Finite type invariants of knots in homology 3–spheres with respect to null LP–surgeries

Moussard, Delphine¹

¹ Institut de Mathématiques de Bourgogne, Université de Bourgogne, Dijon, France

Geometry & topology, Tome 23 (2019) no. 4, pp. 2005-2050.

Résumé

We study a theory of finite type invariants for nullhomologous knots in rational homology $3$–spheres with respect to null Lagrangian-preserving surgeries. It is an analogue in the setting of the rational homology of the Garoufalidis–Rozansky theory for knots in integral homology $3$–spheres. We give a partial combinatorial description of the graded space associated with our theory and determine some cases when this description is complete. For nullhomologous knots in rational homology $3$–spheres with a trivial Alexander polynomial, we show that the Kricker lift of the Kontsevich integral and the Lescop equivariant invariant built from integrals in configuration spaces are universal finite type invariants for this theory; in particular, this implies that they are equivalent for such knots.

Détail

DOI : 10.2140/gt.2019.23.2005

Classification : 57M27
Keywords: 3-manifold, knot, homology sphere, beaded Jacobi diagram, Kontsevich integral, Borromean surgery, null-move, Lagrangian-preserving surgery, finite type invariant

Bibliographie

[1] E Auclair, Surfaces et invariants de type fini en dimension 3, PhD thesis, Université Joseph-Fourier – Grenoble I (2006)

[2] E Auclair, C Lescop, Clover calculus for homology 3–spheres via basic algebraic topology, Algebr. Geom. Topol. 5 (2005) 71 | DOI

[3] B Audoux, D Moussard, Toward universality in degree 2 of the Kricker lift of the Kontsevich integral and the Lescop equivariant invariant, preprint (2017)

[4] R C Blanchfield, Intersection theory of manifolds with operators with applications to knot theory, Ann. of Math. 65 (1957) 340 | DOI

[5] T D Cochran, P Melvin, Finite type invariants of 3–manifolds, Invent. Math. 140 (2000) 45 | DOI

[6] S Garoufalidis, M Goussarov, M Polyak, Calculus of clovers and finite type invariants of 3–manifolds, Geom. Topol. 5 (2001) 75 | DOI

[7] S Garoufalidis, A Kricker, A rational noncommutative invariant of boundary links, Geom. Topol. 8 (2004) 115 | DOI

[8] S Garoufalidis, L Rozansky, The loop expansion of the Kontsevich integral, the null-move and S–equivalence, Topology 43 (2004) 1183 | DOI

[9] N Habegger, Milnor, Johnson, and tree level perturbative invariants, preprint (2000)

[10] K Habiro, Claspers and finite type invariants of links, Geom. Topol. 4 (2000) 1 | DOI

[11] A Kricker, The lines of the Kontsevich integral and Rozansky’s rationality conjecture, preprint (2000)

[12] C Lescop, Invariants of knots and 3–manifolds derived from the equivariant linking pairing, from: "Chern–Simons gauge theory : years after" (editors J E Andersen, H U Boden, A Hahn, B Himpel), AMS/IP Stud. Adv. Math. 50, Amer. Math. Soc. (2011) 217

[13] C Lescop, A universal equivariant finite type knot invariant defined from configuration space integrals, preprint (2013)

[14] S V Matveev, Generalized surgeries of three-dimensional manifolds and representations of homology spheres, Mat. Zametki 42 (1987) 268

[15] D Moussard, Finite type invariants of rational homology 3–spheres, Algebr. Geom. Topol. 12 (2012) 2389 | DOI

[16] D Moussard, On Alexander modules and Blanchfield forms of null-homologous knots in rational homology spheres, J. Knot Theory Ramifications 21 (2012) | DOI

[17] D Moussard, Rational Blanchfield forms, S–equivalence, and null LP–surgeries, Bull. Soc. Math. France 143 (2015) 403 | DOI

[18] D Moussard, Splitting formulas for the rational lift of the Kontsevich integral, preprint (2017)

[19] T Ohtsuki, Finite type invariants of integral homology 3–spheres, J. Knot Theory Ramifications 5 (1996) 101 | DOI