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 of lattice vectors in terms of arithmetically defined local quantities. The formula has been originally derived by A. Higashitani and M. Masuda (arXiv:1204.0088v2 [math.CO]) with the aid of the Riemann-Roch formula applied in the context of toric topology. They also demonstrated that a generalized versions 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' \mu(a,b,c), where {a,b} and {b,c} are bases of the lattice Z^2, as fundamental discrete invariants of 'modular lattice geometry'.