Gukov's asymptotic expansion conjecture for colored Jones polynomials

From papers

Let $K$ be a knot and let $J_N(K;q)$ be its colored Jones polynomial. Let $u:=2\pi\sqrt{-1}(N/k-1)$ remain fixed as $N,k\to\infty$ . Let $S(u)$ be the classical Chern–Simons action, let $T_K(u)$ be the Ray–Singer torsion twisted by the flat connection associated with $\rho:\pi_1(S^3\setminus K)\to SL(2,\mathbb{C})$ determined by $u$ , let $\delta_K(u)\in\mathbb{Z}$ be determined by the topology and representation, and let $S_n(u)$ denote the $n$ -loop contribution.

Asymptotic expansion conjecture. As $N\to\infty$ and $k\to\infty$ with $u$ fixed,

\log J_N\left(K;\exp(2\pi\sqrt{-1}/k)\right)\underset{N,k\to\infty}{\sim}\frac{k}{\sqrt{-1}}S(u)+\frac{1}{2}\delta_K(u)\log k+\frac{1}{2}\log\left(\frac{T_K(u)}{2\pi^2}\right)+\sum_{n=1}^{\infty}\left(\frac{2\pi}{k}\right)^nS_{n+1}(u).

This is the proposed perturbative $SL(2,\mathbb{C})$ Chern–Simons interpretation of the colored Jones polynomial. The supplied text describes it as a conjectural prediction and does not give evidence of a complete proof.

Progress summary

Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

Gukov's asymptotic expansion conjecture for colored Jones polynomials
Let $K$ be a hyperbolic knot in $S^3$ . Let $U\subset\mathbb{C}$ be a neighborhood of $0$ , let $u\in U\setminus\pi\sqrt{-1}\mathbb{Q}$ , and put
$\xi:=2\pi\sqrt{-1}+u.$
Let $T(K;u)$ be the cohomological twisted Reidemeister torsion and $S(K;u)$ the $\operatorname{SL}(2;\mathbb{C})$ Chern--Simons invariant associated with an irreducible representation of $\pi_1(S^3\setminus K)$ into $\operatorname{SL}(2;\mathbb{C})$ sending a meridian to an element with eigenvalues $\exp(u/2)$ and $\exp(-u/2)$ . Gukov's asymptotic expansion conjecture. There exists such a neighborhood $U$ for which
$J_N(K;\exp(\xi/N))\underset{N\to\infty}{\sim}\frac{\sqrt{-\pi}}{2\sinh(u/2)}T(K;u)^{1/2}\left(\frac{N}{\xi}\right)^{1/2}\exp\left(\frac{N}{\xi}S(K;u)\right).$
The paper states that its theorem confirms this conjecture for the figure-eight knot when $u$ is real and $0<u<\log((3+\sqrt{5})/2)$ . The general hyperbolic-knot statement remains open on the supplied evidence.
source: Hitoshi Murakami, “The colored Jones polynomial, the Chern–Simons invariant, and the Reidemeister torsion of the figure-eight knot”, arXiv:1102.3530 (2011).

Sources & referencesView supporting material

Primary source

Sergei Gukov and Hitoshi Murakami, “SL(2,C) Chern-Simons theory and the asymptotic behavior of the colored Jones polynomial”, arXiv:math/0608324 (2007).

1 week ago