Lattice multi-polygons

Details Lattice multi-polygons Lattice multi-polygons

2012-03-31

arxiv.org/abs/1204.0088v2

Mathematics

Combinatorics

20 pages, 11 figures

Scientific paper

We discuss generalizations of some results on lattice polygons to certain piecewise linear loops which may have a self-intersection but have vertices in the lattice $\Z^2$. We first prove a formula on the rotation number of a unimodular sequence in $\Z^2$ using toric topology. This formula implies the generalized twelve-point theorem. We then introduce the notion of lattice multi-polygons which is a generalization of lattice polygons, state the generalized Pick's formula and discuss the classification of Ehrhart polynomials of lattice multi-polygons and also of several natural subfamilies of lattice multi-polygons.

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