LDP polygons and the number 12 revisited

TL;DR

This paper provides a combinatorial proof of a lattice point identity involving polygons and their duals, extending a known area formula for reflexive polygons and linking it to the stringy Libgober-Wood identity in algebraic geometry.

Contribution

It introduces a new combinatorial proof of a lattice point identity that generalizes a classical area formula and connects to stringy invariants of toric surfaces.

Findings

Proves a generalized lattice point identity for polygons and their duals.

Establishes the equivalence of the identity with the stringy Libgober-Wood identity.

Extends classical results in lattice geometry to a broader class of polygons.

Abstract

We give a combinatorial proof of a lattice point identity involving a lattice polygon and its dual, generalizing the formula $area(\Delta) + area(\Delta^*) = 6$ for reflexive $\Delta$. The identity is equivalent to the stringy Libgober-Wood identity for toric log del Pezzo surfaces.