1.1. GENERIC TEMPERLEY-LIEB-JONES ALGEBROIDS 11 i j k m l = n F ijk l nm i j k n l This change of basis matrix (F ijk l )nm = F ijk l nm is called an F -matrix, and the {F ijk l nm } are called 6j symbols. In general graphical calculus for RFCs, we only draw graphs whose edges are transversal to the x-direction, i.e., no horizontal edges. But in TLJ theories, this subtlety is unnecessary. Therefore we will draw the F -matrix as i l m j k i l n = ∑ n F ijk l nm j k As a special case, we have i j = k Δk Δ(i, j, k) i j k i j Finally, we are ready to see that each TL diagram in TLn(A) is an explicit matrix. Consider labeling the following uni-trivalent tree Γ: i bn−2 ... · · · b1 b0 an−2 ... · · · a1 a0 Lemma 1.17. The admissible labelings of Γ form a basis of TLn(A), denoted as {eB A i }, where B = (b0,...,bn−2), A = (a0,...,an−2). Lemma 1.18. tr(eB A i ) = δAB Δ(i, 1,an−2) Δa n−2 Δ(an−2, 1,an−1) Δa n−3 · · · Δ(a2, 1,a1) Δa 1 Δ(a1, 1, 1). Note a0 = b0 = 1. Proof. Use Prop. 1.16(7) repeatedly. For i ∈ {0, 1,...,n} and i = n mod 2, fix a basis {eC i } of Hom(i, 1n), where C = (c0,c1,...,cn−2) and eC i is the following labeled tree:

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