Equivariant triple intersections

International audienceGiven a null-homologous knot $K$ in a rational homology 3-sphere $M$, and the standard infinite cyclic covering $\widetilde{X}$ of $(M,K)$, we define an invariant of triples of curves in $\widetilde{X}$ by means of equivariant triple intersections of surfaces. We prove that this invariant provides a map $\phi $ on $\mathfrak{A}^{\otimes 3}$, where $\mathfrak{A}$ 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 $(\mathfrak{A},\mathfrak{b})$, we consider pairs $(M,K)$ whose Blanchfield modules are isomorphic to $(\mathfrak{A},\mathfrak{b})$ equipped with a marking, i.e. a fixed isomorphism from $(\mathfrak{A},\mathfrak{b})$ 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 with rational values of marked pairs $(M,K)$