Lattice multi-polygons

Abstract

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 ${\mathbb{Z}}^2$. We first prove a formula on the rotation number of a unimodular sequence in ${\mathbb{Z}}^2$. This formula implies the generalized twelve-point theorem in [12]. 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.

21 pages, 7 figures, Kyoto J. Math. to appear

References in corpus (2)

• Reflexive polytopes of higher index and the number 12
• A short proof of the Twelve points theorem

Cited by in corpus (2)

• Toric origami manifolds and multi-fans
• Rotation number of primitive vector sequences