Rotation number of a unimodular cycle: An elementary approach

We give an elementary proof of a formula expressing the rotation number of a cyclic unimodular sequence L = u 1 u 2 … u d of lattice vectors u i ∈ Z 2 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 Z 2 , as fundamental discrete invariants of modular lattice geometry.