The 6j symbols
Allow me to speak very loosely here. I’m still learning this algebra.
From one perspective, the 6j symbols describe how to translate from one splitting of a morphism to another. You may care to view a morphism as a directed graph. Here we have two morphisms viewed as trees; one directed upwards and one directed downwards. Though they branch differently, their ends correspond as shown below.
Though the two trees branch differently, there is a cobordism between them. As we slide the oddball branch from one position to the other, we sweep out 6 faces. This forms a pleasant sort of 2-complex with only one triple point.
Let us deform this cobordism version of the 2-complex to something more symmetric.
Here’s a larger picture of the end result.
This 2-complex is the cone of the closed graph below. And the 6j symbol can be seen as the trace of the corresponding morphism.
Here it is again in a different context.
Lol and behold: the spine of a tetrahedron!