Functorial, operadic and modular operadic combinatorics of circuit algebras
Abstract
Circuit algebras are a symmetric analogue of Jones's planar algebras introduced to study finite-type invariants of virtual knotted objects. Circuit algebra structures appear, in different forms, across mathematics. This paper provides a dictionary for translating between their diverse incarnations and describing their wider context. A formal definition of a broad class of circuit algebras is established and three equivalent descriptions of circuit algebras are provided: in terms of operads of wiring diagrams, modular operads and categories of Brauer diagrams. As an application, circuit algebra characterisations of algebras over the orthogonal and symplectic groups are given.
Cite
@article{arxiv.2412.20260,
title = {Functorial, operadic and modular operadic combinatorics of circuit algebras},
author = {Sophie Raynor},
journal= {arXiv preprint arXiv:2412.20260},
year = {2025}
}
Comments
36 pages, many figures. Relative to version 1, the title has been changed, the introduction has been tidied and a table added. Other changes are minor. This paper and "Modular operads, distributive laws and a nerve theorem for circuit algebras" supercede "Brauer diagrams, modular operads, and a graphical nerve theorem for circuit algebras" arXiv:2108.04557. Comments welcome