M Theory Lesson 179

Recall that Bar-Natan's 1998 paper on the Grothendieck Teichmuller group discusses associated braids. On forgetting the crossing information, a braid in $B_n$ becomes a permutation on $n$ letters. So with bracketed endpoint sets, it marks a vertex of the $(n-1)$ dimensional permutoassociahedron for $S_n$.

As well as the category of bracketed permutations, Bar-Natan considers the category of bracketed braids with linearised morphisms of the form $\sum {\alpha }_j{B}^j$, where the $B^j$ are allowable braids corresponding to a given element $P\in S_n$ and the ${\alpha }_j$ are numerical coefficients. For example, the Pauli permutation ${\sigma }_x$ gives morphisms of the form where $a$ and $b$ are usually rational numbers. The functor to the permutation category that forgets the braid structure is an example of a fibration of a very nice kind. The GT group is a certain group of endofunctors (from bracketed braids to bracketed braids) which fix ${\sigma }_x$ (with a choice of crossing). Bar-Natan shows that the bracketed braids are generated by the two basic diagrams shown above (ie. associator and ${\sigma }_x$ and inverses).