Skew parallelogram nets and universal factorization

We obtain many objects of discrete differential geometry as reductions of skew parallelogram nets, a system of lattice equations that may be formulated for any unital associative algebra. The Lax representation is linear in the spectral parameter, and paths in the lattice give rise to polynomial dependencies. We prove that generic polynomials in complex 2 × 2 matrices factorize, implying that skew parallelogram nets encompass all systems with such a polynomial representation. We demonstrate factorization in the context of discrete curves by constructing pairs of Bäcklund transformations that induce Euclidean motions on discrete elastic rods. More generally, we define a hierarchy of discrete curves by requiring such an invariance after an integer number of Bäcklund transformations. Moreover, we provide the factorization explicitly for discrete constant curvature surfaces and reveal that they are slices in certain 4D cross-ratio systems. Encompassing the discrete DPW method, this interpretation constructs such surfaces from given discrete holomorphic maps.