12 1. TEMPERLEY-LIEB-JONES THEORIES i cn−2 ... · · · c1 c0 Lemma 1.19. The basis {eC i } is orthogonal with respect to the Markov pairing. Proof. eC i , eC i = δi,i δC,C Δ(1, 1,c1) Δc 1 Δ(c1, 1,c2) Δc 2 · · · Δ(cn−3, 1,cn−2) Δc n−2 Δ(cn−2, 1,i). Theorem 1.20. (1) Each eB A i is a matrix unit up to a scalar on Hom(i, 1n) with respect to the basis {eC i }. Explicitly, eA B i (eC i ) = δii δAC tr(eA A i ) Δi eB i so that eB A i has exactly one nonzero entry. (2) Given a braid σ Bn, the Jones representation ρA(σ) as a matrix is ob- tained by stacking σ onto the top of each eC i , resolving the crossings with the Kauffman bracket, and then expanding the resulting formal diagrams in the basis {eC i } for each irreducible sector i {0, 1,...,n} and i = n mod 2. This theorem follows from existing works on TL algebras and Jones represen- tations. 1.1.7. Colored Jones polynomials. More convenient for our applications is the Kauffman bracket for framed unoriented links, which is a variation of the Jones polynomial for oriented links. The Jones polynomial for oriented links can be obtained from the Kauffman bracket by multiplying by a power of A depending on the writhe. A coloring of a link L is a labeling of each component by a natural number. This natural number is different from the framing. We always use the blackboard framing for link diagrams, i.e., the framing from a parallel copy of each component in the plane. Suppose LD is a link diagram of L. Then LD is in Hom(0, 0), hence a scalar multiple of id0. This scalar LD A will be called the colored Kauffman bracket of L. If L is oriented with all components colored by a and w(L) is the writhe of the link diagram LD, then Ja(L t) = (−Aa(a+2))−w(L) LD A is the colored Jones polynomial at t = A−4. When a = 1, Ja(L t) is the usual Jones polynomial. 1.1.8. Colored Jones representations. The Jones representation can be extended to colored Jones representations of braid groups Bn for any coloring. If there is more than one color, then we get a representation of a subgroup of Bn. For example, if all colors are pairwise distinct, then we get a representation of the pure braid group PBn. How those braid group representations decompose into irreducibles seems to be unknown.
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