8 1. TEMPERLEY-LIEB-JONES THEORIES for any formal diagram in Hom(x, y), we will not draw the square R with the understanding that that the univalent vertices are representing some objects. Also a natural number l beside an edge always means the presence of the Jones-Wenzl projector pl. We will consider many relations among formal diagrams, so we remark that one relation can lead to many new relations by the following principle. Lemma 1.11 (Principle of annular consequence). Suppose the square R is inside a bigger square S. In the annulus between R and S, suppose there are formal diagrams connecting objects on R and S. Then any relation r of formal diagrams supported in R induces one supported in S by including the relation r into S, and deleting the boundary of the old R. The resulting new relation r will be called an annular consequence of r (Fig. 1.8). More generally, S can be any compact surface, in which case we will call r a generalized annular consequence of r. = = d = d2 Figure 1.8. Annular consequence. Proposition 1.12. Let x, y be two objects such that |x| + |y| = 2m. Then (1) dim Hom(x, y) = 1 m+1 ( 2m m ) . (2) Let G be a (connected) uni-trivalent tree connecting x and y. Then the collection of all admissible colorings of G forms a basis of Hom(x, y). Proof. (1) Without loss of generality, we may assume |x| ≥ |y|. By bending arms down (Fig. 1.9), we see that Hom(x, y) ∼ TLm(A) as vector spaces. → ↔ Figure 1.9. Bending arms down. (2) Counting admissible colorings of G gives the right dimension. For lin- ear independence, note that the Markov pairing on TLn(A) extends to any Hom(x, y), and is nondegenerate. (This will be easier to see after Sec. 1.1.6.)

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