Brauer diagrams, modular operads, and a graphical nerve theorem for circuit algebras
arXiv:2108.04557
Abstract
Circuit algebras, used in the study of finite-type knot invariants, are a symmetric analogue of Jones's planar algebras. They are very closely related to circuit operads, which are a variation of modular operads admitting an extra monoidal product. This paper gives a description of circuit algebras in terms categories of Brauer diagrams. An abstract nerve theorem for circuit operads -- and hence circuit algebras -- is proved using an iterated distributive law, and an existing nerve theorem for modular operads.
As of December 2024, this paper has been superceded by arXiv:2412.20260 (on circuit algebras, Brauer diagrams, modular operads and invariant theory) and arXiv:2412.20262 (on a graphical calculus, iterated distributive laws and nerve theorem for circuit algebras). The version below is from November 2022 (c.f., v2). It comprises 65 pages and many figures