Unimodularity of Induced Toric Tilings

Abstract

Induced tilings \( \mathcal{T}=\mathcal{T}\left|{}_{\mathrm{Kr}}\right. \) of the d-dimensional torus ????^d generated by an embedded karyon Kr are considered. The operations of differentiation \( \sigma :\mathcal{T}\to {\mathcal{T}}^{\sigma } \) are defined; as a result, the induced tilings \( {\mathcal{T}}^{\sigma }=\mathcal{T}\left|{}_{{\mathrm{Kr}}^{\sigma }}\right. \) of the same torus ????^d, generated by the derivative karyon Kr^σ, are obtained. In terms of karyons Kr, the differentiations σ reduce to a combination of geometric transformations of the space ℝ^d. It is proved that if the karyon Kr is unimodular, then it generates an induced tiling \( \mathcal{T}=\mathcal{T}\left|{}_{\mathrm{Kr}}\right. \), and the derivative karyons Kr^σ are unimodular as well, whence the corresponding derivative tilings \( {\mathcal{T}}^{\sigma }=\mathcal{T}\left|{}_{{\mathrm{Kr}}^{\sigma }}\right. \) exist. Using unimodular karyons, one can construct an infinite family of induced tilings \( \mathcal{T}=\mathcal{T} \) (α, Kr_*), depending on a shift vector α of the torus ????^d and an initial karyon Kr_*. Two algorithms for constructing such unimodular karyons Kr_* are presented.