Jacobi identities in low-dimensional topology

TL;DR

This paper explores the role of the Jacobi identity, also known as the IHX-relation, in low-dimensional topology, connecting knot invariants, 3- and 4-manifold theories, and embedding obstructions through topological and algebraic perspectives.

Contribution

It reveals the topological unity of IHX-relations in 3- and 4-dimensional topology, linking knot invariants with embedding obstructions via grope cobordisms.

Findings

Identifies the IHX-relation as a unifying topological principle.

Connects knot invariants with 4-manifold embedding obstructions.

Provides a geometric interpretation using Borromean rings and handlebodies.

Abstract

The Jacobi identity is the key relation in the definition of a Lie algebra. In the last decade, it also appeared at the heart of the theory of finite type invariants of knots, links and 3-manifolds (and is there called the IHX-relation). In addition, this relation was recently found to arise naturally in a theory of embedding obstructions for 2-spheres in 4-manifolds. We expose the underlying topological unity between the 3- and 4-dimensional IHX-relations, deriving from a picture of the Borromean rings embedded on the boundary of an unknotted genus three handlebody in 3-space. This is most naturally related to knot and 3-manifold invariants via the theory of grope cobordisms.