l₁-embeddability of generic quadrilateral Möbius maps

A connected graph G is called l₁-embeddable, if G can be isometrically embedded into the l₁-space. We prove that an l₁-embeddable quadrilateral Möbius map (M,G) contains a unique shortest non-nulhomotopic cycle C, provided that (M,G) is generic, that is, its face cycles are isometric. Moreover, C is convex and orientation-reversing. After cutting (M,G) along C, the map falls apart into a number of quadrilateral plane maps called beads. We analyze the structure of the bead graph in which two beads are adjacent when they share a segment of C. We also introduce the foundation B(M,G) which helps to decide whether a concrete map (M,G) is l₁-embeddable.