# Equivariant triple intersections

Abstract : Given 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 $(M,K)$.
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https://hal.archives-ouvertes.fr/hal-00954767
Contributor : Delphine Moussard <>
Submitted on : Saturday, August 15, 2015 - 4:23:23 PM
Last modification on : Friday, June 8, 2018 - 2:50:07 PM
Long-term archiving on : Monday, November 16, 2015 - 10:45:46 AM

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### Citation

Delphine Moussard. Equivariant triple intersections. Annales de la Faculté des Sciences de Toulouse. Mathématiques., Université Paul Sabatier _ Cellule Mathdoc 2017, 26 (3), pp.601 - 643. ⟨10.5802/afst.1547⟩. ⟨hal-00954767v2⟩

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