Splitting formulas for the rational lift of the Kontsevich integral

Moussard, Delphine¹

¹Japan Society for the Promotion of Science, Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan, Institut de Mathématiques de Marseille, Aix-Marseille University, Marseille, France

Algebraic and Geometric Topology, Tome 20 (2020) no. 1, pp. 303-342

Résumé

Kricker defined an invariant of knots in homology 3–spheres which is a rational lift of the Kontsevich integral and proved with Garoufalidis that this invariant satisfies splitting formulas with respect to a surgery move called null-move. We define a functorial extension of the Kricker invariant and prove splitting formulas for this functorial invariant with respect to null Lagrangian-preserving surgery, a generalization of the null-move. We apply these splitting formulas to the Kricker invariant.

Détail

DOI : 10.2140/agt.2020.20.303

Classification : 57M27, 57M25, 57N10
Keywords: $3$–manifold, knot, homology sphere, cobordism category, Lagrangian cobordism, bottom–top tangle, beaded Jacobi diagram, Kontsevich integral, LMO invariant, Kricker invariant, Lagrangian-preserving surgery, finite type invariant, splitting formula

Bibliographie

[1] D Bar-Natan, On the Vassiliev knot invariants, Topology 34 (1995) 423 | DOI

[2] D Bar-Natan, S Garoufalidis, L Rozansky, D P Thurston, The Århus integral of rational homology 3–spheres, I : A highly non trivial flat connection on S3, Selecta Math. 8 (2002) 315 | DOI

[3] D Bar-Natan, S Garoufalidis, L Rozansky, D P Thurston, The Århus integral of rational homology 3–spheres, II : Invariance and universality, Selecta Math. 8 (2002) 341 | DOI

[4] D Bar-Natan, R Lawrence, A rational surgery formula for the LMO invariant, Israel J. Math. 140 (2004) 29 | DOI

[5] D Cheptea, K Habiro, G Massuyeau, A functorial LMO invariant for Lagrangian cobordisms, Geom. Topol. 12 (2008) 1091 | 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] K Habiro, Claspers and finite type invariants of links, Geom. Topol. 4 (2000) 1 | DOI

[10] K Habiro, T Widmer, On Kirby calculus for null-homotopic framed links in 3–manifolds, Algebr. Geom. Topol. 14 (2014) 115 | DOI

[11] D M Jackson, I Moffatt, A Morales, On the group-like behavior of the Le–Murakami–Ohtsuki invariant, J. Knot Theory Ramifications 16 (2007) 699 | DOI

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

[13] T T Q Le, H Murakami, J Murakami, T Ohtsuki, A three-manifold invariant via the Kontsevich integral, Osaka J. Math. 36 (1999) 365

[14] T Q T Le, J Murakami, Representation of the category of tangles by Kontsevich’s iterated integral, Comm. Math. Phys. 168 (1995) 535 | DOI

[15] T Q T Le, J Murakami, The universal Vassiliev–Kontsevich invariant for framed oriented links, Compositio Math. 102 (1996) 41

[16] T T Q Le, J Murakami, Parallel version of the universal Vassiliev–Kontsevich invariant, J. Pure Appl. Algebra 121 (1997) 271 | DOI

[17] C Lescop, Splitting formulae for the Kontsevich–Kuperberg–Thurston invariant of rational homology 3–spheres, preprint (2004)

[18] C Lescop, Invariants of knots and 3–manifolds derived from the equivariant linking pairing, from: "Chern–Simons gauge theory : 20 years after" (editors J E Andersen, H U Boden, A Hahn, B Himpel), AMS/IP Stud. Adv. Math. 50, Amer. Math. Soc. (2011) 217

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

[20] G Massuyeau, Splitting formulas for the LMO invariant of rational homology three-spheres, Algebr. Geom. Topol. 14 (2014) 3553 | DOI

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

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

[23] D Moussard, Equivariant triple intersections, Ann. Fac. Sci. Toulouse Math. 26 (2017) 601 | DOI

[24] D Moussard, Finite type invariants of knots in homology 3–spheres with respect to null LP-surgeries, Geom. Topol. 23 (2019) 2005 | DOI

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