Circuit algebras are wheeled props

TL;DR

This paper establishes an equivalence between circuit algebras, a generalization of planar algebras used in topology, and linear wheeled props, a type of symmetric tensor category relevant in mathematical physics.

Contribution

It provides the first classification of circuit algebras as linear wheeled props, extending the known classification of planar algebras.

Findings

Circuit algebras are equivalent to linear wheeled props.

The classification connects low-dimensional topology with homotopy and deformation theories.

Establishes a categorical framework for virtual and welded tangles.

Abstract

Circuit algebras, introduced by Bar-Natan and the first author, are a generalization of Jones's planar algebras, in which one drops the planarity condition on "connection diagrams". They provide a useful language for the study of virtual and welded tangles in low-dimensional topology. In this note, we present the circuit algebra analogue of the well-known classification of planar algebras as pivotal categories with a self-dual generator. Our main theorem is that there is an equivalence of categories between circuit algebras and the category of linear wheeled props - a type of strict symmetric tensor category with duals that arises in homotopy theory, deformation theory and the Batalin-Vilkovisky quantization formalism.