Integral invariants of reduced plumbing graphs

Abstract

An algebraic plumbing graph determines an integral trilinear form, a parity class, and a Pontryagin linear form. We prove rank-two rigidity and show that uniqueness fails first in rank three. For each fixed rank $r\geq5$, the numbers of connected reduced spin presentations are unbounded as the invariant system varies, even with nonzero labels and primitive normalized Pontryagin form. Coprime factorizations of an odd integer $N>1$ give $2^{\omega(N)-1}$ distinct reduced classes in one fibre. For each $N$, these graphs present a single oriented diffeomorphism class of six-manifolds admitting positive Ricci curvature. At fixed rank, varying $N$ gives pairwise nonisomorphic graded integral cohomology rings and a common rational homotopy type. They are indecomposable as connected sums and are not sphere-bundle total spaces with positive-dimensional base and fibre. In arbitrary rank, the restricted ambient Euclidean form, together with the trilinear form and parity class, recovers the reduced presentation. The Pontryagin form is then redundant. The proof recovers coordinate covectors from the Voronoi cell and the forest from their closed circuits. For reduced spin graphs with nonzero integer labels, the fifth-order coordinate moment alone also recovers the reduced class.