2012/09/22 by Rade T. Zivaljevic, Zivaljevic, Rade T.
Mathematics · #05A99 #11Axx #52Axx #Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG) #math.CO #math.MG #msc:05A99 #msc:11Axx #msc:52Axx
paper · pdf · doi:10.48550/arxiv.1209.4981
This version is identical to v2. By mistake v2 was replaced by v3 (a different paper) so v4 is just a correction of this mistake
arxiv created 2013/07/19 · arxiv updated 2013/07/22
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' μ(a,b,c), where a,b and b,c are bases of the lattice Z2, as fundamental discrete invariants of 'modular lattice geometry'.