Functorial, operadic and modular operadic combinatorics of circuit algebras

Raynor, Sophie

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.

Journal

Journal of Pure and Applied Algebra

Publication Name

Journal of Pure and Applied Algebra

Volume

229

ISBN/ISSN

1873-1376

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Publisher

Elsevier

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N/A

DOI

10.1016/j.jpaa.2025.108105