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Equivariant triple intersections
Delphine Moussard
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 26 (2017) no. 3, p. 601-643

Given a null-homologous knot K in a rational homology 3-sphere M, and the standard infinite cyclic covering X ˜ of (M,K), we define an invariant of triples of curves in X ˜ by means of equivariant triple intersections of surfaces. We prove that this invariant provides a map φ on 𝔄 3 , where 𝔄 is the Alexander module of (M,K), and that the isomorphism class of φ is an invariant of the pair (M,K). For a fixed Blanchfield module (𝔄,𝔟), we consider pairs (M,K) whose Blanchfield modules are isomorphic to (𝔄,𝔟) equipped with a marking, i.e. a fixed isomorphism from (𝔄,𝔟) to the Blanchfield module of (M,K). In this setting, we compute the variation of φ under null Borromean surgeries and we describe the set of all maps φ. Finally, we prove that the map φ 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).

Étant donné un nœud K dans une sphère d’homologie rationnelle M, et le revêtement infini cyclique standard X ˜ de (M,K), on définit un invariant des triplets de courbes dans X ˜, via des intersections triples équivariantes de surfaces. On montre que cet invariant fournit une application φ sur 𝔄 3 , où 𝔄 est le module d’Alexander de (M,K), et que la classe d’isomorphisme de φ est un invariant de la paire (M,K). Pour un module de Blanchfield (𝔄,𝔟) fixé, on considère les paires (M,K) dont le module de Blanchfield est isomorphe à (𝔄,𝔟), équippées d’un marquage, c’est-à-dire d’un isomorphisme fixé de (𝔄,𝔟) vers le module de Blanchfield de (M,K). Dans ce cadre, on calcule la variation de φ sous l’effet d’une chirurgie borroméenne nulle, et on décrit l’ensemble de toutes les applications φ. Enfin, on montre que l’application φ est un invariant de type fini de degré 1 des paires marquées (M,K) par rapport aux chirurgies LP nulles, et on détermine l’espace de tous les invariants de degré 1 à valeurs rationnelles des paires marquées (M,K).

Received : 2015-09-06
Accepted : 2016-05-24
Published online : 2017-06-13
DOI : https://doi.org/10.5802/afst.1547
Classification:  57M27,  57M25,  57N65,  57N10
Keywords: Knot, Homology sphere, Equivariant intersection, Alexander module, Blanchfield form, Borromean surgery, Null-move, Lagrangian-preserving surgery, Finite type invariant.
@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},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 26},
     number = {3},
     year = {2017},
     pages = {601-643},
     doi = {10.5802/afst.1547},
     language = {en},
     url = {https://afst.centre-mersenne.org/item/AFST_2017_6_26_3_601_0}
}
Moussard, Delphine. Equivariant triple intersections. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 26 (2017) no. 3, pp. 601-643. doi : 10.5802/afst.1547. afst.centre-mersenne.org/item/AFST_2017_6_26_3_601_0/

[1] Emmanuel Auclair; Christine Lescop Clover calculus for homology 3-spheres via basic algebraic topology, Algebr. Geom. Topol., Tome 5 (2005), pp. 71-106

[2] Richard C. Blanchfield Intersection theory of manifolds with operators with applications to knot theory, Ann. Math., Tome 65 (1957), pp. 340-356

[3] Stavros Garoufalidis; Mikhail Goussarov; Michael Polyak Calculus of clovers and finite type invariants of 3-manifolds, Geom. Topol., Tome 5 (2001), pp. 75-108

[4] Stavros Garoufalidis; Andrew Kricker A rational noncommutative invariant of boundary links, Geom. Topol., Tome 8 (2004), pp. 115-204

[5] Stavros Garoufalidis; Lev Rozansky The loop expansion of the Kontsevich integral, the null-move and S-equivalence, Topology, Tome 43 (2004) no. 5, pp. 1183-1210

[6] Andrew Kricker The lines of the Kontsevich integral and Rozansky’s rationality conjecture (2000) (http://arxiv.org/abs/math/0005284)

[7] Christine Lescop On the cube of the equivariant linking pairing for knots and 3-manifolds of rank one (2010) (http://arxiv.org/abs/1008.5026)

[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.) Tome 50, AMS, Providence, RI, 2011, pp. 217-242

[9] W. B. Raymond Lickorish An introduction to knot theory, Graduate Texts in Mathematics, Tome 175, Springer-Verlag, New York, 1997, x+201 pages

[10] Sergei V. Matveev Generalized surgeries of three-dimensional manifolds and representations of homology spheres, Mat. Zametki, Tome 42 (1987) no. 2, pp. 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 (2012) (Ph. D. Thesis)

[12] Delphine Moussard Finite type invariants of rational homology 3-spheres, Algebr. Geom. Topol., Tome 12 (2012) no. 4, pp. 2389-2428

[13] Delphine Moussard On Alexander modules and Blanchfield forms of null-homologous knots in rational homology spheres, J. Knot Theory Ramifications, Tome 21 (2012) no. 5, 1250042, 21 p. pages

[14] Delphine Moussard Rational Blanchfield forms, S-equivalence, and null LP-surgeries, Bull. Soc. Math. Fr., Tome 143 (2015) no. 2, pp. 403-431