Communications in Mathematical Physics - Volume 209 by A. Jaffe (Chief Editor)

By A. Jaffe (Chief Editor)

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6) . · φ(lside 1 ,(n1 ,l2 ),m),((n1 ,l2 ),l1 ,m) The summation over n1 can be suppressed but we should remember that l2 ∈ ( 21 ⊗ 21 ⊗ 21 ). Now that we have determined the states for Fig. 7) l1 ,l2 where l1 = 0, 1 and l2 = 1/2, 1/2, 3/2 for k ≥ 3. 8) ) 1 + q 2 + q 2 + 3q 2 + 4q 2 + 6q 2 −1 2 + 4q −3 2 + 3q −5 2 +q −7 2 +q −9 2 +q −11 2 . The following relation is easy to check by computing the bracket polynomial using the recursive method: P{1},{1} [(2, 3) ∗ DH ]| 1 q 4 =−A = − (2, 3) ∗ DH . 9) This is expected from Theorem 1 for the five component link.

Ln −2sn [D] . 19) equals l1 /2 S(s1 ), s1 =0 46 P. Ramadevi, S. Naik where l1 /2 denotes the greatest integer less than or equal to l1 /2. We split this into min{ l /2 ,1} l /2 and s 1=2 and use Eq. 16) to substitute for the Ali −2ji ,si . two sums s =0 1 1 1 It is easy to see that: min{ l1 /2 ,1} s1 =0 S(s1 ) = Pl1 ,l2 −2s2 ,... ,ln −2sn [D]. 16) we have : s1 l1 − j1 j1 (−1)j1 S(s1 ) = j1 =s1 −1 l1 − 2j1 − 1 Pl1 −2s1 ,... ,ln −2sn [D] s1 − j1 + (l1 − 2s1 + 1) × s1 −2 (l1 − j1 )(l1 − j1 − 1) .

C1 . 44 P. Ramadevi, S. Naik Again writing the states for the mixed composite braiding in a suitable basis (See Fig. 8): | (c1 ,c2 ,... l2m ,t0 ,... ,t2m−4 [P ] l1 ,l2 ,... ,l2m t0 ,t1 ,... ,t2m−4 · |φ˜ l1 ,l2 ,t0 ,l3 ,t1 ,... ,t2m−4 ,l2m−1 ,l2m , (c1 ,c2 ,... ,cn ) | 2 = Bl1 ,... ,l2m ,t0 ,... 12) l1 ,l2 ,... ,l2m t0 ,t1 ,... ,t2m−4 · φ˜ l1 ,l2 ,t0 ,l3 ,t1 ,... ,t2m−4 ,l2m−1 ,l2m |, where ti ∈ (ti−1 ⊗ li−2 ) with t−1 = l2 , t2m−3 = l2m and li ’s as in . The closure of the braid demands: (lm+1 , .

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