2-3 Pachner move and Biedenharn-Elliot identity

[KOSS]

Homework Statement

Show that up to "fudge factors" (such as a few theta-nets an a loop) the 2-3 Pachner move is just the Biedenhard-Elliot identity between 6j symbols.

If you go to the Quantum Gravity Seminar notes of Baez and Alvarez, you can see this problem here: :math.ucr.edu/home/baez/qg-fall2000/QGravity/QGravity.pdf: (sorry can't make that an url since I don't have the 15 post privilege yet) (anyway, it's Ch.21, Ex 16 of that pdf).

Homework Equations

Roughly (imagine this "English" translated into diagrams), given the known relation between the 6j-symbols and the spin network tetrahedron (with a few theta-nets and a loop), this isomorphism,

("the 3-tet-net"*Oj)/(theta-net*theta-net*theta-net) = "the 2-tet-net"/theta-net

is a diagrammatic representation of Biedenhard-Elliot, namely,
$\sum_k \left\{\begin{array}{ccc}a&b&k\\ c&f&e\end{array}\right\}\left\{\begin{array}{ccc}a&k&i\\ g&d&f\end{array}\right\}\left\{\begin{array}{ccc}b&c&j\\ d&i&k\end{array}\right\} = \left\{\begin{array}{ccc}e&c&j\\ d&g&f\end{array}\right\}\left\{\begin{array}{ccc}a&b&i\\ j&g&e\end{array}\right\}$

3. The Attempt at a Solution
If I start with the Biedenhard-Elliot identity and work out all the triangular faces of the equivalent the faces of the tet-net I can obtain the above diagrammatic isomorphism with cancellation of two theta-nets and two loops on each side. But my labels are not the same as written in the text by Alvarez (cited to above). also, what confuses me is that the Alvarez-Baez diagram equation seems like it should have a further theta-net canceled in the denominators on each side. Is this just a misleading typo, or do theta-nets not "cancel" that naively? I'd appreciate any help on this, and since I'm not a student and just an amateur dabbler in physics if anyone wants to take the opportunity to expound on at length about this problem and similar puzzles then please feel free.

 

[Jhero]

First of all, let's clarify some terminology. The 2-3 Pachner move refers to a specific topological transformation that can be applied to a spin network, which is a graph-like structure used in loop quantum gravity to represent spacetime. The Biedenhard-Elliot identity, on the other hand, is a mathematical relation between the 6j symbols, which are used in the calculation of the quantum states of a spin network. So, the goal here is to show that the 2-3 Pachner move and the Biedenhard-Elliot identity are essentially the same thing, up to some "fudge factors" that account for differences in notation and terminology.

To start, let's look at the Biedenhard-Elliot identity:

\sum_k \left\{\begin{array}{ccc}a&b&k\\ c&f&e\end{array}\right\}\left\{\begin{array}{ccc}a&k&i\\ g&d&f\end{array}\right\}\left\{\begin{array}{ccc}b&c&j\\ d&i&k\end{array}\right\} = \left\{\begin{array}{ccc}e&c&j\\ d&g&f\end{array}\right\}\left\{\begin{array}{ccc}a&b&i\\ j&g&e\end{array}\right\}

This equation relates six 6j symbols, which are used to calculate the quantum states of a spin network. Each 6j symbol has four indices, labeled a, b, c, and d, and represents the quantum state of a tetrahedron in the spin network. The indices correspond to the spins of the edges of the tetrahedron, and the 6j symbol gives the amplitude for the tetrahedron to have those specific spins. So, the Biedenhard-Elliot identity essentially says that the amplitude for a specific configuration of four tetrahedra is the same, regardless of how those tetrahedra are grouped together.

Now, let's look at the 2-3 Pachner move. This is a topological transformation that can be applied to a spin network, and it essentially involves merging two tetrahedra into one, or splitting one tetrahedron into two. This transformation can be represented