15-j sybmols

T.M.Tlas@gmail.com

Hi Everyone,

I was hoping that some expert ;) may enlighten me on this issue.

I've been reading a lot about the Crane-Yetter TQFT lately, and it
seems all constructions of it (or of isomorphic TQFT's) use some
ordering of vertices at an intermediate step. The question is Why?

Is this because the graph corresponding to a 4-simplex is not
embeddable in 2d without self intersections? So that if we just embedd
it randomly in some way, the diagramme (number) corresponding to it
will be different from the one obtained by embedding it some other way.

If the answer is yes to above, why that particular convention is
chosen? Are there any other consistent conventions for embedding the
diagramme? (Just to show that i'm totally spoilt, why is the 4 simplex,
for example, put on the "right" of the 0 and the 2 simplex on the left?
I'm talking about the diagramme in the paper by Crane and Yetter "A
categorical construction of 4d topological quantum field theories")

Finally, I've seen MANY books, papers, articles ,etc., discussing the
6-j symbols, their relationship to the tetrahedron, identities among
them etc., but have never seen ANY book which discusses the 15-j ones.
Can anybody point to a reference which does, and ARE there similar
identities in the 15-j case (Biedenharn-Elliot, orthogonality, etc..)

Thank you for your time,
Tim