A topological characterisation of the Kashiwara–Vergne groups

In [Math. Ann. 367 (2017), pp. 1517–1586] Bar-Natan and the first author show that solutions to Kashiwara–Vergne equations are in bijection with certain knot invariants: homomorphic expansions of welded foams. Welded foams a class knotted tubes R 4 \mathbb {R}^4 , which can be finitely presented algebraically as circuit algebra, or equivalently, wheeled prop. this paper we describe Kashiwara-Vergne groups alttext="sans-serif K sans-serif V"> mathvariant="sans-serif">K mathvariant="sans-serif">V encoding="application/x-tex">\mathsf {KV} mathvariant="sans-serif">R {KRV} —the symmetry solutions—as automorphisms completed circuit algebras welded foams, their associated graded arrow diagrams, respectively. Finally, provide description Grothendieck-Teichmüller group G T Subscript 1"> mathvariant="sans-serif">G mathvariant="sans-serif">T 1 {GRT}_1 diagrams.