Morrison and Nieh's paper on su(3) knot homology
uses some relations of Kuperberg's
which involves trivalent vertices that we wish to add to the complexes of the homology. That is, instead of a smooth cobordism between the pieces on the right hand side of the third relation, we allow a 4 vertex diagram >-< at the top or bottom of the bordism. In the skein relation, whereas the smooth pair of lines has a coefficient of $q^2$, the 4 vertex diagram has a coefficient of $q^3$. The objects of their bordism category are webs generated by trivalent vertices with directed edges, either all ingoing or all outgoing.
Oh, that reminds me. I fixed up the Terence Tao
These trivalent vertices satisfy a triangle condition $a + b + c = 0$ on the numbers assigned to each edge of the honeycomb. The semi-infinite edges correspond to eigenvalues of the triangular set of 3x3 Hermitean matrices.