A universal equivariant finite type knot invariant defined from configuration space integrals

Abstract

In a previous article, we constructed an invariant Z for null-homologous knots in rational homology spheres, from equivariant intersections in configuration spaces. Here we present an equivalent definition of Z in terms of configuration space integrals, we prove that Z is multiplicative under connected sum, and we prove null Lagrangian-preserving surgery formulae for Z. Our formulae generalize similar formulae that are satisfied by the Kricker rational lift of the Kontsevich integral for null Borromean surgeries. They imply that Z is universal with respect to a natural filtration. According to results of Garoufalidis and Rozansky, they therefore imply that Z is equivalent to the Kricker lift of the Kontsevich integral for null-homologous knots with trivial Alexander polynomial in integral homology spheres.