A universal finite type invariant of knots in homology 3-spheres

TL;DR

This paper introduces a new, stronger universal finite type invariant for knots in homology 3-spheres, extending the known invariants and providing a complete diagrammatic description of their finite type invariants.

Contribution

It constructs a refined invariant that is proven to be universal for knots in homology 3-spheres, surpassing previous invariants in power.

Findings

The new invariant is strictly stronger than the Garoufalidis-Kricker invariant.

It provides a full diagrammatic description of the graded space of finite type invariants.

The invariant is universal for all knots in homology 3-spheres.

Abstract

An essential goal in the study of finite type invariants of some objects (knots, manifolds) is the construction of a universal finite type invariant, universal in the sense that it contains all finite type invariants of the given objects. Such a universal finite type invariant is known for knots in the 3-sphere -- the Kontsevich integral -- and for homology 3-spheres -- the Le-Murakami-Ohtsuki invariant. For knots in homology 3-spheres, an invariant constructed by Garoufalidis and Kricker as a lift of the Kontsevich integral has been considered for the last two decades as the best candidate to be a universal finite type invariant. Although this invariant is eventually universal in restriction to knots whose Alexander polynomial is trivial, we prove here that it is not powerful enough in general. For that we provide a refinement of its construction which produces a strictly stronger invariant, and we prove that this new invariant is a universal finite type invariant of knots in homology 3-spheres. This provides a full diagrammatic description of the graded space of finite type invariants of knots in homology 3-spheres.