Gordon, Power and Street have proven that every monoidal 2-category is equivalent to a Gray monoid. This means that the only coherence 2-isomorphisms we have to be concerned about are the interchangers.
Similarly, Joyal and Street have proven a coherence theorem for braided monoidal categories. Basically, composites of braiding isomorphisms describe a braid diagram, and if two such braid diagrams are equivalent, then so are the composites of braiding isomorphisms they came from.
Of course, following the periodic table, one wonders whether a generalization of the geometric result of Joyal and Street exists for Gray monoids? Namely, a geometric(ish) way of determining when two parallel composites of interchangers are equal.
My guess is that this is the case. To elaborate on this a little, I think that instead of having only braids, one has to consider sheets and braids.
Has something like this already been worked out? In my opinion, a good place to start is the graphical calculus of Bruce Bartlett.
