Welded Tangles and a Topological Interpretation of the Kashiwara-Vergne Group

Explore the fascinating world of welded tangles and their connection to the Kashiwara-Vergne group in this illuminating lecture by Iva Halacheva from Northeastern University. Delve into the higher-dimensional analogue of classical tangles known as welded or w-tangles, and discover their generalization to welded foams or w-trivalent graphs, representing knotted tubes in 4-dimensional space. Learn how welded foams can be algebraically presented as a circuit algebra and uncover the intriguing relationship between their automorphisms and the Kashiwara-Vergne group, which plays a crucial role in the Baker-Campbell-Hausdorff series. Examine the groundbreaking result by Bar-Natan and Dancso that identifies homomorphic expansions for welded foams with solutions to the Kashiwara-Vergne equations, providing a powerful framework for knot invariants. Gain insights into this captivating intersection of topology, algebra, and Lie theory in this hour-long presentation from the University of Miami.