Chandan Singh: A hidden symmetry of Grothendieck–Teichmüller Group

Abstract.

In 2008, Budney showed that the operad of framed little disks admits a cyclic structure. In this work, we show that this translates to a cyclic structure on a groupoid model for the framed little disks: the operad of parenthesized ribbons braids. We extend the known action of the Grothendieck–Teichmüller group GT on the operad of parenthesised ribbon braids to its cyclic version. This provides a new characterization of GT as an automorphism group of the prounipotent cyclic operad of parenthesized ribbons. As a consequence, we exploit this GT action to provide a simple proof of the formality of the cyclic framed little disk operad. Finally, this action extends to the category of parenthesized tangles and verifies the conjecture of Kessel-Turaev about Galois actions on framed tangles.    The work discussed here sets the foundation for exploring GT actions on a broader class of knotted objects such as B-tangles, virtual tangles, and welded tangles, which will be addressed in our future work. This is a joint work with Marcy Robertson.