Rationality theorems for curvature invariants of 2-complexes

TL;DR

This paper proves that certain curvature invariants of finite 2-complexes are rational and can be computed via explicit linear programming problems, establishing their foundational properties and computability.

Contribution

It demonstrates that the curvature invariants are the extrema of rational linear programs, confirming their rationality and algorithmic computability.

Findings

Curvature invariants are rational numbers.

They are realized as extrema of explicit linear programs.

The results enable algorithmic computation of these invariants.

Abstract

Let $X$ be a finite, 2-dimensional cell complex. The curvature invariants $\rho_\pm(X)$ and $\sigma_\pm(X)$ were defined in [13], and a programme of conjectures was outlined. Here, we prove the foundational result that the quantities $\rho_\pm(X)$ and $\sigma_\pm(X)$ are the extrema of explicit rational linear-programming problems. As a result they are rational, realised, and can be computed algorithmically.