In this paper we investigate the asymptotic behavior of the colored Jones
polynomials and the Turaev-Viro invariants for the figure eight knot. More
precisely, we consider the M-th colored Jones polynomials evaluated at
(N+1/2)-th root of unity with a fixed limiting ratio, s, of M and
(N+1/2). We find out the asymptotic expansion formula (AEF) of the colored
Jones polynomials of the figure eight knot with s close to 1. Nonetheless,
we show that the exponential growth rate of the colored Jones polynomials of
the figure eight knot with s close to 1/2 is strictly less than those with
s close to 1. It is known that the Turaev Viro invariant of the figure
eight knot can be expressed in terms of a sum of its colored Jones polynomials.
Our results show that this sum is asymptotically equal to the sum of the terms
with s close to 1. As an application of the asymptotic behavior of the
colored Jones polynomials, we obtain the asymptotic expansion formula for the
Turaev-Viro invariants of the figure eight knot. Finally, we suggest a possible
generalization of our approach so as to relate the AEF for the colored Jones
polynomials and the AEF for the Turaev-Viro invariants for general hyperbolic
knots.Comment: 40 pages, 0 figure