Knots and Classical 3-Geometries

TL;DR

This paper establishes and verifies Rovelli's conjecture linking link classes in Riemannian 3-manifolds to 3-geometries, providing a mathematical foundation for a proposed correspondence.

Contribution

It provides an exact statement and proof of Rovelli's conjecture for compact, orientable, closed 3-manifolds, advancing understanding of geometric and topological relations.

Findings

Confirmed the conjecture for specific 3-manifolds

Established a precise mathematical correspondence

Validated the conjecture's applicability in certain cases

Abstract

It has been conjectured by Rovelli that there is a correspondence between the space of link classes of a Riemannian 3-manifold and the space of 3-geometries (on the same manifold). An exact statement of his conjecture will be established and then verified for the case when the 3-manifold is compact, orientable and closed.