Rotation number of a unimodular cycle: An elementary approach

Abstract

We give an elementary proof of a formula expressing the rotation number of a cyclic unimodular sequence L = u1u2 . . . ud of lattice vectors ui2 in terms of arithmetically defined local quantities. The formula has been originally derived by A. Higashitani and M. Masuda [A. Higashitani, M. Masuda, Lattice multi-polygons, arXiv:1204.0088v2 [math.CO], [v2] Apr 2012; [v3] Dec 2012] with the aid of the Riemann-Roch formula applied in the context of toric topology. These authors also demonstrated that a generalized version of the 'Twelve-point theorem' and a generalized Pick's formula are among the consequences or relatives of their result. Our approach emphasizes the role of 'discrete curvature invariants' μ(a, b, c), where {a, b} and {b, c} are bases of 2, as fundamental discrete invariants of modular lattice geometry.