Cyclic Symmetries of Chord Diagrams

TL;DR

This paper proves an isomorphism between the Grothendieck-Teichmüller group and automorphisms of a cyclic operad of chord diagrams, revealing new symmetries and actions relevant to knot theory.

Contribution

It provides a direct proof of the isomorphism and describes a GRT_K-action on framed chord diagrams related to the Kontsevich integral.

Findings

GRT_K is isomorphic to automorphisms of the cyclic operad of chord diagrams.

A GRT_K-action on the category of framed chord diagrams is constructed.

The results connect GRT_K with symmetries in knot and tangle invariants.

Abstract

We give a direct proof that the proalgebraic graded Grothendieck-Teichm\"uller group $\mathsf{GRT}_{\mathbb{K}}$ 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 $\mathsf{GRT}_{\mathbb{K}}$-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.