With cedram.org All online volumes Latest online issues Advanced Search Table of contents for this issue | Previous article | Next article Delphine MoussardEquivariant triple intersectionsAnnales de la faculté des sciences de Toulouse Sér. 6, 26 no. 3 (2017), p. 601-643, doi: 10.5802/afst.1547 Article PDF Class. Math.: 57M27, 57M25, 57N65, 57N10Keywords: Knot, Homology sphere, Equivariant intersection, Alexander module, Blanchfield form, Borromean surgery, Null-move, Lagrangian-preserving surgery, Finite type invariant. Résumé - AbstractGiven 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 $(M,K)$. Bibliography[1] Emmanuel Auclair & Christine Lescop, “Clover calculus for homology 3-spheres via basic algebraic topology”, Algebr. Geom. Topol. 5 (2005), p. 71-106 [2] Richard C. Blanchfield, “Intersection theory of manifolds with operators with applications to knot theory”, Ann. Math. 65 (1957), p. 340-356 [3] Stavros Garoufalidis, Mikhail Goussarov & Michael Polyak, “Calculus of clovers and finite type invariants of 3-manifolds”, Geom. Topol. 5 (2001), p. 75-108 [4] Stavros Garoufalidis & Andrew Kricker, “A rational noncommutative invariant of boundary links”, Geom. Topol. 8 (2004), p. 115-204 [5] Stavros Garoufalidis & Lev Rozansky, “The loop expansion of the Kontsevich integral, the null-move and $S$-equivalence”, Topology 43 (2004) no. 5, p. 1183-1210 [6] Andrew Kricker, “The lines of the Kontsevich integral and Rozansky’s rationality conjecture”, http://arxiv.org/abs/math/0005284, 2000 [7] Christine Lescop, “On the cube of the equivariant linking pairing for knots and 3-manifolds of rank one”, http://arxiv.org/abs/1008.5026, 2010  MR 2809454[8] Christine Lescop, Invariants of knots and 3-manifolds derived from the equivariant linking pairing, Chern-Simons gauge theory: 20 years after, AMS/IP Stud. Adv. Math. 50, AMS, Providence, RI, 2011, p. 217–242 [9] W. B. Raymond Lickorish, An introduction to knot theory, Graduate Texts in Mathematics 175, Springer-Verlag, New York, 1997 [10] Sergei V. Matveev, “Generalized surgeries of three-dimensional manifolds and representations of homology spheres”, Mat. Zametki 42 (1987) no. 2, p. 268-278, (in Russian), English transl.: Math. Notes, 42 (1987), p. 651-656 [11] Delphine Moussard, Équivariance et invariants de type fini en dimension trois, Ph. D. Thesis, University of Grenoble (France), 2012 [12] Delphine Moussard, “Finite type invariants of rational homology 3-spheres”, Algebr. Geom. Topol. 12 (2012) no. 4, p. 2389-2428 [13] Delphine Moussard, “On Alexander modules and Blanchfield forms of null-homologous knots in rational homology spheres”, J. Knot Theory Ramifications 21 (2012) no. 5 [14] Delphine Moussard, “Rational Blanchfield forms, S-equivalence, and null LP-surgeries”, Bull. Soc. Math. Fr. 143 (2015) no. 2, p. 403-431 Search for an article Search within the site