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Asymptotic Behavior of Colored Jones polynomial and Turaev-Viro Invariant of figure eight knot. (arXiv:1711.11290v1 [math.GT])
来源于:arXiv
In this paper we investigate the asymptotic behavior of the colored Jones
polynomial and the Turaev-Viro invariant for the figure eight knot. More
precisely, we consider the $M$-th colored Jones polynomial evaluated at
$(N+1/2)$-th root of unity with a fixed limiting ratio, $s$, of $M$ and
$(N+1/2)$. Generalizing the work of \cite{WA17} and \cite{HM13}, we obtain the
asymptotic expansion formula (AEF) of the colored Jones polynomial of figure
eight knot with $s$ close to $1$. An upper bound for the asymptotic expansion
formula of the colored Jones polynomial of figure eight knot with $s$ close to
$1/2$ is also obtained. From the result in \cite{DKY17}, the Turaev Viro
invariant of 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 f 查看全文>>