Planar Graphs as Minimal Resolutions of Trivariate Monomial Ideals

We introduce the notion of rigid embedding in a grid surface, a new kind of plane drawing for simple triconnected planar graphs. Rigid embeddings provide methods to (1) find well-structured (cellular, here) minimal free resolutions for arbitrary monomial ideals in three variables; (2) strengthen the Brightwell–Trotter bound on the order dimension of triconnected planar maps by giving a geometric reformulation; and (3) generalize Schnyder’s angle coloring of planar triangulations to arbitrary triconnected planar maps via geometry. The notion of rigid embedding is stable under duality for planar maps, and has certain uniqueness properties. 2000 Mathematics Subject Classification: 05C10, 13D02, 06A07, 13F55, 68R10, 52Cxx