Grothendieck--Teichmüller Symmetries of Cyclic Operads and Tangles

We characterise the profinite Grothendieck-Teichmüller group $\widehat{\mathsf{G}\mathsf{T}}$ 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 $\widehat{\mathsf{G}\mathsf{T}}$ acts naturally on tangles and provide an alternative proof of the formality of the cyclic framed little disks operad.