Toward universality in degree 2 of the Kricker lift of the Kontsevich integral and the Lescop equivariant invariant

TL;DR

This paper investigates the universality of two finite type invariants for null-homologous knots in rational homology 3-spheres, focusing on degree 2 cases and establishing injectivity results with explicit kernel descriptions.

Contribution

It proves injectivity of the relevant maps for degree 2 invariants in most cases and explicitly characterizes the kernel in the exceptional case.

Findings

Injectivity holds for degree 2 invariants in most cases.

Explicit description of the kernel in the exceptional case.

Supports the conjecture that Kricker and Lescop invariants are universal.

Abstract

In the setting of finite type invariants for null-homologous knots in rational homology 3-spheres with respect to null Lagrangian-preserving surgeries, there are two candidates to be universal invariants, defined respectively by Kricker and Lescop. In a previous paper, the second author defined maps between spaces of Jacobi diagrams. Injectivity for these maps would imply that Kricker and Lescop invariants are indeed universal invariants; this would prove in particular that these two invariants are equivalent. In the present paper, we investigate the injectivity status of these maps for degree 2 invariants, in the case of knots whose Blanchfield modules are direct sums of isomorphic Blanchfield modules of Q-dimension two. We prove that they are always injective except in one case, for which we determine explicitly the kernel.