6 1. TEMPERLEY-LIEB-JONES THEORIES Existence is completed by an inductive construction of pn+1 from pn, which also reveals the exact nature of the “generic” restriction on the loop variable d. The induction is as follows, where μn = Δn−1(d) Δn(d) . (1.10) p1 = p2 = − 1 d pn+1 = pn · · · · · · − μn pn pn · · · · · · It is not difficult to check that Uipn = pnUi = 0, 0 i n. (The most interesting case is Un−1.) Tracing the inductive definition of pn+1 yields tr(p1) = d and tr(pn+1) = tr(pn) − Δn−1 Δn tr(pn), showing that tr(pn+1) satisfies the Chebyshev recursion (and the initial data). Thus tr(pn) = Δn. Jones-Wenzl idempotents were discovered by V. Jones [Jo1], and their induc- tive construction is due to H. Wenzl [Wenz]. We list the explicit formulas for p2,p3,p4,p5. p2 = 2 = − 1 d p3 = 3 = + 1 d2 − 1 Ä + ä − d d2 − 1 Ä + ä p4 = 4 = − d d2 − 2 + 1 d2 − 2 Ä + + + ä + −d2 + 1 d3 − 2d Ä + ä − 1 d3 − 2d Ä + ä + d2 d4 − 3d2 + 2 Ä ä − d d4 − 3d2 + 2 Ä + ä + 1 d4 − 3d2 + 2
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