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

Delphine Moussard

Geometry & Topology 23 (2019) 2005–2050

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.

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3-manifold, knot, homology sphere, beaded Jacobi diagram, Kontsevich integral, Borromean surgery, null-move, Lagrangian-preserving surgery, finite type invariant
Mathematical Subject Classification 2010
Primary: 57M27
Received: 5 November 2017
Revised: 13 September 2018
Accepted: 15 November 2018
Published: 17 June 2019
Proposed: Cameron Gordon
Seconded: András I Stipsicz, Walter Neumann
Delphine Moussard
Institut de Mathématiques de Bourgogne
Université de Bourgogne