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

TL;DR

This paper explores the structure of circuit algebras and operads using Brauer diagrams and establishes a nerve theorem connecting these concepts, advancing the understanding of algebraic frameworks in knot theory.

Contribution

It provides a categorical description of circuit algebras via Brauer diagrams and proves a nerve theorem for circuit operads using distributive laws and modular operad results.

Findings

Categorical description of circuit algebras through Brauer diagrams

Proved a nerve theorem for circuit operads and algebras

Connected circuit algebra structures to modular operad theory

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.