Given a null-homologous knot $K$ in a rational homology 3-sphere $M$, and the standard infinite cyclic covering $tilde{X}$ of $(M,K)$, we define an invariant of triples of curves in $tilde{X}$, by means of equivariant triple intersections of surfaces
. We prove that this invariant provides a map $phi$ on $Al^{otimes 3}$, where $Al$ is the Alexander module of $(M,K)$, and that the isomorphism class of $phi$ is an invariant of the pair $(M,K)$. For a fixed Blanchfield module $(Al,bl)$, we consider pairs $(M,K)$ whose Blanchfield modules are isomorphic to $(Al,bl)$, equipped with a marking, {em i.e.} a fixed isomorphism from $(Al,bl)$ to the Blanchfield module of $(M,K)$. In this setting, we compute the variation of $phi$ under null borromean surgeries, and we describe the set of all maps $phi$. Finally, we prove that the map $phi$ is a finite type invariant of degree 1 of marked pairs $(M,K)$ with respect to null Lagrangian-preserving surgeries, and we determine the space of all degree 1 invariants of marked pairs $(M,K)$ with rational values.
Null Lagrangian-preserving surgeries are a generalization of the Garoufalidis and Rozansky null-moves, that these authors introduced to study the Kricker lift of the Kontsevich integral, in the setting of pairs (M,K) composed of a rational homology s
phere M and a null-homologous knot K in M. They are defined as replacements of null-homologous rational homology handlebodies of MK by other such handlebodies with identical Lagrangian. A null Lagrangian-preserving surgery induces a canonical isomorphism between the rational Alexander modules of the involved pairs, which preserves the Blanchfield form. Conversely, we prove that a fixed isomorphism between rational Alexander modules which preserves the Blanchfield form can be realized, up to multiplication by a power of t, by a finite sequence of null Lagrangian-preserving surgeries. We also prove that such classes of isomorphisms can be realized by rational S-equivalences. In the case of integral homology spheres, we prove similar realization results for a fixed isomorphism between integral Alexander modules.
