Cyclic Symmetries of Chord Diagrams

We give a direct proof that the proalgebraic graded Grothendieck-Teichmüller group ${\mathsf{G}\mathsf{R}\mathsf{T}}_{\mathbb{K}}$ is isomorphic to the group of automorphisms of the prounipotent cyclic operad of parenthesized ribbon chord diagrams based on Furusho’s $5$-cycle reformulation of the pentagon equation. As an application, we describe a ${\mathsf{G}\mathsf{R}\mathsf{T}}_{\mathbb{K}}$-action on the category of framed chord diagrams with self-dual objects, which is closely related to the target category of the Kontsevich integral for framed tangles.