Modules of the Temperley-Lieb algebra at zero

We explicitly describe the category of modules of the Temperley-Lieb algebra ${\mathrm{T}\mathrm{L}}_n(\beta )$ 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 ${\mathrm{T}\mathrm{L}}_n(0)$, 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.