Rotation number of a unimodular cycle: an elementary approach

TL;DR

This paper presents an elementary proof for a formula that calculates the rotation number of unimodular lattice vector cycles using local arithmetic invariants, simplifying previous complex derivations.

Contribution

It introduces a new elementary approach emphasizing discrete curvature invariants, providing a simpler proof and deeper insight into the rotation number formula for unimodular cycles.

Findings

Derived an elementary proof of the rotation number formula

Connected the formula to discrete curvature invariants

Highlighted the role of modular lattice geometry

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'.