The –invariant is the simplest –manifold invariant defined by configuration space integrals. It is actually an invariant of rational homology spheres equipped with a combing over the complement of a point. It can be computed as the algebraic intersection of three propagators associated to a given combing in the –point configuration space of a –sphere . These propagators represent the linking form of so that can be thought of as the cube of the linking form of with respect to the combing . The invariant is the sum of and , where denotes the Casson–Walker invariant, and is an invariant of combings, which is an extension of a first relative Pontrjagin class. In this article, we present explicit propagators associated with Heegaard diagrams of a manifold, and we use these “Morse propagators”, constructed with Greg Kuperberg, to prove a combinatorial formula for the –invariant in terms of Heegaard diagrams.
"A formula for the $\Theta$–invariant from Heegaard diagrams." Geom. Topol. 19 (3) 1205 - 1248, 2015. https://doi.org/10.2140/gt.2015.19.1205