Rationality theorems for curvature invariants of 2-complexes

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.

39 pages, 3 figures, companion to arXiv:2210.09841. v3 corrects an error in Section 2. v4 is the final version accepted for publication