Brauer diagrams, modular operads, and a graphical nerve theorem for circuit algebras

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.

Categories of graphs for operadic structures

“…Moreover, the active-inert factorisation systems on these categories, which are used to formulate the Segal condition, should arise in a natural and transparent way (see Remark 4•16). This would all play a role in developments related to [50], which concerns the circuit algebras of Bar-Natan and Dansco [2]. These circuit algebras are a generalisation of the planar algebras of Jones [38], and turn out to be the same thing as wheeled props [18].…”

Modular operads as modules over the Brauer properad

“…Remark 4.41. A version of the first half of Theorem 4.38 for a kind of "ungraded nonconnected modular operads" (which first appeared in work of Schwarz [Sch,§2] and are mentioned in [KW,§2.3.2]) is implicitly contained in work of Raynor [Ray,Corollary 4.12 and Proposition 4.6]. They are described as lax symmetric monoidal functors out of a category of "downward Brauer diagrams" (cf.…”