We obtain a formula for the Turaev-Viro invariants of a link complement in terms of values of the colored Jones polynomials of the link. As an application, we give the first examples of 3-manifolds where the "large r" asymptotics of the Turaev-Viro invariants determine the hyperbolic volume. We verify the volume conjecture of Chen and the third named author [7] for the figure-eight knot and the Borromean rings. Our calculations also exhibit new phenomena of asymptotic behavior of values of the colored Jones polynomials that seem to be predicted neither by the Kashaev-Murakami-Murakami volume conjecture and its generalizations nor by Zagier's quantum modularity conjecture. We conjecture that the asymptotics of the Turaev-Viro invariants of any link complement determine the simplicial volume of the link, and verify this conjecture for all knots with zero simplicial volume. Finally, we observe that our simplicial volume conjecture is compatible with connected summations and split unions of links. of degree r. Throughout this paper, we will consider the case that q = A 2 , where A is either a primitive 4r-th root for any integer r or a primitive 2r-th root for any odd integer r.We use the notation i = (i 1 , . . . , i n ) for a multi-integer of n components (an n-tuple of integers) and use the notation 1 i m to describe all such multi-integers with 1 i k m for each k ∈ {1, . . . , n}. Given a link L with n components, let J L,i (t) denote the i-th colored Jones polynomial of L whose kth component is colored by i k [23,21]. If all the components of L are colored by the same integer i, then we simply denote J L,(i,...,i) (t) by J L,i (t). If L is a knot, then J L,i (t) is the usual i-th colored Jones polynomial. The polynomials are indexed so that J L,1 (t) = 1 and J L,2 (t) is the ordinary Jones polynomial, and are normalized so that
We establish the geometry behind the quantum 6j-symbols under only the admissibility conditions as in the definition of the Turaev-Viro invariants of 3-manifolds. As a classification, we show that the 6-tuples in the quantum 6j-symbols give in a precise way to the dihedral angles of (1) a spherical tetrahedron, (2) a generalized Euclidean tetrahedron, (3) a generalized hyperbolic tetrahedron or (4) in the degenerate case the angles between four oriented straight lines in the Euclidean plane. We also show that for a large proportion of the cases, the 6-tuples always give the dihedral angles of a generalized hyperbolic tetrahedron and the exponential growth rate of the corresponding quantum 6j-symbols equals the suitably defined volume of this generalized hyperbolic tetrahedron. It is worth mentioning that the volume of a generalized hyperbolic tetrahedron can be negative, hence the corresponding sequence of the quantum 6j-symbols could decay exponentially. This is a phenomenon that has never been aware of before.
Combining the work of Bonahon-Wong and Frohman-Kania-Bartoszyńska-Lê with arguments from algebraic geometry, we develop a novel method for computing the Kauffman bracket skein modules of closed 3-manifolds with coefficients in Q(A).We show that if the skein module S(M,is the number of its closed points.We prove a criterion for reducibility and finiteness of character varieties of closed 3manifolds. Combining it with the above methods we compute skein modules of Dehn fillings of the figure-eight knot and of (2, 2n + 1)-torus knots.We also prove that the skein modules of rational homology spheres have dimension at least 1 over Q(A).
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