Grothendieck--Teichm\"uller Symmetries of Cyclic Operads and Tangles

TL;DR

This paper characterizes the profinite Grothendieck-Teichmüller group as automorphisms of a cyclic operad related to tangles, revealing deep symmetries and providing new insights into their algebraic and topological structures.

Contribution

It introduces an operadic model for profinite tangles and their symmetries, connecting the Grothendieck-Teichmüller group to cyclic operads and framed tangles.

Findings

$\widehat{ ext{GT}}$ acts naturally on tangles.

Provides an alternative proof of formality of cyclic framed little disks operad.

Establishes an operadic framework linking symmetries to tangles.

Abstract

We characterise the profinite Grothendieck-Teichm\"uller group $\widehat{\mathsf{GT}}$ as the group of automorphisms of the profinite completion of a cyclic operad of parenthesised ribbon braids. This operad generates a symmetric monoidal category which is equivalent to the category of framed, oriented tangles, thereby providing an operadic model for profinite tangles and their arithmetic symmetries. As applications, we show that $\widehat{\mathsf{GT}}$ acts naturally on tangles and provide an alternative proof of the formality of the cyclic framed little disks operad.