Lattice multi-polygons

TL;DR

This paper extends classical results on lattice polygons to self-intersecting lattice loops called lattice multi-polygons, establishing new formulas for rotation number, generalized Pick's theorem, and classifying Ehrhart polynomials.

Contribution

It introduces lattice multi-polygons, proves a rotation number formula, and generalizes Pick's theorem and Ehrhart polynomial classification for these objects.

Findings

Proved a formula for the rotation number of unimodular sequences.

Established a generalized twelve-point theorem.

Classified Ehrhart polynomials of 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.