Elliptic Associators and the LMO Functor

TL;DR

This paper explores the relationship between elliptic associators and the LMO functor, extending the Kontsevich integral to tangles in the thickened torus and providing insights into higher genus cases.

Contribution

It establishes an elliptic structure on the category of Jacobi diagrams via the LMO functor, relating it to Enriquez's elliptic associator, and offers an alternative proof of its properties.

Findings

Established a link between the LMO functor and elliptic associators.

Provided an alternative proof for properties of Enriquez's elliptic associator.

Suggested potential for constructing associators for higher genus surfaces.

Abstract

The elliptic associator of Enriquez can be used to define an invariant of tangles embedded in the thickened torus, which extends the Kontsevich integral. This construction by Humbert uses the formulation of categories with elliptic structures. In this work we show that an extension of the LMO functor also leads to an elliptic structure on the category of Jacobi diagrams which is used by the Kontsevich integral, and find the relation between the two structures. We use this relation to give an alternative proof for the properties of the elliptic associator of Enriquez. Those results can lead the way to finding associators for higher genra.