Étant donné un nœud dans une sphère d’homologie rationnelle , et le revêtement infini cyclique standard de , on définit un invariant des triplets de courbes dans , via des intersections triples équivariantes de surfaces. On montre que cet invariant fournit une application sur , où est le module d’Alexander de , et que la classe d’isomorphisme de est un invariant de la paire . Pour un module de Blanchfield fixé, on considère les paires dont le module de Blanchfield est isomorphe à , équippées d’un marquage, c’est-à-dire d’un isomorphisme fixé de vers le module de Blanchfield de . Dans ce cadre, on calcule la variation de sous l’effet d’une chirurgie borroméenne nulle, et on décrit l’ensemble de toutes les applications . Enfin, on montre que l’application est un invariant de type fini de degré 1 des paires marquées par rapport aux chirurgies LP nulles, et on détermine l’espace de tous les invariants de degré 1 à valeurs rationnelles des paires marquées .
Given a null-homologous knot in a rational homology 3-sphere , and the standard infinite cyclic covering of , we define an invariant of triples of curves in by means of equivariant triple intersections of surfaces. We prove that this invariant provides a map on , where is the Alexander module of , and that the isomorphism class of is an invariant of the pair . For a fixed Blanchfield module , we consider pairs whose Blanchfield modules are isomorphic to equipped with a marking, i.e. a fixed isomorphism from to the Blanchfield module of . In this setting, we compute the variation of under null Borromean surgeries and we describe the set of all maps . Finally, we prove that the map is a finite type invariant of degree 1 of marked pairs with respect to null Lagrangian-preserving surgeries, and we determine the space of all degree 1 invariants with rational values of marked pairs .
Accepté le :
Publié le :
DOI : 10.5802/afst.1547
Mots clés : Knot, Homology sphere, Equivariant intersection, Alexander module, Blanchfield form, Borromean surgery, Null-move, Lagrangian-preserving surgery, Finite type invariant.
Delphine Moussard 1
CC-BY 4.0
@article{AFST_2017_6_26_3_601_0,
author = {Delphine Moussard},
title = {Equivariant triple intersections},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {601--643},
publisher = {Universit\'e Paul Sabatier, Toulouse},
volume = {Ser. 6, 26},
number = {3},
year = {2017},
doi = {10.5802/afst.1547},
mrnumber = {3669967},
zbl = {1411.57030},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1547/}
}
TY - JOUR AU - Delphine Moussard TI - Equivariant triple intersections JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2017 SP - 601 EP - 643 VL - 26 IS - 3 PB - Université Paul Sabatier, Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1547/ DO - 10.5802/afst.1547 LA - en ID - AFST_2017_6_26_3_601_0 ER -
%0 Journal Article %A Delphine Moussard %T Equivariant triple intersections %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2017 %P 601-643 %V 26 %N 3 %I Université Paul Sabatier, Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1547/ %R 10.5802/afst.1547 %G en %F AFST_2017_6_26_3_601_0
Delphine Moussard. Equivariant triple intersections. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 26 (2017) no. 3, pp. 601-643. doi : 10.5802/afst.1547. https://afst.centre-mersenne.org/articles/10.5802/afst.1547/
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