Abstract: Going beyond the world of usual tangles, one important family we encounter is that of welded or w-tangles. Work of Satoh shows that they (conjecturally) capture the topology of certain knotted surfaces in 4-space. W-foams are a generalization of w-tangles which involves adding trivalent vertices. Bar-Natan and Dancso showed that a collection of w-foam invariants called homomorphic expansions are essentially in one-to-one correspondence with solutions to the Kashiwara–Vergne (KV) problem, thus relating them to a question in Lie theory.
I will present joint work with Dancso and Robertson, where we show that the circuit algebra structure of w-tangles (and w-foams) makes them a wheeled prop, as well as discuss some progress we’ve made towards identifying the group of automorphisms of w-foams with the group acting on KV solutions.