Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Cyclic Symmetries of Chord Diagrams

Chandan Singh

Abstract

We give a direct proof that the proalgebraic graded Grothendieck-Teichmüller 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.

Cyclic Symmetries of Chord Diagrams

Abstract

We give a direct proof that the proalgebraic graded Grothendieck-Teichmüller group is isomorphic to the group of automorphisms of the prounipotent cyclic operad of parenthesized ribbon chord diagrams based on Furusho's -cycle reformulation of the pentagon equation. As an application, we describe a -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.

Paper Structure

This paper contains 10 sections, 19 theorems, 109 equations, 2 figures.

Table of Contents

  1. A cyclic operad of chord diagrams
  2. Cyclic Operads
  3. Cyclic Structure on Ribbon Chord Diagrams
  4. Drinfeld Associators
  5. The graded Grothendieck--Teichmüller group and cyclic automorphisms
  6. Action on Chord Diagrams
  7. Infinitesimal Symmetric Monoidal Categories
  8. The category $\mathbf{A}(\mathbb{K})$
  9. Envelop and Metric Prop
  10. GRT action on Chord diagrams

Key Result

Theorem 1

${\mathsf{GRT}}_{\mathbb{K}} \cong \operatorname{Aut}_{\mathbf{Cyc}}^{+}({\mathsf{PaRCD}}_{\mathbb{K}}^{\mathbf{cyc}})$. $\blacktriangleleft$$\blacktriangleleft$

Figures (2)

  • Figure 1: Generating morphisms of the operad $\mathsf{PaRCD}$.
  • Figure 2: The 4T relation

Theorems & Definitions (49)

  • Theorem 1
  • Theorem 2
  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Remark 1.4
  • Definition 1.5
  • Proposition 1.6
  • proof
  • Remark 1.7
  • ...and 39 more