UNIMODULARITY OF INDUCED TORIC TILINGS

摘要

Induced tilings T = T∣_(Kr) of the d-dimensional torus T~d generated by an embedded karyon Kr are considered. The operations of differentiation δ : T → T~δ are defined; as a result, the induced tilings T~δ = T∣_(Kr~δ) of the same torus T~d, generated by the derivative karyon Kr~δ, are obtained. In terms of karyons Kr, the differentiations a 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 T = T∣_(Kr), and the derivative karyons Kr~δ are unimodular as well, whence the corresponding derivative tilings T~δ = ∣_(Kr~δ) exist. Using unimodular karyons, one can construct an infinite family of induced tilings T = T(α,Kr_*), depending on a shift vector a of the torus T~d and an initial karyon Kr_*. Two algorithms for constructing such unimodular karyons Kr_* are presented. Bibliography: 23 titles.