2 1. TEMPERLEY-LIEB-JONES THEORIES following, when F is clear from the context or F = C, we will refer to an F-algebroid just as an algebroid. Let A be an indeterminant over C, and d = −A2 − A−2. We will call A the Kauffman variable, and d the loop variable. Let F = C[A, A−1] be the quotient field of the ring of polynomials in A. Let I = [0, 1] be the unit interval, and R = I × I be the square in the plane. The generic Temperley-Lieb (TL) algebroid TL(A) is defined as follows. An object of TL(A) is the unit interval with a finite set of points in the interior of I, allowing the empty set. The object I with no interior points is denoted as 0. We use |x| to denote the cardinality of points in x for x ∈ TL(A)0. Given x, y ∈ TL(A)0, the set of morphisms Hom(x, y) is the following F-vector space: If |x| + |y| is odd, then Hom(x, y) is the 0-vector space. If |x| + |y| is even, first we define an (x, y)-TL diagram. Identify x with the bottom of R and y with the top of R. A TL-diagram or just a diagram D is the square R with a collection of |x|+|y| 2 smooth arcs in the interior of R joining the |x| + |y| points on the boundary of R, plus any number of smooth simple closed loops in R. All arcs and simple loops are pairwise non-intersecting, and moreover, all arcs meet the boundary of R perpendicularly. Note that when |x| = |y| = 0, TL diagrams are just disjoint simple closed loops in R, including the empty diagram. The square with the empty diagram is denoted by 10. For examples, see the diagrams below. Two diagrams D1,D2 are d-isotopic if they induce the same pairing of the |x|+|y| boundary points (Fig. 1.1). Note that D1,D2 might have different numbers of simple closed loops. Finally, we define Hom(x, y) to be the F-vector space with basis the set of (x, y)-TL diagrams modulo the subspace spanned by all elements of the form D1 − dmD2, where D1 is d-isotopic to D2 and m is the number of simple closed loops in D1 minus the number in D2. Note that any diagram D in Hom(0, 0) is d-isotopic to the empty diagram. Hence a diagram D with m simple closed loops as a vector is equal to dm10. ∼ d-isotopy Figure 1.1. d-isotopic diagrams. Composition of morphisms is given first for diagrams. Suppose D1,D2 are diagrams in Hom(y, z) and Hom(x, y), respectively. The composition of D1 and D2 is the diagram D1D2 in Hom(x, z) obtained by stacking D1 on top of D2, rescaling the resulting rectangle back to R, and deleting the middle horizontal line (Fig. 1.2). = Figure 1.2. Composition of diagrams. Composition preserves d-isotopy, and extends uniquely to a bilinear product Hom(y, z) × Hom(x, y) → Hom(x, z).

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