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
colored Jones polynomials
evaluated at
roots of unity
with a fixed limiting ratio,
,
of
and
. We find out
the asymptotic expansion formula (AEF) of the colored Jones polynomials of the figure eight
knot with
close
to
. We show
that the exponential growth rate of the colored Jones polynomials of the figure eight knot
with
close to
is strictly less
than those with
close to
.
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
close
to
. 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.
Keywords
volume conjecture, figure eight knot, colored Jones
polynomials, Turaev–Viro invariants
