The profinite Grothendieck-Teichmüller group (GT), originally introduced by Drinfeld, is a powerful mathematical object characterized by its action on a tower of appropriately completed braid groups. It holds deep arithmetic significance, most notably containing the absolute Galois group of rationals. Traditionally, these rich symmetries have been encoded using the operad of parenthesised braids.
In this talk, I will demonstrate how GT naturally acts on the cyclic operad of parenthesised ribbon braids. By introducing this cyclic symmetry, we obtain an operad that generates a symmetric monoidal category equivalent to Furusho's category of profinite tangles. We will conclude by exploring how this framework allows us to extend GT (and absolute Galois) symmetries directly to these tangles, elegantly bridging the topological behavior of configuration spaces with deep arithmetic symmetries.