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

TL;DR

This paper develops a finite type invariant theory for null-homologous knots in rational homology 3-spheres, extending Goussarov-Rozansky theory, and identifies universal invariants in specific cases.

Contribution

It introduces a new finite type invariant framework for knots in rational homology spheres and relates known invariants as universal cases.

Findings

Kricker lift of Kontsevich integral is universal for trivial Alexander polynomial knots.

Lescop invariant from configuration space integrals is also universal in this setting.

Partial combinatorial description of the graded space is provided.

Abstract

We study a theory of finite type invariants for null-homologous 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 Goussarov-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 null-homologous 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 it implies that they are equivalent for such knots.