Fibonacci anyons

I'm trying to understand Claire's quantum gates for Fibonacci anyons. There's one kind of particle (not counting the vacuum), and trivalent vertices. She told me the F-matrix, and I get the key relation

Claire: H = sqrt(phi) cup-cap - 1/sqrt(phi) ||

I fudge my vertex = phi^(-1/4) times Claire's. I get

Then I get the Jones-Wenzl at q=exp(i pi/5). A crossing is

sigma = q^3 || + q^-3 cup-cap

An fuse-split (like a letter I) is equal to the Jones-Wenzl idempotent p_2.

Kuperberg-esque relations:

bubble = phi
digon = 1
triangle = -1/phi
square = 1/phi^2 (cup-cap + ||)
pentagon = -1/phi (break-one-side + break-the-two-adjacent-sides-instead)

It's a graph planar algebra. The colors are 0 and 1, with weights phi and 1 respectively. For a vertex with colors 001, the 1 gets 180 degrees. There are fudge constants

000: -1/phi
001: 1