Authors: Živaljević, Rade 
Affiliations: Mathematical Institute of the Serbian Academy of Sciences and Arts 
Title: Rotation number of a unimodular cycle: An elementary approach
Journal: Discrete Mathematics
Volume: 313
Issue: 20
First page: 2253
Last page: 2261
Issue Date: 1-Jan-2013
Rank: M22
ISSN: 0012-365X
DOI: 10.1016/j.disc.2013.06.003
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.
Keywords: Lattice points | Rotation number | Toric topology | Unimodular sequence
Publisher: Elsevier
Project: Geometry and Topology of Manifolds, Classical Mechanics and Integrable Dynamical Systems 
Topology, geometry and global analysis on manifolds and discrete structures 

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