10 1. TEMPERLEY-LIEB-JONES THEORIES Proof. (1) Exercise. (2) Recall that an edge of color l harbors a JWP pl. JWPs kill any turn-backs, therefore every morphism in Hom(a, b) is proportional to the identity if a = b and 0 otherwise. (3) If a, b, c are not admissible, then there will be turn-backs. (4) Same as the TL(A) case. So far everything is in the plane, but we are interested in links. The bridge is provided by the Kauffman bracket. The Kauffman bracket resolves a crossing of a link diagram into a linear combination of TL diagrams, hence a link diagram is just a formal diagram. In particular, any link diagram LD is in Hom(0, 0), and hence = λid0 as a morphism. The scalar λ, a Laurent polynomial of A, is called the Kauffman bracket of LD, denoted as LD A or LD . Proposition 1.16. Let [m]A = A2m−A−2m A2−A−2 be the quantum integer and [m]! be the quantum factorial [m]A[m − 1]A · · · [1]A. Note the loop variable d = −[2]A. (1) i = Δi (2) (no tadpole) b a = δa,0Δb (3) ˜ij = i j = (−1)i+j[(i + 1)(j + 1)]A (4) i = (−1)iAi(i+2) i (5) j k i = (−1) i+j+k 2 A i(i+2)−j(j+2)−k(k+2) 2 j k i (6) Δ(a, b, c) = b a c = (−1)m+n+p[m+n+p+1]![m]![n]![p]! [m+n]![n+p]![p+m]! (7) b c a a = δa,a Δ(a,b,c) Δa a Proof. We leave them as exercises, or see [KL]. Given any i, j, k, l, we have two different bases of Hom(l, i ⊗ j ⊗ k) by labeling two different trees:

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