A topological characterisation of the Kashiwara-Vergne groups

TL;DR

This paper characterizes the Kashiwara-Vergne groups using automorphisms of algebraic structures related to welded foams and arrow diagrams, linking knot invariants with topological and algebraic symmetries.

Contribution

It provides a topological and algebraic description of the Kashiwara-Vergne groups as automorphisms of circuit algebras and arrow diagrams, connecting them to the Grothendieck-Teichmüller group.

Findings

Kashiwara-Vergne groups are automorphisms of welded foam circuit algebras.

The graded Grothendieck-Teichmüller group is described as automorphisms of arrow diagrams.

Topological interpretation of Kashiwara-Vergne solutions via knot invariants.

Abstract

In 2017 Bar-Natan and the first author showed that solutions to the Kashiwara--Vergne equations are in bijection with certain knot invariants: homomorphic expansions of welded foams. Welded foams are a class of knotted tubes in $\mathbb{R}^4$, which can be finitely presented algebraically as a circuit algebra, or, equivalently, a wheeled prop. In this paper we describe the Kashiwara-Vergne groups $\mathsf{KV}$ and $\mathsf{KRV}$ -- the symmetry groups of Kashiwara-Vergne solutions -- as automorphisms of the completed circuit algebras of welded foams, and their associated graded circuit algebra of arrow diagrams, respectively. Finally, we provide a description of the graded Grothendieck-Teichm\"uller group $\mathsf{GRT}_1$ as automorphisms of arrow diagrams.