2021/06/04 by Dancso, Zsuzsanna, Halacheva, Iva, Robertson, Marcy
#Algebraic Topology (math.AT) #FOS: Mathematics #Geometric Topology (math.GT) #Quantum Algebra (math.QA)
paper · doi:10.48550/arxiv.2106.02373
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 ℝ4, which can be finitely presented algebraically as a circuit algebra, or, equivalently, a wheeled prop. In this paper we describe the Kashiwara-Vergne groups KV and 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-Teichmüller group GRT1 as automorphisms of arrow diagrams.