The asymptotic behaviors of the colored Jones polynomials of the figure eight-knot, and an affine representation

Abstract

We study the asymptotic behavior of the $N$-dimensional colored Jones polynomial of the figure-eight knot evaluated at $\exp\bigl((κ+2pπı/N\bigr)$, where $κ:=\arccosh(3/2)$ and $p$ is a positive integer. We can prove that it grows exponentially with growth rate determined by the Chern--Simons invariant of an affine representation from the fundamental group of the knot complement to the Lie group $\SL(2;\C)$.

Figures (21)

• Figure 1: Contour plots of the functions $\operatorname{Re}{z^3}$ (left) and $\operatorname{Re}{z^2}$ (right) around their saddle points. The saddle point $O$ of $\operatorname{Re}{z^3}$ is of order two, and that of $\operatorname{Re}{z^2}$ is of order one.
• Figure 2: The region \ref{['eq:ell01']} is between the two thick dotted lines minus the two red lines $\ell_{0}^{+}$ and $\ell_{1}^{-}$.
• Figure 3: The region \ref{['eq:ell00']} is between the two thick dotted lines minus the two red lines $\ell_0^{+}$ and $\ell_0^{-}$.
• Figure 4: The yellow region is $\Omega$. The green triangles are $\Delta_{0}^{+}$ and $\Delta_{1}^{-}$.
• Figure 5: The yellow region is $\Omega^{\ast}_{\nu}$. The green triangles are $\Delta^{+}_{0,\nu}$ and $\Delta^{-}_{1,\nu}$.

Theorems & Definitions (88)

• Conjecture 1.1: Volume conjecture
• Conjecture 1.2: Complexification of the volume conjecture
• Conjecture 1.3: Refined volume conjecture
• Conjecture 1.4: Generalized volume conjecture
• Theorem 1.5
• Theorem 1.6: Murakami:JPJGT2007
• Theorem 1.7: Hikami/Murakami:COMCM2008
• Theorem 1.8
• Corollary 1.9
• Remark 1.10