A three-manifold invariant from graph configurations

The logarithm of the Kontsevich–Kuperberg–Thurston invariant counts embeddings of connected trivalent graphs in an oriented rational homology sphere, using integrals on configuration spaces of points in the given manifold. It is a universal finite type invariant of oriented rational homology spheres. The exponential of this invariant is often called the perturbative expansion of the Chern–Simons theory. In this article, we give an independent original definition of the degree two part of the logarithm of the Kontsevich–Kuperberg–Thurston invariant appropriate for concrete computations. This article can also serve as an introduction to the general definition of the Kontsevich–Kuperberg–Thurston invariant.