$\ell_{1}$-Embeddability of Regular Quadrilateral Maps on the Torus and the Klein Bottle

A connected graph is called $\ell_1$-embeddable if its shortest-path metric admits an isometric embedding into the $\ell_1$-space; equivalently, in the finite case, it admits a scale embedding into a hypercube. This paper gives a complete parameter-level classification of $\ell_1$-embeddability for