Taylor Y. U.'s 6j symbols for Uq (sl2 ) and non-Euclidean tetrahedra PDF

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G) If τ exists, but all edge lengths are vanishing, then s(k) = 1. 10. Questions (a) Is there a uniform formula which includes all cases, just as the Schulten– Gordon formula (36) includes cases (a) and (b)? Such a formula (or least uniform estimates) would be needed to apply these results to the asymptotics of the quantum invariants such as Turaev–Viro and Jones, since one would presumably have to show that the contributions from the non-degenerate case are dominant. (b) Is there a geometric description of the tensor category of representations of Uq (sl2 ) in terms of the moduli space of hyperbolic bundles [2], for q positive real, similar to the description for q a primitive root of unity using unitary bundles, which gives a geometric explanation of the hyperbolic formulas?

Reˇsetikhin. Quantum linear problem for the sine-Gordon equation and higher representations. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 101 (1981), 101–110, 207 (in Russian). [24] J. Marakami and A. Ushijima. A volume formula for hyperbolic tetrahedra in terms of edge lengths. MG/0402087. [25] V. B. Mehta and C. S. Seshadri. Moduli of vector bundles on curves with parabolic structure. Math. Ann. 248 (1980), 205–239. [26] J. Milnor. Collected Papers. Vol. 1. Publish or Perish, Houston, TX, 1994.

568 Y. U. Taylor and C. T. Woodward Sel. , New ser. 6. Let jab , 1 ≤ a < b ≤ 4, be non-negative half-integers, r > 2 and s(k) := kj12 kj34 kj23 kj14 kj13 kj24 . q=exp(π/r(k)) (a) If τ exists and is non-degenerate, then s(k) ∼ where φ(k) = r(k) 2π 2π cos(φ(k) + π/4) r(k)3/2 |det(− cosh(lab ))|1/4 θab (k)lab (k) + 2 Vol(τ (k)) . a

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