numerical coincidence regarding the evaluation of the Jones polynomial at $t=-1$. We furthermore deduce a consequence in the representation theory of symmetric groups over characteristic two.