  The problem of detecting auxetic behavior, originating in materials science
and mathematical crystallography, refers to the property of a flexible periodic
bar-and-joint framework to widen, rather than shrink, when stretched in some
direction. The only known algorithmic solution for detecting infinitesimal
auxeticity is based on the rather heavy machinery of fixed-dimension
semi-definite programming. In this paper we present a new, simpler algorithmic
approach which is applicable to a natural family of 3D periodic bar-and-joint
frameworks with 3 degrees-of-freedom. This class includes most zeolite
structures, which are important for applications in computational materials
science. We show that the existence of auxetic deformations is related to
properties of an associated elliptic curve. A fast algorithm for recognizing
auxetic capabilities is obtained via the classical Aronhold invariants of the
cubic form defining the curve.
