Circuit algebras and wheeled props
Circuit algebras are algebraic structures defined by Bar-Natan and Dancso in the context of virtual and welded tangles to construct fundamental and powerful topological invariants. They are generalizations of Jones's planar algebras where one drops the planarity condition, and can often be presented in terms of a small number of generators and relations. Wheeled props, which usually arise in geometry, were introduced by Markl, Merkulov, and Shadrin as a convenient way to encode axioms for operations with many inputs and outputs. We show that every directed circuit algebra is a wheeled prop, thus bridging several notions from algebraic knot theory and homotopy theory. I will discuss this result and several important examples on both sides. This is joint work with Zsuzsanna Dancso and Marcy Robertson.
Dr Iva Halacheva, University of Melbourne