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Rotation number of a unimodular cycle: an elementary approach. (English) Zbl 1281.05032
Summary: We give an elementary proof of a formula expressing the rotation number of a cyclic unimodular sequence \(L=u_1u_2\dots u_d\) of lattice vectors \(u_i\in\mathbb Z^2\) in terms of arithmetically defined local quantities. The formula has been originally derived by A. Higashitani and M. Masuda [“Lattice multi-polygons”, Preprint, arXiv:1204.0088] 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’ \(\mu (a,b,c)\), where \(\{a,b\}\) and \(\{b,c\}\) are bases of \(\mathbb Z^2\), as fundamental discrete invariants of modular lattice geometry.

MSC:
05B25 Combinatorial aspects of finite geometries
05B35 Combinatorial aspects of matroids and geometric lattices
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